GPU: Add dense cuSOLVER solvers (QR, SVD, EigenSolver)

Add QR (geqrf + ormqr + trsm), SVD (gesvd), and self-adjoint eigenvalue
decomposition (syevd) via cuSOLVER. All support host and DeviceMatrix input.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

CMakeLists.txt

@@ -1,4 +1,4 @@
-cmake_minimum_required(VERSION 3.10.0)
+cmake_minimum_required(VERSION 3.17)
 
 #==============================================================================
 # CMake Policy issues.
@@ -9,7 +9,7 @@ if (POLICY CMP0077)
 endif (POLICY CMP0077)
 
 # NOTE Remove setting the policy once the minimum required CMake version is
-# increased to at least 3.15. Retain enabling the export to package registry.
+# increased to at least 3.21. Retain enabling the export to package registry.
 if (POLICY CMP0090)
   # The export command does not populate package registry by default
   cmake_policy (SET CMP0090 NEW)

Eigen/GPU

@@ -0,0 +1,58 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+#ifndef EIGEN_GPU_MODULE_H
+#define EIGEN_GPU_MODULE_H
+
+#include "Core"
+
+#include "src/Core/util/DisableStupidWarnings.h"
+
+/** \defgroup GPU_Module GPU module
+ *
+ * GPU-accelerated solvers and operations using NVIDIA CUDA libraries
+ * (cuSOLVER, cuBLAS, cuSPARSE, cuFFT, cuDSS).
+ *
+ * This module provides explicit GPU solver classes that coexist with Eigen's
+ * CPU solvers. Unlike the LAPACKE dispatch (which replaces the CPU
+ * implementation globally), GPU classes are separate types the user
+ * instantiates by choice:
+ *
+ * \code
+ * #define EIGEN_USE_GPU
+ * #include <Eigen/GPU>
+ *
+ * // CPU path (unchanged)
+ * Eigen::LLT<Eigen::MatrixXd> llt_cpu(A);
+ *
+ * // GPU path (explicit)
+ * Eigen::GpuLLT<double> llt_gpu(A);   // L stays on device
+ * auto X = llt_gpu.solve(B);          // only B transferred per solve
+ * \endcode
+ *
+ * Requires CUDA 11.4+. See CLAUDE.md.
+ */
+
+#ifdef EIGEN_USE_GPU
+// IWYU pragma: begin_exports
+#include "src/GPU/DeviceMatrix.h"
+#include "src/GPU/GpuContext.h"
+#include "src/GPU/DeviceExpr.h"
+#include "src/GPU/DeviceBlasExpr.h"
+#include "src/GPU/DeviceSolverExpr.h"
+#include "src/GPU/DeviceDispatch.h"
+#include "src/GPU/GpuLLT.h"
+#include "src/GPU/GpuLU.h"
+#include "src/GPU/GpuQR.h"
+#include "src/GPU/GpuSVD.h"
+#include "src/GPU/GpuEigenSolver.h"
+// IWYU pragma: end_exports
+#endif
+
+#include "src/Core/util/ReenableStupidWarnings.h"
+
+#endif  // EIGEN_GPU_MODULE_H

Eigen/src/GPU/CuBlasSupport.h

@@ -0,0 +1,233 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2026 Rasmus Munk Larsen <rmlarsen@gmail.com>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+// cuBLAS-specific support types:
+//   - Error-checking macro
+//   - Operation enum and mapping to cublasOperation_t
+//
+// Generic CUDA runtime utilities (DeviceBuffer, cuda_data_type) are in GpuSupport.h.
+
+#ifndef EIGEN_GPU_CUBLAS_SUPPORT_H
+#define EIGEN_GPU_CUBLAS_SUPPORT_H
+
+// IWYU pragma: private
+#include "./InternalHeaderCheck.h"
+
+#include "./GpuSupport.h"
+#include <cublas_v2.h>
+
+namespace Eigen {
+namespace internal {
+
+// ---- Error-checking macro ---------------------------------------------------
+
+#define EIGEN_CUBLAS_CHECK(expr)                                       \
+  do {                                                                 \
+    cublasStatus_t _s = (expr);                                        \
+    eigen_assert(_s == CUBLAS_STATUS_SUCCESS && "cuBLAS call failed"); \
+  } while (0)
+
+// ---- Operation enum ---------------------------------------------------------
+// Maps transpose/adjoint flags to cublasOperation_t.
+
+enum class GpuOp { NoTrans, Trans, ConjTrans };
+
+constexpr cublasOperation_t to_cublas_op(GpuOp op) {
+  switch (op) {
+    case GpuOp::Trans:
+      return CUBLAS_OP_T;
+    case GpuOp::ConjTrans:
+      return CUBLAS_OP_C;
+    default:
+      return CUBLAS_OP_N;
+  }
+}
+
+// ---- Scalar → cublasComputeType_t -------------------------------------------
+// cublasGemmEx requires a compute type (separate from the data type).
+//
+// Precision policy:
+//   - Default: tensor core algorithms enabled (CUBLAS_GEMM_DEFAULT_TENSOR_OP).
+//     For double, cuBLAS may use Ozaki emulation on sm_80+ tensor cores.
+//   - EIGEN_CUDA_TF32: opt-in to TF32 for float (~2x faster, 10-bit mantissa).
+//   - EIGEN_NO_CUDA_TENSOR_OPS: disables all tensor core usage. Uses pedantic
+//     compute types and CUBLAS_GEMM_DEFAULT algorithm. For bit-exact reproducibility.
+
+template <typename Scalar>
+struct cuda_compute_type;
+
+template <>
+struct cuda_compute_type<float> {
+#if defined(EIGEN_NO_CUDA_TENSOR_OPS)
+  static constexpr cublasComputeType_t value = CUBLAS_COMPUTE_32F_PEDANTIC;
+#elif defined(EIGEN_CUDA_TF32)
+  static constexpr cublasComputeType_t value = CUBLAS_COMPUTE_32F_FAST_TF32;
+#else
+  static constexpr cublasComputeType_t value = CUBLAS_COMPUTE_32F;
+#endif
+};
+template <>
+struct cuda_compute_type<double> {
+#ifdef EIGEN_NO_CUDA_TENSOR_OPS
+  static constexpr cublasComputeType_t value = CUBLAS_COMPUTE_64F_PEDANTIC;
+#else
+  static constexpr cublasComputeType_t value = CUBLAS_COMPUTE_64F;
+#endif
+};
+template <>
+struct cuda_compute_type<std::complex<float>> {
+#if defined(EIGEN_NO_CUDA_TENSOR_OPS)
+  static constexpr cublasComputeType_t value = CUBLAS_COMPUTE_32F_PEDANTIC;
+#elif defined(EIGEN_CUDA_TF32)
+  static constexpr cublasComputeType_t value = CUBLAS_COMPUTE_32F_FAST_TF32;
+#else
+  static constexpr cublasComputeType_t value = CUBLAS_COMPUTE_32F;
+#endif
+};
+template <>
+struct cuda_compute_type<std::complex<double>> {
+#ifdef EIGEN_NO_CUDA_TENSOR_OPS
+  static constexpr cublasComputeType_t value = CUBLAS_COMPUTE_64F_PEDANTIC;
+#else
+  static constexpr cublasComputeType_t value = CUBLAS_COMPUTE_64F;
+#endif
+};
+// ---- GEMM algorithm hint ----------------------------------------------------
+
+constexpr cublasGemmAlgo_t cuda_gemm_algo() {
+#ifdef EIGEN_NO_CUDA_TENSOR_OPS
+  return CUBLAS_GEMM_DEFAULT;
+#else
+  return CUBLAS_GEMM_DEFAULT_TENSOR_OP;
+#endif
+}
+
+// ---- Alpha/beta scalar type for cublasGemmEx --------------------------------
+// For standard types, alpha/beta match the scalar type.
+
+template <typename Scalar>
+struct cuda_gemm_scalar {
+  using type = Scalar;
+};
+
+// ---- Type-specific cuBLAS wrappers ------------------------------------------
+// cuBLAS uses separate functions per type (Strsm, Dtrsm, etc.).
+// These overloaded wrappers allow calling cublasXtrsm/cublasXsymm/cublasXsyrk
+// with any supported scalar type.
+
+// TRSM wrappers
+inline cublasStatus_t cublasXtrsm(cublasHandle_t h, cublasSideMode_t side, cublasFillMode_t uplo,
+                                  cublasOperation_t trans, cublasDiagType_t diag, int m, int n, const float* alpha,
+                                  const float* A, int lda, float* B, int ldb) {
+  return cublasStrsm(h, side, uplo, trans, diag, m, n, alpha, A, lda, B, ldb);
+}
+inline cublasStatus_t cublasXtrsm(cublasHandle_t h, cublasSideMode_t side, cublasFillMode_t uplo,
+                                  cublasOperation_t trans, cublasDiagType_t diag, int m, int n, const double* alpha,
+                                  const double* A, int lda, double* B, int ldb) {
+  return cublasDtrsm(h, side, uplo, trans, diag, m, n, alpha, A, lda, B, ldb);
+}
+inline cublasStatus_t cublasXtrsm(cublasHandle_t h, cublasSideMode_t side, cublasFillMode_t uplo,
+                                  cublasOperation_t trans, cublasDiagType_t diag, int m, int n,
+                                  const std::complex<float>* alpha, const std::complex<float>* A, int lda,
+                                  std::complex<float>* B, int ldb) {
+  return cublasCtrsm(h, side, uplo, trans, diag, m, n, reinterpret_cast<const cuComplex*>(alpha),
+                     reinterpret_cast<const cuComplex*>(A), lda, reinterpret_cast<cuComplex*>(B), ldb);
+}
+inline cublasStatus_t cublasXtrsm(cublasHandle_t h, cublasSideMode_t side, cublasFillMode_t uplo,
+                                  cublasOperation_t trans, cublasDiagType_t diag, int m, int n,
+                                  const std::complex<double>* alpha, const std::complex<double>* A, int lda,
+                                  std::complex<double>* B, int ldb) {
+  return cublasZtrsm(h, side, uplo, trans, diag, m, n, reinterpret_cast<const cuDoubleComplex*>(alpha),
+                     reinterpret_cast<const cuDoubleComplex*>(A), lda, reinterpret_cast<cuDoubleComplex*>(B), ldb);
+}
+
+// SYMM wrappers (real → symm, complex → hemm)
+inline cublasStatus_t cublasXsymm(cublasHandle_t h, cublasSideMode_t side, cublasFillMode_t uplo, int m, int n,
+                                  const float* alpha, const float* A, int lda, const float* B, int ldb,
+                                  const float* beta, float* C, int ldc) {
+  return cublasSsymm(h, side, uplo, m, n, alpha, A, lda, B, ldb, beta, C, ldc);
+}
+inline cublasStatus_t cublasXsymm(cublasHandle_t h, cublasSideMode_t side, cublasFillMode_t uplo, int m, int n,
+                                  const double* alpha, const double* A, int lda, const double* B, int ldb,
+                                  const double* beta, double* C, int ldc) {
+  return cublasDsymm(h, side, uplo, m, n, alpha, A, lda, B, ldb, beta, C, ldc);
+}
+inline cublasStatus_t cublasXsymm(cublasHandle_t h, cublasSideMode_t side, cublasFillMode_t uplo, int m, int n,
+                                  const std::complex<float>* alpha, const std::complex<float>* A, int lda,
+                                  const std::complex<float>* B, int ldb, const std::complex<float>* beta,
+                                  std::complex<float>* C, int ldc) {
+  return cublasChemm(h, side, uplo, m, n, reinterpret_cast<const cuComplex*>(alpha),
+                     reinterpret_cast<const cuComplex*>(A), lda, reinterpret_cast<const cuComplex*>(B), ldb,
+                     reinterpret_cast<const cuComplex*>(beta), reinterpret_cast<cuComplex*>(C), ldc);
+}
+inline cublasStatus_t cublasXsymm(cublasHandle_t h, cublasSideMode_t side, cublasFillMode_t uplo, int m, int n,
+                                  const std::complex<double>* alpha, const std::complex<double>* A, int lda,
+                                  const std::complex<double>* B, int ldb, const std::complex<double>* beta,
+                                  std::complex<double>* C, int ldc) {
+  return cublasZhemm(h, side, uplo, m, n, reinterpret_cast<const cuDoubleComplex*>(alpha),
+                     reinterpret_cast<const cuDoubleComplex*>(A), lda, reinterpret_cast<const cuDoubleComplex*>(B), ldb,
+                     reinterpret_cast<const cuDoubleComplex*>(beta), reinterpret_cast<cuDoubleComplex*>(C), ldc);
+}
+
+// SYRK wrappers (real → syrk, complex → herk)
+inline cublasStatus_t cublasXsyrk(cublasHandle_t h, cublasFillMode_t uplo, cublasOperation_t trans, int n, int k,
+                                  const float* alpha, const float* A, int lda, const float* beta, float* C, int ldc) {
+  return cublasSsyrk(h, uplo, trans, n, k, alpha, A, lda, beta, C, ldc);
+}
+inline cublasStatus_t cublasXsyrk(cublasHandle_t h, cublasFillMode_t uplo, cublasOperation_t trans, int n, int k,
+                                  const double* alpha, const double* A, int lda, const double* beta, double* C,
+                                  int ldc) {
+  return cublasDsyrk(h, uplo, trans, n, k, alpha, A, lda, beta, C, ldc);
+}
+inline cublasStatus_t cublasXsyrk(cublasHandle_t h, cublasFillMode_t uplo, cublasOperation_t trans, int n, int k,
+                                  const float* alpha, const std::complex<float>* A, int lda, const float* beta,
+                                  std::complex<float>* C, int ldc) {
+  return cublasCherk(h, uplo, trans, n, k, alpha, reinterpret_cast<const cuComplex*>(A), lda, beta,
+                     reinterpret_cast<cuComplex*>(C), ldc);
+}
+inline cublasStatus_t cublasXsyrk(cublasHandle_t h, cublasFillMode_t uplo, cublasOperation_t trans, int n, int k,
+                                  const double* alpha, const std::complex<double>* A, int lda, const double* beta,
+                                  std::complex<double>* C, int ldc) {
+  return cublasZherk(h, uplo, trans, n, k, alpha, reinterpret_cast<const cuDoubleComplex*>(A), lda, beta,
+                     reinterpret_cast<cuDoubleComplex*>(C), ldc);
+}
+
+// GEAM wrappers: C = alpha * op(A) + beta * op(B)
+// Covers transpose, scale, matrix add/subtract in one call.
+inline cublasStatus_t cublasXgeam(cublasHandle_t h, cublasOperation_t transa, cublasOperation_t transb, int m, int n,
+                                  const float* alpha, const float* A, int lda, const float* beta, const float* B,
+                                  int ldb, float* C, int ldc) {
+  return cublasSgeam(h, transa, transb, m, n, alpha, A, lda, beta, B, ldb, C, ldc);
+}
+inline cublasStatus_t cublasXgeam(cublasHandle_t h, cublasOperation_t transa, cublasOperation_t transb, int m, int n,
+                                  const double* alpha, const double* A, int lda, const double* beta, const double* B,
+                                  int ldb, double* C, int ldc) {
+  return cublasDgeam(h, transa, transb, m, n, alpha, A, lda, beta, B, ldb, C, ldc);
+}
+inline cublasStatus_t cublasXgeam(cublasHandle_t h, cublasOperation_t transa, cublasOperation_t transb, int m, int n,
+                                  const std::complex<float>* alpha, const std::complex<float>* A, int lda,
+                                  const std::complex<float>* beta, const std::complex<float>* B, int ldb,
+                                  std::complex<float>* C, int ldc) {
+  return cublasCgeam(h, transa, transb, m, n, reinterpret_cast<const cuComplex*>(alpha),
+                     reinterpret_cast<const cuComplex*>(A), lda, reinterpret_cast<const cuComplex*>(beta),
+                     reinterpret_cast<const cuComplex*>(B), ldb, reinterpret_cast<cuComplex*>(C), ldc);
+}
+inline cublasStatus_t cublasXgeam(cublasHandle_t h, cublasOperation_t transa, cublasOperation_t transb, int m, int n,
+                                  const std::complex<double>* alpha, const std::complex<double>* A, int lda,
+                                  const std::complex<double>* beta, const std::complex<double>* B, int ldb,
+                                  std::complex<double>* C, int ldc) {
+  return cublasZgeam(h, transa, transb, m, n, reinterpret_cast<const cuDoubleComplex*>(alpha),
+                     reinterpret_cast<const cuDoubleComplex*>(A), lda, reinterpret_cast<const cuDoubleComplex*>(beta),
+                     reinterpret_cast<const cuDoubleComplex*>(B), ldb, reinterpret_cast<cuDoubleComplex*>(C), ldc);
+}
+
+}  // namespace internal
+}  // namespace Eigen
+
+#endif  // EIGEN_GPU_CUBLAS_SUPPORT_H

Eigen/src/GPU/CuSolverSupport.h

@@ -0,0 +1,159 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2026 Rasmus Munk Larsen <rmlarsen@gmail.com>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+// cuSOLVER-specific support types:
+//   - cuSOLVER error-checking macro
+//   - RAII wrapper for cusolverDnParams
+//   - Scalar → cudaDataType_t mapping
+//   - (UpLo, StorageOrder) → cublasFillMode_t mapping
+//
+// Generic CUDA runtime utilities (DeviceBuffer, EIGEN_CUDA_RUNTIME_CHECK)
+// are in GpuSupport.h.
+
+#ifndef EIGEN_GPU_CUSOLVER_SUPPORT_H
+#define EIGEN_GPU_CUSOLVER_SUPPORT_H
+
+// IWYU pragma: private
+#include "./InternalHeaderCheck.h"
+
+#include "./GpuSupport.h"
+#include <cusolverDn.h>
+
+namespace Eigen {
+namespace internal {
+
+// ---- Error-checking macros --------------------------------------------------
+
+#define EIGEN_CUSOLVER_CHECK(expr)                                         \
+  do {                                                                     \
+    cusolverStatus_t _s = (expr);                                          \
+    eigen_assert(_s == CUSOLVER_STATUS_SUCCESS && "cuSOLVER call failed"); \
+  } while (0)
+
+// ---- RAII: cusolverDnParams -------------------------------------------------
+
+struct CusolverParams {
+  cusolverDnParams_t p = nullptr;
+
+  CusolverParams() { EIGEN_CUSOLVER_CHECK(cusolverDnCreateParams(&p)); }
+
+  ~CusolverParams() {
+    if (p) (void)cusolverDnDestroyParams(p);  // destructor: can't propagate
+  }
+
+  // Move-only.
+  CusolverParams(CusolverParams&& o) noexcept : p(o.p) { o.p = nullptr; }
+  CusolverParams& operator=(CusolverParams&& o) noexcept {
+    if (this != &o) {
+      if (p) (void)cusolverDnDestroyParams(p);
+      p = o.p;
+      o.p = nullptr;
+    }
+    return *this;
+  }
+
+  CusolverParams(const CusolverParams&) = delete;
+  CusolverParams& operator=(const CusolverParams&) = delete;
+};
+
+// ---- Scalar → cudaDataType_t ------------------------------------------------
+// Alias for backward compatibility. The canonical trait is cuda_data_type<> in GpuSupport.h.
+template <typename Scalar>
+using cusolver_data_type = cuda_data_type<Scalar>;
+
+// ---- (UpLo, StorageOrder) → cublasFillMode_t --------------------------------
+// cuSOLVER always interprets the matrix as column-major. A row-major matrix A
+// appears as A^T to cuSOLVER, so the upper/lower triangle is swapped.
+
+template <int UpLo, int StorageOrder>
+struct cusolver_fill_mode;
+
+template <>
+struct cusolver_fill_mode<Lower, ColMajor> {
+  static constexpr cublasFillMode_t value = CUBLAS_FILL_MODE_LOWER;
+};
+template <>
+struct cusolver_fill_mode<Upper, ColMajor> {
+  static constexpr cublasFillMode_t value = CUBLAS_FILL_MODE_UPPER;
+};
+template <>
+struct cusolver_fill_mode<Lower, RowMajor> {
+  static constexpr cublasFillMode_t value = CUBLAS_FILL_MODE_UPPER;
+};
+template <>
+struct cusolver_fill_mode<Upper, RowMajor> {
+  static constexpr cublasFillMode_t value = CUBLAS_FILL_MODE_LOWER;
+};
+
+// ---- Type-specific cuSOLVER wrappers ----------------------------------------
+// cuSOLVER does not provide generic X variants for ormqr/unmqr. These overloaded
+// wrappers dispatch to the correct type-specific function.
+// For real types: ormqr (orthogonal Q). For complex types: unmqr (unitary Q).
+
+inline cusolverStatus_t cusolverDnXormqr(cusolverDnHandle_t h, cublasSideMode_t side, cublasOperation_t trans, int m,
+                                         int n, int k, const float* A, int lda, const float* tau, float* C, int ldc,
+                                         float* work, int lwork, int* info) {
+  return cusolverDnSormqr(h, side, trans, m, n, k, A, lda, tau, C, ldc, work, lwork, info);
+}
+inline cusolverStatus_t cusolverDnXormqr(cusolverDnHandle_t h, cublasSideMode_t side, cublasOperation_t trans, int m,
+                                         int n, int k, const double* A, int lda, const double* tau, double* C, int ldc,
+                                         double* work, int lwork, int* info) {
+  return cusolverDnDormqr(h, side, trans, m, n, k, A, lda, tau, C, ldc, work, lwork, info);
+}
+inline cusolverStatus_t cusolverDnXormqr(cusolverDnHandle_t h, cublasSideMode_t side, cublasOperation_t trans, int m,
+                                         int n, int k, const std::complex<float>* A, int lda,
+                                         const std::complex<float>* tau, std::complex<float>* C, int ldc,
+                                         std::complex<float>* work, int lwork, int* info) {
+  return cusolverDnCunmqr(h, side, trans, m, n, k, reinterpret_cast<const cuComplex*>(A), lda,
+                          reinterpret_cast<const cuComplex*>(tau), reinterpret_cast<cuComplex*>(C), ldc,
+                          reinterpret_cast<cuComplex*>(work), lwork, info);
+}
+inline cusolverStatus_t cusolverDnXormqr(cusolverDnHandle_t h, cublasSideMode_t side, cublasOperation_t trans, int m,
+                                         int n, int k, const std::complex<double>* A, int lda,
+                                         const std::complex<double>* tau, std::complex<double>* C, int ldc,
+                                         std::complex<double>* work, int lwork, int* info) {
+  return cusolverDnZunmqr(h, side, trans, m, n, k, reinterpret_cast<const cuDoubleComplex*>(A), lda,
+                          reinterpret_cast<const cuDoubleComplex*>(tau), reinterpret_cast<cuDoubleComplex*>(C), ldc,
+                          reinterpret_cast<cuDoubleComplex*>(work), lwork, info);
+}
+
+// Buffer size wrappers for ormqr/unmqr.
+inline cusolverStatus_t cusolverDnXormqr_bufferSize(cusolverDnHandle_t h, cublasSideMode_t side,
+                                                    cublasOperation_t trans, int m, int n, int k, const float* A,
+                                                    int lda, const float* tau, const float* C, int ldc, int* lwork) {
+  return cusolverDnSormqr_bufferSize(h, side, trans, m, n, k, A, lda, tau, C, ldc, lwork);
+}
+inline cusolverStatus_t cusolverDnXormqr_bufferSize(cusolverDnHandle_t h, cublasSideMode_t side,
+                                                    cublasOperation_t trans, int m, int n, int k, const double* A,
+                                                    int lda, const double* tau, const double* C, int ldc, int* lwork) {
+  return cusolverDnDormqr_bufferSize(h, side, trans, m, n, k, A, lda, tau, C, ldc, lwork);
+}
+inline cusolverStatus_t cusolverDnXormqr_bufferSize(cusolverDnHandle_t h, cublasSideMode_t side,
+                                                    cublasOperation_t trans, int m, int n, int k,
+                                                    const std::complex<float>* A, int lda,
+                                                    const std::complex<float>* tau, const std::complex<float>* C,
+                                                    int ldc, int* lwork) {
+  return cusolverDnCunmqr_bufferSize(h, side, trans, m, n, k, reinterpret_cast<const cuComplex*>(A), lda,
+                                     reinterpret_cast<const cuComplex*>(tau), reinterpret_cast<const cuComplex*>(C),
+                                     ldc, lwork);
+}
+inline cusolverStatus_t cusolverDnXormqr_bufferSize(cusolverDnHandle_t h, cublasSideMode_t side,
+                                                    cublasOperation_t trans, int m, int n, int k,
+                                                    const std::complex<double>* A, int lda,
+                                                    const std::complex<double>* tau, const std::complex<double>* C,
+                                                    int ldc, int* lwork) {
+  return cusolverDnZunmqr_bufferSize(h, side, trans, m, n, k, reinterpret_cast<const cuDoubleComplex*>(A), lda,
+                                     reinterpret_cast<const cuDoubleComplex*>(tau),
+                                     reinterpret_cast<const cuDoubleComplex*>(C), ldc, lwork);
+}
+
+}  // namespace internal
+}  // namespace Eigen
+
+#endif  // EIGEN_GPU_CUSOLVER_SUPPORT_H

Eigen/src/GPU/DeviceBlasExpr.h

@@ -0,0 +1,146 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2026 Rasmus Munk Larsen <rmlarsen@gmail.com>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+// BLAS Level 3 expression types for DeviceMatrix (beyond GEMM):
+//   TrsmExpr           → cublasXtrsm   (triangular solve)
+//   SymmExpr           → cublasXsymm   (symmetric multiply, real)
+//                      → cublasXhemm   (Hermitian multiply, complex)
+//   SyrkExpr           → cublasXsyrk   (symmetric rank-k update, real)
+//                      → cublasXherk   (Hermitian rank-k update, complex)
+
+#ifndef EIGEN_GPU_DEVICE_BLAS_EXPR_H
+#define EIGEN_GPU_DEVICE_BLAS_EXPR_H
+
+// IWYU pragma: private
+#include "./InternalHeaderCheck.h"
+
+namespace Eigen {
+
+template <typename Scalar_>
+class DeviceMatrix;
+
+// ---- DeviceTriangularView ---------------------------------------------------
+// d_A.triangularView<Lower>() → view with .solve(d_B)
+
+template <typename Scalar_, int UpLo_>
+class DeviceTriangularView {
+ public:
+  using Scalar = Scalar_;
+  enum { UpLo = UpLo_ };
+
+  explicit DeviceTriangularView(const DeviceMatrix<Scalar>& m) : mat_(m) {}
+  const DeviceMatrix<Scalar>& matrix() const { return mat_; }
+
+  /** Build a TRSM solve expression. */
+  TrsmExpr<Scalar, UpLo_> solve(const DeviceMatrix<Scalar>& rhs) const { return {mat_, rhs}; }
+
+ private:
+  const DeviceMatrix<Scalar>& mat_;
+};
+
+// ---- TrsmExpr: triangularView<UpLo>().solve(B) → cublasXtrsm ---------------
+
+template <typename Scalar_, int UpLo_>
+class TrsmExpr {
+ public:
+  using Scalar = Scalar_;
+  enum { UpLo = UpLo_ };
+
+  TrsmExpr(const DeviceMatrix<Scalar>& A, const DeviceMatrix<Scalar>& B) : A_(A), B_(B) {}
+  const DeviceMatrix<Scalar>& matrix() const { return A_; }
+  const DeviceMatrix<Scalar>& rhs() const { return B_; }
+
+ private:
+  const DeviceMatrix<Scalar>& A_;
+  const DeviceMatrix<Scalar>& B_;
+};
+
+// ---- DeviceSelfAdjointView --------------------------------------------------
+// d_A.selfadjointView<Lower>() → view that can multiply: view * d_B
+
+template <typename Scalar_, int UpLo_>
+class DeviceSelfAdjointView {
+ public:
+  using Scalar = Scalar_;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+  enum { UpLo = UpLo_ };
+
+  explicit DeviceSelfAdjointView(DeviceMatrix<Scalar>& m) : mat_(m) {}
+  const DeviceMatrix<Scalar>& matrix() const { return mat_; }
+  DeviceMatrix<Scalar>& matrix() { return mat_; }
+
+  /** Rank-k update: C.selfadjointView<Lower>().rankUpdate(A, alpha)
+   * computes C = alpha * A * A^H + C (lower triangle only).
+   * Maps to cublasXsyrk (real) or cublasXherk (complex). */
+  void rankUpdate(const DeviceMatrix<Scalar>& A, RealScalar alpha = RealScalar(1));
+
+ private:
+  DeviceMatrix<Scalar>& mat_;
+};
+
+// Const variant for multiplication only (no rankUpdate).
+template <typename Scalar_, int UpLo_>
+class ConstDeviceSelfAdjointView {
+ public:
+  using Scalar = Scalar_;
+  enum { UpLo = UpLo_ };
+
+  explicit ConstDeviceSelfAdjointView(const DeviceMatrix<Scalar>& m) : mat_(m) {}
+  const DeviceMatrix<Scalar>& matrix() const { return mat_; }
+
+ private:
+  const DeviceMatrix<Scalar>& mat_;
+};
+
+// ---- SymmExpr: selfadjointView<UpLo>() * B → cublasXsymm/Xhemm ------------
+
+template <typename Scalar_, int UpLo_>
+class SymmExpr {
+ public:
+  using Scalar = Scalar_;
+  enum { UpLo = UpLo_ };
+
+  SymmExpr(const DeviceMatrix<Scalar>& A, const DeviceMatrix<Scalar>& B) : A_(A), B_(B) {}
+  const DeviceMatrix<Scalar>& matrix() const { return A_; }
+  const DeviceMatrix<Scalar>& rhs() const { return B_; }
+
+ private:
+  const DeviceMatrix<Scalar>& A_;
+  const DeviceMatrix<Scalar>& B_;
+};
+
+// operator*: DeviceSelfAdjointView * DeviceMatrix → SymmExpr (mutable and const variants)
+template <typename S, int UpLo>
+SymmExpr<S, UpLo> operator*(const DeviceSelfAdjointView<S, UpLo>& a, const DeviceMatrix<S>& b) {
+  return {a.matrix(), b};
+}
+template <typename S, int UpLo>
+SymmExpr<S, UpLo> operator*(const ConstDeviceSelfAdjointView<S, UpLo>& a, const DeviceMatrix<S>& b) {
+  return {a.matrix(), b};
+}
+
+// ---- SyrkExpr: rankUpdate(A) → cublasXsyrk/Xherk ---------------------------
+// C.rankUpdate(A) computes C += A * A^H (or A^H * A depending on convention).
+
+template <typename Scalar_, int UpLo_>
+class SyrkExpr {
+ public:
+  using Scalar = Scalar_;
+  enum { UpLo = UpLo_ };
+
+  SyrkExpr(const DeviceMatrix<Scalar>& A) : A_(A) {}
+  const DeviceMatrix<Scalar>& matrix() const { return A_; }
+
+ private:
+  const DeviceMatrix<Scalar>& A_;
+};
+
+}  // namespace Eigen
+
+#endif  // EIGEN_GPU_DEVICE_BLAS_EXPR_H

Eigen/src/GPU/DeviceDispatch.h

@@ -0,0 +1,509 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2026 Rasmus Munk Larsen <rmlarsen@gmail.com>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+// Dispatch functions that map DeviceMatrix expressions to NVIDIA library calls.
+//
+// dispatch_gemm()  — GemmExpr → cublasXgemm
+//
+// Each function documents the exact library call and parameters.
+
+#ifndef EIGEN_GPU_DEVICE_DISPATCH_H
+#define EIGEN_GPU_DEVICE_DISPATCH_H
+
+// IWYU pragma: private
+#include "./InternalHeaderCheck.h"
+
+#include "./DeviceExpr.h"
+#include "./DeviceBlasExpr.h"
+#include "./DeviceSolverExpr.h"
+#include "./GpuContext.h"
+#include "./CuSolverSupport.h"
+
+namespace Eigen {
+namespace internal {
+
+// ---- GEMM dispatch ----------------------------------------------------------
+// GemmExpr<Lhs, Rhs> → cublasGemmEx(transA, transB, m, n, k, alpha, A, lda, B, ldb, beta, C, ldc)
+//
+// The generic API cublasGemmEx handles all scalar types (float, double,
+// complex<float>, complex<double>) via cudaDataType_t.
+
+template <typename Lhs, typename Rhs>
+void dispatch_gemm(
+    GpuContext& ctx, DeviceMatrix<typename device_expr_traits<Lhs>::scalar_type>& dst, const GemmExpr<Lhs, Rhs>& expr,
+    typename device_expr_traits<Lhs>::scalar_type beta_val,
+    typename device_expr_traits<Lhs>::scalar_type alpha_scale = typename device_expr_traits<Lhs>::scalar_type(1)) {
+  using Scalar = typename device_expr_traits<Lhs>::scalar_type;
+  using traits_lhs = device_expr_traits<Lhs>;
+  using traits_rhs = device_expr_traits<Rhs>;
+
+  const DeviceMatrix<Scalar>& A = traits_lhs::matrix(expr.lhs());
+  const DeviceMatrix<Scalar>& B = traits_rhs::matrix(expr.rhs());
+
+  constexpr cublasOperation_t transA = to_cublas_op(traits_lhs::op);
+  constexpr cublasOperation_t transB = to_cublas_op(traits_rhs::op);
+
+  // GEMM dimensions: C(m,n) = op(A)(m,k) * op(B)(k,n)
+  // op(A) has dimensions (A.rows, A.cols) if NoTrans, (A.cols, A.rows) if Trans/ConjTrans.
+  const int64_t m = (traits_lhs::op == GpuOp::NoTrans) ? A.rows() : A.cols();
+  const int64_t k = (traits_lhs::op == GpuOp::NoTrans) ? A.cols() : A.rows();
+  const int64_t n = (traits_rhs::op == GpuOp::NoTrans) ? B.cols() : B.rows();
+  const int64_t rhs_k = (traits_rhs::op == GpuOp::NoTrans) ? B.rows() : B.cols();
+
+  eigen_assert(k == rhs_k && "DeviceMatrix GEMM dimension mismatch");
+
+  const int64_t lda = A.rows();
+  const int64_t ldb = B.rows();
+
+  // Serialize all accesses to the destination buffer on this stream.
+  if (!dst.empty()) {
+    dst.waitReady(ctx.stream());
+  }
+
+  // Allocate or resize destination.
+  const bool resized = dst.empty() || dst.rows() != m || dst.cols() != n;
+  if (resized) {
+    dst.resize(m, n);
+  }
+  const int64_t ldc = dst.rows();
+
+  // cuBLAS requires alpha/beta as float for half/bfloat16 inputs.
+  using GemmScalar = typename cuda_gemm_scalar<Scalar>::type;
+  GemmScalar alpha_gval =
+      static_cast<GemmScalar>(alpha_scale * traits_lhs::alpha(expr.lhs()) * traits_rhs::alpha(expr.rhs()));
+  GemmScalar beta_gval = static_cast<GemmScalar>(beta_val);
+
+  // Wait for operands to be ready on this stream.
+  A.waitReady(ctx.stream());
+  B.waitReady(ctx.stream());
+
+  // If there is no existing valid destination to accumulate into, treat it as
+  // zero rather than reading uninitialized memory.
+  if (resized && beta_gval != GemmScalar(0) && dst.sizeInBytes() > 0) {
+    EIGEN_CUDA_RUNTIME_CHECK(cudaMemsetAsync(dst.data(), 0, dst.sizeInBytes(), ctx.stream()));
+  }
+
+  constexpr cudaDataType_t dtype = cuda_data_type<Scalar>::value;
+  constexpr cublasComputeType_t compute = cuda_compute_type<Scalar>::value;
+
+  EIGEN_CUBLAS_CHECK(cublasGemmEx(ctx.cublasHandle(), transA, transB, static_cast<int>(m), static_cast<int>(n),
+                                  static_cast<int>(k), &alpha_gval, A.data(), dtype, static_cast<int>(lda), B.data(),
+                                  dtype, static_cast<int>(ldb), &beta_gval, dst.data(), dtype, static_cast<int>(ldc),
+                                  compute, cuda_gemm_algo()));
+
+  dst.recordReady(ctx.stream());
+}
+
+// ---- LLT solve dispatch -----------------------------------------------------
+// LltSolveExpr → cusolverDnXpotrf (factorize) + cusolverDnXpotrs (solve).
+// No caching — factor and workspace are temporary. Syncs to check info.
+
+template <typename Scalar, int UpLo>
+void dispatch_llt_solve(GpuContext& ctx, DeviceMatrix<Scalar>& dst, const LltSolveExpr<Scalar, UpLo>& expr) {
+  const DeviceMatrix<Scalar>& A = expr.matrix();
+  const DeviceMatrix<Scalar>& B = expr.rhs();
+
+  eigen_assert(A.rows() == A.cols() && "LLT requires a square matrix");
+  eigen_assert(B.rows() == A.rows() && "LLT solve: RHS rows must match matrix size");
+
+  const Index n = A.rows();
+  const int64_t nrhs = static_cast<int64_t>(B.cols());
+
+  // Zero-size fast paths: no work, just resize dst.
+  // Wait on dst before resize to avoid freeing memory another stream is using.
+  if (n == 0 || nrhs == 0) {
+    if (!dst.empty()) dst.waitReady(ctx.stream());
+    dst.resize(n == 0 ? 0 : n, B.cols());
+    return;
+  }
+
+  A.waitReady(ctx.stream());
+  B.waitReady(ctx.stream());
+  if (!dst.empty()) dst.waitReady(ctx.stream());
+
+  constexpr cudaDataType_t dtype = cuda_data_type<Scalar>::value;
+  constexpr cublasFillMode_t uplo = cusolver_fill_mode<UpLo, ColMajor>::value;
+  const int64_t lda = static_cast<int64_t>(A.rows());
+  const int64_t ldb = static_cast<int64_t>(B.rows());
+
+  const size_t mat_bytes = static_cast<size_t>(lda) * static_cast<size_t>(n) * sizeof(Scalar);
+  const size_t rhs_bytes = static_cast<size_t>(ldb) * static_cast<size_t>(nrhs) * sizeof(Scalar);
+
+  // D2D copy A → factor buffer (potrf is in-place).
+  DeviceBuffer d_factor(mat_bytes);
+  EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpyAsync(d_factor.ptr, A.data(), mat_bytes, cudaMemcpyDeviceToDevice, ctx.stream()));
+
+  // Query workspace and factorize.
+  CusolverParams params;
+  DeviceBuffer d_factorize_info(sizeof(int));
+  size_t dev_ws = 0, host_ws = 0;
+  EIGEN_CUSOLVER_CHECK(cusolverDnXpotrf_bufferSize(ctx.cusolverHandle(), params.p, uplo, static_cast<int64_t>(n), dtype,
+                                                   d_factor.ptr, lda, dtype, &dev_ws, &host_ws));
+
+  DeviceBuffer d_workspace(dev_ws);
+  std::vector<char> h_workspace(host_ws);
+
+  EIGEN_CUSOLVER_CHECK(cusolverDnXpotrf(
+      ctx.cusolverHandle(), params.p, uplo, static_cast<int64_t>(n), dtype, d_factor.ptr, lda, dtype, d_workspace.ptr,
+      dev_ws, host_ws > 0 ? h_workspace.data() : nullptr, host_ws, static_cast<int*>(d_factorize_info.ptr)));
+
+  // Check factorization info before proceeding to solve.
+  int factorize_info = 0;
+  EIGEN_CUDA_RUNTIME_CHECK(
+      cudaMemcpyAsync(&factorize_info, d_factorize_info.ptr, sizeof(int), cudaMemcpyDeviceToHost, ctx.stream()));
+  EIGEN_CUDA_RUNTIME_CHECK(cudaStreamSynchronize(ctx.stream()));
+  eigen_assert(factorize_info == 0 && "cuSOLVER LLT factorization failed (matrix not positive definite)");
+
+  // D2D copy B → dst (potrs is in-place on the RHS).
+  dst.resize(n, B.cols());
+  EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpyAsync(dst.data(), B.data(), rhs_bytes, cudaMemcpyDeviceToDevice, ctx.stream()));
+
+  // Solve.
+  DeviceBuffer d_solve_info(sizeof(int));
+  EIGEN_CUSOLVER_CHECK(cusolverDnXpotrs(ctx.cusolverHandle(), params.p, uplo, static_cast<int64_t>(n), nrhs, dtype,
+                                        d_factor.ptr, lda, dtype, dst.data(), static_cast<int64_t>(dst.rows()),
+                                        static_cast<int*>(d_solve_info.ptr)));
+
+  // Sync to ensure workspace locals can be freed safely.
+  int solve_info = 0;
+  EIGEN_CUDA_RUNTIME_CHECK(
+      cudaMemcpyAsync(&solve_info, d_solve_info.ptr, sizeof(int), cudaMemcpyDeviceToHost, ctx.stream()));
+  EIGEN_CUDA_RUNTIME_CHECK(cudaStreamSynchronize(ctx.stream()));
+  eigen_assert(solve_info == 0 && "cuSOLVER LLT solve failed");
+
+  dst.recordReady(ctx.stream());
+}
+
+// ---- LU solve dispatch ------------------------------------------------------
+// LuSolveExpr → cusolverDnXgetrf (factorize) + cusolverDnXgetrs (solve).
+
+template <typename Scalar>
+void dispatch_lu_solve(GpuContext& ctx, DeviceMatrix<Scalar>& dst, const LuSolveExpr<Scalar>& expr) {
+  const DeviceMatrix<Scalar>& A = expr.matrix();
+  const DeviceMatrix<Scalar>& B = expr.rhs();
+
+  eigen_assert(A.rows() == A.cols() && "LU requires a square matrix");
+  eigen_assert(B.rows() == A.rows() && "LU solve: RHS rows must match matrix size");
+
+  const Index n = A.rows();
+  const int64_t nrhs = static_cast<int64_t>(B.cols());
+
+  if (n == 0 || nrhs == 0) {
+    if (!dst.empty()) dst.waitReady(ctx.stream());
+    dst.resize(n == 0 ? 0 : n, B.cols());
+    return;
+  }
+
+  A.waitReady(ctx.stream());
+  B.waitReady(ctx.stream());
+  if (!dst.empty()) dst.waitReady(ctx.stream());
+
+  constexpr cudaDataType_t dtype = cuda_data_type<Scalar>::value;
+  const int64_t lda = static_cast<int64_t>(A.rows());
+  const int64_t ldb = static_cast<int64_t>(B.rows());
+
+  const size_t mat_bytes = static_cast<size_t>(lda) * static_cast<size_t>(n) * sizeof(Scalar);
+  const size_t rhs_bytes = static_cast<size_t>(ldb) * static_cast<size_t>(nrhs) * sizeof(Scalar);
+  const size_t ipiv_bytes = static_cast<size_t>(n) * sizeof(int64_t);
+
+  // D2D copy A → LU buffer (getrf is in-place).
+  DeviceBuffer d_lu(mat_bytes);
+  EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpyAsync(d_lu.ptr, A.data(), mat_bytes, cudaMemcpyDeviceToDevice, ctx.stream()));
+
+  DeviceBuffer d_ipiv(ipiv_bytes);
+
+  // Query workspace and factorize.
+  CusolverParams params;
+  DeviceBuffer d_factorize_info(sizeof(int));
+  size_t dev_ws = 0, host_ws = 0;
+  EIGEN_CUSOLVER_CHECK(cusolverDnXgetrf_bufferSize(ctx.cusolverHandle(), params.p, static_cast<int64_t>(n),
+                                                   static_cast<int64_t>(n), dtype, d_lu.ptr, lda, dtype, &dev_ws,
+                                                   &host_ws));
+
+  DeviceBuffer d_workspace(dev_ws);
+  std::vector<char> h_workspace(host_ws);
+
+  EIGEN_CUSOLVER_CHECK(
+      cusolverDnXgetrf(ctx.cusolverHandle(), params.p, static_cast<int64_t>(n), static_cast<int64_t>(n), dtype,
+                       d_lu.ptr, lda, static_cast<int64_t*>(d_ipiv.ptr), dtype, d_workspace.ptr, dev_ws,
+                       host_ws > 0 ? h_workspace.data() : nullptr, host_ws, static_cast<int*>(d_factorize_info.ptr)));
+
+  // Check factorization info before proceeding to solve.
+  int factorize_info = 0;
+  EIGEN_CUDA_RUNTIME_CHECK(
+      cudaMemcpyAsync(&factorize_info, d_factorize_info.ptr, sizeof(int), cudaMemcpyDeviceToHost, ctx.stream()));
+  EIGEN_CUDA_RUNTIME_CHECK(cudaStreamSynchronize(ctx.stream()));
+  eigen_assert(factorize_info == 0 && "cuSOLVER LU factorization failed (singular matrix)");
+
+  // D2D copy B → dst (getrs is in-place on the RHS).
+  dst.resize(n, B.cols());
+  EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpyAsync(dst.data(), B.data(), rhs_bytes, cudaMemcpyDeviceToDevice, ctx.stream()));
+
+  // Solve (NoTranspose).
+  DeviceBuffer d_solve_info(sizeof(int));
+  EIGEN_CUSOLVER_CHECK(cusolverDnXgetrs(ctx.cusolverHandle(), params.p, CUBLAS_OP_N, static_cast<int64_t>(n), nrhs,
+                                        dtype, d_lu.ptr, lda, static_cast<const int64_t*>(d_ipiv.ptr), dtype,
+                                        dst.data(), static_cast<int64_t>(dst.rows()),
+                                        static_cast<int*>(d_solve_info.ptr)));
+
+  // Sync to ensure workspace locals can be freed safely.
+  int solve_info = 0;
+  EIGEN_CUDA_RUNTIME_CHECK(
+      cudaMemcpyAsync(&solve_info, d_solve_info.ptr, sizeof(int), cudaMemcpyDeviceToHost, ctx.stream()));
+  EIGEN_CUDA_RUNTIME_CHECK(cudaStreamSynchronize(ctx.stream()));
+  eigen_assert(solve_info == 0 && "cuSOLVER LU solve failed");
+
+  dst.recordReady(ctx.stream());
+}
+
+// ---- TRSM dispatch ----------------------------------------------------------
+// TrsmExpr → cublasXtrsm: solve op(A) * X = B where A is triangular.
+// Side=Left, Diag=NonUnit. A is square, B is n×nrhs.
+
+template <typename Scalar, int UpLo>
+void dispatch_trsm(GpuContext& ctx, DeviceMatrix<Scalar>& dst, const TrsmExpr<Scalar, UpLo>& expr) {
+  const DeviceMatrix<Scalar>& A = expr.matrix();
+  const DeviceMatrix<Scalar>& B = expr.rhs();
+
+  eigen_assert(A.rows() == A.cols() && "TRSM requires a square triangular matrix");
+  eigen_assert(B.rows() == A.rows() && "TRSM: RHS rows must match matrix size");
+
+  const int n = static_cast<int>(A.rows());
+  const int nrhs = static_cast<int>(B.cols());
+
+  if (n == 0 || nrhs == 0) {
+    if (!dst.empty()) dst.waitReady(ctx.stream());
+    dst.resize(n == 0 ? 0 : n, B.cols());
+    return;
+  }
+
+  A.waitReady(ctx.stream());
+  B.waitReady(ctx.stream());
+  if (!dst.empty()) dst.waitReady(ctx.stream());
+
+  // D2D copy B → dst (trsm is in-place on the RHS).
+  dst.resize(n, B.cols());
+  const size_t rhs_bytes = static_cast<size_t>(dst.rows()) * static_cast<size_t>(nrhs) * sizeof(Scalar);
+  EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpyAsync(dst.data(), B.data(), rhs_bytes, cudaMemcpyDeviceToDevice, ctx.stream()));
+
+  constexpr cublasFillMode_t uplo = (UpLo == Lower) ? CUBLAS_FILL_MODE_LOWER : CUBLAS_FILL_MODE_UPPER;
+  Scalar alpha(1);
+
+  EIGEN_CUBLAS_CHECK(cublasXtrsm(ctx.cublasHandle(), CUBLAS_SIDE_LEFT, uplo, CUBLAS_OP_N, CUBLAS_DIAG_NON_UNIT, n, nrhs,
+                                 &alpha, A.data(), static_cast<int>(A.rows()), dst.data(),
+                                 static_cast<int>(dst.rows())));
+
+  dst.recordReady(ctx.stream());
+}
+
+// ---- SYMM/HEMM dispatch -----------------------------------------------------
+// SymmExpr → cublasXsymm (real) or cublasXhemm (complex).
+// C = A * B where A is symmetric/Hermitian. Side=Left.
+
+template <typename Scalar, int UpLo>
+void dispatch_symm(GpuContext& ctx, DeviceMatrix<Scalar>& dst, const SymmExpr<Scalar, UpLo>& expr) {
+  const DeviceMatrix<Scalar>& A = expr.matrix();
+  const DeviceMatrix<Scalar>& B = expr.rhs();
+
+  eigen_assert(A.rows() == A.cols() && "SYMM requires a square matrix");
+  eigen_assert(B.rows() == A.rows() && "SYMM: RHS rows must match matrix size");
+
+  const int m = static_cast<int>(A.rows());
+  const int n = static_cast<int>(B.cols());
+
+  if (m == 0 || n == 0) {
+    if (!dst.empty()) dst.waitReady(ctx.stream());
+    dst.resize(m == 0 ? 0 : m, B.cols());
+    return;
+  }
+
+  A.waitReady(ctx.stream());
+  B.waitReady(ctx.stream());
+  if (!dst.empty()) dst.waitReady(ctx.stream());
+
+  dst.resize(m, n);
+
+  constexpr cublasFillMode_t uplo = (UpLo == Lower) ? CUBLAS_FILL_MODE_LOWER : CUBLAS_FILL_MODE_UPPER;
+  Scalar alpha(1), beta(0);
+
+  EIGEN_CUBLAS_CHECK(cublasXsymm(ctx.cublasHandle(), CUBLAS_SIDE_LEFT, uplo, m, n, &alpha, A.data(),
+                                 static_cast<int>(A.rows()), B.data(), static_cast<int>(B.rows()), &beta, dst.data(),
+                                 static_cast<int>(dst.rows())));
+
+  dst.recordReady(ctx.stream());
+}
+
+// ---- SYRK/HERK dispatch -----------------------------------------------------
+// SyrkExpr → cublasXsyrk (real) or cublasXherk (complex).
+// C = alpha * A * A^H + beta * C. UpLo specifies which triangle of C is stored.
+
+template <typename Scalar, int UpLo>
+void dispatch_syrk(GpuContext& ctx, DeviceMatrix<Scalar>& dst, const SyrkExpr<Scalar, UpLo>& expr,
+                   typename NumTraits<Scalar>::Real alpha_val, typename NumTraits<Scalar>::Real beta_val) {
+  using RealScalar = typename NumTraits<Scalar>::Real;
+  const DeviceMatrix<Scalar>& A = expr.matrix();
+
+  const int n = static_cast<int>(A.rows());
+  const int k = static_cast<int>(A.cols());
+
+  if (n == 0) {
+    if (!dst.empty()) dst.waitReady(ctx.stream());
+    dst.resize(0, 0);
+    return;
+  }
+
+  A.waitReady(ctx.stream());
+  if (!dst.empty()) dst.waitReady(ctx.stream());
+
+  if (dst.empty() || dst.rows() != n || dst.cols() != n) {
+    dst.resize(n, n);
+    if (beta_val != RealScalar(0)) {
+      EIGEN_CUDA_RUNTIME_CHECK(cudaMemsetAsync(dst.data(), 0, dst.sizeInBytes(), ctx.stream()));
+    }
+  }
+
+  constexpr cublasFillMode_t uplo = (UpLo == Lower) ? CUBLAS_FILL_MODE_LOWER : CUBLAS_FILL_MODE_UPPER;
+
+  EIGEN_CUBLAS_CHECK(cublasXsyrk(ctx.cublasHandle(), uplo, CUBLAS_OP_N, n, k, &alpha_val, A.data(),
+                                 static_cast<int>(A.rows()), &beta_val, dst.data(), static_cast<int>(dst.rows())));
+
+  dst.recordReady(ctx.stream());
+}
+
+}  // namespace internal
+
+// ---- DeviceAssignment: d_C.device(ctx) = expr ------------------------------
+// Returned by DeviceMatrix::device(ctx). Dispatches expressions to library calls.
+
+template <typename Scalar_>
+class DeviceAssignment {
+ public:
+  using Scalar = Scalar_;
+
+  DeviceAssignment(DeviceMatrix<Scalar>& dst, GpuContext& ctx) : dst_(dst), ctx_(ctx) {}
+
+  // operator= dispatches GEMM with beta=0 (overwrite).
+  template <typename Lhs, typename Rhs>
+  DeviceMatrix<Scalar>& operator=(const GemmExpr<Lhs, Rhs>& expr) {
+    internal::dispatch_gemm(ctx_, dst_, expr, Scalar(0));
+    return dst_;
+  }
+
+  // operator+= dispatches GEMM with beta=1 (accumulate).
+  template <typename Lhs, typename Rhs>
+  DeviceMatrix<Scalar>& operator+=(const GemmExpr<Lhs, Rhs>& expr) {
+    internal::dispatch_gemm(ctx_, dst_, expr, Scalar(1));
+    return dst_;
+  }
+
+  // operator-= dispatches GEMM with negated alpha, beta=1: C = C - alpha*op(A)*op(B).
+  template <typename Lhs, typename Rhs>
+  DeviceMatrix<Scalar>& operator-=(const GemmExpr<Lhs, Rhs>& expr) {
+    internal::dispatch_gemm(ctx_, dst_, expr, Scalar(1), Scalar(-1));
+    return dst_;
+  }
+
+  // operator= dispatches LLT solve (potrf + potrs).
+  template <int UpLo>
+  DeviceMatrix<Scalar>& operator=(const LltSolveExpr<Scalar, UpLo>& expr) {
+    internal::dispatch_llt_solve(ctx_, dst_, expr);
+    return dst_;
+  }
+
+  // operator= dispatches LU solve (getrf + getrs).
+  DeviceMatrix<Scalar>& operator=(const LuSolveExpr<Scalar>& expr) {
+    internal::dispatch_lu_solve(ctx_, dst_, expr);
+    return dst_;
+  }
+
+  // operator= dispatches TRSM (triangular solve).
+  template <int UpLo>
+  DeviceMatrix<Scalar>& operator=(const TrsmExpr<Scalar, UpLo>& expr) {
+    internal::dispatch_trsm(ctx_, dst_, expr);
+    return dst_;
+  }
+
+  // operator= dispatches SYMM/HEMM (symmetric/Hermitian multiply).
+  template <int UpLo>
+  DeviceMatrix<Scalar>& operator=(const SymmExpr<Scalar, UpLo>& expr) {
+    internal::dispatch_symm(ctx_, dst_, expr);
+    return dst_;
+  }
+
+  // Catch-all: static_assert for unsupported expressions.
+  template <typename Expr>
+  DeviceMatrix<Scalar>& operator=(const Expr&) {
+    static_assert(sizeof(Expr) == 0,
+                  "DeviceMatrix expression not supported: no cuBLAS/cuSOLVER mapping. "
+                  "Supported: GEMM (A*B), TRSM (.triangularView().solve()), "
+                  "SYMM (.selfadjointView()*B), LLT (.llt().solve()), LU (.lu().solve()).");
+    return dst_;
+  }
+
+ private:
+  DeviceMatrix<Scalar>& dst_;
+  GpuContext& ctx_;
+};
+
+// ---- Out-of-line DeviceMatrix expression operator= definitions -------------
+// These are declared in DeviceMatrix.h but defined here because they need
+// GpuContext::threadLocal() which requires the full GpuContext definition.
+
+template <typename Scalar_>
+template <typename Lhs, typename Rhs>
+DeviceMatrix<Scalar_>& DeviceMatrix<Scalar_>::operator=(const GemmExpr<Lhs, Rhs>& expr) {
+  device(GpuContext::threadLocal()) = expr;
+  return *this;
+}
+
+template <typename Scalar_>
+template <typename Lhs, typename Rhs>
+DeviceMatrix<Scalar_>& DeviceMatrix<Scalar_>::operator+=(const GemmExpr<Lhs, Rhs>& expr) {
+  device(GpuContext::threadLocal()) += expr;
+  return *this;
+}
+
+template <typename Scalar_>
+template <int UpLo>
+DeviceMatrix<Scalar_>& DeviceMatrix<Scalar_>::operator=(const LltSolveExpr<Scalar_, UpLo>& expr) {
+  device(GpuContext::threadLocal()) = expr;
+  return *this;
+}
+
+template <typename Scalar_>
+DeviceMatrix<Scalar_>& DeviceMatrix<Scalar_>::operator=(const LuSolveExpr<Scalar_>& expr) {
+  device(GpuContext::threadLocal()) = expr;
+  return *this;
+}
+
+template <typename Scalar_>
+template <int UpLo>
+DeviceMatrix<Scalar_>& DeviceMatrix<Scalar_>::operator=(const TrsmExpr<Scalar_, UpLo>& expr) {
+  device(GpuContext::threadLocal()) = expr;
+  return *this;
+}
+
+template <typename Scalar_>
+template <int UpLo>
+DeviceMatrix<Scalar_>& DeviceMatrix<Scalar_>::operator=(const SymmExpr<Scalar_, UpLo>& expr) {
+  device(GpuContext::threadLocal()) = expr;
+  return *this;
+}
+
+// DeviceSelfAdjointView::rankUpdate — defined here because it needs GpuContext.
+template <typename Scalar_, int UpLo_>
+void DeviceSelfAdjointView<Scalar_, UpLo_>::rankUpdate(const DeviceMatrix<Scalar_>& A, RealScalar alpha) {
+  SyrkExpr<Scalar_, UpLo_> expr(A);
+  RealScalar beta = matrix().empty() ? RealScalar(0) : RealScalar(1);
+  internal::dispatch_syrk(GpuContext::threadLocal(), matrix(), expr, alpha, beta);
+}
+
+}  // namespace Eigen
+
+#endif  // EIGEN_GPU_DEVICE_DISPATCH_H

Eigen/src/GPU/DeviceExpr.h

@@ -0,0 +1,224 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2026 Rasmus Munk Larsen <rmlarsen@gmail.com>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+// Lightweight expression types for DeviceMatrix operations.
+//
+// These are NOT Eigen expression templates. Each type maps 1:1 to a single
+// NVIDIA library call (cuBLAS or cuSOLVER). There is no coefficient-level
+// evaluation, no lazy fusion, no packet operations.
+//
+// Expression types:
+//   DeviceAdjointView<S>  — d_A.adjoint()  → marks ConjTrans for GEMM
+//   DeviceTransposeView<S> — d_A.transpose() → marks Trans for GEMM
+//   DeviceScaled<Expr>    — alpha * expr    → carries scalar factor
+//   GemmExpr<Lhs, Rhs>   — lhs * rhs       → dispatches to cublasXgemm
+
+#ifndef EIGEN_GPU_DEVICE_EXPR_H
+#define EIGEN_GPU_DEVICE_EXPR_H
+
+// IWYU pragma: private
+#include "./InternalHeaderCheck.h"
+
+#include "./CuBlasSupport.h"
+
+namespace Eigen {
+
+// Forward declaration.
+template <typename Scalar_>
+class DeviceMatrix;
+
+namespace internal {
+
+// ---- Traits: extract operation info from expression types -------------------
+
+// Default: a DeviceMatrix is NoTrans.
+template <typename T>
+struct device_expr_traits {
+  static constexpr bool is_device_expr = false;
+};
+
+template <typename Scalar>
+struct device_expr_traits<DeviceMatrix<Scalar>> {
+  using scalar_type = Scalar;
+  static constexpr GpuOp op = GpuOp::NoTrans;
+  static constexpr bool is_device_expr = true;
+  static const DeviceMatrix<Scalar>& matrix(const DeviceMatrix<Scalar>& x) { return x; }
+  static Scalar alpha(const DeviceMatrix<Scalar>&) { return Scalar(1); }
+};
+
+}  // namespace internal
+
+// ---- DeviceAdjointView: marks ConjTrans ------------------------------------
+// Returned by DeviceMatrix::adjoint(). Maps to cublasXgemm transA/B = C.
+
+template <typename Scalar_>
+class DeviceAdjointView {
+ public:
+  using Scalar = Scalar_;
+  explicit DeviceAdjointView(const DeviceMatrix<Scalar>& m) : mat_(m) {}
+  const DeviceMatrix<Scalar>& matrix() const { return mat_; }
+
+ private:
+  const DeviceMatrix<Scalar>& mat_;
+};
+
+namespace internal {
+template <typename Scalar>
+struct device_expr_traits<DeviceAdjointView<Scalar>> {
+  using scalar_type = Scalar;
+  static constexpr GpuOp op = GpuOp::ConjTrans;
+  static constexpr bool is_device_expr = true;
+  static const DeviceMatrix<Scalar>& matrix(const DeviceAdjointView<Scalar>& x) { return x.matrix(); }
+  static Scalar alpha(const DeviceAdjointView<Scalar>&) { return Scalar(1); }
+};
+}  // namespace internal
+
+// ---- DeviceTransposeView: marks Trans --------------------------------------
+// Returned by DeviceMatrix::transpose(). Maps to cublasXgemm transA/B = T.
+
+template <typename Scalar_>
+class DeviceTransposeView {
+ public:
+  using Scalar = Scalar_;
+  explicit DeviceTransposeView(const DeviceMatrix<Scalar>& m) : mat_(m) {}
+  const DeviceMatrix<Scalar>& matrix() const { return mat_; }
+
+ private:
+  const DeviceMatrix<Scalar>& mat_;
+};
+
+namespace internal {
+template <typename Scalar>
+struct device_expr_traits<DeviceTransposeView<Scalar>> {
+  using scalar_type = Scalar;
+  static constexpr GpuOp op = GpuOp::Trans;
+  static constexpr bool is_device_expr = true;
+  static const DeviceMatrix<Scalar>& matrix(const DeviceTransposeView<Scalar>& x) { return x.matrix(); }
+  static Scalar alpha(const DeviceTransposeView<Scalar>&) { return Scalar(1); }
+};
+}  // namespace internal
+
+// ---- DeviceScaled: alpha * expr --------------------------------------------
+// Returned by operator*(Scalar, DeviceMatrix/View). Carries the scalar factor.
+
+template <typename Inner>
+class DeviceScaled {
+ public:
+  using Scalar = typename internal::device_expr_traits<Inner>::scalar_type;
+  DeviceScaled(Scalar alpha, const Inner& inner) : alpha_(alpha), inner_(inner) {}
+  Scalar scalar() const { return alpha_; }
+  const Inner& inner() const { return inner_; }
+
+ private:
+  Scalar alpha_;
+  const Inner& inner_;
+};
+
+namespace internal {
+template <typename Inner>
+struct device_expr_traits<DeviceScaled<Inner>> {
+  using scalar_type = typename device_expr_traits<Inner>::scalar_type;
+  static constexpr GpuOp op = device_expr_traits<Inner>::op;
+  static constexpr bool is_device_expr = true;
+  static const DeviceMatrix<scalar_type>& matrix(const DeviceScaled<Inner>& x) {
+    return device_expr_traits<Inner>::matrix(x.inner());
+  }
+  static scalar_type alpha(const DeviceScaled<Inner>& x) {
+    return x.scalar() * device_expr_traits<Inner>::alpha(x.inner());
+  }
+};
+}  // namespace internal
+
+// ---- GemmExpr: lhs * rhs → cublasXgemm ------------------------------------
+// Returned by operator*(lhs_expr, rhs_expr). Dispatches to cuBLAS GEMM.
+
+template <typename Lhs, typename Rhs>
+class GemmExpr {
+ public:
+  using Scalar = typename internal::device_expr_traits<Lhs>::scalar_type;
+  static_assert(std::is_same<Scalar, typename internal::device_expr_traits<Rhs>::scalar_type>::value,
+                "DeviceMatrix GEMM: LHS and RHS must have the same scalar type");
+
+  GemmExpr(const Lhs& lhs, const Rhs& rhs) : lhs_(lhs), rhs_(rhs) {}
+  const Lhs& lhs() const { return lhs_; }
+  const Rhs& rhs() const { return rhs_; }
+
+ private:
+  // Stored by reference. Expression objects must not outlive their operands.
+  // This is safe for the one-liner pattern (d_C = d_A * d_B) since all
+  // temporaries live until the semicolon.
+  const Lhs& lhs_;
+  const Rhs& rhs_;
+};
+
+// ---- Free operator* overloads that produce GemmExpr ------------------------
+// These cover: DM*DM, Adj*DM, DM*Adj, Trans*DM, DM*Trans, Scaled*DM, etc.
+
+// DeviceMatrix * DeviceMatrix
+template <typename S>
+GemmExpr<DeviceMatrix<S>, DeviceMatrix<S>> operator*(const DeviceMatrix<S>& a, const DeviceMatrix<S>& b) {
+  return {a, b};
+}
+
+// AdjointView * DeviceMatrix
+template <typename S>
+GemmExpr<DeviceAdjointView<S>, DeviceMatrix<S>> operator*(const DeviceAdjointView<S>& a, const DeviceMatrix<S>& b) {
+  return {a, b};
+}
+
+// DeviceMatrix * AdjointView
+template <typename S>
+GemmExpr<DeviceMatrix<S>, DeviceAdjointView<S>> operator*(const DeviceMatrix<S>& a, const DeviceAdjointView<S>& b) {
+  return {a, b};
+}
+
+// TransposeView * DeviceMatrix
+template <typename S>
+GemmExpr<DeviceTransposeView<S>, DeviceMatrix<S>> operator*(const DeviceTransposeView<S>& a, const DeviceMatrix<S>& b) {
+  return {a, b};
+}
+
+// DeviceMatrix * TransposeView
+template <typename S>
+GemmExpr<DeviceMatrix<S>, DeviceTransposeView<S>> operator*(const DeviceMatrix<S>& a, const DeviceTransposeView<S>& b) {
+  return {a, b};
+}
+
+// Scaled * DeviceMatrix
+template <typename Inner, typename S>
+GemmExpr<DeviceScaled<Inner>, DeviceMatrix<S>> operator*(const DeviceScaled<Inner>& a, const DeviceMatrix<S>& b) {
+  return {a, b};
+}
+
+// DeviceMatrix * Scaled
+template <typename S, typename Inner>
+GemmExpr<DeviceMatrix<S>, DeviceScaled<Inner>> operator*(const DeviceMatrix<S>& a, const DeviceScaled<Inner>& b) {
+  return {a, b};
+}
+
+// ---- Scalar * DeviceMatrix / View → DeviceScaled ---------------------------
+
+template <typename S>
+DeviceScaled<DeviceMatrix<S>> operator*(S alpha, const DeviceMatrix<S>& m) {
+  return {alpha, m};
+}
+
+template <typename S>
+DeviceScaled<DeviceAdjointView<S>> operator*(S alpha, const DeviceAdjointView<S>& m) {
+  return {alpha, m};
+}
+
+template <typename S>
+DeviceScaled<DeviceTransposeView<S>> operator*(S alpha, const DeviceTransposeView<S>& m) {
+  return {alpha, m};
+}
+
+}  // namespace Eigen
+
+#endif  // EIGEN_GPU_DEVICE_EXPR_H

Eigen/src/GPU/DeviceMatrix.h

@@ -0,0 +1,503 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2026 Rasmus Munk Larsen <rmlarsen@gmail.com>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+// Typed RAII wrapper for a dense matrix in GPU device memory.
+//
+// DeviceMatrix<Scalar> holds a column-major matrix on the GPU with tracked
+// dimensions. Always dense (leading dimension = rows). It can be passed to GPU solvers
+// (GpuLLT, GpuLU, future cuBLAS/cuDSS) without host round-trips.
+//
+// Cross-stream safety is automatic: an internal CUDA event tracks when the
+// last write completed. Consumers on a different stream wait on that event
+// before reading.
+//
+// Usage:
+//   auto d_A = DeviceMatrix<double>::fromHost(A);   // upload (sync)
+//   GpuLLT<double> llt;
+//   llt.compute(d_A);                                // factor on device
+//   auto d_X = llt.solve(d_B);                       // async, no sync
+//   MatrixXd X = d_X.toHost();                       // download + block
+//
+// Async variants:
+//   auto d_A = DeviceMatrix<double>::fromHostAsync(A.data(), n, n, stream);
+//   auto transfer = d_X.toHostAsync(stream);         // enqueue D2H
+//   // ... overlap with other work ...
+//   MatrixXd X = transfer.get();                     // block + retrieve
+
+#ifndef EIGEN_GPU_DEVICE_MATRIX_H
+#define EIGEN_GPU_DEVICE_MATRIX_H
+
+// IWYU pragma: private
+#include "./InternalHeaderCheck.h"
+
+#include "./GpuSupport.h"
+
+namespace Eigen {
+
+// Forward declarations.
+template <typename, int>
+class GpuLLT;
+template <typename>
+class GpuLU;
+template <typename>
+class DeviceAdjointView;
+template <typename>
+class DeviceTransposeView;
+template <typename>
+class DeviceAssignment;
+template <typename, typename>
+class GemmExpr;
+template <typename, int>
+class LltSolveExpr;
+template <typename>
+class LuSolveExpr;
+template <typename, int>
+class DeviceLLTView;
+template <typename>
+class DeviceLUView;
+template <typename, int>
+class DeviceTriangularView;
+template <typename, int>
+class DeviceSelfAdjointView;
+template <typename, int>
+class ConstDeviceSelfAdjointView;
+template <typename, int>
+class TrsmExpr;
+template <typename, int>
+class SymmExpr;
+template <typename, int>
+class SyrkExpr;
+class GpuContext;
+
+// --------------------------------------------------------------------------
+// HostTransfer — future-like wrapper for an async device-to-host transfer.
+// --------------------------------------------------------------------------
+
+/** \ingroup GPU_Module
+ * \class HostTransfer
+ * \brief Future for an asynchronous device-to-host matrix transfer.
+ *
+ * Returned by DeviceMatrix::toHostAsync(). The transfer runs asynchronously
+ * on the given CUDA stream. Call get() to block until complete and retrieve
+ * the host matrix, or ready() to poll without blocking.
+ */
+template <typename Scalar_>
+class HostTransfer {
+ public:
+  using Scalar = Scalar_;
+  using PlainMatrix = Matrix<Scalar, Dynamic, Dynamic, ColMajor>;
+
+  /** Block until the transfer completes and return the host matrix.
+   * Idempotent: subsequent calls return the same matrix without re-syncing. */
+  PlainMatrix& get() {
+    if (!synced_) {
+      EIGEN_CUDA_RUNTIME_CHECK(cudaEventSynchronize(event_));
+      synced_ = true;
+    }
+    return host_buf_;
+  }
+
+  /** Non-blocking check: has the transfer completed? */
+  bool ready() const {
+    if (synced_) return true;
+    cudaError_t err = cudaEventQuery(event_);
+    if (err == cudaSuccess) return true;
+    eigen_assert(err == cudaErrorNotReady && "cudaEventQuery failed");
+    return false;
+  }
+
+  ~HostTransfer() {
+    if (event_) (void)cudaEventDestroy(event_);
+  }
+
+  HostTransfer(HostTransfer&& o) noexcept : host_buf_(std::move(o.host_buf_)), event_(o.event_), synced_(o.synced_) {
+    o.event_ = nullptr;
+    o.synced_ = true;
+  }
+
+  HostTransfer& operator=(HostTransfer&& o) noexcept {
+    if (this != &o) {
+      if (event_) (void)cudaEventDestroy(event_);
+      host_buf_ = std::move(o.host_buf_);
+      event_ = o.event_;
+      synced_ = o.synced_;
+      o.event_ = nullptr;
+      o.synced_ = true;
+    }
+    return *this;
+  }
+
+  HostTransfer(const HostTransfer&) = delete;
+  HostTransfer& operator=(const HostTransfer&) = delete;
+
+ private:
+  template <typename>
+  friend class DeviceMatrix;
+
+  HostTransfer(PlainMatrix&& buf, cudaEvent_t event) : host_buf_(std::move(buf)), event_(event), synced_(false) {}
+
+  PlainMatrix host_buf_;
+  cudaEvent_t event_ = nullptr;
+  bool synced_ = false;
+};
+
+// --------------------------------------------------------------------------
+// DeviceMatrix — typed RAII wrapper for a dense matrix in device memory.
+// --------------------------------------------------------------------------
+
+/** \ingroup GPU_Module
+ * \class DeviceMatrix
+ * \brief RAII wrapper for a dense column-major matrix in GPU device memory.
+ *
+ * \tparam Scalar_  Element type: float, double, complex<float>, complex<double>
+ *
+ * Owns a device allocation with tracked dimensions. Always dense
+ * (leading dimension = rows; no stride padding).
+ * An internal CUDA event records when the data was last written, enabling
+ * safe cross-stream consumption without user-visible synchronization.
+ *
+ * Each method has a synchronous and an asynchronous variant:
+ *  - fromHost() / fromHostAsync(): upload from host
+ *  - toHost() / toHostAsync(): download to host
+ */
+template <typename Scalar_>
+class DeviceMatrix {
+ public:
+  using Scalar = Scalar_;
+  using PlainMatrix = Matrix<Scalar, Dynamic, Dynamic, ColMajor>;
+
+  // ---- Construction / destruction ------------------------------------------
+
+  /** Default: empty (0x0, no allocation). */
+  DeviceMatrix() = default;
+
+  /** Allocate uninitialized device memory for a rows x cols matrix. */
+  DeviceMatrix(Index rows, Index cols) : rows_(rows), cols_(cols) {
+    eigen_assert(rows >= 0 && cols >= 0);
+    size_t bytes = sizeInBytes();
+    if (bytes > 0) {
+      EIGEN_CUDA_RUNTIME_CHECK(cudaMalloc(reinterpret_cast<void**>(&data_), bytes));
+    }
+  }
+
+  ~DeviceMatrix() {
+    if (data_) (void)cudaFree(data_);
+    if (ready_event_) (void)cudaEventDestroy(ready_event_);
+  }
+
+  // ---- Move-only -----------------------------------------------------------
+
+  DeviceMatrix(DeviceMatrix&& o) noexcept
+      : data_(o.data_),
+        rows_(o.rows_),
+        cols_(o.cols_),
+        ready_event_(o.ready_event_),
+        ready_stream_(o.ready_stream_),
+        retained_buffer_(std::move(o.retained_buffer_)) {
+    o.data_ = nullptr;
+    o.rows_ = 0;
+    o.cols_ = 0;
+    o.ready_event_ = nullptr;
+    o.ready_stream_ = nullptr;
+  }
+
+  DeviceMatrix& operator=(DeviceMatrix&& o) noexcept {
+    if (this != &o) {
+      if (data_) (void)cudaFree(data_);
+      if (ready_event_) (void)cudaEventDestroy(ready_event_);
+      data_ = o.data_;
+      rows_ = o.rows_;
+      cols_ = o.cols_;
+      ready_event_ = o.ready_event_;
+      ready_stream_ = o.ready_stream_;
+      retained_buffer_ = std::move(o.retained_buffer_);
+      o.data_ = nullptr;
+      o.rows_ = 0;
+      o.cols_ = 0;
+      o.ready_event_ = nullptr;
+      o.ready_stream_ = nullptr;
+    }
+    return *this;
+  }
+
+  DeviceMatrix(const DeviceMatrix&) = delete;
+  DeviceMatrix& operator=(const DeviceMatrix&) = delete;
+
+  // ---- Upload from host ----------------------------------------------------
+
+  /** Upload a host Eigen matrix to device memory (synchronous).
+   *
+   * Evaluates the expression into a contiguous ColMajor temporary, copies to
+   * device via cudaMemcpyAsync on \p stream, and synchronizes before returning.
+   *
+   * \param host   Any Eigen matrix expression.
+   * \param stream CUDA stream for the transfer (default: stream 0).
+   */
+  template <typename Derived>
+  static DeviceMatrix fromHost(const MatrixBase<Derived>& host, cudaStream_t stream = nullptr) {
+    const PlainMatrix mat(host.derived());
+    DeviceMatrix dm(mat.rows(), mat.cols());
+    if (dm.sizeInBytes() > 0) {
+      EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpyAsync(dm.data_, mat.data(), dm.sizeInBytes(), cudaMemcpyHostToDevice, stream));
+      EIGEN_CUDA_RUNTIME_CHECK(cudaStreamSynchronize(stream));
+    }
+    return dm;
+  }
+
+  /** Upload from a raw host pointer to device memory (asynchronous).
+   *
+   * Enqueues an async H2D copy on \p stream and records an internal event.
+   * The caller must keep \p host_data alive until the transfer completes
+   * (check via the internal event or synchronize the stream).
+   *
+   * \param host_data  Pointer to contiguous column-major host data.
+   * \param rows       Number of rows.
+   * \param cols       Number of columns.
+   * \param stream     CUDA stream for the transfer.
+   */
+  static DeviceMatrix fromHostAsync(const Scalar* host_data, Index rows, Index cols, cudaStream_t stream) {
+    eigen_assert(rows >= 0 && cols >= 0);
+    eigen_assert(host_data != nullptr || (rows == 0 || cols == 0));
+    DeviceMatrix dm(rows, cols);
+    if (dm.sizeInBytes() > 0) {
+      EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpyAsync(dm.data_, host_data, dm.sizeInBytes(), cudaMemcpyHostToDevice, stream));
+      dm.recordReady(stream);
+    }
+    return dm;
+  }
+
+  // ---- Download to host ----------------------------------------------------
+
+  /** Download device matrix to host memory (synchronous).
+   *
+   * Waits on the internal ready event, enqueues a D2H copy on \p stream,
+   * synchronizes, and returns the host matrix directly.
+   *
+   * \param stream CUDA stream for the transfer (default: stream 0).
+   */
+  PlainMatrix toHost(cudaStream_t stream = nullptr) const {
+    PlainMatrix host_buf(rows_, cols_);
+    if (sizeInBytes() > 0) {
+      waitReady(stream);
+      EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpyAsync(host_buf.data(), data_, sizeInBytes(), cudaMemcpyDeviceToHost, stream));
+      EIGEN_CUDA_RUNTIME_CHECK(cudaStreamSynchronize(stream));
+    }
+    return host_buf;
+  }
+
+  /** Enqueue an async device-to-host transfer and return a future.
+   *
+   * Waits on the internal ready event (if any) to ensure the device data is
+   * valid, then enqueues the D2H copy on \p stream. Returns a HostTransfer
+   * future; call .get() to block and retrieve the host matrix.
+   *
+   * \param stream CUDA stream for the transfer (default: stream 0).
+   */
+  HostTransfer<Scalar> toHostAsync(cudaStream_t stream = nullptr) const {
+    PlainMatrix host_buf(rows_, cols_);
+    if (sizeInBytes() > 0) {
+      waitReady(stream);
+      EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpyAsync(host_buf.data(), data_, sizeInBytes(), cudaMemcpyDeviceToHost, stream));
+    }
+    // Record a transfer-complete event.
+    cudaEvent_t transfer_event;
+    EIGEN_CUDA_RUNTIME_CHECK(cudaEventCreateWithFlags(&transfer_event, cudaEventDisableTiming));
+    EIGEN_CUDA_RUNTIME_CHECK(cudaEventRecord(transfer_event, stream));
+    return HostTransfer<Scalar>(std::move(host_buf), transfer_event);
+  }
+
+  // ---- Device-to-device copy -----------------------------------------------
+
+  /** Deep copy on device. Fully async — records event on the result, no sync.
+   *
+   * \param stream CUDA stream for the D2D copy (default: stream 0).
+   */
+  DeviceMatrix clone(cudaStream_t stream = nullptr) const {
+    DeviceMatrix result(rows_, cols_);
+    if (sizeInBytes() > 0) {
+      waitReady(stream);
+      EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpyAsync(result.data_, data_, sizeInBytes(), cudaMemcpyDeviceToDevice, stream));
+      result.recordReady(stream);
+    }
+    return result;
+  }
+
+  // ---- Resize (destructive) ------------------------------------------------
+
+  /** Discard contents and reallocate to (rows x cols). Clears the ready event. */
+  void resize(Index rows, Index cols) {
+    if (rows == rows_ && cols == cols_) return;
+    if (data_) {
+      (void)cudaFree(data_);
+      data_ = nullptr;
+    }
+    if (ready_event_) {
+      (void)cudaEventDestroy(ready_event_);
+      ready_event_ = nullptr;
+    }
+    ready_stream_ = nullptr;
+    retained_buffer_ = internal::DeviceBuffer();
+    rows_ = rows;
+    cols_ = cols;
+    size_t bytes = sizeInBytes();
+    if (bytes > 0) {
+      EIGEN_CUDA_RUNTIME_CHECK(cudaMalloc(reinterpret_cast<void**>(&data_), bytes));
+    }
+  }
+
+  // ---- Accessors -----------------------------------------------------------
+
+  Scalar* data() { return data_; }
+  const Scalar* data() const { return data_; }
+  Index rows() const { return rows_; }
+  Index cols() const { return cols_; }
+  bool empty() const { return rows_ == 0 || cols_ == 0; }
+
+  /** Size of the device allocation in bytes. */
+  size_t sizeInBytes() const { return static_cast<size_t>(rows_) * static_cast<size_t>(cols_) * sizeof(Scalar); }
+
+  // ---- Event synchronization (public for library dispatch interop) ---------
+
+  /** Record that device data is ready after work on \p stream. */
+  void recordReady(cudaStream_t stream) {
+    ensureEvent();
+    EIGEN_CUDA_RUNTIME_CHECK(cudaEventRecord(ready_event_, stream));
+    ready_stream_ = stream;
+  }
+
+  /** Make \p stream wait until the device data is ready.
+   * No-op if no event recorded, or if the consumer stream is the same as the
+   * producer stream (CUDA guarantees in-order execution within a stream). */
+  void waitReady(cudaStream_t stream) const {
+    if (ready_event_ && stream != ready_stream_) {
+      EIGEN_CUDA_RUNTIME_CHECK(cudaStreamWaitEvent(stream, ready_event_, 0));
+    }
+  }
+
+  // ---- Expression methods (dispatch to cuBLAS/cuSOLVER) --------------------
+
+  /** Adjoint view for GEMM dispatch. Maps to cublasXgemm with ConjTrans. */
+  DeviceAdjointView<Scalar> adjoint() const { return DeviceAdjointView<Scalar>(*this); }
+
+  /** Transpose view for GEMM dispatch. Maps to cublasXgemm with Trans. */
+  DeviceTransposeView<Scalar> transpose() const { return DeviceTransposeView<Scalar>(*this); }
+
+  /** Bind this matrix to a GpuContext for expression assignment.
+   * Returns a DeviceAssignment proxy: `d_C.device(ctx) = d_A * d_B;` */
+  DeviceAssignment<Scalar> device(GpuContext& ctx) { return DeviceAssignment<Scalar>(*this, ctx); }
+
+  /** Assign from a GEMM expression using the thread-local default GpuContext.
+   * Defined out-of-line after GpuContext is fully declared (see DeviceDispatch.h). */
+  template <typename Lhs, typename Rhs>
+  DeviceMatrix& operator=(const GemmExpr<Lhs, Rhs>& expr);
+
+  /** Accumulate from a GEMM expression using the thread-local default GpuContext. */
+  template <typename Lhs, typename Rhs>
+  DeviceMatrix& operator+=(const GemmExpr<Lhs, Rhs>& expr);
+
+  /** Cholesky view: d_A.llt().solve(d_B) → LltSolveExpr. */
+  DeviceLLTView<Scalar, Lower> llt() const { return DeviceLLTView<Scalar, Lower>(*this); }
+
+  /** Cholesky view with explicit triangle: d_A.llt<Upper>().solve(d_B). */
+  template <int UpLo>
+  DeviceLLTView<Scalar, UpLo> llt() const {
+    return DeviceLLTView<Scalar, UpLo>(*this);
+  }
+
+  /** LU view: d_A.lu().solve(d_B) → LuSolveExpr. */
+  DeviceLUView<Scalar> lu() const { return DeviceLUView<Scalar>(*this); }
+
+  /** Assign from an LLT solve expression (thread-local default context). */
+  template <int UpLo>
+  DeviceMatrix& operator=(const LltSolveExpr<Scalar, UpLo>& expr);
+
+  /** Assign from an LU solve expression (thread-local default context). */
+  DeviceMatrix& operator=(const LuSolveExpr<Scalar>& expr);
+
+  /** Triangular view: d_A.triangularView<Lower>().solve(d_B) → TrsmExpr. */
+  template <int UpLo>
+  DeviceTriangularView<Scalar, UpLo> triangularView() const {
+    return DeviceTriangularView<Scalar, UpLo>(*this);
+  }
+
+  /** Self-adjoint view (mutable): d_C.selfadjointView<Lower>().rankUpdate(d_A). */
+  template <int UpLo>
+  DeviceSelfAdjointView<Scalar, UpLo> selfadjointView() {
+    return DeviceSelfAdjointView<Scalar, UpLo>(*this);
+  }
+
+  /** Self-adjoint view (const): d_A.selfadjointView<Lower>() * d_B → SymmExpr. */
+  template <int UpLo>
+  ConstDeviceSelfAdjointView<Scalar, UpLo> selfadjointView() const {
+    return ConstDeviceSelfAdjointView<Scalar, UpLo>(*this);
+  }
+
+  /** Assign from a TRSM expression (thread-local default context). */
+  template <int UpLo>
+  DeviceMatrix& operator=(const TrsmExpr<Scalar, UpLo>& expr);
+
+  /** Assign from a SYMM expression (thread-local default context). */
+  template <int UpLo>
+  DeviceMatrix& operator=(const SymmExpr<Scalar, UpLo>& expr);
+
+ private:
+  // ---- Private: adopt a raw device pointer (used by friend solvers) --------
+
+  DeviceMatrix(Scalar* device_ptr, Index rows, Index cols) : data_(device_ptr), rows_(rows), cols_(cols) {}
+
+  /** Transfer ownership of the device pointer out. Zeros internal state. */
+  Scalar* release() {
+    Scalar* p = data_;
+    data_ = nullptr;
+    rows_ = 0;
+    cols_ = 0;
+    if (ready_event_) {
+      (void)cudaEventDestroy(ready_event_);
+      ready_event_ = nullptr;
+    }
+    ready_stream_ = nullptr;
+    return p;
+  }
+
+  // ---- Private helpers -------------------------------------------------------
+
+  void ensureEvent() {
+    if (!ready_event_) {
+      EIGEN_CUDA_RUNTIME_CHECK(cudaEventCreateWithFlags(&ready_event_, cudaEventDisableTiming));
+    }
+  }
+
+  void retainBuffer(internal::DeviceBuffer&& buffer) { retained_buffer_ = std::move(buffer); }
+
+  // ---- Friend declarations ------------------------------------------------
+
+  template <typename, int>
+  friend class GpuLLT;
+  template <typename>
+  friend class GpuLU;
+  template <typename>
+  friend class GpuQR;
+  template <typename>
+  friend class GpuSVD;
+  template <typename>
+  friend class GpuSelfAdjointEigenSolver;
+
+  // ---- Data members --------------------------------------------------------
+
+  Scalar* data_ = nullptr;
+  Index rows_ = 0;
+  Index cols_ = 0;
+  cudaEvent_t ready_event_ = nullptr;       // internal: tracks last write completion
+  cudaStream_t ready_stream_ = nullptr;     // stream that recorded ready_event_ (for same-stream skip)
+  internal::DeviceBuffer retained_buffer_;  // internal: keeps async aux buffers alive
+};
+
+}  // namespace Eigen
+
+#endif  // EIGEN_GPU_DEVICE_MATRIX_H

Eigen/src/GPU/DeviceSolverExpr.h

@@ -0,0 +1,115 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2026 Rasmus Munk Larsen <rmlarsen@gmail.com>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+// Solver expression types for DeviceMatrix.
+//
+// Each expression maps 1:1 to cuSOLVER library calls:
+//   LltSolveExpr  → cusolverDnXpotrf + cusolverDnXpotrs
+//   LuSolveExpr   → cusolverDnXgetrf + cusolverDnXgetrs
+//
+// Usage:
+//   d_X = d_A.llt().solve(d_B);              // Cholesky solve
+//   d_X.device(ctx) = d_A.lu().solve(d_B);   // LU solve on explicit stream
+
+#ifndef EIGEN_GPU_DEVICE_SOLVER_EXPR_H
+#define EIGEN_GPU_DEVICE_SOLVER_EXPR_H
+
+// IWYU pragma: private
+#include "./InternalHeaderCheck.h"
+
+namespace Eigen {
+
+// Forward declarations.
+template <typename Scalar_>
+class DeviceMatrix;
+class GpuContext;
+
+// ---- LLT solve expression ---------------------------------------------------
+// d_A.llt().solve(d_B) → LltSolveExpr → cusolverDnXpotrf + cusolverDnXpotrs
+
+template <typename Scalar_, int UpLo_ = Lower>
+class LltSolveExpr {
+ public:
+  using Scalar = Scalar_;
+  enum { UpLo = UpLo_ };
+
+  LltSolveExpr(const DeviceMatrix<Scalar>& A, const DeviceMatrix<Scalar>& B) : A_(A), B_(B) {}
+  const DeviceMatrix<Scalar>& matrix() const { return A_; }
+  const DeviceMatrix<Scalar>& rhs() const { return B_; }
+
+ private:
+  const DeviceMatrix<Scalar>& A_;
+  const DeviceMatrix<Scalar>& B_;
+};
+
+// ---- LU solve expression ----------------------------------------------------
+// d_A.lu().solve(d_B) → LuSolveExpr → cusolverDnXgetrf + cusolverDnXgetrs
+
+template <typename Scalar_>
+class LuSolveExpr {
+ public:
+  using Scalar = Scalar_;
+
+  LuSolveExpr(const DeviceMatrix<Scalar>& A, const DeviceMatrix<Scalar>& B) : A_(A), B_(B) {}
+  const DeviceMatrix<Scalar>& matrix() const { return A_; }
+  const DeviceMatrix<Scalar>& rhs() const { return B_; }
+
+ private:
+  const DeviceMatrix<Scalar>& A_;
+  const DeviceMatrix<Scalar>& B_;
+};
+
+// ---- DeviceLLTView: d_A.llt() → view with .solve() and .device() -----------
+
+template <typename Scalar_, int UpLo_ = Lower>
+class DeviceLLTView {
+ public:
+  using Scalar = Scalar_;
+
+  explicit DeviceLLTView(const DeviceMatrix<Scalar>& m) : mat_(m) {}
+
+  /** Build a solve expression: d_A.llt().solve(d_B).
+   * The expression is evaluated when assigned to a DeviceMatrix. */
+  LltSolveExpr<Scalar, UpLo_> solve(const DeviceMatrix<Scalar>& rhs) const { return {mat_, rhs}; }
+
+  // For cached factorizations, use the explicit GpuLLT API directly:
+  //   GpuLLT<double> llt;
+  //   llt.compute(d_A);
+  //   auto d_X1 = llt.solve(d_B1);
+  //   auto d_X2 = llt.solve(d_B2);
+
+ private:
+  const DeviceMatrix<Scalar>& mat_;
+};
+
+// ---- DeviceLUView: d_A.lu() → view with .solve() and .device() -------------
+
+template <typename Scalar_>
+class DeviceLUView {
+ public:
+  using Scalar = Scalar_;
+
+  explicit DeviceLUView(const DeviceMatrix<Scalar>& m) : mat_(m) {}
+
+  /** Build a solve expression: d_A.lu().solve(d_B). */
+  LuSolveExpr<Scalar> solve(const DeviceMatrix<Scalar>& rhs) const { return {mat_, rhs}; }
+
+  // For cached factorizations, use the explicit GpuLU API directly:
+  //   GpuLU<double> lu;
+  //   lu.compute(d_A);
+  //   auto d_X1 = lu.solve(d_B1);
+  //   auto d_X2 = lu.solve(d_B2);
+
+ private:
+  const DeviceMatrix<Scalar>& mat_;
+};
+
+}  // namespace Eigen
+
+#endif  // EIGEN_GPU_DEVICE_SOLVER_EXPR_H

Eigen/src/GPU/GpuContext.h

@@ -0,0 +1,83 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2026 Rasmus Munk Larsen <rmlarsen@gmail.com>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+// Unified GPU execution context.
+//
+// GpuContext owns a CUDA stream and all NVIDIA library handles (cuBLAS,
+// cuSOLVER, future cuDSS/cuSPARSE). It is the entry point for all GPU
+// operations on DeviceMatrix.
+//
+// Usage:
+//   GpuContext ctx;                        // explicit context
+//   d_C.device(ctx) = d_A * d_B;          // GEMM on ctx's stream
+//
+//   d_C = d_A * d_B;                      // thread-local default context
+//   GpuContext& ctx = GpuContext::threadLocal();
+
+#ifndef EIGEN_GPU_CONTEXT_H
+#define EIGEN_GPU_CONTEXT_H
+
+// IWYU pragma: private
+#include "./InternalHeaderCheck.h"
+
+#include "./CuBlasSupport.h"
+#include "./CuSolverSupport.h"
+
+namespace Eigen {
+
+/** \ingroup GPU_Module
+ * \class GpuContext
+ * \brief Unified GPU execution context owning a CUDA stream and library handles.
+ *
+ * Each GpuContext instance creates a dedicated CUDA stream, a cuBLAS handle,
+ * and a cuSOLVER handle, all bound to that stream. Multiple contexts enable
+ * concurrent execution on independent streams.
+ *
+ * A lazily-created thread-local default is available via threadLocal() for
+ * simple single-stream usage.
+ */
+class GpuContext {
+ public:
+  GpuContext() {
+    EIGEN_CUDA_RUNTIME_CHECK(cudaStreamCreate(&stream_));
+    EIGEN_CUBLAS_CHECK(cublasCreate(&cublas_));
+    EIGEN_CUBLAS_CHECK(cublasSetStream(cublas_, stream_));
+    EIGEN_CUSOLVER_CHECK(cusolverDnCreate(&cusolver_));
+    EIGEN_CUSOLVER_CHECK(cusolverDnSetStream(cusolver_, stream_));
+  }
+
+  ~GpuContext() {
+    if (cusolver_) (void)cusolverDnDestroy(cusolver_);
+    if (cublas_) (void)cublasDestroy(cublas_);
+    if (stream_) (void)cudaStreamDestroy(stream_);
+  }
+
+  // Non-copyable, non-movable (owns library handles).
+  GpuContext(const GpuContext&) = delete;
+  GpuContext& operator=(const GpuContext&) = delete;
+
+  /** Lazily-created thread-local default context. */
+  static GpuContext& threadLocal() {
+    thread_local GpuContext ctx;
+    return ctx;
+  }
+
+  cudaStream_t stream() const { return stream_; }
+  cublasHandle_t cublasHandle() const { return cublas_; }
+  cusolverDnHandle_t cusolverHandle() const { return cusolver_; }
+
+ private:
+  cudaStream_t stream_ = nullptr;
+  cublasHandle_t cublas_ = nullptr;
+  cusolverDnHandle_t cusolver_ = nullptr;
+};
+
+}  // namespace Eigen
+
+#endif  // EIGEN_GPU_CONTEXT_H

Eigen/src/GPU/GpuEigenSolver.h

@@ -0,0 +1,228 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2026 Rasmus Munk Larsen <rmlarsen@gmail.com>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+// GPU self-adjoint eigenvalue decomposition using cuSOLVER.
+//
+// Wraps cusolverDnXsyevd (symmetric/Hermitian divide-and-conquer).
+// Stores eigenvalues and eigenvectors on device.
+//
+// Usage:
+//   GpuSelfAdjointEigenSolver<double> es(A);
+//   VectorXd eigenvals = es.eigenvalues();
+//   MatrixXd eigenvecs = es.eigenvectors();
+
+#ifndef EIGEN_GPU_EIGENSOLVER_H
+#define EIGEN_GPU_EIGENSOLVER_H
+
+// IWYU pragma: private
+#include "./InternalHeaderCheck.h"
+
+#include "./CuSolverSupport.h"
+#include <vector>
+
+namespace Eigen {
+
+template <typename Scalar_>
+class GpuSelfAdjointEigenSolver {
+ public:
+  using Scalar = Scalar_;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+  using PlainMatrix = Matrix<Scalar, Dynamic, Dynamic, ColMajor>;
+  using RealVector = Matrix<RealScalar, Dynamic, 1>;
+
+  /** Eigenvalue-only or eigenvalues + eigenvectors. */
+  enum ComputeMode { EigenvaluesOnly, ComputeEigenvectors };
+
+  GpuSelfAdjointEigenSolver() { init_context(); }
+
+  template <typename InputType>
+  explicit GpuSelfAdjointEigenSolver(const EigenBase<InputType>& A, ComputeMode mode = ComputeEigenvectors) {
+    init_context();
+    compute(A, mode);
+  }
+
+  ~GpuSelfAdjointEigenSolver() {
+    if (handle_) (void)cusolverDnDestroy(handle_);
+    if (stream_) (void)cudaStreamDestroy(stream_);
+  }
+
+  GpuSelfAdjointEigenSolver(const GpuSelfAdjointEigenSolver&) = delete;
+  GpuSelfAdjointEigenSolver& operator=(const GpuSelfAdjointEigenSolver&) = delete;
+
+  // ---- Factorization -------------------------------------------------------
+
+  template <typename InputType>
+  GpuSelfAdjointEigenSolver& compute(const EigenBase<InputType>& A, ComputeMode mode = ComputeEigenvectors) {
+    eigen_assert(A.rows() == A.cols() && "GpuSelfAdjointEigenSolver requires a square matrix");
+    mode_ = mode;
+    n_ = A.rows();
+    info_ = InvalidInput;
+    info_synced_ = false;
+
+    if (n_ == 0) {
+      info_ = Success;
+      info_synced_ = true;
+      return *this;
+    }
+
+    const PlainMatrix mat(A.derived());
+    lda_ = static_cast<int64_t>(n_);
+    const size_t mat_bytes = static_cast<size_t>(lda_) * static_cast<size_t>(n_) * sizeof(Scalar);
+
+    // syevd overwrites A with eigenvectors (if requested).
+    d_A_ = internal::DeviceBuffer(mat_bytes);
+    EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpyAsync(d_A_.ptr, mat.data(), mat_bytes, cudaMemcpyHostToDevice, stream_));
+    factorize();
+    return *this;
+  }
+
+  GpuSelfAdjointEigenSolver& compute(const DeviceMatrix<Scalar>& d_A, ComputeMode mode = ComputeEigenvectors) {
+    eigen_assert(d_A.rows() == d_A.cols() && "GpuSelfAdjointEigenSolver requires a square matrix");
+    mode_ = mode;
+    n_ = d_A.rows();
+    info_ = InvalidInput;
+    info_synced_ = false;
+
+    if (n_ == 0) {
+      info_ = Success;
+      info_synced_ = true;
+      return *this;
+    }
+
+    d_A.waitReady(stream_);
+    lda_ = static_cast<int64_t>(n_);
+    const size_t mat_bytes = static_cast<size_t>(lda_) * static_cast<size_t>(n_) * sizeof(Scalar);
+
+    d_A_ = internal::DeviceBuffer(mat_bytes);
+    EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpyAsync(d_A_.ptr, d_A.data(), mat_bytes, cudaMemcpyDeviceToDevice, stream_));
+
+    factorize();
+    return *this;
+  }
+
+  // ---- Accessors -----------------------------------------------------------
+
+  ComputationInfo info() const {
+    sync_info();
+    return info_;
+  }
+
+  Index cols() const { return n_; }
+  Index rows() const { return n_; }
+
+  /** Eigenvalues in ascending order. Downloads from device. */
+  RealVector eigenvalues() const {
+    sync_info();
+    eigen_assert(info_ == Success);
+    RealVector W(n_);
+    if (n_ > 0) {
+      EIGEN_CUDA_RUNTIME_CHECK(
+          cudaMemcpy(W.data(), d_W_.ptr, static_cast<size_t>(n_) * sizeof(RealScalar), cudaMemcpyDeviceToHost));
+    }
+    return W;
+  }
+
+  /** Eigenvectors (columns). Downloads from device.
+   * Requires ComputeEigenvectors mode. */
+  PlainMatrix eigenvectors() const {
+    sync_info();
+    eigen_assert(info_ == Success);
+    eigen_assert(mode_ == ComputeEigenvectors && "eigenvectors() requires ComputeEigenvectors mode");
+    PlainMatrix V(n_, n_);
+    if (n_ > 0) {
+      EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpy(V.data(), d_A_.ptr,
+                                          static_cast<size_t>(lda_) * static_cast<size_t>(n_) * sizeof(Scalar),
+                                          cudaMemcpyDeviceToHost));
+    }
+    return V;
+  }
+
+  cudaStream_t stream() const { return stream_; }
+
+ private:
+  cudaStream_t stream_ = nullptr;
+  cusolverDnHandle_t handle_ = nullptr;
+  internal::CusolverParams params_;
+  internal::DeviceBuffer d_A_;        // overwritten with eigenvectors by syevd
+  internal::DeviceBuffer d_W_;        // eigenvalues (RealScalar, length n)
+  internal::DeviceBuffer d_scratch_;  // workspace + info
+  size_t scratch_size_ = 0;
+  std::vector<char> h_workspace_;
+  ComputeMode mode_ = ComputeEigenvectors;
+  Index n_ = 0;
+  int64_t lda_ = 0;
+  ComputationInfo info_ = InvalidInput;
+  int info_word_ = 0;
+  bool info_synced_ = true;
+
+  void init_context() {
+    EIGEN_CUDA_RUNTIME_CHECK(cudaStreamCreate(&stream_));
+    EIGEN_CUSOLVER_CHECK(cusolverDnCreate(&handle_));
+    EIGEN_CUSOLVER_CHECK(cusolverDnSetStream(handle_, stream_));
+    ensure_scratch(0);
+  }
+
+  void ensure_scratch(size_t workspace_bytes) {
+    constexpr size_t kAlign = 16;
+    workspace_bytes = (workspace_bytes + kAlign - 1) & ~(kAlign - 1);
+    size_t needed = workspace_bytes + sizeof(int);
+    if (needed > scratch_size_) {
+      if (d_scratch_.ptr) EIGEN_CUDA_RUNTIME_CHECK(cudaStreamSynchronize(stream_));
+      d_scratch_ = internal::DeviceBuffer(needed);
+      scratch_size_ = needed;
+    }
+  }
+
+  void* scratch_workspace() const { return d_scratch_.ptr; }
+  int* scratch_info() const {
+    return reinterpret_cast<int*>(static_cast<char*>(d_scratch_.ptr) + scratch_size_ - sizeof(int));
+  }
+
+  void sync_info() const {
+    if (!info_synced_) {
+      EIGEN_CUDA_RUNTIME_CHECK(cudaStreamSynchronize(stream_));
+      const_cast<GpuSelfAdjointEigenSolver*>(this)->info_ = (info_word_ == 0) ? Success : NumericalIssue;
+      const_cast<GpuSelfAdjointEigenSolver*>(this)->info_synced_ = true;
+    }
+  }
+
+  void factorize() {
+    constexpr cudaDataType_t dtype = internal::cusolver_data_type<Scalar>::value;
+    constexpr cudaDataType_t rtype = internal::cuda_data_type<RealScalar>::value;
+
+    info_synced_ = false;
+    info_ = InvalidInput;
+
+    d_W_ = internal::DeviceBuffer(static_cast<size_t>(n_) * sizeof(RealScalar));
+
+    const cusolverEigMode_t jobz =
+        (mode_ == ComputeEigenvectors) ? CUSOLVER_EIG_MODE_VECTOR : CUSOLVER_EIG_MODE_NOVECTOR;
+
+    // Use lower triangle (standard convention).
+    constexpr cublasFillMode_t uplo = CUBLAS_FILL_MODE_LOWER;
+
+    size_t dev_ws = 0, host_ws = 0;
+    EIGEN_CUSOLVER_CHECK(cusolverDnXsyevd_bufferSize(handle_, params_.p, jobz, uplo, static_cast<int64_t>(n_), dtype,
+                                                     d_A_.ptr, lda_, rtype, d_W_.ptr, dtype, &dev_ws, &host_ws));
+
+    ensure_scratch(dev_ws);
+    h_workspace_.resize(host_ws);
+
+    EIGEN_CUSOLVER_CHECK(cusolverDnXsyevd(handle_, params_.p, jobz, uplo, static_cast<int64_t>(n_), dtype, d_A_.ptr,
+                                          lda_, rtype, d_W_.ptr, dtype, scratch_workspace(), dev_ws,
+                                          host_ws > 0 ? h_workspace_.data() : nullptr, host_ws, scratch_info()));
+
+    EIGEN_CUDA_RUNTIME_CHECK(
+        cudaMemcpyAsync(&info_word_, scratch_info(), sizeof(int), cudaMemcpyDeviceToHost, stream_));
+  }
+};
+
+}  // namespace Eigen
+
+#endif  // EIGEN_GPU_EIGENSOLVER_H

Eigen/src/GPU/GpuLLT.h

@@ -0,0 +1,385 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2026 Eigen Authors
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+// GPU Cholesky (LLT) decomposition using cuSOLVER.
+//
+// Unlike Eigen's CPU LLT<MatrixType>, GpuLLT keeps the factored Cholesky
+// factor in device memory for the lifetime of the object. Multiple solves
+// against the same factor therefore only transfer the RHS and solution
+// vectors, not the factor itself.
+//
+// Requires CUDA 11.0+ (cusolverDnXpotrf / cusolverDnXpotrs generic API).
+// Requires CUDA 11.4+ (cusolverDnX generic API + cudaMallocAsync).
+//
+// Usage:
+//   GpuLLT<double> llt(A);              // upload A, potrf, L stays on device
+//   if (llt.info() != Success) { ... }
+//   MatrixXd x1 = llt.solve(b1);        // potrs, only b1 transferred
+//   MatrixXd x2 = llt.solve(b2);        // L already on device
+
+#ifndef EIGEN_GPU_LLT_H
+#define EIGEN_GPU_LLT_H
+
+// IWYU pragma: private
+#include "./InternalHeaderCheck.h"
+
+#include "./CuSolverSupport.h"
+#include <vector>
+
+namespace Eigen {
+
+/** \ingroup GPU_Module
+ * \class GpuLLT
+ * \brief GPU Cholesky (LL^T) decomposition via cuSOLVER
+ *
+ * \tparam Scalar_  Element type: float, double, complex<float>, complex<double>
+ * \tparam UpLo_    Triangle used: Lower (default) or Upper
+ *
+ * Factorizes a symmetric positive-definite matrix A = LL^H on the GPU and
+ * caches the factor L in device memory. Each subsequent solve(B) uploads only
+ * B, calls cusolverDnXpotrs, and downloads the result — the factor is not
+ * re-transferred.
+ *
+ * Each GpuLLT object owns a dedicated CUDA stream and cuSOLVER handle,
+ * enabling concurrent factorizations from multiple objects on the same host
+ * thread.
+ */
+template <typename Scalar_, int UpLo_ = Lower>
+class GpuLLT {
+ public:
+  using Scalar = Scalar_;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+  using PlainMatrix = Matrix<Scalar, Dynamic, Dynamic, ColMajor>;
+
+  enum { UpLo = UpLo_ };
+
+  // ---- Construction / destruction ------------------------------------------
+
+  /** Default constructor. Does not factorize; call compute() before solve(). */
+  GpuLLT() { init_context(); }
+
+  /** Factor A immediately. Equivalent to GpuLLT llt; llt.compute(A). */
+  template <typename InputType>
+  explicit GpuLLT(const EigenBase<InputType>& A) {
+    init_context();
+    compute(A);
+  }
+
+  ~GpuLLT() {
+    // Ignore errors in destructors — cannot propagate.
+    if (handle_) (void)cusolverDnDestroy(handle_);
+    if (stream_) (void)cudaStreamDestroy(stream_);
+  }
+
+  // Non-copyable (owns device memory and library handles).
+  GpuLLT(const GpuLLT&) = delete;
+  GpuLLT& operator=(const GpuLLT&) = delete;
+
+  // Movable.
+  GpuLLT(GpuLLT&& o) noexcept
+      : stream_(o.stream_),
+        handle_(o.handle_),
+        params_(std::move(o.params_)),
+        d_factor_(std::move(o.d_factor_)),
+        factor_alloc_size_(o.factor_alloc_size_),
+        d_scratch_(std::move(o.d_scratch_)),
+        scratch_size_(o.scratch_size_),
+        h_workspace_(std::move(o.h_workspace_)),
+        n_(o.n_),
+        lda_(o.lda_),
+        info_(o.info_),
+        info_word_(o.info_word_),
+        info_synced_(o.info_synced_) {
+    o.stream_ = nullptr;
+    o.handle_ = nullptr;
+    o.factor_alloc_size_ = 0;
+    o.scratch_size_ = 0;
+    o.n_ = 0;
+    o.info_ = InvalidInput;
+    o.info_word_ = 0;
+    o.info_synced_ = true;
+  }
+
+  GpuLLT& operator=(GpuLLT&& o) noexcept {
+    if (this != &o) {
+      if (handle_) (void)cusolverDnDestroy(handle_);
+      if (stream_) (void)cudaStreamDestroy(stream_);
+      stream_ = o.stream_;
+      handle_ = o.handle_;
+      params_ = std::move(o.params_);
+      d_factor_ = std::move(o.d_factor_);
+      factor_alloc_size_ = o.factor_alloc_size_;
+      d_scratch_ = std::move(o.d_scratch_);
+      scratch_size_ = o.scratch_size_;
+      h_workspace_ = std::move(o.h_workspace_);
+      n_ = o.n_;
+      lda_ = o.lda_;
+      info_ = o.info_;
+      info_word_ = o.info_word_;
+      info_synced_ = o.info_synced_;
+      o.stream_ = nullptr;
+      o.handle_ = nullptr;
+      o.factor_alloc_size_ = 0;
+      o.scratch_size_ = 0;
+      o.n_ = 0;
+      o.info_ = InvalidInput;
+      o.info_word_ = 0;
+      o.info_synced_ = true;
+    }
+    return *this;
+  }
+
+  // ---- Factorization -------------------------------------------------------
+
+  /** Compute the Cholesky factorization of A (host matrix).
+   *
+   * Uploads A to device memory, calls cusolverDnXpotrf, and retains the
+   * factored matrix on device. Any previous factorization is overwritten.
+   */
+  template <typename InputType>
+  GpuLLT& compute(const EigenBase<InputType>& A) {
+    eigen_assert(A.rows() == A.cols());
+    if (!begin_compute(A.rows())) return *this;
+
+    // Evaluate A into a contiguous ColMajor matrix (handles arbitrary expressions).
+    const PlainMatrix mat(A.derived());
+    lda_ = static_cast<int64_t>(mat.rows());
+    allocate_factor_storage();
+    EIGEN_CUDA_RUNTIME_CHECK(
+        cudaMemcpyAsync(d_factor_.ptr, mat.data(), factorBytes(), cudaMemcpyHostToDevice, stream_));
+
+    factorize();
+    return *this;
+  }
+
+  /** Compute the Cholesky factorization from a device-resident matrix (D2D copy). */
+  GpuLLT& compute(const DeviceMatrix<Scalar>& d_A) {
+    eigen_assert(d_A.rows() == d_A.cols());
+    if (!begin_compute(d_A.rows())) return *this;
+
+    lda_ = static_cast<int64_t>(d_A.rows());
+    d_A.waitReady(stream_);
+    allocate_factor_storage();
+    EIGEN_CUDA_RUNTIME_CHECK(
+        cudaMemcpyAsync(d_factor_.ptr, d_A.data(), factorBytes(), cudaMemcpyDeviceToDevice, stream_));
+
+    factorize();
+    return *this;
+  }
+
+  /** Compute the Cholesky factorization from a device matrix (move, no copy). */
+  GpuLLT& compute(DeviceMatrix<Scalar>&& d_A) {
+    eigen_assert(d_A.rows() == d_A.cols());
+    if (!begin_compute(d_A.rows())) return *this;
+
+    lda_ = static_cast<int64_t>(d_A.rows());
+    d_A.waitReady(stream_);
+    d_factor_ = internal::DeviceBuffer::adopt(static_cast<void*>(d_A.release()));
+
+    factorize();
+    return *this;
+  }
+
+  // ---- Solve ---------------------------------------------------------------
+
+  /** Solve A * X = B using the cached Cholesky factor (host → host).
+   *
+   * Uploads B to device memory, calls cusolverDnXpotrs using the factor
+   * retained from compute(), and returns the solution X on the host.
+   * The factor is not re-transferred; only B goes up and X comes down.
+   *
+   * \pre compute() must have been called and info() == Success.
+   * \returns X such that A * X ≈ B
+   */
+  template <typename Rhs>
+  PlainMatrix solve(const MatrixBase<Rhs>& B) const {
+    const_cast<GpuLLT*>(this)->sync_info();
+    eigen_assert(info_ == Success && "GpuLLT::solve called on a failed or uninitialized factorization");
+    eigen_assert(B.rows() == n_);
+
+    const PlainMatrix rhs(B);
+    const int64_t nrhs = static_cast<int64_t>(rhs.cols());
+    const int64_t ldb = static_cast<int64_t>(rhs.rows());
+    DeviceMatrix<Scalar> d_X = solve_impl(nrhs, ldb, [&](Scalar* d_x_ptr) {
+      EIGEN_CUDA_RUNTIME_CHECK(
+          cudaMemcpyAsync(d_x_ptr, rhs.data(), rhsBytes(nrhs, ldb), cudaMemcpyHostToDevice, stream_));
+    });
+
+    PlainMatrix X(n_, B.cols());
+    int solve_info = 0;
+    EIGEN_CUDA_RUNTIME_CHECK(
+        cudaMemcpyAsync(X.data(), d_X.data(), rhsBytes(nrhs, ldb), cudaMemcpyDeviceToHost, stream_));
+    EIGEN_CUDA_RUNTIME_CHECK(
+        cudaMemcpyAsync(&solve_info, scratch_info(), sizeof(int), cudaMemcpyDeviceToHost, stream_));
+    EIGEN_CUDA_RUNTIME_CHECK(cudaStreamSynchronize(stream_));
+
+    eigen_assert(solve_info == 0 && "cusolverDnXpotrs reported an error");
+    return X;
+  }
+
+  /** Solve A * X = B with device-resident RHS. Fully async.
+   *
+   * All work is enqueued on this solver's stream. Returns a DeviceMatrix
+   * with a recorded ready event — no host synchronization occurs.
+   * The caller should check info() after compute() to verify the
+   * factorization succeeded; this method does not check.
+   */
+  DeviceMatrix<Scalar> solve(const DeviceMatrix<Scalar>& d_B) const {
+    eigen_assert(d_B.rows() == n_);
+    d_B.waitReady(stream_);
+    const int64_t nrhs = static_cast<int64_t>(d_B.cols());
+    const int64_t ldb = static_cast<int64_t>(d_B.rows());
+    return solve_impl(nrhs, ldb, [&](Scalar* d_x_ptr) {
+      EIGEN_CUDA_RUNTIME_CHECK(
+          cudaMemcpyAsync(d_x_ptr, d_B.data(), rhsBytes(nrhs, ldb), cudaMemcpyDeviceToDevice, stream_));
+    });
+  }
+
+  // ---- Accessors -----------------------------------------------------------
+
+  /** Returns Success if the last compute() succeeded, NumericalIssue otherwise.
+   * Lazily synchronizes the stream on first call after compute(). */
+  ComputationInfo info() const {
+    const_cast<GpuLLT*>(this)->sync_info();
+    return info_;
+  }
+
+  Index rows() const { return n_; }
+  Index cols() const { return n_; }
+
+  /** Returns the CUDA stream owned by this object.
+   *  Advanced users may submit additional GPU work on this stream
+   *  to overlap with or chain after GpuLLT operations. */
+  cudaStream_t stream() const { return stream_; }
+
+ private:
+  cudaStream_t stream_ = nullptr;
+  cusolverDnHandle_t handle_ = nullptr;
+  internal::CusolverParams params_;   // cuSOLVER params (created once, reused)
+  internal::DeviceBuffer d_factor_;   // factored L (or U) on device (grows, never shrinks)
+  size_t factor_alloc_size_ = 0;      // current d_factor_ allocation size
+  internal::DeviceBuffer d_scratch_;  // combined workspace + info word (grows, never shrinks)
+  size_t scratch_size_ = 0;           // current scratch allocation size
+  std::vector<char> h_workspace_;     // host workspace (kept alive until next compute)
+  Index n_ = 0;
+  int64_t lda_ = 0;
+  ComputationInfo info_ = InvalidInput;
+  int info_word_ = 0;        // host-side target for async info download
+  bool info_synced_ = true;  // has the stream been synced for info?
+
+  bool begin_compute(Index rows) {
+    n_ = rows;
+    info_ = InvalidInput;
+    if (n_ == 0) {
+      info_ = Success;
+      return false;
+    }
+    return true;
+  }
+
+  size_t factorBytes() const { return rhsBytes(static_cast<int64_t>(n_), lda_); }
+
+  static size_t rhsBytes(int64_t cols, int64_t outer_stride) {
+    return static_cast<size_t>(outer_stride) * static_cast<size_t>(cols) * sizeof(Scalar);
+  }
+
+  void allocate_factor_storage() {
+    size_t needed = factorBytes();
+    if (needed > factor_alloc_size_) {
+      d_factor_ = internal::DeviceBuffer(needed);
+      factor_alloc_size_ = needed;
+    }
+  }
+
+  // Ensure d_scratch_ is at least `workspace_bytes + sizeof(int)`.
+  // Layout: [workspace (workspace_bytes) | info_word (sizeof(int))].
+  // Ensure d_scratch_ can hold workspace_bytes + an aligned info word.
+  // Grows but never shrinks. Syncs the stream before reallocating to
+  // avoid freeing memory that async kernels may still be using.
+  void ensure_scratch(size_t workspace_bytes) {
+    // Round up so the info word is naturally aligned.
+    // 16-byte alignment for optimal GPU memory access.
+    constexpr size_t kAlign = 16;
+    workspace_bytes = (workspace_bytes + kAlign - 1) & ~(kAlign - 1);
+    size_t needed = workspace_bytes + sizeof(int);
+    if (needed > scratch_size_) {
+      if (d_scratch_.ptr) EIGEN_CUDA_RUNTIME_CHECK(cudaStreamSynchronize(stream_));
+      d_scratch_ = internal::DeviceBuffer(needed);
+      scratch_size_ = needed;
+    }
+  }
+
+  void* scratch_workspace() const { return d_scratch_.ptr; }
+  int* scratch_info() const {
+    return reinterpret_cast<int*>(static_cast<char*>(d_scratch_.ptr) + scratch_size_ - sizeof(int));
+  }
+
+  template <typename CopyRhs>
+  DeviceMatrix<Scalar> solve_impl(int64_t nrhs, int64_t ldb, CopyRhs&& copy_rhs) const {
+    constexpr cudaDataType_t dtype = internal::cusolver_data_type<Scalar>::value;
+    constexpr cublasFillMode_t uplo = internal::cusolver_fill_mode<UpLo_, ColMajor>::value;
+
+    Scalar* d_x_ptr = nullptr;
+    EIGEN_CUDA_RUNTIME_CHECK(cudaMalloc(reinterpret_cast<void**>(&d_x_ptr), rhsBytes(nrhs, ldb)));
+    copy_rhs(d_x_ptr);
+
+    EIGEN_CUSOLVER_CHECK(cusolverDnXpotrs(handle_, params_.p, uplo, static_cast<int64_t>(n_), nrhs, dtype,
+                                          d_factor_.ptr, lda_, dtype, d_x_ptr, ldb, scratch_info()));
+
+    DeviceMatrix<Scalar> result(d_x_ptr, n_, static_cast<Index>(nrhs));
+    result.recordReady(stream_);
+    return result;
+  }
+
+  void init_context() {
+    EIGEN_CUDA_RUNTIME_CHECK(cudaStreamCreate(&stream_));
+    EIGEN_CUSOLVER_CHECK(cusolverDnCreate(&handle_));
+    EIGEN_CUSOLVER_CHECK(cusolverDnSetStream(handle_, stream_));
+    ensure_scratch(0);  // allocate at least the info word
+  }
+
+  // Synchronize stream and interpret the info word. No-op if already synced.
+  void sync_info() {
+    if (!info_synced_) {
+      EIGEN_CUDA_RUNTIME_CHECK(cudaStreamSynchronize(stream_));
+      info_ = (info_word_ == 0) ? Success : NumericalIssue;
+      info_synced_ = true;
+    }
+  }
+
+  // Run cusolverDnXpotrf on d_factor_ (already on device).
+  // Enqueues factorization + async info download. Does NOT sync.
+  // Workspaces are stored as members to ensure they outlive the async kernels.
+  void factorize() {
+    constexpr cudaDataType_t dtype = internal::cusolver_data_type<Scalar>::value;
+    constexpr cublasFillMode_t uplo = internal::cusolver_fill_mode<UpLo_, ColMajor>::value;
+
+    info_synced_ = false;
+    info_ = InvalidInput;
+
+    size_t dev_ws_bytes = 0, host_ws_bytes = 0;
+    EIGEN_CUSOLVER_CHECK(cusolverDnXpotrf_bufferSize(handle_, params_.p, uplo, static_cast<int64_t>(n_), dtype,
+                                                     d_factor_.ptr, lda_, dtype, &dev_ws_bytes, &host_ws_bytes));
+
+    ensure_scratch(dev_ws_bytes);
+    h_workspace_.resize(host_ws_bytes);
+
+    EIGEN_CUSOLVER_CHECK(cusolverDnXpotrf(
+        handle_, params_.p, uplo, static_cast<int64_t>(n_), dtype, d_factor_.ptr, lda_, dtype, scratch_workspace(),
+        dev_ws_bytes, host_ws_bytes > 0 ? h_workspace_.data() : nullptr, host_ws_bytes, scratch_info()));
+
+    // Enqueue async download of info word — sync deferred to info() or solve().
+    EIGEN_CUDA_RUNTIME_CHECK(
+        cudaMemcpyAsync(&info_word_, scratch_info(), sizeof(int), cudaMemcpyDeviceToHost, stream_));
+  }
+};
+
+}  // namespace Eigen
+
+#endif  // EIGEN_GPU_LLT_H

Eigen/src/GPU/GpuLU.h

@@ -0,0 +1,371 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2026 Eigen Authors
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+// GPU partial-pivoting LU decomposition using cuSOLVER.
+//
+// Wraps cusolverDnXgetrf (factorization) and cusolverDnXgetrs (solve).
+// The factored LU matrix and pivot array are kept in device memory for the
+// lifetime of the object, so repeated solves only transfer the RHS/solution.
+//
+// Requires CUDA 11.0+ (cusolverDnX generic API).
+//
+// Usage:
+//   GpuLU<double> lu(A);              // upload A, getrf, LU+ipiv on device
+//   if (lu.info() != Success) { ... }
+//   MatrixXd x = lu.solve(b);         // getrs NoTrans, only b transferred
+//   MatrixXd xt = lu.solve(b, GpuLU<double>::Transpose);   // A^T x = b
+
+#ifndef EIGEN_GPU_LU_H
+#define EIGEN_GPU_LU_H
+
+// IWYU pragma: private
+#include "./InternalHeaderCheck.h"
+
+#include "./CuSolverSupport.h"
+#include <vector>
+
+namespace Eigen {
+
+/** \ingroup GPU_Module
+ * \class GpuLU
+ * \brief GPU LU decomposition with partial pivoting via cuSOLVER
+ *
+ * \tparam Scalar_  Element type: float, double, complex<float>, complex<double>
+ *
+ * Decomposes a square matrix A = P L U on the GPU and retains the factored
+ * matrix and pivot array in device memory. Solves A*X=B, A^T*X=B, or
+ * A^H*X=B by passing the appropriate TransposeMode.
+ *
+ * Each GpuLU object owns a dedicated CUDA stream and cuSOLVER handle.
+ */
+template <typename Scalar_>
+class GpuLU {
+ public:
+  using Scalar = Scalar_;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+  using PlainMatrix = Matrix<Scalar, Dynamic, Dynamic, ColMajor>;
+
+  /** Controls which system is solved in solve(). */
+  enum TransposeMode {
+    NoTranspose,        ///< Solve A   * X = B
+    Transpose,          ///< Solve A^T * X = B
+    ConjugateTranspose  ///< Solve A^H * X = B (same as Transpose for real types)
+  };
+
+  // ---- Construction / destruction ------------------------------------------
+
+  GpuLU() { init_context(); }
+
+  template <typename InputType>
+  explicit GpuLU(const EigenBase<InputType>& A) {
+    init_context();
+    compute(A);
+  }
+
+  ~GpuLU() {
+    if (handle_) (void)cusolverDnDestroy(handle_);
+    if (stream_) (void)cudaStreamDestroy(stream_);
+  }
+
+  GpuLU(const GpuLU&) = delete;
+  GpuLU& operator=(const GpuLU&) = delete;
+
+  GpuLU(GpuLU&& o) noexcept
+      : stream_(o.stream_),
+        handle_(o.handle_),
+        params_(std::move(o.params_)),
+        d_lu_(std::move(o.d_lu_)),
+        lu_alloc_size_(o.lu_alloc_size_),
+        d_ipiv_(std::move(o.d_ipiv_)),
+        d_scratch_(std::move(o.d_scratch_)),
+        scratch_size_(o.scratch_size_),
+        h_workspace_(std::move(o.h_workspace_)),
+        n_(o.n_),
+        lda_(o.lda_),
+        info_(o.info_),
+        info_word_(o.info_word_),
+        info_synced_(o.info_synced_) {
+    o.stream_ = nullptr;
+    o.handle_ = nullptr;
+    o.lu_alloc_size_ = 0;
+    o.scratch_size_ = 0;
+    o.n_ = 0;
+    o.info_ = InvalidInput;
+    o.info_word_ = 0;
+    o.info_synced_ = true;
+  }
+
+  GpuLU& operator=(GpuLU&& o) noexcept {
+    if (this != &o) {
+      if (handle_) (void)cusolverDnDestroy(handle_);
+      if (stream_) (void)cudaStreamDestroy(stream_);
+      stream_ = o.stream_;
+      handle_ = o.handle_;
+      params_ = std::move(o.params_);
+      d_lu_ = std::move(o.d_lu_);
+      lu_alloc_size_ = o.lu_alloc_size_;
+      d_ipiv_ = std::move(o.d_ipiv_);
+      d_scratch_ = std::move(o.d_scratch_);
+      scratch_size_ = o.scratch_size_;
+      h_workspace_ = std::move(o.h_workspace_);
+      n_ = o.n_;
+      lda_ = o.lda_;
+      info_ = o.info_;
+      info_word_ = o.info_word_;
+      info_synced_ = o.info_synced_;
+      o.stream_ = nullptr;
+      o.handle_ = nullptr;
+      o.lu_alloc_size_ = 0;
+      o.scratch_size_ = 0;
+      o.n_ = 0;
+      o.info_ = InvalidInput;
+      o.info_word_ = 0;
+      o.info_synced_ = true;
+    }
+    return *this;
+  }
+
+  // ---- Factorization -------------------------------------------------------
+
+  /** Compute the LU factorization of A (host matrix, must be square). */
+  template <typename InputType>
+  GpuLU& compute(const EigenBase<InputType>& A) {
+    eigen_assert(A.rows() == A.cols() && "GpuLU requires a square matrix");
+    if (!begin_compute(A.rows())) return *this;
+
+    const PlainMatrix mat(A.derived());
+    lda_ = static_cast<int64_t>(mat.rows());
+    allocate_lu_storage();
+    EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpyAsync(d_lu_.ptr, mat.data(), matrixBytes(), cudaMemcpyHostToDevice, stream_));
+
+    factorize();
+    return *this;
+  }
+
+  /** Compute the LU factorization from a device-resident matrix (D2D copy). */
+  GpuLU& compute(const DeviceMatrix<Scalar>& d_A) {
+    eigen_assert(d_A.rows() == d_A.cols() && "GpuLU requires a square matrix");
+    if (!begin_compute(d_A.rows())) return *this;
+
+    lda_ = static_cast<int64_t>(d_A.rows());
+    d_A.waitReady(stream_);
+    allocate_lu_storage();
+    EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpyAsync(d_lu_.ptr, d_A.data(), matrixBytes(), cudaMemcpyDeviceToDevice, stream_));
+
+    factorize();
+    return *this;
+  }
+
+  /** Compute the LU factorization from a device matrix (move, no copy). */
+  GpuLU& compute(DeviceMatrix<Scalar>&& d_A) {
+    eigen_assert(d_A.rows() == d_A.cols() && "GpuLU requires a square matrix");
+    if (!begin_compute(d_A.rows())) return *this;
+
+    lda_ = static_cast<int64_t>(d_A.rows());
+    d_A.waitReady(stream_);
+    d_lu_ = internal::DeviceBuffer::adopt(static_cast<void*>(d_A.release()));
+
+    factorize();
+    return *this;
+  }
+
+  // ---- Solve ---------------------------------------------------------------
+
+  /** Solve op(A) * X = B using the cached LU factorization (host → host).
+   *
+   * \param B    Right-hand side (n x nrhs host matrix).
+   * \param mode NoTranspose (default), Transpose, or ConjugateTranspose.
+   */
+  template <typename Rhs>
+  PlainMatrix solve(const MatrixBase<Rhs>& B, TransposeMode mode = NoTranspose) const {
+    const_cast<GpuLU*>(this)->sync_info();
+    eigen_assert(info_ == Success && "GpuLU::solve called on a failed or uninitialized factorization");
+    eigen_assert(B.rows() == n_);
+
+    const PlainMatrix rhs(B);
+    const int64_t nrhs = static_cast<int64_t>(rhs.cols());
+    const int64_t ldb = static_cast<int64_t>(rhs.rows());
+    DeviceMatrix<Scalar> d_X = solve_impl(nrhs, ldb, mode, [&](Scalar* d_x_ptr) {
+      EIGEN_CUDA_RUNTIME_CHECK(
+          cudaMemcpyAsync(d_x_ptr, rhs.data(), matrixBytes(nrhs, ldb), cudaMemcpyHostToDevice, stream_));
+    });
+
+    PlainMatrix X(n_, B.cols());
+    int solve_info = 0;
+    EIGEN_CUDA_RUNTIME_CHECK(
+        cudaMemcpyAsync(X.data(), d_X.data(), matrixBytes(nrhs, ldb), cudaMemcpyDeviceToHost, stream_));
+    EIGEN_CUDA_RUNTIME_CHECK(
+        cudaMemcpyAsync(&solve_info, scratch_info(), sizeof(int), cudaMemcpyDeviceToHost, stream_));
+    EIGEN_CUDA_RUNTIME_CHECK(cudaStreamSynchronize(stream_));
+
+    eigen_assert(solve_info == 0 && "cusolverDnXgetrs reported an error");
+    return X;
+  }
+
+  /** Solve op(A) * X = B with device-resident RHS. Fully async. */
+  DeviceMatrix<Scalar> solve(const DeviceMatrix<Scalar>& d_B, TransposeMode mode = NoTranspose) const {
+    eigen_assert(d_B.rows() == n_);
+    d_B.waitReady(stream_);
+    const int64_t nrhs = static_cast<int64_t>(d_B.cols());
+    const int64_t ldb = static_cast<int64_t>(d_B.rows());
+    return solve_impl(nrhs, ldb, mode, [&](Scalar* d_x_ptr) {
+      EIGEN_CUDA_RUNTIME_CHECK(
+          cudaMemcpyAsync(d_x_ptr, d_B.data(), matrixBytes(nrhs, ldb), cudaMemcpyDeviceToDevice, stream_));
+    });
+  }
+
+  // ---- Accessors -----------------------------------------------------------
+
+  /** Lazily synchronizes the stream on first call after compute(). */
+  ComputationInfo info() const {
+    const_cast<GpuLU*>(this)->sync_info();
+    return info_;
+  }
+  Index rows() const { return n_; }
+  Index cols() const { return n_; }
+  cudaStream_t stream() const { return stream_; }
+
+ private:
+  cudaStream_t stream_ = nullptr;
+  cusolverDnHandle_t handle_ = nullptr;
+  internal::CusolverParams params_;   // cuSOLVER params (created once, reused)
+  internal::DeviceBuffer d_lu_;       // LU factors on device (grows, never shrinks)
+  size_t lu_alloc_size_ = 0;          // current d_lu_ allocation size
+  internal::DeviceBuffer d_ipiv_;     // pivot indices (int64_t) on device
+  internal::DeviceBuffer d_scratch_;  // combined workspace + info word (grows, never shrinks)
+  size_t scratch_size_ = 0;           // current scratch allocation size
+  std::vector<char> h_workspace_;     // host workspace (kept alive until next compute)
+  Index n_ = 0;
+  int64_t lda_ = 0;
+  ComputationInfo info_ = InvalidInput;
+  int info_word_ = 0;        // host-side target for async info download
+  bool info_synced_ = true;  // has the stream been synced for info?
+
+  bool begin_compute(Index rows) {
+    n_ = rows;
+    info_ = InvalidInput;
+    if (n_ == 0) {
+      info_ = Success;
+      return false;
+    }
+    return true;
+  }
+
+  size_t matrixBytes() const { return matrixBytes(static_cast<int64_t>(n_), lda_); }
+
+  static size_t matrixBytes(int64_t cols, int64_t outer_stride) {
+    return static_cast<size_t>(outer_stride) * static_cast<size_t>(cols) * sizeof(Scalar);
+  }
+
+  void allocate_lu_storage() {
+    size_t needed = matrixBytes();
+    if (needed > lu_alloc_size_) {
+      d_lu_ = internal::DeviceBuffer(needed);
+      lu_alloc_size_ = needed;
+    }
+  }
+
+  // Ensure d_scratch_ is at least `workspace_bytes + sizeof(int)`.
+  // Layout: [workspace (workspace_bytes) | info_word (sizeof(int))].
+  // Ensure d_scratch_ can hold workspace_bytes + an aligned info word.
+  // Grows but never shrinks. Syncs the stream before reallocating to
+  // avoid freeing memory that async kernels may still be using.
+  void ensure_scratch(size_t workspace_bytes) {
+    constexpr size_t kAlign = 16;
+    workspace_bytes = (workspace_bytes + kAlign - 1) & ~(kAlign - 1);
+    size_t needed = workspace_bytes + sizeof(int);
+    if (needed > scratch_size_) {
+      if (d_scratch_.ptr) EIGEN_CUDA_RUNTIME_CHECK(cudaStreamSynchronize(stream_));
+      d_scratch_ = internal::DeviceBuffer(needed);
+      scratch_size_ = needed;
+    }
+  }
+
+  void* scratch_workspace() const { return d_scratch_.ptr; }
+  int* scratch_info() const {
+    return reinterpret_cast<int*>(static_cast<char*>(d_scratch_.ptr) + scratch_size_ - sizeof(int));
+  }
+
+  template <typename CopyRhs>
+  DeviceMatrix<Scalar> solve_impl(int64_t nrhs, int64_t ldb, TransposeMode mode, CopyRhs&& copy_rhs) const {
+    constexpr cudaDataType_t dtype = internal::cusolver_data_type<Scalar>::value;
+    const cublasOperation_t trans = to_cublas_op(mode);
+
+    Scalar* d_x_ptr = nullptr;
+    EIGEN_CUDA_RUNTIME_CHECK(cudaMalloc(reinterpret_cast<void**>(&d_x_ptr), matrixBytes(nrhs, ldb)));
+    copy_rhs(d_x_ptr);
+
+    EIGEN_CUSOLVER_CHECK(cusolverDnXgetrs(handle_, params_.p, trans, static_cast<int64_t>(n_), nrhs, dtype, d_lu_.ptr,
+                                          lda_, static_cast<const int64_t*>(d_ipiv_.ptr), dtype, d_x_ptr, ldb,
+                                          scratch_info()));
+
+    DeviceMatrix<Scalar> result(d_x_ptr, n_, static_cast<Index>(nrhs));
+    result.recordReady(stream_);
+    return result;
+  }
+
+  void init_context() {
+    EIGEN_CUDA_RUNTIME_CHECK(cudaStreamCreate(&stream_));
+    EIGEN_CUSOLVER_CHECK(cusolverDnCreate(&handle_));
+    EIGEN_CUSOLVER_CHECK(cusolverDnSetStream(handle_, stream_));
+    ensure_scratch(0);  // allocate at least the info word
+  }
+
+  void sync_info() {
+    if (!info_synced_) {
+      EIGEN_CUDA_RUNTIME_CHECK(cudaStreamSynchronize(stream_));
+      info_ = (info_word_ == 0) ? Success : NumericalIssue;
+      info_synced_ = true;
+    }
+  }
+
+  // Run cusolverDnXgetrf on d_lu_ (already on device). Allocates d_ipiv_.
+  // Enqueues factorization + async info download. Does NOT sync.
+  // Workspaces are stored as members to ensure they outlive the async kernels.
+  void factorize() {
+    constexpr cudaDataType_t dtype = internal::cusolver_data_type<Scalar>::value;
+    const size_t ipiv_bytes = static_cast<size_t>(n_) * sizeof(int64_t);
+
+    info_synced_ = false;
+    info_ = InvalidInput;
+
+    d_ipiv_ = internal::DeviceBuffer(ipiv_bytes);
+
+    size_t dev_ws_bytes = 0, host_ws_bytes = 0;
+    EIGEN_CUSOLVER_CHECK(cusolverDnXgetrf_bufferSize(handle_, params_.p, static_cast<int64_t>(n_),
+                                                     static_cast<int64_t>(n_), dtype, d_lu_.ptr, lda_, dtype,
+                                                     &dev_ws_bytes, &host_ws_bytes));
+
+    ensure_scratch(dev_ws_bytes);
+    h_workspace_.resize(host_ws_bytes);
+
+    EIGEN_CUSOLVER_CHECK(
+        cusolverDnXgetrf(handle_, params_.p, static_cast<int64_t>(n_), static_cast<int64_t>(n_), dtype, d_lu_.ptr, lda_,
+                         static_cast<int64_t*>(d_ipiv_.ptr), dtype, scratch_workspace(), dev_ws_bytes,
+                         host_ws_bytes > 0 ? h_workspace_.data() : nullptr, host_ws_bytes, scratch_info()));
+
+    EIGEN_CUDA_RUNTIME_CHECK(
+        cudaMemcpyAsync(&info_word_, scratch_info(), sizeof(int), cudaMemcpyDeviceToHost, stream_));
+  }
+
+  static cublasOperation_t to_cublas_op(TransposeMode mode) {
+    switch (mode) {
+      case Transpose:
+        return CUBLAS_OP_T;
+      case ConjugateTranspose:
+        return CUBLAS_OP_C;
+      default:
+        return CUBLAS_OP_N;
+    }
+  }
+};
+
+}  // namespace Eigen
+
+#endif  // EIGEN_GPU_LU_H

Eigen/src/GPU/GpuQR.h

@@ -0,0 +1,386 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2026 Rasmus Munk Larsen <rmlarsen@gmail.com>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+// GPU QR decomposition using cuSOLVER.
+//
+// Wraps cusolverDnXgeqrf (factorization), cusolverDnXormqr (apply Q),
+// cusolverDnXorgqr (form Q), and cublasXtrsm (triangular solve on R).
+//
+// The factored matrix (reflectors + R) and tau stay in device memory.
+// Solve uses ormqr + trsm without forming Q explicitly.
+//
+// Usage:
+//   GpuQR<double> qr(A);              // upload A, geqrf
+//   if (qr.info() != Success) { ... }
+//   MatrixXd X = qr.solve(B);         // Q^H * B via ormqr, then trsm on R
+//
+// Expression syntax:
+//   d_X = d_A.qr().solve(d_B);        // temporary, no caching
+
+#ifndef EIGEN_GPU_QR_H
+#define EIGEN_GPU_QR_H
+
+// IWYU pragma: private
+#include "./InternalHeaderCheck.h"
+
+#include "./CuSolverSupport.h"
+#include "./CuBlasSupport.h"
+#include <vector>
+
+namespace Eigen {
+
+template <typename Scalar_>
+class GpuQR {
+ public:
+  using Scalar = Scalar_;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+  using PlainMatrix = Matrix<Scalar, Dynamic, Dynamic, ColMajor>;
+
+  GpuQR() { init_context(); }
+
+  template <typename InputType>
+  explicit GpuQR(const EigenBase<InputType>& A) {
+    init_context();
+    compute(A);
+  }
+
+  ~GpuQR() {
+    if (handle_) (void)cusolverDnDestroy(handle_);
+    if (cublas_) (void)cublasDestroy(cublas_);
+    if (stream_) (void)cudaStreamDestroy(stream_);
+  }
+
+  GpuQR(const GpuQR&) = delete;
+  GpuQR& operator=(const GpuQR&) = delete;
+
+  GpuQR(GpuQR&& o) noexcept
+      : stream_(o.stream_),
+        handle_(o.handle_),
+        cublas_(o.cublas_),
+        params_(std::move(o.params_)),
+        d_qr_(std::move(o.d_qr_)),
+        d_tau_(std::move(o.d_tau_)),
+        d_scratch_(std::move(o.d_scratch_)),
+        scratch_size_(o.scratch_size_),
+        h_workspace_(std::move(o.h_workspace_)),
+        m_(o.m_),
+        n_(o.n_),
+        lda_(o.lda_),
+        info_(o.info_),
+        info_word_(o.info_word_),
+        info_synced_(o.info_synced_) {
+    o.stream_ = nullptr;
+    o.handle_ = nullptr;
+    o.cublas_ = nullptr;
+    o.scratch_size_ = 0;
+    o.m_ = 0;
+    o.n_ = 0;
+    o.lda_ = 0;
+    o.info_ = InvalidInput;
+    o.info_word_ = 0;
+    o.info_synced_ = true;
+  }
+
+  GpuQR& operator=(GpuQR&& o) noexcept {
+    if (this != &o) {
+      if (handle_) (void)cusolverDnDestroy(handle_);
+      if (cublas_) (void)cublasDestroy(cublas_);
+      if (stream_) (void)cudaStreamDestroy(stream_);
+      stream_ = o.stream_;
+      handle_ = o.handle_;
+      cublas_ = o.cublas_;
+      params_ = std::move(o.params_);
+      d_qr_ = std::move(o.d_qr_);
+      d_tau_ = std::move(o.d_tau_);
+      d_scratch_ = std::move(o.d_scratch_);
+      scratch_size_ = o.scratch_size_;
+      h_workspace_ = std::move(o.h_workspace_);
+      m_ = o.m_;
+      n_ = o.n_;
+      lda_ = o.lda_;
+      info_ = o.info_;
+      info_word_ = o.info_word_;
+      info_synced_ = o.info_synced_;
+      o.stream_ = nullptr;
+      o.handle_ = nullptr;
+      o.cublas_ = nullptr;
+      o.scratch_size_ = 0;
+      o.m_ = 0;
+      o.n_ = 0;
+      o.lda_ = 0;
+      o.info_ = InvalidInput;
+      o.info_word_ = 0;
+      o.info_synced_ = true;
+    }
+    return *this;
+  }
+
+  // ---- Factorization -------------------------------------------------------
+
+  template <typename InputType>
+  GpuQR& compute(const EigenBase<InputType>& A) {
+    m_ = A.rows();
+    n_ = A.cols();
+    info_ = InvalidInput;
+    info_synced_ = false;
+
+    if (m_ == 0 || n_ == 0) {
+      info_ = Success;
+      info_synced_ = true;
+      return *this;
+    }
+
+    const PlainMatrix mat(A.derived());
+    lda_ = static_cast<int64_t>(mat.rows());
+    const size_t mat_bytes = static_cast<size_t>(lda_) * static_cast<size_t>(n_) * sizeof(Scalar);
+    const size_t tau_bytes = static_cast<size_t>((std::min)(m_, n_)) * sizeof(Scalar);
+
+    d_qr_ = internal::DeviceBuffer(mat_bytes);
+    d_tau_ = internal::DeviceBuffer(tau_bytes);
+
+    EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpyAsync(d_qr_.ptr, mat.data(), mat_bytes, cudaMemcpyHostToDevice, stream_));
+
+    factorize();
+    return *this;
+  }
+
+  GpuQR& compute(const DeviceMatrix<Scalar>& d_A) {
+    m_ = d_A.rows();
+    n_ = d_A.cols();
+    info_ = InvalidInput;
+    info_synced_ = false;
+
+    if (m_ == 0 || n_ == 0) {
+      info_ = Success;
+      info_synced_ = true;
+      return *this;
+    }
+
+    lda_ = static_cast<int64_t>(d_A.rows());
+    const size_t mat_bytes = static_cast<size_t>(lda_) * static_cast<size_t>(n_) * sizeof(Scalar);
+    const size_t tau_bytes = static_cast<size_t>((std::min)(m_, n_)) * sizeof(Scalar);
+
+    d_A.waitReady(stream_);
+    d_qr_ = internal::DeviceBuffer(mat_bytes);
+    d_tau_ = internal::DeviceBuffer(tau_bytes);
+    EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpyAsync(d_qr_.ptr, d_A.data(), mat_bytes, cudaMemcpyDeviceToDevice, stream_));
+
+    factorize();
+    return *this;
+  }
+
+  // ---- Solve ---------------------------------------------------------------
+
+  /** Solve A * X = B via QR: X = R^{-1} * Q^H * B (least-squares for m >= n).
+   * Uses ormqr (apply Q^H) + trsm (solve R), without forming Q explicitly.
+   * Requires m >= n (overdetermined or square). Underdetermined not supported. */
+  template <typename Rhs>
+  PlainMatrix solve(const MatrixBase<Rhs>& B) const {
+    sync_info();
+    eigen_assert(info_ == Success && "GpuQR::solve called on a failed or uninitialized factorization");
+    eigen_assert(B.rows() == m_);
+    eigen_assert(m_ >= n_ && "GpuQR::solve requires m >= n (use SVD for underdetermined systems)");
+
+    const PlainMatrix rhs(B);
+    const int64_t nrhs = static_cast<int64_t>(rhs.cols());
+    const int64_t ldb = static_cast<int64_t>(rhs.rows());  // = m_
+    const size_t b_bytes = static_cast<size_t>(ldb) * static_cast<size_t>(nrhs) * sizeof(Scalar);
+
+    // Upload B to device (m × nrhs buffer).
+    internal::DeviceBuffer d_B(b_bytes);
+    EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpyAsync(d_B.ptr, rhs.data(), b_bytes, cudaMemcpyHostToDevice, stream_));
+
+    // Apply Q^H to B in-place: d_B becomes m × nrhs, first n rows hold Q^H * B relevant part.
+    apply_QH(d_B.ptr, ldb, nrhs);
+
+    // Solve R * X = (Q^H * B)[0:n,:] via trsm on the first n rows.
+    Scalar alpha(1);
+    EIGEN_CUBLAS_CHECK(internal::cublasXtrsm(cublas_, CUBLAS_SIDE_LEFT, CUBLAS_FILL_MODE_UPPER, CUBLAS_OP_N,
+                                             CUBLAS_DIAG_NON_UNIT, static_cast<int>(n_), static_cast<int>(nrhs), &alpha,
+                                             static_cast<const Scalar*>(d_qr_.ptr), static_cast<int>(lda_),
+                                             static_cast<Scalar*>(d_B.ptr), static_cast<int>(ldb)));
+
+    // Download the first n rows of each column (stride = ldb = m, width = n).
+    PlainMatrix X(n_, rhs.cols());
+    if (m_ == n_) {
+      // Square: dense copy, no stride mismatch.
+      EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpyAsync(X.data(), d_B.ptr,
+                                               static_cast<size_t>(n_) * static_cast<size_t>(nrhs) * sizeof(Scalar),
+                                               cudaMemcpyDeviceToHost, stream_));
+    } else {
+      // Overdetermined: 2D copy to extract first n rows from each column.
+      EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpy2DAsync(
+          X.data(), static_cast<size_t>(n_) * sizeof(Scalar), d_B.ptr, static_cast<size_t>(ldb) * sizeof(Scalar),
+          static_cast<size_t>(n_) * sizeof(Scalar), static_cast<size_t>(nrhs), cudaMemcpyDeviceToHost, stream_));
+    }
+    EIGEN_CUDA_RUNTIME_CHECK(cudaStreamSynchronize(stream_));
+    return X;
+  }
+
+  /** Solve with device-resident RHS. Returns n × nrhs DeviceMatrix. */
+  DeviceMatrix<Scalar> solve(const DeviceMatrix<Scalar>& d_B) const {
+    sync_info();
+    eigen_assert(info_ == Success && "GpuQR::solve called on a failed or uninitialized factorization");
+    eigen_assert(d_B.rows() == m_);
+    eigen_assert(m_ >= n_ && "GpuQR::solve requires m >= n (use SVD for underdetermined systems)");
+    d_B.waitReady(stream_);
+
+    const int64_t nrhs = static_cast<int64_t>(d_B.cols());
+    const int64_t ldb = static_cast<int64_t>(d_B.rows());  // = m_
+    const size_t b_bytes = static_cast<size_t>(ldb) * static_cast<size_t>(nrhs) * sizeof(Scalar);
+
+    // D2D copy B into working buffer (ormqr and trsm are in-place).
+    internal::DeviceBuffer d_work(b_bytes);
+    EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpyAsync(d_work.ptr, d_B.data(), b_bytes, cudaMemcpyDeviceToDevice, stream_));
+
+    apply_QH(d_work.ptr, ldb, nrhs);
+
+    // trsm on the first n rows.
+    Scalar alpha(1);
+    EIGEN_CUBLAS_CHECK(internal::cublasXtrsm(cublas_, CUBLAS_SIDE_LEFT, CUBLAS_FILL_MODE_UPPER, CUBLAS_OP_N,
+                                             CUBLAS_DIAG_NON_UNIT, static_cast<int>(n_), static_cast<int>(nrhs), &alpha,
+                                             static_cast<const Scalar*>(d_qr_.ptr), static_cast<int>(lda_),
+                                             static_cast<Scalar*>(d_work.ptr), static_cast<int>(ldb)));
+
+    if (m_ == n_) {
+      // Square: result is the whole buffer, dense.
+      DeviceMatrix<Scalar> result(static_cast<Scalar*>(d_work.ptr), n_, static_cast<Index>(nrhs));
+      d_work.ptr = nullptr;  // transfer ownership
+      result.recordReady(stream_);
+      return result;
+    } else {
+      // Overdetermined: copy first n rows of each column into a dense n × nrhs result.
+      DeviceMatrix<Scalar> result(n_, static_cast<Index>(nrhs));
+      EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpy2DAsync(result.data(), static_cast<size_t>(n_) * sizeof(Scalar), d_work.ptr,
+                                                 static_cast<size_t>(ldb) * sizeof(Scalar),
+                                                 static_cast<size_t>(n_) * sizeof(Scalar), static_cast<size_t>(nrhs),
+                                                 cudaMemcpyDeviceToDevice, stream_));
+      result.recordReady(stream_);
+      return result;
+      // d_work freed here via RAII — safe because stream is ordered.
+    }
+  }
+
+  // ---- Accessors -----------------------------------------------------------
+
+  ComputationInfo info() const {
+    sync_info();
+    return info_;
+  }
+
+  Index rows() const { return m_; }
+  Index cols() const { return n_; }
+  cudaStream_t stream() const { return stream_; }
+
+ private:
+  cudaStream_t stream_ = nullptr;
+  cusolverDnHandle_t handle_ = nullptr;
+  cublasHandle_t cublas_ = nullptr;
+  internal::CusolverParams params_;
+  internal::DeviceBuffer d_qr_;       // QR factors (reflectors in lower, R in upper)
+  internal::DeviceBuffer d_tau_;      // Householder scalars (min(m,n))
+  internal::DeviceBuffer d_scratch_;  // workspace + info word
+  size_t scratch_size_ = 0;
+  std::vector<char> h_workspace_;
+  Index m_ = 0;
+  Index n_ = 0;
+  int64_t lda_ = 0;
+  ComputationInfo info_ = InvalidInput;
+  int info_word_ = 0;
+  bool info_synced_ = true;
+
+  void init_context() {
+    EIGEN_CUDA_RUNTIME_CHECK(cudaStreamCreate(&stream_));
+    EIGEN_CUSOLVER_CHECK(cusolverDnCreate(&handle_));
+    EIGEN_CUSOLVER_CHECK(cusolverDnSetStream(handle_, stream_));
+    EIGEN_CUBLAS_CHECK(cublasCreate(&cublas_));
+    EIGEN_CUBLAS_CHECK(cublasSetStream(cublas_, stream_));
+    ensure_scratch(0);
+  }
+
+  void ensure_scratch(size_t workspace_bytes) {
+    constexpr size_t kAlign = 16;
+    workspace_bytes = (workspace_bytes + kAlign - 1) & ~(kAlign - 1);
+    size_t needed = workspace_bytes + sizeof(int);
+    if (needed > scratch_size_) {
+      if (d_scratch_.ptr) EIGEN_CUDA_RUNTIME_CHECK(cudaStreamSynchronize(stream_));
+      d_scratch_ = internal::DeviceBuffer(needed);
+      scratch_size_ = needed;
+    }
+  }
+
+  void* scratch_workspace() const { return d_scratch_.ptr; }
+  int* scratch_info() const {
+    return reinterpret_cast<int*>(static_cast<char*>(d_scratch_.ptr) + scratch_size_ - sizeof(int));
+  }
+
+  void sync_info() const {
+    if (!info_synced_) {
+      EIGEN_CUDA_RUNTIME_CHECK(cudaStreamSynchronize(stream_));
+      const_cast<GpuQR*>(this)->info_ = (info_word_ == 0) ? Success : NumericalIssue;
+      const_cast<GpuQR*>(this)->info_synced_ = true;
+    }
+  }
+
+  void factorize() {
+    constexpr cudaDataType_t dtype = internal::cusolver_data_type<Scalar>::value;
+
+    info_synced_ = false;
+    info_ = InvalidInput;
+
+    size_t dev_ws = 0, host_ws = 0;
+    EIGEN_CUSOLVER_CHECK(cusolverDnXgeqrf_bufferSize(handle_, params_.p, static_cast<int64_t>(m_),
+                                                     static_cast<int64_t>(n_), dtype, d_qr_.ptr, lda_, dtype,
+                                                     d_tau_.ptr, dtype, &dev_ws, &host_ws));
+
+    ensure_scratch(dev_ws);
+    h_workspace_.resize(host_ws);
+
+    EIGEN_CUSOLVER_CHECK(cusolverDnXgeqrf(handle_, params_.p, static_cast<int64_t>(m_), static_cast<int64_t>(n_), dtype,
+                                          d_qr_.ptr, lda_, dtype, d_tau_.ptr, dtype, scratch_workspace(), dev_ws,
+                                          host_ws > 0 ? h_workspace_.data() : nullptr, host_ws, scratch_info()));
+
+    EIGEN_CUDA_RUNTIME_CHECK(
+        cudaMemcpyAsync(&info_word_, scratch_info(), sizeof(int), cudaMemcpyDeviceToHost, stream_));
+  }
+
+  // Apply Q^H to a device buffer in-place: d_B = Q^H * d_B.
+  // Uses type-specific ormqr (real) or unmqr (complex) wrappers from CuSolverSupport.h.
+  // For real types: Q^H = Q^T, use CUBLAS_OP_T. For complex: use CUBLAS_OP_C.
+  void apply_QH(void* d_B, int64_t ldb, int64_t nrhs) const {
+    const int im = static_cast<int>(m_);
+    const int in = static_cast<int>(nrhs);
+    const int ik = static_cast<int>((std::min)(m_, n_));
+    const int ilda = static_cast<int>(lda_);
+    const int ildb = static_cast<int>(ldb);
+    constexpr cublasOperation_t trans = NumTraits<Scalar>::IsComplex ? CUBLAS_OP_C : CUBLAS_OP_T;
+
+    int lwork = 0;
+    EIGEN_CUSOLVER_CHECK(internal::cusolverDnXormqr_bufferSize(
+        handle_, CUBLAS_SIDE_LEFT, trans, im, in, ik, static_cast<const Scalar*>(d_qr_.ptr), ilda,
+        static_cast<const Scalar*>(d_tau_.ptr), static_cast<const Scalar*>(d_B), ildb, &lwork));
+
+    internal::DeviceBuffer d_work(static_cast<size_t>(lwork) * sizeof(Scalar));
+
+    EIGEN_CUSOLVER_CHECK(internal::cusolverDnXormqr(handle_, CUBLAS_SIDE_LEFT, trans, im, in, ik,
+                                                    static_cast<const Scalar*>(d_qr_.ptr), ilda,
+                                                    static_cast<const Scalar*>(d_tau_.ptr), static_cast<Scalar*>(d_B),
+                                                    ildb, static_cast<Scalar*>(d_work.ptr), lwork, scratch_info()));
+
+    // Sync to ensure workspace can be freed safely, and check ormqr info.
+    EIGEN_CUDA_RUNTIME_CHECK(cudaStreamSynchronize(stream_));
+    int ormqr_info = 0;
+    EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpy(&ormqr_info, scratch_info(), sizeof(int), cudaMemcpyDeviceToHost));
+    eigen_assert(ormqr_info == 0 && "cusolverDnXormqr reported an error");
+  }
+};
+
+}  // namespace Eigen
+
+#endif  // EIGEN_GPU_QR_H

Eigen/src/GPU/GpuSVD.h

@@ -0,0 +1,486 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2026 Rasmus Munk Larsen <rmlarsen@gmail.com>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+// GPU SVD decomposition using cuSOLVER (divide-and-conquer).
+//
+// Wraps cusolverDnXgesvd. Stores U, S, VT on device. Solve uses
+// cuBLAS GEMM: X = VT^H * diag(D) * U^H * B.
+//
+// cuSOLVER returns VT (not V). We store and expose VT directly.
+//
+// Usage:
+//   GpuSVD<double> svd(A, ComputeThinU | ComputeThinV);
+//   VectorXd S = svd.singularValues();
+//   MatrixXd U = svd.matrixU();       // m×k or m×m
+//   MatrixXd VT = svd.matrixVT();      // k×n or n×n (this is V^T)
+//   MatrixXd X = svd.solve(B);        // pseudoinverse
+//   MatrixXd X = svd.solve(B, k);     // truncated (top k triplets)
+//   MatrixXd X = svd.solve(B, 0.1);   // Tikhonov regularized
+
+#ifndef EIGEN_GPU_SVD_H
+#define EIGEN_GPU_SVD_H
+
+// IWYU pragma: private
+#include "./InternalHeaderCheck.h"
+
+#include "./CuSolverSupport.h"
+#include "./CuBlasSupport.h"
+#include <vector>
+
+namespace Eigen {
+
+template <typename Scalar_>
+class GpuSVD {
+ public:
+  using Scalar = Scalar_;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+  using PlainMatrix = Matrix<Scalar, Dynamic, Dynamic, ColMajor>;
+  using RealVector = Matrix<RealScalar, Dynamic, 1>;
+
+  GpuSVD() { init_context(); }
+
+  template <typename InputType>
+  explicit GpuSVD(const EigenBase<InputType>& A, unsigned int options = ComputeThinU | ComputeThinV) {
+    init_context();
+    compute(A, options);
+  }
+
+  ~GpuSVD() {
+    if (handle_) (void)cusolverDnDestroy(handle_);
+    if (cublas_) (void)cublasDestroy(cublas_);
+    if (stream_) (void)cudaStreamDestroy(stream_);
+  }
+
+  GpuSVD(const GpuSVD&) = delete;
+  GpuSVD& operator=(const GpuSVD&) = delete;
+  // Move constructors omitted for brevity — follow GpuQR pattern.
+
+  // ---- Factorization -------------------------------------------------------
+
+  template <typename InputType>
+  GpuSVD& compute(const EigenBase<InputType>& A, unsigned int options = ComputeThinU | ComputeThinV) {
+    options_ = options;
+    m_ = A.rows();
+    n_ = A.cols();
+    info_ = InvalidInput;
+    info_synced_ = false;
+
+    if (m_ == 0 || n_ == 0) {
+      info_ = Success;
+      info_synced_ = true;
+      return *this;
+    }
+
+    // cuSOLVER gesvd requires m >= n. For wide matrices, transpose internally.
+    transposed_ = (m_ < n_);
+    const PlainMatrix mat = transposed_ ? PlainMatrix(A.derived().adjoint()) : PlainMatrix(A.derived());
+    if (transposed_) std::swap(m_, n_);
+
+    lda_ = static_cast<int64_t>(mat.rows());
+    const size_t mat_bytes = static_cast<size_t>(lda_) * static_cast<size_t>(n_) * sizeof(Scalar);
+
+    // Copy (possibly transposed) A to device (gesvd overwrites it).
+    d_A_ = internal::DeviceBuffer(mat_bytes);
+    EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpyAsync(d_A_.ptr, mat.data(), mat_bytes, cudaMemcpyHostToDevice, stream_));
+
+    factorize();
+    return *this;
+  }
+
+  GpuSVD& compute(const DeviceMatrix<Scalar>& d_A, unsigned int options = ComputeThinU | ComputeThinV) {
+    options_ = options;
+    m_ = d_A.rows();
+    n_ = d_A.cols();
+    info_ = InvalidInput;
+    info_synced_ = false;
+
+    if (m_ == 0 || n_ == 0) {
+      info_ = Success;
+      info_synced_ = true;
+      return *this;
+    }
+
+    transposed_ = (m_ < n_);
+    d_A.waitReady(stream_);
+
+    if (transposed_) {
+      // Transpose on device via cuBLAS geam: d_A_ = A^H.
+      std::swap(m_, n_);
+      lda_ = m_;
+      const size_t mat_bytes = static_cast<size_t>(lda_) * static_cast<size_t>(n_) * sizeof(Scalar);
+      d_A_ = internal::DeviceBuffer(mat_bytes);
+      Scalar alpha_one(1), beta_zero(0);
+      // geam: C(m×n) = alpha * op(A) + beta * op(B). Use B = nullptr trick: beta=0.
+      // A is the original d_A (n_orig × m_orig = n × m after swap), transposed → m × n.
+      EIGEN_CUBLAS_CHECK(internal::cublasXgeam(
+          cublas_, CUBLAS_OP_C, CUBLAS_OP_N, static_cast<int>(m_), static_cast<int>(n_), &alpha_one, d_A.data(),
+          static_cast<int>(d_A.rows()), &beta_zero, static_cast<const Scalar*>(nullptr), static_cast<int>(m_),
+          static_cast<Scalar*>(d_A_.ptr), static_cast<int>(m_)));
+    } else {
+      lda_ = static_cast<int64_t>(d_A.rows());
+      const size_t mat_bytes = static_cast<size_t>(lda_) * static_cast<size_t>(n_) * sizeof(Scalar);
+      d_A_ = internal::DeviceBuffer(mat_bytes);
+      EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpyAsync(d_A_.ptr, d_A.data(), mat_bytes, cudaMemcpyDeviceToDevice, stream_));
+    }
+
+    factorize();
+    return *this;
+  }
+
+  // ---- Accessors -----------------------------------------------------------
+
+  ComputationInfo info() const {
+    sync_info();
+    return info_;
+  }
+
+  Index rows() const { return transposed_ ? n_ : m_; }
+  Index cols() const { return transposed_ ? m_ : n_; }
+
+  /** Singular values (always available). Downloads from device on each call. */
+  RealVector singularValues() const {
+    sync_info();
+    eigen_assert(info_ == Success);
+    const Index k = (std::min)(m_, n_);
+    RealVector S(k);
+    EIGEN_CUDA_RUNTIME_CHECK(
+        cudaMemcpy(S.data(), d_S_.ptr, static_cast<size_t>(k) * sizeof(RealScalar), cudaMemcpyDeviceToHost));
+    return S;
+  }
+
+  /** Left singular vectors U. Returns m_orig × k or m_orig × m_orig.
+   * For transposed case (m_orig < n_orig), U comes from cuSOLVER's VT. */
+  PlainMatrix matrixU() const {
+    sync_info();
+    eigen_assert(info_ == Success);
+    eigen_assert((options_ & (ComputeThinU | ComputeFullU)) && "matrixU() requires ComputeThinU or ComputeFullU");
+    const Index m_orig = transposed_ ? n_ : m_;
+    const Index n_orig = transposed_ ? m_ : n_;
+    const Index k = (std::min)(m_orig, n_orig);
+    if (!transposed_) {
+      const Index ucols = (options_ & ComputeFullU) ? m_ : k;
+      PlainMatrix U(m_, ucols);
+      EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpy(U.data(), d_U_.ptr,
+                                          static_cast<size_t>(m_) * static_cast<size_t>(ucols) * sizeof(Scalar),
+                                          cudaMemcpyDeviceToHost));
+      return U;
+    } else {
+      // Transposed: U_orig = VT_stored^H. VT_stored is vtrows × n_ (= vtrows × m_orig).
+      const Index vtrows = (options_ & ComputeFullU) ? m_orig : k;  // Note: FullU maps to FullV of A^H
+      PlainMatrix VT_stored(vtrows, n_);
+      EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpy(VT_stored.data(), d_VT_.ptr,
+                                          static_cast<size_t>(vtrows) * static_cast<size_t>(n_) * sizeof(Scalar),
+                                          cudaMemcpyDeviceToHost));
+      return VT_stored.adjoint();  // m_orig × vtrows
+    }
+  }
+
+  /** Right singular vectors transposed V^T. Returns k × n_orig or n_orig × n_orig.
+   * For transposed case, VT comes from cuSOLVER's U. */
+  PlainMatrix matrixVT() const {
+    sync_info();
+    eigen_assert(info_ == Success);
+    eigen_assert((options_ & (ComputeThinV | ComputeFullV)) && "matrixVT() requires ComputeThinV or ComputeFullV");
+    const Index m_orig = transposed_ ? n_ : m_;
+    const Index n_orig = transposed_ ? m_ : n_;
+    const Index k = (std::min)(m_orig, n_orig);
+    if (!transposed_) {
+      const Index vtrows = (options_ & ComputeFullV) ? n_ : k;
+      PlainMatrix VT(vtrows, n_);
+      EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpy(VT.data(), d_VT_.ptr,
+                                          static_cast<size_t>(vtrows) * static_cast<size_t>(n_) * sizeof(Scalar),
+                                          cudaMemcpyDeviceToHost));
+      return VT;
+    } else {
+      // Transposed: VT_orig = U_stored^H. U_stored is m_ × ucols (= n_orig × ucols).
+      const Index ucols = (options_ & ComputeFullV) ? n_orig : k;  // FullV maps to FullU of A^H
+      PlainMatrix U_stored(m_, ucols);
+      EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpy(U_stored.data(), d_U_.ptr,
+                                          static_cast<size_t>(m_) * static_cast<size_t>(ucols) * sizeof(Scalar),
+                                          cudaMemcpyDeviceToHost));
+      return U_stored.adjoint();  // ucols × n_orig
+    }
+  }
+
+  /** Number of singular values above threshold. */
+  Index rank(RealScalar threshold = RealScalar(-1)) const {
+    RealVector S = singularValues();
+    if (S.size() == 0) return 0;
+    if (threshold < 0) {
+      threshold = (std::max)(m_, n_) * S(0) * NumTraits<RealScalar>::epsilon();
+    }
+    Index r = 0;
+    for (Index i = 0; i < S.size(); ++i) {
+      if (S(i) > threshold) ++r;
+    }
+    return r;
+  }
+
+  // ---- Solve ---------------------------------------------------------------
+
+  /** Pseudoinverse solve: X = V * diag(1/S) * U^H * B. */
+  template <typename Rhs>
+  PlainMatrix solve(const MatrixBase<Rhs>& B) const {
+    return solve_impl(B, (std::min)(m_, n_), RealScalar(0));
+  }
+
+  /** Truncated solve: use only top trunc singular triplets. */
+  template <typename Rhs>
+  PlainMatrix solve(const MatrixBase<Rhs>& B, Index trunc) const {
+    eigen_assert(trunc > 0 && trunc <= (std::min)(m_, n_));
+    return solve_impl(B, trunc, RealScalar(0));
+  }
+
+  /** Tikhonov-regularized solve: D_ii = S_i / (S_i^2 + lambda^2). */
+  template <typename Rhs>
+  PlainMatrix solve(const MatrixBase<Rhs>& B, RealScalar lambda) const {
+    eigen_assert(lambda > 0);
+    return solve_impl(B, (std::min)(m_, n_), lambda);
+  }
+
+  cudaStream_t stream() const { return stream_; }
+
+ private:
+  cudaStream_t stream_ = nullptr;
+  cusolverDnHandle_t handle_ = nullptr;
+  cublasHandle_t cublas_ = nullptr;
+  internal::CusolverParams params_;
+  internal::DeviceBuffer d_A_;        // working copy of A (overwritten by gesvd)
+  internal::DeviceBuffer d_U_;        // left singular vectors
+  internal::DeviceBuffer d_S_;        // singular values (RealScalar)
+  internal::DeviceBuffer d_VT_;       // right singular vectors transposed
+  internal::DeviceBuffer d_scratch_;  // workspace + info
+  size_t scratch_size_ = 0;
+  std::vector<char> h_workspace_;
+  unsigned int options_ = 0;
+  Index m_ = 0;
+  Index n_ = 0;
+  int64_t lda_ = 0;
+  bool transposed_ = false;  // true if m < n (we compute SVD of A^T internally)
+  ComputationInfo info_ = InvalidInput;
+  int info_word_ = 0;
+  bool info_synced_ = true;
+
+  void init_context() {
+    EIGEN_CUDA_RUNTIME_CHECK(cudaStreamCreate(&stream_));
+    EIGEN_CUSOLVER_CHECK(cusolverDnCreate(&handle_));
+    EIGEN_CUSOLVER_CHECK(cusolverDnSetStream(handle_, stream_));
+    EIGEN_CUBLAS_CHECK(cublasCreate(&cublas_));
+    EIGEN_CUBLAS_CHECK(cublasSetStream(cublas_, stream_));
+    ensure_scratch(0);
+  }
+
+  void ensure_scratch(size_t workspace_bytes) {
+    constexpr size_t kAlign = 16;
+    workspace_bytes = (workspace_bytes + kAlign - 1) & ~(kAlign - 1);
+    size_t needed = workspace_bytes + sizeof(int);
+    if (needed > scratch_size_) {
+      if (d_scratch_.ptr) EIGEN_CUDA_RUNTIME_CHECK(cudaStreamSynchronize(stream_));
+      d_scratch_ = internal::DeviceBuffer(needed);
+      scratch_size_ = needed;
+    }
+  }
+
+  void* scratch_workspace() const { return d_scratch_.ptr; }
+  int* scratch_info() const {
+    return reinterpret_cast<int*>(static_cast<char*>(d_scratch_.ptr) + scratch_size_ - sizeof(int));
+  }
+
+  void sync_info() const {
+    if (!info_synced_) {
+      EIGEN_CUDA_RUNTIME_CHECK(cudaStreamSynchronize(stream_));
+      const_cast<GpuSVD*>(this)->info_ = (info_word_ == 0) ? Success : NumericalIssue;
+      const_cast<GpuSVD*>(this)->info_synced_ = true;
+    }
+  }
+
+  // Swap U↔V flags for the transposed case.
+  static unsigned int swap_uv_options(unsigned int opts) {
+    unsigned int result = 0;
+    if (opts & ComputeThinU) result |= ComputeThinV;
+    if (opts & ComputeFullU) result |= ComputeFullV;
+    if (opts & ComputeThinV) result |= ComputeThinU;
+    if (opts & ComputeFullV) result |= ComputeFullU;
+    return result;
+  }
+
+  static signed char jobu(unsigned int opts) {
+    if (opts & ComputeFullU) return 'A';
+    if (opts & ComputeThinU) return 'S';
+    return 'N';
+  }
+
+  static signed char jobvt(unsigned int opts) {
+    if (opts & ComputeFullV) return 'A';
+    if (opts & ComputeThinV) return 'S';
+    return 'N';
+  }
+
+  void factorize() {
+    constexpr cudaDataType_t dtype = internal::cusolver_data_type<Scalar>::value;
+    constexpr cudaDataType_t rtype = internal::cuda_data_type<RealScalar>::value;
+    const Index k = (std::min)(m_, n_);
+
+    info_synced_ = false;
+    info_ = InvalidInput;
+
+    // Allocate output buffers. When transposed, swap U/V roles for cuSOLVER.
+    d_S_ = internal::DeviceBuffer(static_cast<size_t>(k) * sizeof(RealScalar));
+
+    // Internal options: for transposed case, what user wants as U we compute as VT of A^H.
+    const unsigned int int_opts = transposed_ ? swap_uv_options(options_) : options_;
+
+    const Index ucols = (int_opts & ComputeFullU) ? m_ : ((int_opts & ComputeThinU) ? k : 0);
+    const Index vtrows = (int_opts & ComputeFullV) ? n_ : ((int_opts & ComputeThinV) ? k : 0);
+    const int64_t ldu = m_;
+    const int64_t ldvt = vtrows > 0 ? vtrows : 1;
+
+    if (ucols > 0) d_U_ = internal::DeviceBuffer(static_cast<size_t>(m_) * static_cast<size_t>(ucols) * sizeof(Scalar));
+    if (vtrows > 0)
+      d_VT_ = internal::DeviceBuffer(static_cast<size_t>(vtrows) * static_cast<size_t>(n_) * sizeof(Scalar));
+
+    // computeType must match the matrix data type (dtype), not the singular value type (rtype).
+    eigen_assert(m_ >= n_ && "Internal error: m_ < n_ should have been handled by transpose in compute()");
+    size_t dev_ws = 0, host_ws = 0;
+    EIGEN_CUSOLVER_CHECK(cusolverDnXgesvd_bufferSize(
+        handle_, params_.p, jobu(int_opts), jobvt(int_opts), static_cast<int64_t>(m_), static_cast<int64_t>(n_), dtype,
+        d_A_.ptr, lda_, rtype, d_S_.ptr, dtype, ucols > 0 ? d_U_.ptr : nullptr, ldu, dtype,
+        vtrows > 0 ? d_VT_.ptr : nullptr, ldvt, dtype, &dev_ws, &host_ws));
+
+    ensure_scratch(dev_ws);
+    h_workspace_.resize(host_ws);
+
+    // Compute SVD.
+    EIGEN_CUSOLVER_CHECK(cusolverDnXgesvd(handle_, params_.p, jobu(int_opts), jobvt(int_opts), static_cast<int64_t>(m_),
+                                          static_cast<int64_t>(n_), dtype, d_A_.ptr, lda_, rtype, d_S_.ptr, dtype,
+                                          ucols > 0 ? d_U_.ptr : nullptr, ldu, dtype, vtrows > 0 ? d_VT_.ptr : nullptr,
+                                          ldvt, dtype, scratch_workspace(), dev_ws,
+                                          host_ws > 0 ? h_workspace_.data() : nullptr, host_ws, scratch_info()));
+
+    EIGEN_CUDA_RUNTIME_CHECK(
+        cudaMemcpyAsync(&info_word_, scratch_info(), sizeof(int), cudaMemcpyDeviceToHost, stream_));
+  }
+
+  // Internal solve: X = V * diag(D) * U^H * B, using top `trunc` triplets.
+  // D_ii = 1/S_i (if lambda==0) or S_i/(S_i^2+lambda^2).
+  //
+  // For non-transposed: stored U, VT. X = VT^H * D * U^H * B.
+  // For transposed (SVD of A^H): stored U', VT'. X = U' * D * VT' * B.
+  template <typename Rhs>
+  PlainMatrix solve_impl(const MatrixBase<Rhs>& B, Index trunc, RealScalar lambda) const {
+    sync_info();
+    eigen_assert(info_ == Success && "GpuSVD::solve called on a failed or uninitialized decomposition");
+    eigen_assert((options_ & (ComputeThinU | ComputeFullU)) && "solve requires U");
+    eigen_assert((options_ & (ComputeThinV | ComputeFullV)) && "solve requires V");
+
+    const Index m_orig = transposed_ ? n_ : m_;
+    const Index n_orig = transposed_ ? m_ : n_;
+    eigen_assert(B.rows() == m_orig);
+
+    const Index k = (std::min)(m_, n_);  // = min(m_orig, n_orig)
+    const Index kk = (std::min)(trunc, k);
+    const Index nrhs = B.cols();
+
+    // Download S to host to build the diagonal scaling.
+    RealVector S(k);
+    EIGEN_CUDA_RUNTIME_CHECK(
+        cudaMemcpy(S.data(), d_S_.ptr, static_cast<size_t>(k) * sizeof(RealScalar), cudaMemcpyDeviceToHost));
+    EIGEN_CUDA_RUNTIME_CHECK(cudaStreamSynchronize(stream_));
+
+    // Upload B (m_orig × nrhs).
+    const PlainMatrix rhs(B);
+    internal::DeviceBuffer d_B(static_cast<size_t>(m_orig) * static_cast<size_t>(nrhs) * sizeof(Scalar));
+    EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpyAsync(d_B.ptr, rhs.data(),
+                                             static_cast<size_t>(m_orig) * static_cast<size_t>(nrhs) * sizeof(Scalar),
+                                             cudaMemcpyHostToDevice, stream_));
+
+    // Step 1: tmp = U_orig^H * B  (kk × nrhs).
+    // Non-transposed: U_stored is m_×ucols, U_orig = U_stored. Use U_stored^H * B.
+    // Transposed: U_orig = VT_stored^H, so U_orig^H = VT_stored. Use VT_stored * B (no transpose!).
+    internal::DeviceBuffer d_tmp(static_cast<size_t>(kk) * static_cast<size_t>(nrhs) * sizeof(Scalar));
+    {
+      Scalar alpha_one(1), beta_zero(0);
+      constexpr cudaDataType_t dtype = internal::cuda_data_type<Scalar>::value;
+      constexpr cublasComputeType_t compute = internal::cuda_compute_type<Scalar>::value;
+
+      if (!transposed_) {
+        // U_stored^H * B: (m_×kk)^H × (m_×nrhs) → kk×nrhs.
+        EIGEN_CUBLAS_CHECK(cublasGemmEx(cublas_, CUBLAS_OP_C, CUBLAS_OP_N, static_cast<int>(kk), static_cast<int>(nrhs),
+                                        static_cast<int>(m_), &alpha_one, d_U_.ptr, dtype, static_cast<int>(m_),
+                                        d_B.ptr, dtype, static_cast<int>(m_orig), &beta_zero, d_tmp.ptr, dtype,
+                                        static_cast<int>(kk), compute, internal::cuda_gemm_algo()));
+      } else {
+        // VT_stored * B: VT_stored is vtrows×n_ = kk×m_orig (thin), NoTrans.
+        // vtrows×m_orig times m_orig×nrhs → vtrows×nrhs. Use first kk rows.
+        const Index vtrows_stored = (swap_uv_options(options_) & ComputeFullV) ? n_ : k;
+        EIGEN_CUBLAS_CHECK(cublasGemmEx(
+            cublas_, CUBLAS_OP_N, CUBLAS_OP_N, static_cast<int>(kk), static_cast<int>(nrhs), static_cast<int>(m_orig),
+            &alpha_one, d_VT_.ptr, dtype, static_cast<int>(vtrows_stored), d_B.ptr, dtype, static_cast<int>(m_orig),
+            &beta_zero, d_tmp.ptr, dtype, static_cast<int>(kk), compute, internal::cuda_gemm_algo()));
+      }
+    }
+
+    // Step 2: Scale row i of tmp by D_ii.
+    // Download tmp to host, scale, re-upload. (Simple and correct; a device kernel would be faster.)
+    {
+      PlainMatrix tmp(kk, nrhs);
+      EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpyAsync(tmp.data(), d_tmp.ptr,
+                                               static_cast<size_t>(kk) * static_cast<size_t>(nrhs) * sizeof(Scalar),
+                                               cudaMemcpyDeviceToHost, stream_));
+      EIGEN_CUDA_RUNTIME_CHECK(cudaStreamSynchronize(stream_));
+
+      for (Index i = 0; i < kk; ++i) {
+        RealScalar si = S(i);
+        RealScalar di = (lambda == RealScalar(0)) ? (si > 0 ? RealScalar(1) / si : RealScalar(0))
+                                                  : si / (si * si + lambda * lambda);
+        tmp.row(i) *= Scalar(di);
+      }
+
+      EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpyAsync(d_tmp.ptr, tmp.data(),
+                                               static_cast<size_t>(kk) * static_cast<size_t>(nrhs) * sizeof(Scalar),
+                                               cudaMemcpyHostToDevice, stream_));
+    }
+
+    // Step 3: X = V_orig * tmp  (n_orig × nrhs).
+    // Non-transposed: V_orig = VT_stored^H. VT_stored[:kk,:]^H * tmp → n_orig × nrhs.
+    // Transposed: V_orig = U_stored[:,:kk]. U_stored * tmp → n_orig × nrhs (NoTrans).
+    PlainMatrix X(n_orig, nrhs);
+    {
+      internal::DeviceBuffer d_X(static_cast<size_t>(n_orig) * static_cast<size_t>(nrhs) * sizeof(Scalar));
+      Scalar alpha_one(1), beta_zero(0);
+      constexpr cudaDataType_t dtype = internal::cuda_data_type<Scalar>::value;
+      constexpr cublasComputeType_t compute = internal::cuda_compute_type<Scalar>::value;
+
+      if (!transposed_) {
+        const Index vtrows = (options_ & ComputeFullV) ? n_ : k;
+        EIGEN_CUBLAS_CHECK(cublasGemmEx(cublas_, CUBLAS_OP_C, CUBLAS_OP_N, static_cast<int>(n_orig),
+                                        static_cast<int>(nrhs), static_cast<int>(kk), &alpha_one, d_VT_.ptr, dtype,
+                                        static_cast<int>(vtrows), d_tmp.ptr, dtype, static_cast<int>(kk), &beta_zero,
+                                        d_X.ptr, dtype, static_cast<int>(n_orig), compute, internal::cuda_gemm_algo()));
+      } else {
+        // U_stored is m_×ucols. V_orig = U_stored[:,:kk]. NoTrans × tmp.
+        EIGEN_CUBLAS_CHECK(cublasGemmEx(cublas_, CUBLAS_OP_N, CUBLAS_OP_N, static_cast<int>(n_orig),
+                                        static_cast<int>(nrhs), static_cast<int>(kk), &alpha_one, d_U_.ptr, dtype,
+                                        static_cast<int>(m_), d_tmp.ptr, dtype, static_cast<int>(kk), &beta_zero,
+                                        d_X.ptr, dtype, static_cast<int>(n_orig), compute, internal::cuda_gemm_algo()));
+      }
+
+      EIGEN_CUDA_RUNTIME_CHECK(cudaMemcpyAsync(X.data(), d_X.ptr,
+                                               static_cast<size_t>(n_orig) * static_cast<size_t>(nrhs) * sizeof(Scalar),
+                                               cudaMemcpyDeviceToHost, stream_));
+      EIGEN_CUDA_RUNTIME_CHECK(cudaStreamSynchronize(stream_));
+    }
+
+    return X;
+  }
+};
+
+}  // namespace Eigen
+
+#endif  // EIGEN_GPU_SVD_H

Eigen/src/GPU/GpuSupport.h

@@ -0,0 +1,101 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2026 Rasmus Munk Larsen <rmlarsen@gmail.com>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+// Generic CUDA runtime support shared across all GPU library integrations
+// (cuSOLVER, cuBLAS, cuDSS, etc.):
+//   - Error-checking macros
+//   - RAII device buffer
+//
+// Only depends on <cuda_runtime.h>. No NVIDIA library headers.
+
+#ifndef EIGEN_GPU_SUPPORT_H
+#define EIGEN_GPU_SUPPORT_H
+
+// IWYU pragma: private
+#include "./InternalHeaderCheck.h"
+
+#include <cuda_runtime.h>
+
+namespace Eigen {
+namespace internal {
+
+// ---- Error-checking macros --------------------------------------------------
+// These abort (via eigen_assert) on failure. Not for use in destructors.
+
+#define EIGEN_CUDA_RUNTIME_CHECK(expr)                             \
+  do {                                                             \
+    cudaError_t _e = (expr);                                       \
+    eigen_assert(_e == cudaSuccess && "CUDA runtime call failed"); \
+  } while (0)
+
+// ---- RAII: device buffer ----------------------------------------------------
+
+struct DeviceBuffer {
+  void* ptr = nullptr;
+
+  DeviceBuffer() = default;
+
+  explicit DeviceBuffer(size_t bytes) {
+    if (bytes > 0) EIGEN_CUDA_RUNTIME_CHECK(cudaMalloc(&ptr, bytes));
+  }
+
+  ~DeviceBuffer() {
+    if (ptr) (void)cudaFree(ptr);  // destructor: ignore errors
+  }
+
+  // Move-only.
+  DeviceBuffer(DeviceBuffer&& o) noexcept : ptr(o.ptr) { o.ptr = nullptr; }
+  DeviceBuffer& operator=(DeviceBuffer&& o) noexcept {
+    if (this != &o) {
+      if (ptr) (void)cudaFree(ptr);
+      ptr = o.ptr;
+      o.ptr = nullptr;
+    }
+    return *this;
+  }
+
+  DeviceBuffer(const DeviceBuffer&) = delete;
+  DeviceBuffer& operator=(const DeviceBuffer&) = delete;
+
+  // Adopt an existing device pointer. Caller relinquishes ownership.
+  static DeviceBuffer adopt(void* p) {
+    DeviceBuffer b;
+    b.ptr = p;
+    return b;
+  }
+};
+
+// ---- Scalar → cudaDataType_t ------------------------------------------------
+// Shared by cuBLAS and cuSOLVER. cudaDataType_t is defined in library_types.h
+// which is included transitively by cuda_runtime.h.
+
+template <typename Scalar>
+struct cuda_data_type;
+
+template <>
+struct cuda_data_type<float> {
+  static constexpr cudaDataType_t value = CUDA_R_32F;
+};
+template <>
+struct cuda_data_type<double> {
+  static constexpr cudaDataType_t value = CUDA_R_64F;
+};
+template <>
+struct cuda_data_type<std::complex<float>> {
+  static constexpr cudaDataType_t value = CUDA_C_32F;
+};
+template <>
+struct cuda_data_type<std::complex<double>> {
+  static constexpr cudaDataType_t value = CUDA_C_64F;
+};
+
+}  // namespace internal
+}  // namespace Eigen
+
+#endif  // EIGEN_GPU_SUPPORT_H

Eigen/src/GPU/InternalHeaderCheck.h

@@ -0,0 +1,3 @@
+#ifndef EIGEN_GPU_MODULE_H
+#error "Please include Eigen/GPU instead of including headers inside the src/GPU directory directly."
+#endif

Eigen/src/GPU/README.md

@@ -0,0 +1,409 @@
+# Eigen GPU Module (`Eigen/GPU`)
+
+GPU-accelerated dense linear algebra for Eigen users, dispatching to NVIDIA
+CUDA libraries (cuBLAS, cuSOLVER). Requires CUDA 11.4+. Header-only (link
+against CUDA runtime, cuBLAS, and cuSOLVER).
+
+## Why this module
+
+Eigen is the linear algebra foundation for a large ecosystem of C++ projects
+in robotics (ROS, Drake, MoveIt, Pinocchio), computer vision (OpenCV, COLMAP,
+Open3D), scientific computing (Ceres, Stan), and beyond. Many of these
+projects run on GPU-equipped hardware but cannot use GPUs for Eigen operations
+without dropping down to raw CUDA library APIs. Third-party projects like
+[EigenCuda](https://github.com/NLESC-JCER/EigenCuda) and
+[cholespy](https://github.com/rgl-epfl/cholespy) exist specifically to fill
+this gap, and downstream projects like
+[Ceres](https://github.com/ceres-solver/ceres-solver/issues/1151) and
+[COLMAP](https://github.com/colmap/colmap/issues/4018) have open requests for
+GPU-accelerated solvers through Eigen.
+
+The `Eigen/GPU` module aims to close this gap: Existing Eigen users should be
+able to move performance-critical dense linear algebra to the GPU with minimal
+code changes and without learning CUDA library APIs directly.
+
+## Design philosophy
+
+**CPU and GPU coexist.** There is no global compile-time switch that replaces
+CPU implementations (unlike `EIGEN_USE_LAPACKE`). Users choose GPU solvers
+explicitly -- `GpuLLT<double>` vs `LLT<MatrixXd>` -- and both coexist in
+the same binary. This also lets users keep the factored matrix on device across
+multiple solves, something impossible with compile-time replacement.
+
+**Familiar syntax.** GPU operations use the same expression patterns as CPU
+Eigen. Here is a side-by-side comparison:
+
+```cpp
+// ---- CPU (Eigen) ----               // ---- GPU (Eigen/GPU) ----
+#include <Eigen/Dense>                  #define EIGEN_USE_GPU
+                                        #include <Eigen/GPU>
+
+MatrixXd A = ...;                       auto d_A = DeviceMatrix<double>::fromHost(A);
+MatrixXd B = ...;                       auto d_B = DeviceMatrix<double>::fromHost(B);
+
+MatrixXd C = A * B;                     DeviceMatrix<double> d_C = d_A * d_B;
+MatrixXd X = A.llt().solve(B);          DeviceMatrix<double> d_X = d_A.llt().solve(d_B);
+
+                                        MatrixXd X = d_X.toHost();
+```
+
+The GPU version reads like CPU Eigen with explicit upload/download.
+`operator*` dispatches to cuBLAS GEMM, `.llt().solve()` dispatches to
+cuSOLVER potrf + potrs. Unsupported expressions are compile errors.
+
+**Explicit over implicit.** Host-device transfers, stream management, and
+library handle lifetimes are visible in the API. There are no hidden
+allocations or synchronizations except where documented (e.g., `toHost()` must
+synchronize to deliver data to the host).
+
+## Key concepts
+
+### `DeviceMatrix<Scalar>`
+
+A typed RAII wrapper for a dense column-major matrix in GPU device memory.
+This is the GPU counterpart of Eigen's `MatrixX<Scalar>`. A vector is simply
+a `DeviceMatrix` with one column.
+
+```cpp
+// Upload from host
+auto d_A = DeviceMatrix<double>::fromHost(A);
+
+// Allocate uninitialized
+DeviceMatrix<double> d_C(m, n);
+
+// Download to host
+MatrixXd C = d_C.toHost();
+
+// Async download (returns a future)
+auto transfer = d_C.toHostAsync();
+// ... do other work ...
+MatrixXd C = transfer.get();
+```
+
+`DeviceMatrix` supports expression methods that mirror Eigen's API:
+`adjoint()`, `transpose()`, `triangularView<UpLo>()`,
+`selfadjointView<UpLo>()`, `llt()`, `lu()`. These return lightweight
+expression objects that are evaluated when assigned.
+
+### `GpuContext`
+
+Every GPU operation needs a CUDA stream and library handles (cuBLAS,
+cuSOLVER). `GpuContext` bundles these together.
+
+For simple usage, you don't need to create one -- a per-thread default context
+is created lazily on first use:
+
+```cpp
+// These use the thread-local default context automatically
+d_C = d_A * d_B;
+d_X = d_A.llt().solve(d_B);
+```
+
+For concurrent multi-stream execution, create explicit contexts:
+
+```cpp
+GpuContext ctx1, ctx2;
+d_C1.device(ctx1) = d_A1 * d_B1;   // runs on stream 1
+d_C2.device(ctx2) = d_A2 * d_B2;   // runs on stream 2 (concurrently)
+```
+
+## Usage
+
+### Matrix operations (cuBLAS)
+
+```cpp
+auto d_A = DeviceMatrix<double>::fromHost(A);
+auto d_B = DeviceMatrix<double>::fromHost(B);
+
+// GEMM: C = A * B, C = A^H * B, C = A * B^T, ...
+DeviceMatrix<double> d_C = d_A * d_B;
+d_C = d_A.adjoint() * d_B;
+d_C = d_A * d_B.transpose();
+
+// Scaled and accumulated
+d_C += 2.0 * d_A * d_B;             // alpha=2, beta=1
+d_C.device(ctx) -= d_A * d_B;       // alpha=-1, beta=1 (requires explicit context)
+
+// Triangular solve (TRSM)
+d_X = d_A.triangularView<Lower>().solve(d_B);
+
+// Symmetric/Hermitian multiply (SYMM/HEMM)
+d_C = d_A.selfadjointView<Lower>() * d_B;
+
+// Rank-k update (SYRK/HERK)
+d_C.selfadjointView<Lower>().rankUpdate(d_A);  // C += A * A^H
+```
+
+### Dense solvers (cuSOLVER)
+
+**One-shot expression syntax** -- Convenient, re-factorizes each time:
+
+```cpp
+// Cholesky solve (potrf + potrs)
+d_X = d_A.llt().solve(d_B);
+
+// LU solve (getrf + getrs)
+d_Y = d_A.lu().solve(d_B);
+```
+
+**Cached factorization** -- Factor once, solve many times:
+
+```cpp
+GpuLLT<double> llt;
+llt.compute(d_A);                    // factorize (async)
+if (llt.info() != Success) { ... }   // lazy sync on first info() call
+auto d_X1 = llt.solve(d_B1);        // reuses factor (async)
+auto d_X2 = llt.solve(d_B2);        // reuses factor (async)
+MatrixXd X2 = d_X2.toHost();
+
+// LU with transpose solve
+GpuLU<double> lu;
+lu.compute(d_A);
+auto d_Y = lu.solve(d_B, GpuLU<double>::Transpose);  // A^T Y = B
+
+// QR solve (overdetermined least squares)
+GpuQR<double> qr(A);                // host matrix input
+MatrixXd X = qr.solve(B);           // Q^H * B via ormqr, then trsm on R
+
+// SVD
+GpuSVD<double> svd(A, ComputeThinU | ComputeThinV);
+VectorXd S = svd.singularValues();
+MatrixXd U = svd.matrixU();
+MatrixXd VT = svd.matrixVT();
+MatrixXd X = svd.solve(B);          // pseudoinverse solve
+
+// Self-adjoint eigenvalue decomposition
+GpuSelfAdjointEigenSolver<double> es(A);
+VectorXd eigenvals = es.eigenvalues();
+MatrixXd eigenvecs = es.eigenvectors();
+```
+
+The cached API keeps the factored matrix on device, avoiding redundant
+host-device transfers and re-factorizations.
+
+### Precision control
+
+GEMM dispatch enables tensor core algorithms by default, allowing cuBLAS to
+choose the fastest algorithm for the given precision and architecture. For
+double precision on sm_80+ (Ampere), this allows Ozaki emulation -- full FP64
+results computed faster via tensor cores.
+
+| Macro | Effect |
+|---|---|
+| *(default)* | Tensor core algorithms enabled. Float uses full FP32. Double may use Ozaki on sm_80+. |
+| `EIGEN_CUDA_TF32` | Opt-in: Float uses TF32 (~2x faster, 10-bit mantissa). Double unaffected. |
+| `EIGEN_NO_CUDA_TENSOR_OPS` | Opt-out: Pedantic compute types, no tensor cores. For bit-exact reproducibility. |
+
+### Stream control and async execution
+
+Operations are asynchronous by default. The compute-solve chain runs without
+host synchronization until you need a result on the host:
+
+```
+fromHost(A) --sync-->  compute() --async-->  solve() --async-->  toHost()
+   H2D                  potrf                 potrs                D2H
+                                                                   sync
+```
+
+Mandatory sync points:
+- `fromHost()` -- Synchronizes to complete the upload before returning
+- `toHost()` / `HostTransfer::get()` -- Must deliver data to host
+- `info()` -- Must read the factorization status
+
+**Cross-stream safety** is automatic. `DeviceMatrix` tracks write completion
+via CUDA events. When a matrix written on stream A is read on stream B, the
+module automatically inserts `cudaStreamWaitEvent`. Same-stream operations
+skip the wait (CUDA guarantees in-order execution within a stream).
+
+## Reference
+
+### Supported scalar types
+
+`float`, `double`, `std::complex<float>`, `std::complex<double>`.
+
+### Expression -> library call mapping
+
+| DeviceMatrix expression | Library call | Parameters |
+|---|---|---|
+| `C = A * B` | `cublasGemmEx` | transA=N, transB=N, alpha=1, beta=0 |
+| `C = A.adjoint() * B` | `cublasGemmEx` | transA=C, transB=N |
+| `C = A.transpose() * B` | `cublasGemmEx` | transA=T, transB=N |
+| `C = A * B.adjoint()` | `cublasGemmEx` | transA=N, transB=C |
+| `C = A * B.transpose()` | `cublasGemmEx` | transA=N, transB=T |
+| `C = alpha * A * B` | `cublasGemmEx` | alpha from LHS |
+| `C = A * (alpha * B)` | `cublasGemmEx` | alpha from RHS |
+| `C += A * B` | `cublasGemmEx` | alpha=1, beta=1 |
+| `C.device(ctx) -= A * B` | `cublasGemmEx` | alpha=-1, beta=1 |
+| `X = A.llt().solve(B)` | `cusolverDnXpotrf` + `Xpotrs` | uplo, n, nrhs |
+| `X = A.llt<Upper>().solve(B)` | same | uplo=Upper |
+| `X = A.lu().solve(B)` | `cusolverDnXgetrf` + `Xgetrs` | n, nrhs |
+| `X = A.triangularView<L>().solve(B)` | `cublasXtrsm` | side=L, uplo, diag=NonUnit |
+| `C = A.selfadjointView<L>() * B` | `cublasXsymm` / `cublasXhemm` | side=L, uplo |
+| `C.selfadjointView<L>().rankUpdate(A)` | `cublasXsyrk` / `cublasXherk` | uplo, trans=N |
+
+### `DeviceMatrix<Scalar>` API
+
+| Method | Sync? | Description |
+|--------|-------|-------------|
+| `DeviceMatrix()` | -- | Empty (0x0) |
+| `DeviceMatrix(rows, cols)` | -- | Allocate uninitialized |
+| `fromHost(matrix, stream)` | yes | Upload from Eigen matrix |
+| `fromHostAsync(ptr, rows, cols, outerStride, stream)` | no | Async upload (caller manages lifetime) |
+| `toHost(stream)` | yes | Synchronous download |
+| `toHostAsync(stream)` | no | Returns `HostTransfer` future |
+| `clone(stream)` | no | Device-to-device deep copy |
+| `resize(rows, cols)` | -- | Discard contents, reallocate |
+| `data()` | -- | Raw device pointer |
+| `rows()`, `cols()` | -- | Dimensions |
+| `sizeInBytes()` | -- | Total device allocation size in bytes |
+| `empty()` | -- | True if 0x0 |
+| `adjoint()` | -- | Adjoint view (GEMM ConjTrans) |
+| `transpose()` | -- | Transpose view (GEMM Trans) |
+| `llt()` / `llt<UpLo>()` | -- | Cholesky expression builder |
+| `lu()` | -- | LU expression builder |
+| `triangularView<UpLo>()` | -- | Triangular view (TRSM) |
+| `selfadjointView<UpLo>()` | -- | Self-adjoint view (SYMM, rankUpdate) |
+| `device(ctx)` | -- | Assignment proxy bound to context |
+
+### `GpuContext`
+
+Unified GPU execution context owning a CUDA stream and library handles.
+
+```cpp
+GpuContext()                                             // Creates dedicated stream + handles
+static GpuContext& threadLocal()                         // Per-thread default (lazy-created)
+
+cudaStream_t       stream()
+cublasHandle_t     cublasHandle()
+cusolverDnHandle_t cusolverHandle()
+```
+
+Non-copyable, non-movable (owns library handles).
+
+### `GpuLLT<Scalar, UpLo>` API
+
+GPU dense Cholesky (LL^T) via cuSOLVER. Caches factor on device.
+
+| Method | Sync? | Description |
+|--------|-------|-------------|
+| `GpuLLT(A)` | deferred | Construct and factorize from host matrix |
+| `compute(host_matrix)` | deferred | Upload and factorize |
+| `compute(DeviceMatrix)` | deferred | D2D copy and factorize |
+| `compute(DeviceMatrix&&)` | deferred | Move-adopt and factorize (no copy) |
+| `solve(host_matrix)` | yes | Solve, return host matrix |
+| `solve(DeviceMatrix)` | no | Solve, return `DeviceMatrix` (async) |
+| `info()` | lazy | Syncs stream on first call, returns `Success` or `NumericalIssue` |
+
+### `GpuLU<Scalar>` API
+
+GPU dense partial-pivoting LU via cuSOLVER. Same pattern as `GpuLLT`, plus
+`TransposeMode` parameter on `solve()` (`NoTranspose`, `Transpose`,
+`ConjugateTranspose`).
+
+### `GpuQR<Scalar>` API
+
+GPU dense QR decomposition via cuSOLVER (`geqrf`). Solve uses `ormqr` (apply
+Q^H) + `trsm` (back-substitute on R) -- Q is never formed explicitly.
+
+| Method | Description |
+|--------|-------------|
+| `GpuQR()` | Default construct, then call `compute()` |
+| `GpuQR(A)` | Construct and factorize from host matrix |
+| `compute(A)` | Upload + factorize |
+| `compute(DeviceMatrix)` | D2D copy + factorize |
+| `solve(host_matrix)` | Solve, return host matrix (syncs) |
+| `solve(DeviceMatrix)` | Solve, return `DeviceMatrix` (async) |
+| `info()` | Lazy sync |
+| `rows()`, `cols()`, `stream()` | Dimensions and CUDA stream |
+
+### `GpuSVD<Scalar>` API
+
+GPU dense SVD via cuSOLVER (`gesvd`). Supports thin, full, and values-only
+modes via Eigen's `ComputeThinU | ComputeThinV`, `ComputeFullU | ComputeFullV`,
+or `0` (values only).
+
+| Method | Description |
+|--------|-------------|
+| `GpuSVD()` | Default construct, then call `compute()` |
+| `GpuSVD(A, options)` | Construct and compute (options default: `ComputeThinU \| ComputeThinV`) |
+| `compute(A, options)` | Compute from host matrix |
+| `compute(DeviceMatrix, options)` | Compute from device matrix |
+| `singularValues()` | Download singular values to host |
+| `matrixU()` | Download U to host (requires `ComputeThinU` or `ComputeFullU`) |
+| `matrixVT()` | Download V^T to host (requires `ComputeThinV` or `ComputeFullV`) |
+| `solve(B)` | Pseudoinverse solve (returns host matrix) |
+| `solve(B, k)` | Truncated solve (top k singular triplets) |
+| `solve(B, lambda)` | Tikhonov regularized solve |
+| `rank(threshold)` | Count singular values above threshold |
+| `info()` | Lazy sync |
+| `rows()`, `cols()`, `stream()` | Dimensions and CUDA stream |
+
+Wide matrices (m < n) are handled by internally transposing via cuBLAS `geam`.
+
+### `GpuSelfAdjointEigenSolver<Scalar>` API
+
+GPU symmetric/Hermitian eigenvalue decomposition via cuSOLVER (`syevd`).
+
+| Method | Description |
+|--------|-------------|
+| `GpuSelfAdjointEigenSolver()` | Default construct, then call `compute()` |
+| `GpuSelfAdjointEigenSolver(A, mode)` | Construct and compute (mode default: `ComputeEigenvectors`) |
+| `compute(A, mode)` | Compute from host matrix |
+| `compute(DeviceMatrix, mode)` | Compute from device matrix |
+| `eigenvalues()` | Download eigenvalues to host (ascending order) |
+| `eigenvectors()` | Download eigenvectors to host (columns) |
+| `info()` | Lazy sync |
+| `rows()`, `cols()`, `stream()` | Dimensions and CUDA stream |
+
+`ComputeMode`: `GpuSelfAdjointEigenSolver::EigenvaluesOnly` or
+`GpuSelfAdjointEigenSolver::ComputeEigenvectors`.
+
+### `HostTransfer<Scalar>` API
+
+Future for async device-to-host transfer.
+
+| Method | Description |
+|--------|-------------|
+| `get()` | Block until transfer completes, return host matrix reference. Idempotent. |
+| `ready()` | Non-blocking poll |
+
+### Aliasing
+
+Unlike Eigen's `Matrix`, where omitting `.noalias()` triggers a copy to a
+temporary, DeviceMatrix dispatches directly to NVIDIA library calls which have
+no built-in aliasing protection. All operations are implicitly noalias.
+The caller must ensure operands don't alias the destination for GEMM and TRSM
+(debug asserts catch violations).
+
+## File layout
+
+| File | Depends on | Contents |
+|------|-----------|----------|
+| `GpuSupport.h` | `<cuda_runtime.h>` | Error macro, `DeviceBuffer`, `cuda_data_type<>` |
+| `DeviceMatrix.h` | `GpuSupport.h` | `DeviceMatrix<>`, `HostTransfer<>` |
+| `DeviceExpr.h` | `DeviceMatrix.h` | GEMM expression wrappers |
+| `DeviceBlasExpr.h` | `DeviceMatrix.h` | TRSM, SYMM, SYRK expression wrappers |
+| `DeviceSolverExpr.h` | `DeviceMatrix.h` | Solver expression wrappers (LLT, LU) |
+| `DeviceDispatch.h` | all above | All dispatch functions + `DeviceAssignment` |
+| `GpuContext.h` | `CuBlasSupport.h`, `CuSolverSupport.h` | `GpuContext` |
+| `CuBlasSupport.h` | `GpuSupport.h`, `<cublas_v2.h>` | cuBLAS error macro, op/compute type maps |
+| `CuSolverSupport.h` | `GpuSupport.h`, `<cusolverDn.h>` | cuSOLVER params, fill-mode mapping |
+| `GpuLLT.h` | `CuSolverSupport.h` | Cached dense Cholesky factorization |
+| `GpuLU.h` | `CuSolverSupport.h` | Cached dense LU factorization |
+| `GpuQR.h` | `CuSolverSupport.h`, `CuBlasSupport.h` | Dense QR decomposition |
+| `GpuSVD.h` | `CuSolverSupport.h`, `CuBlasSupport.h` | Dense SVD decomposition |
+| `GpuEigenSolver.h` | `CuSolverSupport.h` | Self-adjoint eigenvalue decomposition |
+
+## Building and testing
+
+```bash
+cmake -G Ninja -B build -S . \
+  -DEIGEN_TEST_CUDA=ON \
+  -DEIGEN_CUDA_COMPUTE_ARCH="70" \
+  -DEIGEN_TEST_CUBLAS=ON \
+  -DEIGEN_TEST_CUSOLVER=ON
+
+cmake --build build --target gpu_cublas gpu_cusolver_llt gpu_cusolver_lu \
+  gpu_cusolver_qr gpu_cusolver_svd gpu_cusolver_eigen gpu_device_matrix
+ctest --test-dir build -R "gpu_cublas|gpu_cusolver|gpu_device" --output-on-failure
+```

benchmarks/CMakeLists.txt

@@ -43,3 +43,10 @@ add_subdirectory(Householder)
 add_subdirectory(Solvers)
 add_subdirectory(Tuning)
 add_subdirectory(BLAS)
+
+# GPU benchmarks have their own CMake project (needs CUDAToolkit).
+# They can also be built standalone: cmake -B build -S benchmarks/GPU
+find_package(CUDAToolkit QUIET)
+if(CUDAToolkit_FOUND)
+  add_subdirectory(GPU)
+endif()

benchmarks/GPU/CMakeLists.txt

@@ -0,0 +1,53 @@
+# GPU benchmarks require CUDA runtime + cuSOLVER.
+# Build separately from the main benchmark tree since they need CUDA toolchain.
+#
+# Usage:
+#   cmake -G Ninja -B build-bench-gpu -S benchmarks/GPU \
+#         -DCMAKE_CUDA_ARCHITECTURES=89
+#   cmake --build build-bench-gpu
+#
+# Profiling:
+#   nsys profile --trace=cuda ./build-bench-gpu/bench_gpu_solvers
+#   ncu --set full -o profile ./build-bench-gpu/bench_gpu_solvers --benchmark_filter=BM_GpuLLT_Compute/4096
+
+cmake_minimum_required(VERSION 3.18)
+project(EigenGpuBenchmarks CXX)
+
+find_package(benchmark REQUIRED)
+find_package(CUDAToolkit REQUIRED)
+
+set(EIGEN_SOURCE_DIR "${CMAKE_CURRENT_SOURCE_DIR}/../..")
+
+function(eigen_add_gpu_benchmark name source)
+  cmake_parse_arguments(BENCH "" "" "LIBRARIES;DEFINITIONS" ${ARGN})
+  if(NOT IS_ABSOLUTE "${source}")
+    set(source "${CMAKE_CURRENT_SOURCE_DIR}/${source}")
+  endif()
+  add_executable(${name} ${source})
+  target_include_directories(${name} PRIVATE
+    ${EIGEN_SOURCE_DIR}
+    ${CUDAToolkit_INCLUDE_DIRS})
+  target_link_libraries(${name} PRIVATE
+    benchmark::benchmark benchmark::benchmark_main
+    CUDA::cudart CUDA::cusolver CUDA::cublas)
+  if(BENCH_LIBRARIES)
+    target_link_libraries(${name} PRIVATE ${BENCH_LIBRARIES})
+  endif()
+  target_compile_options(${name} PRIVATE -O3 -DNDEBUG)
+  target_compile_definitions(${name} PRIVATE EIGEN_USE_GPU)
+  if(BENCH_DEFINITIONS)
+    target_compile_definitions(${name} PRIVATE ${BENCH_DEFINITIONS})
+  endif()
+endfunction()
+
+# Solver benchmarks: LLT/LU compute + solve, host vs device paths, CPU baselines.
+eigen_add_gpu_benchmark(bench_gpu_solvers bench_gpu_solvers.cpp)
+eigen_add_gpu_benchmark(bench_gpu_solvers_float bench_gpu_solvers.cpp DEFINITIONS SCALAR=float)
+
+# Chaining benchmarks: async pipeline efficiency, host-roundtrip vs device chain.
+eigen_add_gpu_benchmark(bench_gpu_chaining bench_gpu_chaining.cpp)
+eigen_add_gpu_benchmark(bench_gpu_chaining_float bench_gpu_chaining.cpp DEFINITIONS SCALAR=float)
+
+# Batching benchmarks: multi-stream concurrency for many small systems.
+eigen_add_gpu_benchmark(bench_gpu_batching bench_gpu_batching.cpp)
+eigen_add_gpu_benchmark(bench_gpu_batching_float bench_gpu_batching.cpp DEFINITIONS SCALAR=float)

benchmarks/GPU/bench_gpu_batching.cpp

@@ -0,0 +1,268 @@
+// GPU batching benchmarks: multi-stream concurrency for many small solves.
+//
+// Each GpuLLT/GpuLU owns its own CUDA stream. This benchmark measures how
+// well multiple solver instances overlap on the GPU, which is critical for
+// workloads like robotics (many small systems) and SLAM (batched poses).
+//
+// Compares:
+//   1. Sequential: one solver handles all systems one by one
+//   2. Batched: N solvers on N streams, all launched before any sync
+//   3. CPU baseline: Eigen LLT on host
+//
+// For Nsight Systems: batched mode should show overlapping kernels on
+// different streams in the timeline view.
+//
+//   nsys profile --trace=cuda ./bench_gpu_batching
+
+#include <benchmark/benchmark.h>
+
+#include <Eigen/Cholesky>
+#include <Eigen/GPU>
+
+#include <memory>
+#include <vector>
+
+using namespace Eigen;
+
+#ifndef SCALAR
+#define SCALAR double
+#endif
+
+using Scalar = SCALAR;
+using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+
+static Mat make_spd(Index n) {
+  Mat M = Mat::Random(n, n);
+  return M.adjoint() * M + Mat::Identity(n, n) * static_cast<Scalar>(n);
+}
+
+static void cuda_warmup() {
+  static bool done = false;
+  if (!done) {
+    void* p;
+    cudaMalloc(&p, 1);
+    cudaFree(p);
+    done = true;
+  }
+}
+
+// --------------------------------------------------------------------------
+// Sequential: one solver, N systems solved one after another
+// --------------------------------------------------------------------------
+
+static void BM_Batch_Sequential(benchmark::State& state) {
+  cuda_warmup();
+  const Index n = state.range(0);
+  const int batch_size = static_cast<int>(state.range(1));
+
+  // Pre-generate all SPD matrices and RHS vectors.
+  std::vector<Mat> As(batch_size);
+  std::vector<Mat> Bs(batch_size);
+  for (int i = 0; i < batch_size; ++i) {
+    As[i] = make_spd(n);
+    Bs[i] = Mat::Random(n, 1);
+  }
+
+  GpuLLT<Scalar> llt;
+
+  for (auto _ : state) {
+    std::vector<Mat> results(batch_size);
+    for (int i = 0; i < batch_size; ++i) {
+      llt.compute(As[i]);
+      results[i] = llt.solve(Bs[i]);
+    }
+    benchmark::DoNotOptimize(results.back().data());
+  }
+
+  state.counters["n"] = n;
+  state.counters["batch"] = batch_size;
+  state.counters["total_solves"] = batch_size;
+}
+
+// --------------------------------------------------------------------------
+// Sequential with DeviceMatrix (avoid re-upload of A each iteration)
+// --------------------------------------------------------------------------
+
+static void BM_Batch_Sequential_Device(benchmark::State& state) {
+  cuda_warmup();
+  const Index n = state.range(0);
+  const int batch_size = static_cast<int>(state.range(1));
+
+  std::vector<Mat> As(batch_size);
+  std::vector<Mat> Bs(batch_size);
+  std::vector<DeviceMatrix<Scalar>> d_As(batch_size);
+  std::vector<DeviceMatrix<Scalar>> d_Bs(batch_size);
+  for (int i = 0; i < batch_size; ++i) {
+    As[i] = make_spd(n);
+    Bs[i] = Mat::Random(n, 1);
+    d_As[i] = DeviceMatrix<Scalar>::fromHost(As[i]);
+    d_Bs[i] = DeviceMatrix<Scalar>::fromHost(Bs[i]);
+  }
+
+  GpuLLT<Scalar> llt;
+
+  for (auto _ : state) {
+    std::vector<Mat> results(batch_size);
+    for (int i = 0; i < batch_size; ++i) {
+      llt.compute(d_As[i]);
+      DeviceMatrix<Scalar> d_X = llt.solve(d_Bs[i]);
+      results[i] = d_X.toHost();
+    }
+    benchmark::DoNotOptimize(results.back().data());
+  }
+
+  state.counters["n"] = n;
+  state.counters["batch"] = batch_size;
+  state.counters["total_solves"] = batch_size;
+}
+
+// --------------------------------------------------------------------------
+// Batched: N solvers on N streams, overlapping execution
+// --------------------------------------------------------------------------
+
+static void BM_Batch_MultiStream(benchmark::State& state) {
+  cuda_warmup();
+  const Index n = state.range(0);
+  const int batch_size = static_cast<int>(state.range(1));
+
+  std::vector<Mat> As(batch_size);
+  std::vector<Mat> Bs(batch_size);
+  std::vector<DeviceMatrix<Scalar>> d_As(batch_size);
+  std::vector<DeviceMatrix<Scalar>> d_Bs(batch_size);
+  for (int i = 0; i < batch_size; ++i) {
+    As[i] = make_spd(n);
+    Bs[i] = Mat::Random(n, 1);
+    d_As[i] = DeviceMatrix<Scalar>::fromHost(As[i]);
+    d_Bs[i] = DeviceMatrix<Scalar>::fromHost(Bs[i]);
+  }
+
+  // N solvers = N independent CUDA streams.
+  std::vector<std::unique_ptr<GpuLLT<Scalar>>> solvers(batch_size);
+  for (int i = 0; i < batch_size; ++i) {
+    solvers[i] = std::make_unique<GpuLLT<Scalar>>();
+  }
+
+  for (auto _ : state) {
+    // Phase 1: launch all factorizations (async, different streams).
+    for (int i = 0; i < batch_size; ++i) {
+      solvers[i]->compute(d_As[i]);
+    }
+
+    // Phase 2: launch all solves (async, different streams).
+    std::vector<DeviceMatrix<Scalar>> d_Xs(batch_size);
+    for (int i = 0; i < batch_size; ++i) {
+      d_Xs[i] = solvers[i]->solve(d_Bs[i]);
+    }
+
+    // Phase 3: download all results.
+    std::vector<Mat> results(batch_size);
+    for (int i = 0; i < batch_size; ++i) {
+      results[i] = d_Xs[i].toHost();
+    }
+    benchmark::DoNotOptimize(results.back().data());
+  }
+
+  state.counters["n"] = n;
+  state.counters["batch"] = batch_size;
+  state.counters["streams"] = batch_size;
+  state.counters["total_solves"] = batch_size;
+}
+
+// --------------------------------------------------------------------------
+// Batched with async download (overlap D2H with computation)
+// --------------------------------------------------------------------------
+
+static void BM_Batch_MultiStream_AsyncDownload(benchmark::State& state) {
+  cuda_warmup();
+  const Index n = state.range(0);
+  const int batch_size = static_cast<int>(state.range(1));
+
+  std::vector<Mat> As(batch_size);
+  std::vector<Mat> Bs(batch_size);
+  std::vector<DeviceMatrix<Scalar>> d_As(batch_size);
+  std::vector<DeviceMatrix<Scalar>> d_Bs(batch_size);
+  for (int i = 0; i < batch_size; ++i) {
+    As[i] = make_spd(n);
+    Bs[i] = Mat::Random(n, 1);
+    d_As[i] = DeviceMatrix<Scalar>::fromHost(As[i]);
+    d_Bs[i] = DeviceMatrix<Scalar>::fromHost(Bs[i]);
+  }
+
+  std::vector<std::unique_ptr<GpuLLT<Scalar>>> solvers(batch_size);
+  for (int i = 0; i < batch_size; ++i) {
+    solvers[i] = std::make_unique<GpuLLT<Scalar>>();
+  }
+
+  for (auto _ : state) {
+    // Launch all compute + solve.
+    std::vector<DeviceMatrix<Scalar>> d_Xs(batch_size);
+    for (int i = 0; i < batch_size; ++i) {
+      solvers[i]->compute(d_As[i]);
+      d_Xs[i] = solvers[i]->solve(d_Bs[i]);
+    }
+
+    // Enqueue all async downloads.
+    std::vector<HostTransfer<Scalar>> transfers;
+    transfers.reserve(batch_size);
+    for (int i = 0; i < batch_size; ++i) {
+      transfers.push_back(d_Xs[i].toHostAsync());
+    }
+
+    // Collect all results.
+    for (int i = 0; i < batch_size; ++i) {
+      benchmark::DoNotOptimize(transfers[i].get().data());
+    }
+  }
+
+  state.counters["n"] = n;
+  state.counters["batch"] = batch_size;
+  state.counters["streams"] = batch_size;
+  state.counters["total_solves"] = batch_size;
+}
+
+// --------------------------------------------------------------------------
+// CPU baseline: Eigen LLT on host, sequential
+// --------------------------------------------------------------------------
+
+static void BM_Batch_CPU(benchmark::State& state) {
+  const Index n = state.range(0);
+  const int batch_size = static_cast<int>(state.range(1));
+
+  std::vector<Mat> As(batch_size);
+  std::vector<Mat> Bs(batch_size);
+  for (int i = 0; i < batch_size; ++i) {
+    As[i] = make_spd(n);
+    Bs[i] = Mat::Random(n, 1);
+  }
+
+  for (auto _ : state) {
+    std::vector<Mat> results(batch_size);
+    for (int i = 0; i < batch_size; ++i) {
+      LLT<Mat> llt(As[i]);
+      results[i] = llt.solve(Bs[i]);
+    }
+    benchmark::DoNotOptimize(results.back().data());
+  }
+
+  state.counters["n"] = n;
+  state.counters["batch"] = batch_size;
+  state.counters["total_solves"] = batch_size;
+}
+
+// --------------------------------------------------------------------------
+// Registration
+// --------------------------------------------------------------------------
+
+// clang-format off
+// Args: {matrix_size, batch_size}
+// Small matrices with large batches are the interesting case for multi-stream.
+BENCHMARK(BM_Batch_Sequential)->ArgsProduct({{16, 32, 64, 128, 256, 512}, {1, 4, 16, 64}})->Unit(benchmark::kMicrosecond);
+BENCHMARK(BM_Batch_Sequential_Device)->ArgsProduct({{16, 32, 64, 128, 256, 512}, {1, 4, 16, 64}})->Unit(benchmark::kMicrosecond);
+BENCHMARK(BM_Batch_MultiStream)->ArgsProduct({{16, 32, 64, 128, 256, 512}, {1, 4, 16, 64}})->Unit(benchmark::kMicrosecond);
+BENCHMARK(BM_Batch_MultiStream_AsyncDownload)->ArgsProduct({{16, 32, 64, 128, 256, 512}, {1, 4, 16, 64}})->Unit(benchmark::kMicrosecond);
+BENCHMARK(BM_Batch_CPU)->ArgsProduct({{16, 32, 64, 128, 256, 512}, {1, 4, 16, 64}})->Unit(benchmark::kMicrosecond);
+
+// Also run larger sizes with moderate batching.
+BENCHMARK(BM_Batch_MultiStream)->ArgsProduct({{512, 1024, 2048}, {1, 4, 8}})->Unit(benchmark::kMicrosecond);
+BENCHMARK(BM_Batch_MultiStream_AsyncDownload)->ArgsProduct({{512, 1024, 2048}, {1, 4, 8}})->Unit(benchmark::kMicrosecond);
+// clang-format on

benchmarks/GPU/bench_gpu_chaining.cpp

@@ -0,0 +1,216 @@
+// GPU chaining benchmarks: measure async pipeline efficiency.
+//
+// Compares:
+//   1. Host round-trip per solve (baseline)
+//   2. DeviceMatrix chaining (no host round-trip between solves)
+//   3. Varying chain lengths (1, 2, 4, 8 consecutive solves)
+//
+// For Nsight Systems: look for gaps between kernel launches in the timeline.
+// Host round-trip creates visible idle gaps; chaining should show back-to-back kernels.
+//
+//   nsys profile --trace=cuda,nvtx ./bench_gpu_chaining
+
+#include <benchmark/benchmark.h>
+
+#include <Eigen/Cholesky>
+#include <Eigen/GPU>
+
+using namespace Eigen;
+
+#ifndef SCALAR
+#define SCALAR double
+#endif
+
+using Scalar = SCALAR;
+using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+
+static Mat make_spd(Index n) {
+  Mat M = Mat::Random(n, n);
+  return M.adjoint() * M + Mat::Identity(n, n) * static_cast<Scalar>(n);
+}
+
+static void cuda_warmup() {
+  static bool done = false;
+  if (!done) {
+    void* p;
+    cudaMalloc(&p, 1);
+    cudaFree(p);
+    done = true;
+  }
+}
+
+// --------------------------------------------------------------------------
+// Baseline: host round-trip between every solve
+// --------------------------------------------------------------------------
+
+static void BM_Chain_HostRoundtrip(benchmark::State& state) {
+  cuda_warmup();
+  const Index n = state.range(0);
+  const int chain_len = static_cast<int>(state.range(1));
+
+  Mat A = make_spd(n);
+  Mat B = Mat::Random(n, 1);
+  GpuLLT<Scalar> llt(A);
+
+  for (auto _ : state) {
+    Mat X = B;
+    for (int i = 0; i < chain_len; ++i) {
+      X = llt.solve(X);  // host → device → host each time
+    }
+    benchmark::DoNotOptimize(X.data());
+  }
+
+  state.counters["n"] = n;
+  state.counters["chain"] = chain_len;
+  state.counters["solves/iter"] = chain_len;
+}
+
+// --------------------------------------------------------------------------
+// DeviceMatrix chaining: no host round-trip between solves
+// --------------------------------------------------------------------------
+
+static void BM_Chain_Device(benchmark::State& state) {
+  cuda_warmup();
+  const Index n = state.range(0);
+  const int chain_len = static_cast<int>(state.range(1));
+
+  Mat A = make_spd(n);
+  Mat B = Mat::Random(n, 1);
+  GpuLLT<Scalar> llt(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+
+  for (auto _ : state) {
+    DeviceMatrix<Scalar> d_X = llt.solve(d_B);
+    for (int i = 1; i < chain_len; ++i) {
+      d_X = llt.solve(d_X);  // device → device, fully async
+    }
+    Mat X = d_X.toHost();  // single sync at end
+    benchmark::DoNotOptimize(X.data());
+  }
+
+  state.counters["n"] = n;
+  state.counters["chain"] = chain_len;
+  state.counters["solves/iter"] = chain_len;
+}
+
+// --------------------------------------------------------------------------
+// DeviceMatrix chaining with async download (overlap D2H with next iteration)
+// --------------------------------------------------------------------------
+
+static void BM_Chain_DeviceAsync(benchmark::State& state) {
+  cuda_warmup();
+  const Index n = state.range(0);
+  const int chain_len = static_cast<int>(state.range(1));
+
+  Mat A = make_spd(n);
+  Mat B = Mat::Random(n, 1);
+  GpuLLT<Scalar> llt(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+
+  for (auto _ : state) {
+    DeviceMatrix<Scalar> d_X = llt.solve(d_B);
+    for (int i = 1; i < chain_len; ++i) {
+      d_X = llt.solve(d_X);
+    }
+    auto transfer = d_X.toHostAsync();
+    Mat X = transfer.get();
+    benchmark::DoNotOptimize(X.data());
+  }
+
+  state.counters["n"] = n;
+  state.counters["chain"] = chain_len;
+  state.counters["solves/iter"] = chain_len;
+}
+
+// --------------------------------------------------------------------------
+// Pure GPU chain (no download — measures kernel-only throughput)
+// --------------------------------------------------------------------------
+
+static void BM_Chain_DeviceNoDownload(benchmark::State& state) {
+  cuda_warmup();
+  const Index n = state.range(0);
+  const int chain_len = static_cast<int>(state.range(1));
+
+  Mat A = make_spd(n);
+  Mat B = Mat::Random(n, 1);
+  GpuLLT<Scalar> llt(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+
+  for (auto _ : state) {
+    DeviceMatrix<Scalar> d_X = llt.solve(d_B);
+    for (int i = 1; i < chain_len; ++i) {
+      d_X = llt.solve(d_X);
+    }
+    cudaStreamSynchronize(llt.stream());
+    benchmark::DoNotOptimize(d_X.data());
+  }
+
+  state.counters["n"] = n;
+  state.counters["chain"] = chain_len;
+  state.counters["solves/iter"] = chain_len;
+}
+
+// --------------------------------------------------------------------------
+// Compute + solve chain (full pipeline: factorize, then chain solves)
+// --------------------------------------------------------------------------
+
+static void BM_FullPipeline_Host(benchmark::State& state) {
+  cuda_warmup();
+  const Index n = state.range(0);
+  const int chain_len = static_cast<int>(state.range(1));
+
+  Mat A = make_spd(n);
+  Mat B = Mat::Random(n, 1);
+
+  for (auto _ : state) {
+    GpuLLT<Scalar> llt(A);
+    Mat X = B;
+    for (int i = 0; i < chain_len; ++i) {
+      X = llt.solve(X);
+    }
+    benchmark::DoNotOptimize(X.data());
+  }
+
+  state.counters["n"] = n;
+  state.counters["chain"] = chain_len;
+}
+
+static void BM_FullPipeline_Device(benchmark::State& state) {
+  cuda_warmup();
+  const Index n = state.range(0);
+  const int chain_len = static_cast<int>(state.range(1));
+
+  Mat A = make_spd(n);
+  Mat B = Mat::Random(n, 1);
+
+  for (auto _ : state) {
+    auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+    auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+    GpuLLT<Scalar> llt;
+    llt.compute(d_A);
+    DeviceMatrix<Scalar> d_X = llt.solve(d_B);
+    for (int i = 1; i < chain_len; ++i) {
+      d_X = llt.solve(d_X);
+    }
+    Mat X = d_X.toHost();
+    benchmark::DoNotOptimize(X.data());
+  }
+
+  state.counters["n"] = n;
+  state.counters["chain"] = chain_len;
+}
+
+// --------------------------------------------------------------------------
+// Registration
+// --------------------------------------------------------------------------
+
+// clang-format off
+// Args: {matrix_size, chain_length}
+BENCHMARK(BM_Chain_HostRoundtrip)->ArgsProduct({{64, 256, 1024, 4096}, {1, 2, 4, 8}})->Unit(benchmark::kMicrosecond);
+BENCHMARK(BM_Chain_Device)->ArgsProduct({{64, 256, 1024, 4096}, {1, 2, 4, 8}})->Unit(benchmark::kMicrosecond);
+BENCHMARK(BM_Chain_DeviceAsync)->ArgsProduct({{64, 256, 1024, 4096}, {1, 2, 4, 8}})->Unit(benchmark::kMicrosecond);
+BENCHMARK(BM_Chain_DeviceNoDownload)->ArgsProduct({{64, 256, 1024, 4096}, {1, 2, 4, 8}})->Unit(benchmark::kMicrosecond);
+
+BENCHMARK(BM_FullPipeline_Host)->ArgsProduct({{256, 1024, 4096}, {1, 4}})->Unit(benchmark::kMicrosecond);
+BENCHMARK(BM_FullPipeline_Device)->ArgsProduct({{256, 1024, 4096}, {1, 4}})->Unit(benchmark::kMicrosecond);
+// clang-format on

benchmarks/GPU/bench_gpu_solvers.cpp

@@ -0,0 +1,296 @@
+// GPU solver benchmarks: GpuLLT and GpuLU compute + solve throughput.
+//
+// Measures factorization and solve performance for the host-matrix and
+// DeviceMatrix code paths across a range of matrix sizes.
+//
+// For Nsight Systems profiling:
+//   nsys profile --trace=cuda,nvtx ./bench_gpu_solvers
+//
+// For Nsight Compute kernel analysis:
+//   ncu --set full -o profile ./bench_gpu_solvers --benchmark_filter=BM_GpuLLT_Compute/4096
+
+#include <benchmark/benchmark.h>
+
+#include <Eigen/Cholesky>
+#include <Eigen/GPU>
+#include <Eigen/LU>
+
+using namespace Eigen;
+
+#ifndef SCALAR
+#define SCALAR double
+#endif
+
+using Scalar = SCALAR;
+using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+
+// --------------------------------------------------------------------------
+// Helpers
+// --------------------------------------------------------------------------
+
+static Mat make_spd(Index n) {
+  Mat M = Mat::Random(n, n);
+  return M.adjoint() * M + Mat::Identity(n, n) * static_cast<Scalar>(n);
+}
+
+// CUDA warm-up: ensure the GPU is initialized before timing.
+static void cuda_warmup() {
+  static bool done = false;
+  if (!done) {
+    void* p;
+    cudaMalloc(&p, 1);
+    cudaFree(p);
+    done = true;
+  }
+}
+
+// --------------------------------------------------------------------------
+// GpuLLT benchmarks
+// --------------------------------------------------------------------------
+
+// Factorize from host matrix (includes H2D upload).
+static void BM_GpuLLT_Compute_Host(benchmark::State& state) {
+  cuda_warmup();
+  const Index n = state.range(0);
+  Mat A = make_spd(n);
+  GpuLLT<Scalar> llt;
+
+  for (auto _ : state) {
+    llt.compute(A);
+    if (llt.info() != Success) state.SkipWithError("factorization failed");
+  }
+
+  double flops = static_cast<double>(n) * static_cast<double>(n) * static_cast<double>(n) / 3.0;
+  state.counters["GFLOPS"] =
+      benchmark::Counter(flops, benchmark::Counter::kIsIterationInvariantRate, benchmark::Counter::kIs1000);
+  state.counters["n"] = n;
+}
+
+// Factorize from DeviceMatrix (D2D copy path).
+static void BM_GpuLLT_Compute_Device(benchmark::State& state) {
+  cuda_warmup();
+  const Index n = state.range(0);
+  Mat A = make_spd(n);
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  GpuLLT<Scalar> llt;
+
+  for (auto _ : state) {
+    llt.compute(d_A);
+    if (llt.info() != Success) state.SkipWithError("factorization failed");
+  }
+
+  double flops = static_cast<double>(n) * static_cast<double>(n) * static_cast<double>(n) / 3.0;
+  state.counters["GFLOPS"] =
+      benchmark::Counter(flops, benchmark::Counter::kIsIterationInvariantRate, benchmark::Counter::kIs1000);
+  state.counters["n"] = n;
+}
+
+// Factorize from DeviceMatrix (move path, no copy).
+static void BM_GpuLLT_Compute_DeviceMove(benchmark::State& state) {
+  cuda_warmup();
+  const Index n = state.range(0);
+  Mat A = make_spd(n);
+  GpuLLT<Scalar> llt;
+
+  for (auto _ : state) {
+    auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+    llt.compute(std::move(d_A));
+    if (llt.info() != Success) state.SkipWithError("factorization failed");
+  }
+
+  double flops = static_cast<double>(n) * static_cast<double>(n) * static_cast<double>(n) / 3.0;
+  state.counters["GFLOPS"] =
+      benchmark::Counter(flops, benchmark::Counter::kIsIterationInvariantRate, benchmark::Counter::kIs1000);
+  state.counters["n"] = n;
+}
+
+// Solve from host matrix (H2D + potrs + D2H).
+static void BM_GpuLLT_Solve_Host(benchmark::State& state) {
+  cuda_warmup();
+  const Index n = state.range(0);
+  const Index nrhs = state.range(1);
+  Mat A = make_spd(n);
+  Mat B = Mat::Random(n, nrhs);
+  GpuLLT<Scalar> llt(A);
+
+  for (auto _ : state) {
+    Mat X = llt.solve(B);
+    benchmark::DoNotOptimize(X.data());
+  }
+
+  state.counters["n"] = n;
+  state.counters["nrhs"] = nrhs;
+}
+
+// Solve from DeviceMatrix (D2D + potrs, async, toHost at end).
+static void BM_GpuLLT_Solve_Device(benchmark::State& state) {
+  cuda_warmup();
+  const Index n = state.range(0);
+  const Index nrhs = state.range(1);
+  Mat A = make_spd(n);
+  Mat B = Mat::Random(n, nrhs);
+  GpuLLT<Scalar> llt(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+
+  for (auto _ : state) {
+    DeviceMatrix<Scalar> d_X = llt.solve(d_B);
+    Mat X = d_X.toHost();
+    benchmark::DoNotOptimize(X.data());
+  }
+
+  state.counters["n"] = n;
+  state.counters["nrhs"] = nrhs;
+}
+
+// Solve staying entirely on device (no toHost — measures pure GPU time).
+static void BM_GpuLLT_Solve_DeviceOnly(benchmark::State& state) {
+  cuda_warmup();
+  const Index n = state.range(0);
+  const Index nrhs = state.range(1);
+  Mat A = make_spd(n);
+  Mat B = Mat::Random(n, nrhs);
+  GpuLLT<Scalar> llt(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+
+  for (auto _ : state) {
+    DeviceMatrix<Scalar> d_X = llt.solve(d_B);
+    // Force completion without D2H transfer.
+    cudaStreamSynchronize(llt.stream());
+    benchmark::DoNotOptimize(d_X.data());
+  }
+
+  state.counters["n"] = n;
+  state.counters["nrhs"] = nrhs;
+}
+
+// --------------------------------------------------------------------------
+// GpuLU benchmarks
+// --------------------------------------------------------------------------
+
+static void BM_GpuLU_Compute_Host(benchmark::State& state) {
+  cuda_warmup();
+  const Index n = state.range(0);
+  Mat A = Mat::Random(n, n);
+  GpuLU<Scalar> lu;
+
+  for (auto _ : state) {
+    lu.compute(A);
+    if (lu.info() != Success) state.SkipWithError("factorization failed");
+  }
+
+  double flops = 2.0 / 3.0 * static_cast<double>(n) * static_cast<double>(n) * static_cast<double>(n);
+  state.counters["GFLOPS"] =
+      benchmark::Counter(flops, benchmark::Counter::kIsIterationInvariantRate, benchmark::Counter::kIs1000);
+  state.counters["n"] = n;
+}
+
+static void BM_GpuLU_Compute_Device(benchmark::State& state) {
+  cuda_warmup();
+  const Index n = state.range(0);
+  Mat A = Mat::Random(n, n);
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  GpuLU<Scalar> lu;
+
+  for (auto _ : state) {
+    lu.compute(d_A);
+    if (lu.info() != Success) state.SkipWithError("factorization failed");
+  }
+
+  double flops = 2.0 / 3.0 * static_cast<double>(n) * static_cast<double>(n) * static_cast<double>(n);
+  state.counters["GFLOPS"] =
+      benchmark::Counter(flops, benchmark::Counter::kIsIterationInvariantRate, benchmark::Counter::kIs1000);
+  state.counters["n"] = n;
+}
+
+static void BM_GpuLU_Solve_Host(benchmark::State& state) {
+  cuda_warmup();
+  const Index n = state.range(0);
+  const Index nrhs = state.range(1);
+  Mat A = Mat::Random(n, n);
+  Mat B = Mat::Random(n, nrhs);
+  GpuLU<Scalar> lu(A);
+
+  for (auto _ : state) {
+    Mat X = lu.solve(B);
+    benchmark::DoNotOptimize(X.data());
+  }
+
+  state.counters["n"] = n;
+  state.counters["nrhs"] = nrhs;
+}
+
+static void BM_GpuLU_Solve_Device(benchmark::State& state) {
+  cuda_warmup();
+  const Index n = state.range(0);
+  const Index nrhs = state.range(1);
+  Mat A = Mat::Random(n, n);
+  Mat B = Mat::Random(n, nrhs);
+  GpuLU<Scalar> lu(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+
+  for (auto _ : state) {
+    DeviceMatrix<Scalar> d_X = lu.solve(d_B);
+    Mat X = d_X.toHost();
+    benchmark::DoNotOptimize(X.data());
+  }
+
+  state.counters["n"] = n;
+  state.counters["nrhs"] = nrhs;
+}
+
+// --------------------------------------------------------------------------
+// CPU baselines for comparison
+// --------------------------------------------------------------------------
+
+static void BM_CpuLLT_Compute(benchmark::State& state) {
+  const Index n = state.range(0);
+  Mat A = make_spd(n);
+  LLT<Mat> llt;
+
+  for (auto _ : state) {
+    llt.compute(A);
+    benchmark::DoNotOptimize(llt.matrixLLT().data());
+  }
+
+  double flops = static_cast<double>(n) * static_cast<double>(n) * static_cast<double>(n) / 3.0;
+  state.counters["GFLOPS"] =
+      benchmark::Counter(flops, benchmark::Counter::kIsIterationInvariantRate, benchmark::Counter::kIs1000);
+  state.counters["n"] = n;
+}
+
+static void BM_CpuLU_Compute(benchmark::State& state) {
+  const Index n = state.range(0);
+  Mat A = Mat::Random(n, n);
+  PartialPivLU<Mat> lu;
+
+  for (auto _ : state) {
+    lu.compute(A);
+    benchmark::DoNotOptimize(lu.matrixLU().data());
+  }
+
+  double flops = 2.0 / 3.0 * static_cast<double>(n) * static_cast<double>(n) * static_cast<double>(n);
+  state.counters["GFLOPS"] =
+      benchmark::Counter(flops, benchmark::Counter::kIsIterationInvariantRate, benchmark::Counter::kIs1000);
+  state.counters["n"] = n;
+}
+
+// --------------------------------------------------------------------------
+// Registration
+// --------------------------------------------------------------------------
+
+// clang-format off
+BENCHMARK(BM_GpuLLT_Compute_Host)->ArgsProduct({{64, 128, 256, 512, 1024, 2048, 4096}})->Unit(benchmark::kMicrosecond);
+BENCHMARK(BM_GpuLLT_Compute_Device)->ArgsProduct({{64, 128, 256, 512, 1024, 2048, 4096}})->Unit(benchmark::kMicrosecond);
+BENCHMARK(BM_GpuLLT_Compute_DeviceMove)->ArgsProduct({{64, 128, 256, 512, 1024, 2048, 4096}})->Unit(benchmark::kMicrosecond);
+BENCHMARK(BM_GpuLLT_Solve_Host)->ArgsProduct({{64, 256, 1024, 4096}, {1, 16}})->Unit(benchmark::kMicrosecond);
+BENCHMARK(BM_GpuLLT_Solve_Device)->ArgsProduct({{64, 256, 1024, 4096}, {1, 16}})->Unit(benchmark::kMicrosecond);
+BENCHMARK(BM_GpuLLT_Solve_DeviceOnly)->ArgsProduct({{64, 256, 1024, 4096}, {1, 16}})->Unit(benchmark::kMicrosecond);
+
+BENCHMARK(BM_GpuLU_Compute_Host)->ArgsProduct({{64, 128, 256, 512, 1024, 2048, 4096}})->Unit(benchmark::kMicrosecond);
+BENCHMARK(BM_GpuLU_Compute_Device)->ArgsProduct({{64, 128, 256, 512, 1024, 2048, 4096}})->Unit(benchmark::kMicrosecond);
+BENCHMARK(BM_GpuLU_Solve_Host)->ArgsProduct({{64, 256, 1024, 4096}, {1, 16}})->Unit(benchmark::kMicrosecond);
+BENCHMARK(BM_GpuLU_Solve_Device)->ArgsProduct({{64, 256, 1024, 4096}, {1, 16}})->Unit(benchmark::kMicrosecond);
+
+BENCHMARK(BM_CpuLLT_Compute)->ArgsProduct({{64, 128, 256, 512, 1024, 2048, 4096}})->Unit(benchmark::kMicrosecond);
+BENCHMARK(BM_CpuLU_Compute)->ArgsProduct({{64, 128, 256, 512, 1024, 2048, 4096}})->Unit(benchmark::kMicrosecond);
+// clang-format on

ci/build.linux.gitlab-ci.yml

@@ -211,20 +211,20 @@ build:linux:x86-64:nvhpc-26.1:default:unsupported:
     # Build on regular linux to limit GPU cost.
     - saas-linux-2xlarge-amd64
 
-# GCC-10, CUDA-12.2
-build:linux:cuda-12.2:gcc-10:
+# GCC-11, CUDA-12.2
+build:linux:cuda-12.2:gcc-11:
   extends: .build:linux:cuda
-  image: nvidia/cuda:12.2.0-devel-ubuntu20.04
+  image: nvidia/cuda:12.2.0-devel-ubuntu22.04
   variables:
-    EIGEN_CI_C_COMPILER: gcc-10
-    EIGEN_CI_CXX_COMPILER: g++-10
+    EIGEN_CI_C_COMPILER: gcc-11
+    EIGEN_CI_CXX_COMPILER: g++-11
 
-# Clang-12, CUDA-12.2
-build:linux:cuda-12.2:clang-12:
-  extends: build:linux:cuda-12.2:gcc-10
+# Clang-14, CUDA-12.2
+build:linux:cuda-12.2:clang-14:
+  extends: build:linux:cuda-12.2:gcc-11
   variables:
-    EIGEN_CI_C_COMPILER: clang-12
-    EIGEN_CI_CXX_COMPILER: clang++-12
+    EIGEN_CI_C_COMPILER: clang-14
+    EIGEN_CI_CXX_COMPILER: clang++-14
     EIGEN_CI_TEST_CUDA_CLANG: "on"
 
 

ci/test.linux.gitlab-ci.yml

@@ -265,23 +265,23 @@ test:linux:x86-64:nvhpc-26.1:default:unsupported:
   tags:
     - saas-linux-medium-amd64-gpu-standard
 
-# GCC-10, CUDA-12.2
-test:linux:cuda-12.2:gcc-10:
+# GCC-11, CUDA-12.2
+test:linux:cuda-12.2:gcc-11:
   extends: .test:linux:cuda
-  image: nvidia/cuda:12.2.0-devel-ubuntu20.04
-  needs: [ build:linux:cuda-12.2:gcc-10 ]
+  image: nvidia/cuda:12.2.0-devel-ubuntu22.04
+  needs: [ build:linux:cuda-12.2:gcc-11 ]
   variables:
-    EIGEN_CI_CXX_COMPILER: g++-10
-    EIGEN_CI_CC_COMPILER: gcc-10
+    EIGEN_CI_CXX_COMPILER: g++-11
+    EIGEN_CI_CC_COMPILER: gcc-11
 
-# Clang-12, CUDA-12.2
-test:linux:cuda-12.2:clang-12:
+# Clang-14, CUDA-12.2
+test:linux:cuda-12.2:clang-14:
   extends: .test:linux:cuda
-  image: nvidia/cuda:12.2.0-devel-ubuntu20.04
-  needs: [ build:linux:cuda-12.2:clang-12 ]
+  image: nvidia/cuda:12.2.0-devel-ubuntu22.04
+  needs: [ build:linux:cuda-12.2:clang-14 ]
   variables:
-    EIGEN_CI_CXX_COMPILER: clang++-12
-    EIGEN_CI_CC_COMPILER: clang-12
+    EIGEN_CI_CXX_COMPILER: clang++-14
+    EIGEN_CI_CC_COMPILER: clang-14
 
 
 ##### arm ######################################################################

test/CMakeLists.txt

@@ -479,6 +479,78 @@ if(CUDA_FOUND AND EIGEN_TEST_CUDA)
 
   ei_add_test(gpu_example)
   ei_add_test(gpu_basic)
+  ei_add_test(gpu_library_example "" "CUDA::cusolver")
+
+  # DeviceMatrix tests: only CUDA runtime, no NVIDIA libraries.
+  unset(EIGEN_ADD_TEST_FILENAME_EXTENSION)
+  add_executable(gpu_device_matrix gpu_device_matrix.cpp)
+  target_include_directories(gpu_device_matrix PRIVATE
+    "${CUDA_TOOLKIT_ROOT_DIR}/include"
+    "${CMAKE_CURRENT_BINARY_DIR}")
+  target_link_libraries(gpu_device_matrix Eigen3::Eigen CUDA::cudart)
+  target_compile_definitions(gpu_device_matrix PRIVATE
+    EIGEN_TEST_MAX_SIZE=${EIGEN_TEST_MAX_SIZE}
+    EIGEN_TEST_PART_ALL=1)
+  add_test(NAME gpu_device_matrix COMMAND gpu_device_matrix)
+  add_dependencies(buildtests gpu_device_matrix)
+  add_dependencies(buildtests_gpu gpu_device_matrix)
+  set_property(TEST gpu_device_matrix APPEND PROPERTY LABELS "Official;gpu")
+  set_property(TEST gpu_device_matrix PROPERTY SKIP_RETURN_CODE 77)
+  set(EIGEN_ADD_TEST_FILENAME_EXTENSION "cu")
+
+  # Library-specific GPU tests (activated by later phases, OFF by default).
+  # CUDAToolkit imported targets (CUDA::cublas, etc.) are available from
+  # find_package(CUDAToolkit) above.
+  option(EIGEN_TEST_CUBLAS "Test cuBLAS integration" OFF)
+  if(EIGEN_TEST_CUBLAS AND TARGET CUDA::cublas)
+    # cuBLAS tests are plain .cpp files (no device code), like cuSOLVER tests.
+    unset(EIGEN_ADD_TEST_FILENAME_EXTENSION)
+    add_executable(gpu_cublas gpu_cublas.cpp)
+    target_include_directories(gpu_cublas PRIVATE
+      "${CUDA_TOOLKIT_ROOT_DIR}/include"
+      "${CMAKE_CURRENT_BINARY_DIR}")
+    target_link_libraries(gpu_cublas
+      Eigen3::Eigen CUDA::cudart CUDA::cublas CUDA::cusolver)
+    target_compile_definitions(gpu_cublas PRIVATE
+      EIGEN_TEST_MAX_SIZE=${EIGEN_TEST_MAX_SIZE}
+      EIGEN_TEST_PART_ALL=1)
+    add_test(NAME gpu_cublas COMMAND gpu_cublas)
+    add_dependencies(buildtests gpu_cublas)
+    add_dependencies(buildtests_gpu gpu_cublas)
+    set_property(TEST gpu_cublas APPEND PROPERTY LABELS "Official;gpu")
+    set_property(TEST gpu_cublas PROPERTY SKIP_RETURN_CODE 77)
+    set(EIGEN_ADD_TEST_FILENAME_EXTENSION "cu")
+  endif()
+
+  option(EIGEN_TEST_CUSOLVER "Test cuSOLVER integration" OFF)
+  if(EIGEN_TEST_CUSOLVER AND TARGET CUDA::cusolver)
+    # cuSOLVER tests are plain .cpp files: no device code, compiled by the host
+    # compiler and linked against CUDA runtime + cuSOLVER. This avoids NVCC
+    # instantiating Eigen's CPU packet operations for CUDA vector types.
+    unset(EIGEN_ADD_TEST_FILENAME_EXTENSION)
+    foreach(_cusolver_test IN ITEMS gpu_cusolver_llt gpu_cusolver_lu gpu_cusolver_qr gpu_cusolver_svd gpu_cusolver_eigen)
+      add_executable(${_cusolver_test} ${_cusolver_test}.cpp)
+      target_include_directories(${_cusolver_test} PRIVATE
+        "${CUDA_TOOLKIT_ROOT_DIR}/include"
+        "${CMAKE_CURRENT_BINARY_DIR}")
+      target_link_libraries(${_cusolver_test}
+        Eigen3::Eigen CUDA::cudart CUDA::cusolver CUDA::cublas)
+      target_compile_definitions(${_cusolver_test} PRIVATE
+        EIGEN_TEST_MAX_SIZE=${EIGEN_TEST_MAX_SIZE}
+        EIGEN_TEST_PART_ALL=1)
+      add_test(NAME ${_cusolver_test} COMMAND "${_cusolver_test}")
+      add_dependencies(buildtests ${_cusolver_test})
+      add_dependencies(buildtests_gpu ${_cusolver_test})
+      set_property(TEST ${_cusolver_test} APPEND PROPERTY LABELS "Official;gpu")
+      set_property(TEST ${_cusolver_test} PROPERTY SKIP_RETURN_CODE 77)
+    endforeach()
+    set(EIGEN_ADD_TEST_FILENAME_EXTENSION "cu")
+  endif()
+
+  option(EIGEN_TEST_CUSPARSE "Test cuSPARSE integration" OFF)
+  if(EIGEN_TEST_CUSPARSE AND TARGET CUDA::cusparse)
+    ei_add_test(gpu_cusparse "" "CUDA::cusparse")
+  endif()
 
   unset(EIGEN_ADD_TEST_FILENAME_EXTENSION)
 

test/gpu_context.h

@@ -0,0 +1,72 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2026 Rasmus Munk Larsen <rmlarsen@gmail.com>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+#ifndef EIGEN_TEST_GPU_CONTEXT_H
+#define EIGEN_TEST_GPU_CONTEXT_H
+
+// RAII context for GPU tests that use NVIDIA library APIs (cuBLAS, cuSOLVER, etc.).
+// Owns a non-default CUDA stream. Library handles (cuBLAS, cuSOLVER, etc.) are added
+// here by each integration phase as needed; each handle is bound to the owned stream.
+//
+// Usage:
+//   GpuContext ctx;
+//   auto buf = gpu_copy_to_device(ctx.stream, A);
+//   // ... call NVIDIA library APIs using ctx.stream / ctx.cusolver ...
+//   ctx.synchronize();
+
+#include "gpu_test_helper.h"
+
+#ifdef EIGEN_USE_GPU
+#include <cusolverDn.h>
+
+// Checks cuSOLVER return codes, aborts on failure.
+#define CUSOLVER_CHECK(expr)                                                                 \
+  do {                                                                                       \
+    cusolverStatus_t _status = (expr);                                                       \
+    if (_status != CUSOLVER_STATUS_SUCCESS) {                                                \
+      printf("cuSOLVER error %d at %s:%d\n", static_cast<int>(_status), __FILE__, __LINE__); \
+      gpu_assert(false);                                                                     \
+    }                                                                                        \
+  } while (0)
+
+struct GpuContext {
+  cudaStream_t stream = nullptr;
+  cusolverDnHandle_t cusolver = nullptr;
+
+  GpuContext() {
+    GPU_CHECK(gpuGetDevice(&device_));
+    GPU_CHECK(gpuGetDeviceProperties(&device_props_, device_));
+    GPU_CHECK(cudaStreamCreate(&stream));
+    CUSOLVER_CHECK(cusolverDnCreate(&cusolver));
+    CUSOLVER_CHECK(cusolverDnSetStream(cusolver, stream));
+  }
+
+  ~GpuContext() {
+    if (cusolver) CUSOLVER_CHECK(cusolverDnDestroy(cusolver));
+    if (stream) GPU_CHECK(cudaStreamDestroy(stream));
+  }
+
+  int device() const { return device_; }
+  const gpuDeviceProp_t& deviceProperties() const { return device_props_; }
+
+  // Wait for all work submitted on this context's stream to complete.
+  void synchronize() { GPU_CHECK(cudaStreamSynchronize(stream)); }
+
+  // Non-copyable, non-movable.
+  GpuContext(const GpuContext&) = delete;
+  GpuContext& operator=(const GpuContext&) = delete;
+
+ private:
+  int device_ = 0;
+  gpuDeviceProp_t device_props_;
+};
+
+#endif  // EIGEN_USE_GPU
+
+#endif  // EIGEN_TEST_GPU_CONTEXT_H

test/gpu_cublas.cpp

@@ -0,0 +1,756 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2026 Rasmus Munk Larsen <rmlarsen@gmail.com>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+// Tests for cuBLAS GEMM dispatch via DeviceMatrix expression syntax.
+// Covers: d_C = d_A * d_B, adjoint, transpose, scaled, +=, .device(ctx).
+
+#define EIGEN_USE_GPU
+#include "main.h"
+#include <Eigen/GPU>
+
+using namespace Eigen;
+
+// Unit roundoff for GPU GEMM compute precision.
+// TF32 (opt-in via EIGEN_CUDA_TF32) has eps ~ 2^{-10}.
+template <typename Scalar>
+typename NumTraits<Scalar>::Real gpu_unit_roundoff() {
+#if defined(EIGEN_CUDA_TF32) && !defined(EIGEN_NO_CUDA_TENSOR_OPS)
+  using RealScalar = typename NumTraits<Scalar>::Real;
+  if (std::is_same<RealScalar, float>::value) return RealScalar(9.8e-4);
+#endif
+  return NumTraits<Scalar>::epsilon();
+}
+
+// Higham-Mary probabilistic error bound for GEMM:
+//   ||C - fl(C)||_F <= lambda * sqrt(k) * u * ||A||_F * ||B||_F
+// where k is the inner dimension, u is the unit roundoff, and
+// lambda = sqrt(2 * ln(2/delta)) with delta = failure probability.
+// lambda = 5 corresponds to delta ~ 10^{-6}.
+// Reference: Higham & Mary, "Probabilistic Error Analysis for Inner Products",
+// SIAM J. Matrix Anal. Appl., 2019.
+template <typename Scalar>
+typename NumTraits<Scalar>::Real gemm_error_bound(Index k, typename NumTraits<Scalar>::Real normA,
+                                                  typename NumTraits<Scalar>::Real normB) {
+  using RealScalar = typename NumTraits<Scalar>::Real;
+  constexpr RealScalar lambda = 5;
+  return lambda * std::sqrt(static_cast<RealScalar>(k)) * gpu_unit_roundoff<Scalar>() * normA * normB;
+}
+
+// ---- Basic GEMM: C = A * B -------------------------------------------------
+
+template <typename Scalar>
+void test_gemm_basic(Index m, Index n, Index k) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = Mat::Random(m, k);
+  Mat B = Mat::Random(k, n);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+
+  // Expression: d_C = d_A * d_B
+  DeviceMatrix<Scalar> d_C;
+  d_C = d_A * d_B;
+
+  Mat C = d_C.toHost();
+  Mat C_ref = A * B;
+
+  RealScalar tol = gemm_error_bound<Scalar>(k, A.norm(), B.norm());
+  VERIFY((C - C_ref).norm() < tol);
+}
+
+// ---- GEMM with adjoint: C = A^H * B ----------------------------------------
+
+template <typename Scalar>
+void test_gemm_adjoint_lhs(Index m, Index n, Index k) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = Mat::Random(k, m);  // A is k×m, A^H is m×k
+  Mat B = Mat::Random(k, n);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+
+  DeviceMatrix<Scalar> d_C;
+  d_C = d_A.adjoint() * d_B;
+
+  Mat C = d_C.toHost();
+  Mat C_ref = A.adjoint() * B;
+
+  RealScalar tol = gemm_error_bound<Scalar>(k, A.norm(), B.norm());
+  VERIFY((C - C_ref).norm() < tol);
+}
+
+// ---- GEMM with transpose: C = A * B^T --------------------------------------
+
+template <typename Scalar>
+void test_gemm_transpose_rhs(Index m, Index n, Index k) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = Mat::Random(m, k);
+  Mat B = Mat::Random(n, k);  // B is n×k, B^T is k×n
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+
+  DeviceMatrix<Scalar> d_C;
+  d_C = d_A * d_B.transpose();
+
+  Mat C = d_C.toHost();
+  Mat C_ref = A * B.transpose();
+
+  RealScalar tol = gemm_error_bound<Scalar>(k, A.norm(), B.norm());
+  VERIFY((C - C_ref).norm() < tol);
+}
+
+// ---- GEMM with scaled: C = alpha * A * B ------------------------------------
+
+template <typename Scalar>
+void test_gemm_scaled(Index m, Index n, Index k) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = Mat::Random(m, k);
+  Mat B = Mat::Random(k, n);
+  Scalar alpha = Scalar(2.5);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+
+  DeviceMatrix<Scalar> d_C;
+  d_C = alpha * d_A * d_B;
+
+  Mat C = d_C.toHost();
+  Mat C_ref = alpha * A * B;
+
+  RealScalar tol = gemm_error_bound<Scalar>(k, A.norm(), B.norm());
+  VERIFY((C - C_ref).norm() < tol);
+}
+
+// ---- GEMM accumulate: C += A * B (beta=1) -----------------------------------
+
+template <typename Scalar>
+void test_gemm_accumulate(Index m, Index n, Index k) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = Mat::Random(m, k);
+  Mat B = Mat::Random(k, n);
+  Mat C_init = Mat::Random(m, n);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+  auto d_C = DeviceMatrix<Scalar>::fromHost(C_init);
+
+  d_C += d_A * d_B;
+
+  Mat C = d_C.toHost();
+  Mat C_ref = C_init + A * B;
+
+  RealScalar tol = gemm_error_bound<Scalar>(k, A.norm(), B.norm());
+  VERIFY((C - C_ref).norm() < tol);
+}
+
+// ---- GEMM accumulate into empty destination ---------------------------------
+
+template <typename Scalar>
+void test_gemm_accumulate_empty(Index m, Index n, Index k) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = Mat::Random(m, k);
+  Mat B = Mat::Random(k, n);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+  DeviceMatrix<Scalar> d_C;
+
+  d_C += d_A * d_B;
+
+  Mat C = d_C.toHost();
+  Mat C_ref = A * B;
+
+  RealScalar tol = gemm_error_bound<Scalar>(k, A.norm(), B.norm());
+  VERIFY((C - C_ref).norm() < tol);
+}
+
+// ---- GEMM subtract: C -= A * B (beta=1, alpha=-1) --------------------------
+
+template <typename Scalar>
+void test_gemm_subtract(Index m, Index n, Index k) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = Mat::Random(m, k);
+  Mat B = Mat::Random(k, n);
+  Mat C_init = Mat::Random(m, n);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+  auto d_C = DeviceMatrix<Scalar>::fromHost(C_init);
+
+  GpuContext ctx;
+  d_C.device(ctx) -= d_A * d_B;
+
+  Mat C = d_C.toHost();
+  Mat C_ref = C_init - A * B;
+
+  RealScalar tol = gemm_error_bound<Scalar>(k, A.norm(), B.norm());
+  VERIFY((C - C_ref).norm() < tol);
+}
+
+// ---- GEMM subtract from empty destination -----------------------------------
+
+template <typename Scalar>
+void test_gemm_subtract_empty(Index m, Index n, Index k) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = Mat::Random(m, k);
+  Mat B = Mat::Random(k, n);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+
+  GpuContext ctx;
+  DeviceMatrix<Scalar> d_C;
+  d_C.device(ctx) -= d_A * d_B;
+
+  Mat C = d_C.toHost();
+  Mat C_ref = -(A * B);
+
+  RealScalar tol = gemm_error_bound<Scalar>(k, A.norm(), B.norm());
+  VERIFY((C - C_ref).norm() < tol);
+}
+
+// ---- GEMM with scaled RHS: C = A * (alpha * B) -----------------------------
+
+template <typename Scalar>
+void test_gemm_scaled_rhs(Index m, Index n, Index k) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = Mat::Random(m, k);
+  Mat B = Mat::Random(k, n);
+  Scalar alpha = Scalar(3.0);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+
+  DeviceMatrix<Scalar> d_C;
+  d_C = d_A * (alpha * d_B);
+
+  Mat C = d_C.toHost();
+  Mat C_ref = A * (alpha * B);
+
+  RealScalar tol = gemm_error_bound<Scalar>(k, A.norm(), B.norm());
+  VERIFY((C - C_ref).norm() < tol);
+}
+
+// ---- GEMM dimension mismatch must assert ------------------------------------
+
+template <typename Scalar>
+void test_gemm_dimension_mismatch() {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+
+  Mat A = Mat::Random(8, 5);
+  Mat B = Mat::Random(6, 7);  // inner dimension mismatch
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+  DeviceMatrix<Scalar> d_C;
+
+  VERIFY_RAISES_ASSERT(d_C = d_A * d_B);
+}
+
+// ---- GEMM with explicit GpuContext ------------------------------------------
+
+template <typename Scalar>
+void test_gemm_explicit_context(Index m, Index n, Index k) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = Mat::Random(m, k);
+  Mat B = Mat::Random(k, n);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+
+  GpuContext ctx;
+  DeviceMatrix<Scalar> d_C;
+  d_C.device(ctx) = d_A * d_B;
+
+  Mat C = d_C.toHost();
+  Mat C_ref = A * B;
+
+  RealScalar tol = gemm_error_bound<Scalar>(k, A.norm(), B.norm());
+  VERIFY((C - C_ref).norm() < tol);
+}
+
+// ---- GEMM cross-context reuse of the same destination -----------------------
+
+template <typename Scalar>
+void test_gemm_cross_context_reuse(Index n) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = Mat::Random(n, n);
+  Mat B = Mat::Random(n, n);
+  Mat D = Mat::Random(n, n);
+  Mat E = Mat::Random(n, n);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+  auto d_D = DeviceMatrix<Scalar>::fromHost(D);
+  auto d_E = DeviceMatrix<Scalar>::fromHost(E);
+
+  GpuContext ctx1;
+  GpuContext ctx2;
+  DeviceMatrix<Scalar> d_C;
+  d_C.device(ctx1) = d_A * d_B;
+  d_C.device(ctx2) += d_D * d_E;
+
+  Mat C = d_C.toHost();
+  Mat C_ref = A * B + D * E;
+
+  RealScalar tol = gemm_error_bound<Scalar>(n, A.norm(), B.norm()) + gemm_error_bound<Scalar>(n, D.norm(), E.norm());
+  VERIFY((C - C_ref).norm() < tol);
+}
+
+// ---- GEMM cross-context resize of the destination ---------------------------
+
+template <typename Scalar>
+void test_gemm_cross_context_resize() {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = Mat::Random(64, 64);
+  Mat B = Mat::Random(64, 64);
+  Mat D = Mat::Random(32, 16);
+  Mat E = Mat::Random(16, 8);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+  auto d_D = DeviceMatrix<Scalar>::fromHost(D);
+  auto d_E = DeviceMatrix<Scalar>::fromHost(E);
+
+  GpuContext ctx1;
+  GpuContext ctx2;
+  DeviceMatrix<Scalar> d_C;
+  d_C.device(ctx1) = d_A * d_B;
+  d_C.device(ctx2) = d_D * d_E;
+
+  Mat C = d_C.toHost();
+  Mat C_ref = D * E;
+
+  RealScalar tol = gemm_error_bound<Scalar>(16, D.norm(), E.norm());
+  VERIFY((C - C_ref).norm() < tol);
+}
+
+// ---- GEMM chaining: C = (A * B) then D = C * E -----------------------------
+
+template <typename Scalar>
+void test_gemm_chain(Index n) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = Mat::Random(n, n);
+  Mat B = Mat::Random(n, n);
+  Mat E = Mat::Random(n, n);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+  auto d_E = DeviceMatrix<Scalar>::fromHost(E);
+
+  DeviceMatrix<Scalar> d_C;
+  d_C = d_A * d_B;
+  DeviceMatrix<Scalar> d_D;
+  d_D = d_C * d_E;
+
+  Mat D = d_D.toHost();
+  Mat D_ref = (A * B) * E;
+
+  Mat C_ref = A * B;
+  RealScalar tol =
+      gemm_error_bound<Scalar>(n, A.norm(), B.norm()) * E.norm() + gemm_error_bound<Scalar>(n, C_ref.norm(), E.norm());
+  VERIFY((D - D_ref).norm() < tol);
+}
+
+// ---- Square identity check: A * I = A ---------------------------------------
+
+template <typename Scalar>
+void test_gemm_identity(Index n) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+
+  Mat A = Mat::Random(n, n);
+  Mat eye = Mat::Identity(n, n);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_I = DeviceMatrix<Scalar>::fromHost(eye);
+
+  DeviceMatrix<Scalar> d_C;
+  d_C = d_A * d_I;
+
+  Mat C = d_C.toHost();
+  VERIFY_IS_APPROX(C, A);
+}
+
+// ---- LLT solve expression: d_X = d_A.llt().solve(d_B) ----------------------
+
+template <typename MatrixType>
+MatrixType make_spd(Index n) {
+  using Scalar = typename MatrixType::Scalar;
+  MatrixType M = MatrixType::Random(n, n);
+  return M.adjoint() * M + MatrixType::Identity(n, n) * static_cast<Scalar>(n);
+}
+
+template <typename Scalar>
+void test_llt_solve_expr(Index n, Index nrhs) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = make_spd<Mat>(n);
+  Mat B = Mat::Random(n, nrhs);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+
+  DeviceMatrix<Scalar> d_X;
+  d_X = d_A.llt().solve(d_B);
+
+  Mat X = d_X.toHost();
+  RealScalar residual = (A * X - B).norm() / B.norm();
+  VERIFY(residual < RealScalar(n) * gpu_unit_roundoff<Scalar>());
+}
+
+// ---- LLT solve with explicit context ----------------------------------------
+
+template <typename Scalar>
+void test_llt_solve_expr_context(Index n, Index nrhs) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = make_spd<Mat>(n);
+  Mat B = Mat::Random(n, nrhs);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+
+  GpuContext ctx;
+  DeviceMatrix<Scalar> d_X;
+  d_X.device(ctx) = d_A.llt().solve(d_B);
+
+  Mat X = d_X.toHost();
+  RealScalar residual = (A * X - B).norm() / B.norm();
+  VERIFY(residual < RealScalar(n) * gpu_unit_roundoff<Scalar>());
+}
+
+// ---- LU solve expression: d_X = d_A.lu().solve(d_B) ------------------------
+
+template <typename Scalar>
+void test_lu_solve_expr(Index n, Index nrhs) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = Mat::Random(n, n);
+  Mat B = Mat::Random(n, nrhs);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+
+  DeviceMatrix<Scalar> d_X;
+  d_X = d_A.lu().solve(d_B);
+
+  Mat X = d_X.toHost();
+  RealScalar residual = (A * X - B).norm() / (A.norm() * X.norm());
+  VERIFY(residual < RealScalar(10) * RealScalar(n) * gpu_unit_roundoff<Scalar>());
+}
+
+// ---- GEMM + solver chain: C = A * B, X = C.llt().solve(D) ------------------
+
+template <typename Scalar>
+void test_gemm_then_solve(Index n) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = Mat::Random(n, n);
+  Mat D = Mat::Random(n, 1);
+
+  // Make SPD: C = A^H * A + n*I
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  DeviceMatrix<Scalar> d_C;
+  d_C = d_A.adjoint() * d_A;
+
+  // Add n*I on host (no element-wise ops on DeviceMatrix yet).
+  Mat C = d_C.toHost();
+  C += Mat::Identity(n, n) * static_cast<Scalar>(n);
+  d_C = DeviceMatrix<Scalar>::fromHost(C);
+
+  auto d_D = DeviceMatrix<Scalar>::fromHost(D);
+
+  DeviceMatrix<Scalar> d_X;
+  d_X = d_C.llt().solve(d_D);
+
+  Mat X = d_X.toHost();
+  RealScalar residual = (C * X - D).norm() / D.norm();
+  VERIFY(residual < RealScalar(n) * gpu_unit_roundoff<Scalar>());
+}
+
+// ---- LLT solve with Upper triangle -----------------------------------------
+
+template <typename Scalar>
+void test_llt_solve_upper(Index n, Index nrhs) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = make_spd<Mat>(n);
+  Mat B = Mat::Random(n, nrhs);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+
+  DeviceMatrix<Scalar> d_X;
+  d_X = d_A.template llt<Upper>().solve(d_B);
+
+  Mat X = d_X.toHost();
+  RealScalar residual = (A * X - B).norm() / B.norm();
+  VERIFY(residual < RealScalar(n) * gpu_unit_roundoff<Scalar>());
+}
+
+// ---- LU solve with explicit context -----------------------------------------
+
+template <typename Scalar>
+void test_lu_solve_expr_context(Index n, Index nrhs) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = Mat::Random(n, n);
+  Mat B = Mat::Random(n, nrhs);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+
+  GpuContext ctx;
+  DeviceMatrix<Scalar> d_X;
+  d_X.device(ctx) = d_A.lu().solve(d_B);
+
+  Mat X = d_X.toHost();
+  RealScalar residual = (A * X - B).norm() / (A.norm() * X.norm());
+  VERIFY(residual < RealScalar(10) * RealScalar(n) * gpu_unit_roundoff<Scalar>());
+}
+
+// ---- Zero-nrhs solver expressions ------------------------------------------
+
+template <typename Scalar>
+void test_llt_solve_zero_nrhs(Index n) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+
+  Mat A = make_spd<Mat>(n);
+  Mat B = Mat::Random(n, 0);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+
+  DeviceMatrix<Scalar> d_X;
+  d_X = d_A.llt().solve(d_B);
+
+  VERIFY_IS_EQUAL(d_X.rows(), n);
+  VERIFY_IS_EQUAL(d_X.cols(), 0);
+}
+
+template <typename Scalar>
+void test_lu_solve_zero_nrhs(Index n) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+
+  Mat A = Mat::Random(n, n);
+  Mat B = Mat::Random(n, 0);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+
+  DeviceMatrix<Scalar> d_X;
+  d_X = d_A.lu().solve(d_B);
+
+  VERIFY_IS_EQUAL(d_X.rows(), n);
+  VERIFY_IS_EQUAL(d_X.cols(), 0);
+}
+
+// ---- TRSM: triangularView<UpLo>().solve(B) ----------------------------------
+
+template <typename Scalar, int UpLo>
+void test_trsm(Index n, Index nrhs) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  // Build a well-conditioned triangular matrix.
+  Mat A = Mat::Random(n, n);
+  A.diagonal().array() += static_cast<Scalar>(n);  // ensure non-singular
+  if (UpLo == Lower)
+    A = A.template triangularView<Lower>();
+  else
+    A = A.template triangularView<Upper>();
+
+  Mat B = Mat::Random(n, nrhs);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+
+  DeviceMatrix<Scalar> d_X;
+  d_X = d_A.template triangularView<UpLo>().solve(d_B);
+
+  Mat X = d_X.toHost();
+  RealScalar residual = (A * X - B).norm() / B.norm();
+  VERIFY(residual < RealScalar(n) * gpu_unit_roundoff<Scalar>());
+}
+
+// ---- SYMM/HEMM: selfadjointView<UpLo>() * B --------------------------------
+
+template <typename Scalar, int UpLo>
+void test_symm(Index n, Index nrhs) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = make_spd<Mat>(n);  // SPD is also self-adjoint
+  Mat B = Mat::Random(n, nrhs);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+
+  DeviceMatrix<Scalar> d_C;
+  d_C = d_A.template selfadjointView<UpLo>() * d_B;
+
+  Mat C = d_C.toHost();
+  Mat C_ref = A * B;  // A is symmetric, so full multiply == symm
+
+  RealScalar tol = gemm_error_bound<Scalar>(n, A.norm(), B.norm());
+  VERIFY((C - C_ref).norm() < tol);
+}
+
+// ---- SYRK/HERK: rankUpdate(A) → C = A * A^H --------------------------------
+
+template <typename Scalar>
+void test_syrk(Index n, Index k) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = Mat::Random(n, k);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+
+  DeviceMatrix<Scalar> d_C;
+  d_C.template selfadjointView<Lower>().rankUpdate(d_A);
+
+  Mat C = d_C.toHost();
+  // Only lower triangle is meaningful for SYRK. Compare lower triangle.
+  Mat C_ref = A * A.adjoint();
+
+  // Extract lower triangle for comparison.
+  Mat C_lower = C.template triangularView<Lower>();
+  Mat C_ref_lower = C_ref.template triangularView<Lower>();
+
+  RealScalar tol = gemm_error_bound<Scalar>(k, A.norm(), A.norm());
+  VERIFY((C_lower - C_ref_lower).norm() < tol);
+}
+
+// ---- Per-scalar driver ------------------------------------------------------
+
+template <typename Scalar>
+void test_scalar() {
+  CALL_SUBTEST(test_gemm_basic<Scalar>(64, 64, 64));
+  CALL_SUBTEST(test_gemm_basic<Scalar>(128, 64, 32));
+  CALL_SUBTEST(test_gemm_basic<Scalar>(1, 1, 1));
+  CALL_SUBTEST(test_gemm_basic<Scalar>(256, 256, 256));
+
+  CALL_SUBTEST(test_gemm_adjoint_lhs<Scalar>(64, 64, 64));
+  CALL_SUBTEST(test_gemm_adjoint_lhs<Scalar>(128, 32, 64));
+
+  CALL_SUBTEST(test_gemm_transpose_rhs<Scalar>(64, 64, 64));
+  CALL_SUBTEST(test_gemm_transpose_rhs<Scalar>(128, 32, 64));
+
+  CALL_SUBTEST(test_gemm_scaled<Scalar>(64, 64, 64));
+  CALL_SUBTEST(test_gemm_scaled_rhs<Scalar>(64, 64, 64));
+  CALL_SUBTEST(test_gemm_accumulate<Scalar>(64, 64, 64));
+  CALL_SUBTEST(test_gemm_accumulate_empty<Scalar>(64, 64, 64));
+  CALL_SUBTEST(test_gemm_subtract<Scalar>(64, 64, 64));
+  CALL_SUBTEST(test_gemm_subtract_empty<Scalar>(64, 64, 64));
+  CALL_SUBTEST(test_gemm_dimension_mismatch<Scalar>());
+  CALL_SUBTEST(test_gemm_explicit_context<Scalar>(64, 64, 64));
+  CALL_SUBTEST(test_gemm_cross_context_reuse<Scalar>(64));
+  CALL_SUBTEST(test_gemm_cross_context_resize<Scalar>());
+  CALL_SUBTEST(test_gemm_chain<Scalar>(64));
+  CALL_SUBTEST(test_gemm_identity<Scalar>(64));
+
+  // Solver expressions — zero-size edge cases (use dedicated tests, not residual-based)
+
+  // Solver expressions
+  CALL_SUBTEST(test_llt_solve_expr<Scalar>(64, 1));
+  CALL_SUBTEST(test_llt_solve_expr<Scalar>(64, 4));
+  CALL_SUBTEST(test_llt_solve_expr<Scalar>(256, 8));
+  CALL_SUBTEST(test_llt_solve_expr_context<Scalar>(64, 4));
+  CALL_SUBTEST(test_llt_solve_upper<Scalar>(64, 4));
+  CALL_SUBTEST(test_lu_solve_expr<Scalar>(64, 1));
+  CALL_SUBTEST(test_lu_solve_expr<Scalar>(64, 4));
+  CALL_SUBTEST(test_lu_solve_expr<Scalar>(256, 8));
+  CALL_SUBTEST(test_lu_solve_expr_context<Scalar>(64, 4));
+  CALL_SUBTEST(test_llt_solve_zero_nrhs<Scalar>(64));
+  CALL_SUBTEST(test_llt_solve_zero_nrhs<Scalar>(0));
+  CALL_SUBTEST(test_lu_solve_zero_nrhs<Scalar>(64));
+  CALL_SUBTEST(test_lu_solve_zero_nrhs<Scalar>(0));
+  CALL_SUBTEST(test_gemm_then_solve<Scalar>(64));
+
+  // TRSM
+  CALL_SUBTEST((test_trsm<Scalar, Lower>(64, 1)));
+  CALL_SUBTEST((test_trsm<Scalar, Lower>(64, 4)));
+  CALL_SUBTEST((test_trsm<Scalar, Upper>(64, 4)));
+  CALL_SUBTEST((test_trsm<Scalar, Lower>(256, 8)));
+
+  // SYMM/HEMM
+  CALL_SUBTEST((test_symm<Scalar, Lower>(64, 4)));
+  CALL_SUBTEST((test_symm<Scalar, Upper>(64, 4)));
+  CALL_SUBTEST((test_symm<Scalar, Lower>(128, 8)));
+
+  // SYRK/HERK
+  CALL_SUBTEST(test_syrk<Scalar>(64, 64));
+  CALL_SUBTEST(test_syrk<Scalar>(64, 32));
+  CALL_SUBTEST(test_syrk<Scalar>(128, 64));
+}
+
+// ---- Solver failure mode tests (not templated on Scalar) --------------------
+
+void test_llt_not_spd() {
+  // Negative definite matrix — LLT factorization must fail.
+  MatrixXd A = -MatrixXd::Identity(8, 8);
+  MatrixXd B = MatrixXd::Random(8, 1);
+  auto d_A = DeviceMatrix<double>::fromHost(A);
+  auto d_B = DeviceMatrix<double>::fromHost(B);
+  DeviceMatrix<double> d_X;
+  VERIFY_RAISES_ASSERT(d_X = d_A.llt().solve(d_B));
+}
+
+void test_lu_singular() {
+  // Zero matrix — LU factorization must detect singularity.
+  MatrixXd A = MatrixXd::Zero(8, 8);
+  MatrixXd B = MatrixXd::Random(8, 1);
+  auto d_A = DeviceMatrix<double>::fromHost(A);
+  auto d_B = DeviceMatrix<double>::fromHost(B);
+  DeviceMatrix<double> d_X;
+  VERIFY_RAISES_ASSERT(d_X = d_A.lu().solve(d_B));
+}
+
+EIGEN_DECLARE_TEST(gpu_cublas) {
+  CALL_SUBTEST(test_scalar<float>());
+  CALL_SUBTEST(test_scalar<double>());
+  CALL_SUBTEST(test_scalar<std::complex<float>>());
+  CALL_SUBTEST(test_scalar<std::complex<double>>());
+  CALL_SUBTEST(test_llt_not_spd());
+  CALL_SUBTEST(test_lu_singular());
+}

test/gpu_cusolver_eigen.cpp

@@ -0,0 +1,180 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2026 Rasmus Munk Larsen <rmlarsen@gmail.com>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+// Tests for GpuSelfAdjointEigenSolver: GPU symmetric/Hermitian eigenvalue
+// decomposition via cuSOLVER.
+
+#define EIGEN_USE_GPU
+#include "main.h"
+#include <Eigen/Eigenvalues>
+#include <Eigen/GPU>
+
+using namespace Eigen;
+
+// ---- Reconstruction: V * diag(W) * V^H ≈ A ---------------------------------
+
+template <typename Scalar>
+void test_eigen_reconstruction(Index n) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  // Build a symmetric/Hermitian matrix.
+  Mat R = Mat::Random(n, n);
+  Mat A = R + R.adjoint();
+
+  GpuSelfAdjointEigenSolver<Scalar> es(A);
+  VERIFY_IS_EQUAL(es.info(), Success);
+
+  auto W = es.eigenvalues();
+  Mat V = es.eigenvectors();
+
+  VERIFY_IS_EQUAL(W.size(), n);
+  VERIFY_IS_EQUAL(V.rows(), n);
+  VERIFY_IS_EQUAL(V.cols(), n);
+
+  // Reconstruct: A_hat = V * diag(W) * V^H.
+  Mat A_hat = V * W.asDiagonal() * V.adjoint();
+  RealScalar tol = RealScalar(5) * std::sqrt(static_cast<RealScalar>(n)) * NumTraits<Scalar>::epsilon() * A.norm();
+  VERIFY((A_hat - A).norm() < tol);
+
+  // Orthogonality: V^H * V ≈ I.
+  Mat VhV = V.adjoint() * V;
+  Mat eye = Mat::Identity(n, n);
+  VERIFY((VhV - eye).norm() < tol);
+}
+
+// ---- Eigenvalues match CPU SelfAdjointEigenSolver ---------------------------
+
+template <typename Scalar>
+void test_eigen_values(Index n) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat R = Mat::Random(n, n);
+  Mat A = R + R.adjoint();
+
+  GpuSelfAdjointEigenSolver<Scalar> gpu_es(A);
+  VERIFY_IS_EQUAL(gpu_es.info(), Success);
+  auto W_gpu = gpu_es.eigenvalues();
+
+  SelfAdjointEigenSolver<Mat> cpu_es(A);
+  auto W_cpu = cpu_es.eigenvalues();
+
+  RealScalar tol = RealScalar(5) * std::sqrt(static_cast<RealScalar>(n)) * NumTraits<Scalar>::epsilon() *
+                   W_cpu.cwiseAbs().maxCoeff();
+  VERIFY((W_gpu - W_cpu).norm() < tol);
+}
+
+// ---- Eigenvalues-only mode --------------------------------------------------
+
+template <typename Scalar>
+void test_eigen_values_only(Index n) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat R = Mat::Random(n, n);
+  Mat A = R + R.adjoint();
+
+  GpuSelfAdjointEigenSolver<Scalar> gpu_es(A, GpuSelfAdjointEigenSolver<Scalar>::EigenvaluesOnly);
+  VERIFY_IS_EQUAL(gpu_es.info(), Success);
+  auto W_gpu = gpu_es.eigenvalues();
+
+  SelfAdjointEigenSolver<Mat> cpu_es(A, EigenvaluesOnly);
+  auto W_cpu = cpu_es.eigenvalues();
+
+  RealScalar tol = RealScalar(5) * std::sqrt(static_cast<RealScalar>(n)) * NumTraits<Scalar>::epsilon() *
+                   W_cpu.cwiseAbs().maxCoeff();
+  VERIFY((W_gpu - W_cpu).norm() < tol);
+}
+
+// ---- DeviceMatrix input path ------------------------------------------------
+
+template <typename Scalar>
+void test_eigen_device_matrix(Index n) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat R = Mat::Random(n, n);
+  Mat A = R + R.adjoint();
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  GpuSelfAdjointEigenSolver<Scalar> es;
+  es.compute(d_A);
+  VERIFY_IS_EQUAL(es.info(), Success);
+
+  auto W_gpu = es.eigenvalues();
+  Mat V = es.eigenvectors();
+
+  // Verify reconstruction.
+  Mat A_hat = V * W_gpu.asDiagonal() * V.adjoint();
+  RealScalar tol = RealScalar(5) * std::sqrt(static_cast<RealScalar>(n)) * NumTraits<Scalar>::epsilon() * A.norm();
+  VERIFY((A_hat - A).norm() < tol);
+}
+
+// ---- Recompute (reuse solver object) ----------------------------------------
+
+template <typename Scalar>
+void test_eigen_recompute(Index n) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  GpuSelfAdjointEigenSolver<Scalar> es;
+
+  for (int trial = 0; trial < 3; ++trial) {
+    Mat R = Mat::Random(n, n);
+    Mat A = R + R.adjoint();
+    es.compute(A);
+    VERIFY_IS_EQUAL(es.info(), Success);
+
+    auto W = es.eigenvalues();
+    Mat V = es.eigenvectors();
+    Mat A_hat = V * W.asDiagonal() * V.adjoint();
+    RealScalar tol = RealScalar(5) * std::sqrt(static_cast<RealScalar>(n)) * NumTraits<Scalar>::epsilon() * A.norm();
+    VERIFY((A_hat - A).norm() < tol);
+  }
+}
+
+// ---- Empty matrix -----------------------------------------------------------
+
+void test_eigen_empty() {
+  GpuSelfAdjointEigenSolver<double> es(MatrixXd(0, 0));
+  VERIFY_IS_EQUAL(es.info(), Success);
+  VERIFY_IS_EQUAL(es.rows(), 0);
+  VERIFY_IS_EQUAL(es.cols(), 0);
+}
+
+// ---- Per-scalar driver ------------------------------------------------------
+
+template <typename Scalar>
+void test_scalar() {
+  // Reconstruction + orthogonality.
+  CALL_SUBTEST(test_eigen_reconstruction<Scalar>(64));
+  CALL_SUBTEST(test_eigen_reconstruction<Scalar>(128));
+
+  // Eigenvalues match CPU.
+  CALL_SUBTEST(test_eigen_values<Scalar>(64));
+  CALL_SUBTEST(test_eigen_values<Scalar>(128));
+
+  // Values-only mode.
+  CALL_SUBTEST(test_eigen_values_only<Scalar>(64));
+
+  // DeviceMatrix input.
+  CALL_SUBTEST(test_eigen_device_matrix<Scalar>(64));
+
+  // Recompute.
+  CALL_SUBTEST(test_eigen_recompute<Scalar>(32));
+}
+
+EIGEN_DECLARE_TEST(gpu_cusolver_eigen) {
+  CALL_SUBTEST(test_scalar<float>());
+  CALL_SUBTEST(test_scalar<double>());
+  CALL_SUBTEST(test_scalar<std::complex<float>>());
+  CALL_SUBTEST(test_scalar<std::complex<double>>());
+  CALL_SUBTEST(test_eigen_empty());
+}

test/gpu_cusolver_llt.cpp

@@ -0,0 +1,210 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2026 Eigen Authors
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+// Tests for GpuLLT: GPU Cholesky (LL^T) using cuSOLVER.
+// Covers cusolverDnXpotrf (factorization) and cusolverDnXpotrs (solve)
+// for float, double, complex<float>, complex<double>, Lower and Upper.
+
+#define EIGEN_USE_GPU
+#include "main.h"
+#include <Eigen/Cholesky>
+#include <Eigen/GPU>
+
+using namespace Eigen;
+
+// Build a random symmetric positive-definite matrix: A = M^H*M + n*I.
+template <typename MatrixType>
+MatrixType make_spd(Index n) {
+  using Scalar = typename MatrixType::Scalar;
+  MatrixType M = MatrixType::Random(n, n);
+  return M.adjoint() * M + MatrixType::Identity(n, n) * static_cast<Scalar>(n);
+}
+
+// Test factorization: L*L^H must reconstruct A to within floating-point tolerance.
+template <typename Scalar, int UpLo>
+void test_potrf(Index n) {
+  using MatrixType = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  MatrixType A = make_spd<MatrixType>(n);
+
+  GpuLLT<Scalar, UpLo> llt(A);
+  VERIFY_IS_EQUAL(llt.info(), Success);
+
+  // Reconstruct L*L^H and compare to original A.
+  // GpuLLT stores the factor on device; use CPU LLT to get the triangular factor
+  // for reconstruction since GpuLLT does not expose the device-resident factor directly.
+  LLT<MatrixType, UpLo> ref(A);
+  VERIFY_IS_EQUAL(ref.info(), Success);
+  MatrixType A_reconstructed = ref.reconstructedMatrix();
+
+  // Both should equal A to within n*eps*||A||.
+  RealScalar tol = RealScalar(4) * RealScalar(n) * NumTraits<Scalar>::epsilon() * A.norm();
+  VERIFY((A_reconstructed - A).norm() < tol);
+
+  // Smoke-test: llt.solve(b) should return the same result as ref.solve(b).
+  MatrixType b = MatrixType::Random(n, 1);
+  MatrixType x_gpu = llt.solve(b);
+  MatrixType x_cpu = ref.solve(b);
+  VERIFY((x_gpu - x_cpu).norm() < tol);
+}
+
+// Test solve: residual ||A*X - B|| / ||B|| must be small.
+template <typename Scalar, int UpLo>
+void test_potrs(Index n, Index nrhs) {
+  using MatrixType = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  MatrixType A = make_spd<MatrixType>(n);
+  MatrixType B = MatrixType::Random(n, nrhs);
+
+  GpuLLT<Scalar, UpLo> llt(A);
+  VERIFY_IS_EQUAL(llt.info(), Success);
+
+  MatrixType X = llt.solve(B);
+
+  RealScalar residual = (A * X - B).norm() / B.norm();
+  RealScalar tol = RealScalar(n) * NumTraits<Scalar>::epsilon();
+  VERIFY(residual < tol);
+}
+
+// Test that multiple solves against the same factor all produce correct results.
+// This exercises the key design property: L stays on device across calls.
+template <typename Scalar>
+void test_multiple_solves(Index n) {
+  using MatrixType = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  MatrixType A = make_spd<MatrixType>(n);
+  GpuLLT<Scalar, Lower> llt(A);
+  VERIFY_IS_EQUAL(llt.info(), Success);
+
+  RealScalar tol = RealScalar(n) * NumTraits<Scalar>::epsilon();
+  for (int k = 0; k < 5; ++k) {
+    MatrixType B = MatrixType::Random(n, 3);
+    MatrixType X = llt.solve(B);
+    RealScalar residual = (A * X - B).norm() / B.norm();
+    VERIFY(residual < tol);
+  }
+}
+
+// Test that GpuLLT correctly detects a non-SPD matrix.
+void test_not_spd() {
+  MatrixXd A = -MatrixXd::Identity(8, 8);  // negative definite
+  GpuLLT<double> llt(A);
+  VERIFY_IS_EQUAL(llt.info(), NumericalIssue);
+}
+
+// ---- DeviceMatrix integration tests -----------------------------------------
+
+// compute(DeviceMatrix) + solve(DeviceMatrix) → toHost
+template <typename Scalar, int UpLo>
+void test_device_matrix_solve(Index n, Index nrhs) {
+  using MatrixType = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  MatrixType A = make_spd<MatrixType>(n);
+  MatrixType B = MatrixType::Random(n, nrhs);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+
+  GpuLLT<Scalar, UpLo> llt;
+  llt.compute(d_A);
+  VERIFY_IS_EQUAL(llt.info(), Success);
+
+  DeviceMatrix<Scalar> d_X = llt.solve(d_B);
+  MatrixType X = d_X.toHost();
+
+  RealScalar residual = (A * X - B).norm() / B.norm();
+  VERIFY(residual < RealScalar(n) * NumTraits<Scalar>::epsilon());
+}
+
+// compute(DeviceMatrix&&) — move path
+template <typename Scalar>
+void test_device_matrix_move_compute(Index n) {
+  using MatrixType = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  MatrixType A = make_spd<MatrixType>(n);
+  MatrixType B = MatrixType::Random(n, 1);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  GpuLLT<Scalar, Lower> llt;
+  llt.compute(std::move(d_A));
+  VERIFY_IS_EQUAL(llt.info(), Success);
+
+  // d_A should be empty after move.
+  VERIFY(d_A.empty());
+
+  MatrixType X = llt.solve(B);
+  RealScalar residual = (A * X - B).norm() / B.norm();
+  VERIFY(residual < RealScalar(n) * NumTraits<Scalar>::epsilon());
+}
+
+// Full async chain: compute → solve → solve again with result as RHS → toHost
+template <typename Scalar>
+void test_chaining(Index n) {
+  using MatrixType = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  MatrixType A = make_spd<MatrixType>(n);
+  MatrixType B = MatrixType::Random(n, 3);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+
+  GpuLLT<Scalar, Lower> llt;
+  llt.compute(d_A);
+  VERIFY_IS_EQUAL(llt.info(), Success);
+
+  // Chain: solve → use result as RHS for another solve
+  DeviceMatrix<Scalar> d_X = llt.solve(d_B);
+  DeviceMatrix<Scalar> d_Y = llt.solve(d_X);
+
+  // Only sync at the very end.
+  MatrixType Y = d_Y.toHost();
+
+  // Verify: Y = A^{-2} * B
+  MatrixType X_ref = LLT<MatrixType, Lower>(A).solve(B);
+  MatrixType Y_ref = LLT<MatrixType, Lower>(A).solve(X_ref);
+
+  RealScalar tol = RealScalar(4) * RealScalar(n) * NumTraits<Scalar>::epsilon() * Y_ref.norm();
+  VERIFY((Y - Y_ref).norm() < tol);
+}
+
+template <typename Scalar>
+void test_scalar() {
+  CALL_SUBTEST((test_potrf<Scalar, Lower>(1)));
+  CALL_SUBTEST((test_potrf<Scalar, Lower>(64)));
+  CALL_SUBTEST((test_potrf<Scalar, Lower>(256)));
+  CALL_SUBTEST((test_potrf<Scalar, Upper>(64)));
+  CALL_SUBTEST((test_potrf<Scalar, Upper>(256)));
+
+  CALL_SUBTEST((test_potrs<Scalar, Lower>(64, 1)));
+  CALL_SUBTEST((test_potrs<Scalar, Lower>(64, 4)));
+  CALL_SUBTEST((test_potrs<Scalar, Lower>(256, 8)));
+  CALL_SUBTEST((test_potrs<Scalar, Upper>(64, 1)));
+  CALL_SUBTEST((test_potrs<Scalar, Upper>(256, 4)));
+
+  CALL_SUBTEST(test_multiple_solves<Scalar>(128));
+
+  CALL_SUBTEST((test_device_matrix_solve<Scalar, Lower>(64, 4)));
+  CALL_SUBTEST((test_device_matrix_solve<Scalar, Upper>(128, 1)));
+  CALL_SUBTEST(test_device_matrix_move_compute<Scalar>(64));
+  CALL_SUBTEST(test_chaining<Scalar>(64));
+}
+
+EIGEN_DECLARE_TEST(gpu_cusolver_llt) {
+  CALL_SUBTEST(test_scalar<float>());
+  CALL_SUBTEST(test_scalar<double>());
+  CALL_SUBTEST(test_scalar<std::complex<float>>());
+  CALL_SUBTEST(test_scalar<std::complex<double>>());
+  CALL_SUBTEST(test_not_spd());
+}

test/gpu_cusolver_lu.cpp

@@ -0,0 +1,206 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2026 Eigen Authors
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+// Tests for GpuLU: GPU partial-pivoting LU decomposition via cuSOLVER.
+// Covers cusolverDnXgetrf (factorization) and cusolverDnXgetrs (solve)
+// for float, double, complex<float>, complex<double>.
+//
+#define EIGEN_USE_GPU
+#include "main.h"
+#include <Eigen/LU>
+#include <Eigen/GPU>
+
+using namespace Eigen;
+
+// ---- Test factorization + NoTrans solve: residual ||A*X - B|| / ||B|| -------
+
+template <typename Scalar>
+void test_getrf(Index n) {
+  using MatrixType = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  MatrixType A = MatrixType::Random(n, n);
+  MatrixType B = MatrixType::Random(n, 4);
+
+  GpuLU<Scalar> lu(A);
+  VERIFY_IS_EQUAL(lu.info(), Success);
+
+  MatrixType X = lu.solve(B);
+  // Backward error bound for LU: ||A*X - B|| <= O(n*u) * ||A|| * ||X||.
+  // Normalize by ||A||*||X|| rather than ||B|| to be condition-number agnostic.
+  RealScalar residual = (A * X - B).norm() / (A.norm() * X.norm());
+  VERIFY(residual < RealScalar(10) * RealScalar(n) * NumTraits<Scalar>::epsilon());
+}
+
+// ---- Test solve: A^T*X = B and A^H*X = B ------------------------------------
+
+template <typename Scalar>
+void test_getrs_trans(Index n) {
+  using MatrixType = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  MatrixType A = MatrixType::Random(n, n);
+  MatrixType B = MatrixType::Random(n, 3);
+  RealScalar tol = RealScalar(10) * RealScalar(n) * NumTraits<Scalar>::epsilon();
+
+  GpuLU<Scalar> lu(A);
+  VERIFY_IS_EQUAL(lu.info(), Success);
+
+  MatrixType Xt = lu.solve(B, GpuLU<Scalar>::Transpose);
+  VERIFY((A.transpose() * Xt - B).norm() / (A.norm() * Xt.norm()) < tol);
+
+  MatrixType Xc = lu.solve(B, GpuLU<Scalar>::ConjugateTranspose);
+  VERIFY((A.adjoint() * Xc - B).norm() / (A.norm() * Xc.norm()) < tol);
+}
+
+// ---- Test multiple solves reuse the device-resident LU ----------------------
+
+template <typename Scalar>
+void test_multiple_solves(Index n) {
+  using MatrixType = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  MatrixType A = MatrixType::Random(n, n);
+  GpuLU<Scalar> lu(A);
+  VERIFY_IS_EQUAL(lu.info(), Success);
+
+  RealScalar tol = RealScalar(10) * RealScalar(n) * NumTraits<Scalar>::epsilon();
+  for (int k = 0; k < 5; ++k) {
+    MatrixType B = MatrixType::Random(n, 3);
+    MatrixType X = lu.solve(B);
+    VERIFY((A * X - B).norm() / (A.norm() * X.norm()) < tol);
+  }
+}
+
+// ---- Agreement with CPU PartialPivLU ----------------------------------------
+
+template <typename Scalar>
+void test_vs_cpu(Index n) {
+  using MatrixType = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  MatrixType A = MatrixType::Random(n, n);
+  MatrixType B = MatrixType::Random(n, 5);
+
+  GpuLU<Scalar> gpu_lu(A);
+  VERIFY_IS_EQUAL(gpu_lu.info(), Success);
+
+  MatrixType X_gpu = gpu_lu.solve(B);
+  MatrixType X_cpu = PartialPivLU<MatrixType>(A).solve(B);
+
+  RealScalar tol = RealScalar(100) * RealScalar(n) * NumTraits<Scalar>::epsilon();
+  VERIFY((X_gpu - X_cpu).norm() / X_cpu.norm() < tol);
+}
+
+// ---- Singular matrix detection ----------------------------------------------
+
+void test_singular() {
+  MatrixXd A = MatrixXd::Zero(8, 8);
+  GpuLU<double> lu(A);
+  VERIFY_IS_EQUAL(lu.info(), NumericalIssue);
+}
+
+// ---- DeviceMatrix integration tests -----------------------------------------
+
+template <typename Scalar>
+void test_device_matrix_solve(Index n) {
+  using MatrixType = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  MatrixType A = MatrixType::Random(n, n);
+  MatrixType B = MatrixType::Random(n, 4);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+
+  GpuLU<Scalar> lu;
+  lu.compute(d_A);
+  VERIFY_IS_EQUAL(lu.info(), Success);
+
+  DeviceMatrix<Scalar> d_X = lu.solve(d_B);
+  MatrixType X = d_X.toHost();
+
+  RealScalar residual = (A * X - B).norm() / (A.norm() * X.norm());
+  VERIFY(residual < RealScalar(10) * RealScalar(n) * NumTraits<Scalar>::epsilon());
+}
+
+template <typename Scalar>
+void test_device_matrix_move_compute(Index n) {
+  using MatrixType = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  MatrixType A = MatrixType::Random(n, n);
+  MatrixType B = MatrixType::Random(n, 1);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  GpuLU<Scalar> lu;
+  lu.compute(std::move(d_A));
+  VERIFY_IS_EQUAL(lu.info(), Success);
+  VERIFY(d_A.empty());
+
+  MatrixType X = lu.solve(B);
+  RealScalar residual = (A * X - B).norm() / (A.norm() * X.norm());
+  VERIFY(residual < RealScalar(10) * RealScalar(n) * NumTraits<Scalar>::epsilon());
+}
+
+template <typename Scalar>
+void test_chaining(Index n) {
+  using MatrixType = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  MatrixType A = MatrixType::Random(n, n);
+  MatrixType B = MatrixType::Random(n, 3);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+
+  GpuLU<Scalar> lu;
+  lu.compute(d_A);
+  VERIFY_IS_EQUAL(lu.info(), Success);
+
+  // Chain: solve → use result as RHS
+  DeviceMatrix<Scalar> d_X = lu.solve(d_B);
+  DeviceMatrix<Scalar> d_Y = lu.solve(d_X);
+  MatrixType Y = d_Y.toHost();
+
+  MatrixType X_ref = PartialPivLU<MatrixType>(A).solve(B);
+  MatrixType Y_ref = PartialPivLU<MatrixType>(A).solve(X_ref);
+
+  RealScalar tol = RealScalar(100) * RealScalar(n) * NumTraits<Scalar>::epsilon() * Y_ref.norm();
+  VERIFY((Y - Y_ref).norm() < tol);
+}
+
+// ---- Per-scalar driver -------------------------------------------------------
+
+template <typename Scalar>
+void test_scalar() {
+  CALL_SUBTEST(test_getrf<Scalar>(1));
+  CALL_SUBTEST(test_getrf<Scalar>(64));
+  CALL_SUBTEST(test_getrf<Scalar>(256));
+
+  CALL_SUBTEST(test_getrs_trans<Scalar>(64));
+  CALL_SUBTEST(test_getrs_trans<Scalar>(128));
+
+  CALL_SUBTEST(test_multiple_solves<Scalar>(128));
+
+  CALL_SUBTEST(test_vs_cpu<Scalar>(64));
+  CALL_SUBTEST(test_vs_cpu<Scalar>(256));
+
+  CALL_SUBTEST(test_device_matrix_solve<Scalar>(64));
+  CALL_SUBTEST(test_device_matrix_move_compute<Scalar>(64));
+  CALL_SUBTEST(test_chaining<Scalar>(64));
+}
+
+EIGEN_DECLARE_TEST(gpu_cusolver_lu) {
+  CALL_SUBTEST(test_scalar<float>());
+  CALL_SUBTEST(test_scalar<double>());
+  CALL_SUBTEST(test_scalar<std::complex<float>>());
+  CALL_SUBTEST(test_scalar<std::complex<double>>());
+  CALL_SUBTEST(test_singular());
+}

test/gpu_cusolver_qr.cpp

@@ -0,0 +1,185 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2026 Rasmus Munk Larsen <rmlarsen@gmail.com>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+// Tests for GpuQR: GPU QR decomposition via cuSOLVER.
+
+#define EIGEN_USE_GPU
+#include "main.h"
+#include <Eigen/QR>
+#include <Eigen/GPU>
+
+using namespace Eigen;
+
+// ---- Solve square system: A * X = B -----------------------------------------
+
+template <typename Scalar>
+void test_qr_solve_square(Index n, Index nrhs) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = Mat::Random(n, n);
+  Mat B = Mat::Random(n, nrhs);
+
+  GpuQR<Scalar> qr(A);
+  VERIFY_IS_EQUAL(qr.info(), Success);
+
+  Mat X = qr.solve(B);
+  RealScalar residual = (A * X - B).norm() / (A.norm() * X.norm());
+  VERIFY(residual < RealScalar(10) * RealScalar(n) * NumTraits<Scalar>::epsilon());
+}
+
+// ---- Solve overdetermined system: m > n (least-squares) ---------------------
+
+template <typename Scalar>
+void test_qr_solve_overdetermined(Index m, Index n, Index nrhs) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  eigen_assert(m >= n);
+  Mat A = Mat::Random(m, n);
+  Mat B = Mat::Random(m, nrhs);
+
+  GpuQR<Scalar> qr(A);
+  VERIFY_IS_EQUAL(qr.info(), Success);
+
+  Mat X = qr.solve(B);
+  VERIFY_IS_EQUAL(X.rows(), n);
+  VERIFY_IS_EQUAL(X.cols(), nrhs);
+
+  // Compare with CPU QR.
+  Mat X_cpu = HouseholderQR<Mat>(A).solve(B);
+  RealScalar tol = RealScalar(100) * RealScalar(m) * NumTraits<Scalar>::epsilon();
+  VERIFY((X - X_cpu).norm() / X_cpu.norm() < tol);
+}
+
+// ---- Solve with DeviceMatrix input ------------------------------------------
+
+template <typename Scalar>
+void test_qr_solve_device(Index n, Index nrhs) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = Mat::Random(n, n);
+  Mat B = Mat::Random(n, nrhs);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+
+  GpuQR<Scalar> qr;
+  qr.compute(d_A);
+  VERIFY_IS_EQUAL(qr.info(), Success);
+
+  DeviceMatrix<Scalar> d_X = qr.solve(d_B);
+  Mat X = d_X.toHost();
+
+  RealScalar residual = (A * X - B).norm() / (A.norm() * X.norm());
+  VERIFY(residual < RealScalar(10) * RealScalar(n) * NumTraits<Scalar>::epsilon());
+}
+
+// ---- Solve overdetermined via device path -----------------------------------
+
+template <typename Scalar>
+void test_qr_solve_overdetermined_device(Index m, Index n, Index nrhs) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  eigen_assert(m >= n);
+  Mat A = Mat::Random(m, n);
+  Mat B = Mat::Random(m, nrhs);
+
+  auto d_A = DeviceMatrix<Scalar>::fromHost(A);
+  auto d_B = DeviceMatrix<Scalar>::fromHost(B);
+
+  GpuQR<Scalar> qr;
+  qr.compute(d_A);
+  VERIFY_IS_EQUAL(qr.info(), Success);
+
+  DeviceMatrix<Scalar> d_X = qr.solve(d_B);
+  VERIFY_IS_EQUAL(d_X.rows(), n);
+  VERIFY_IS_EQUAL(d_X.cols(), nrhs);
+
+  Mat X = d_X.toHost();
+  Mat X_cpu = HouseholderQR<Mat>(A).solve(B);
+  RealScalar tol = RealScalar(100) * RealScalar(m) * NumTraits<Scalar>::epsilon();
+  VERIFY((X - X_cpu).norm() / X_cpu.norm() < tol);
+}
+
+// ---- Multiple solves reuse the factorization --------------------------------
+
+template <typename Scalar>
+void test_qr_multiple_solves(Index n) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = Mat::Random(n, n);
+  GpuQR<Scalar> qr(A);
+  VERIFY_IS_EQUAL(qr.info(), Success);
+
+  RealScalar tol = RealScalar(10) * RealScalar(n) * NumTraits<Scalar>::epsilon();
+  for (int k = 0; k < 5; ++k) {
+    Mat B = Mat::Random(n, 3);
+    Mat X = qr.solve(B);
+    RealScalar residual = (A * X - B).norm() / (A.norm() * X.norm());
+    VERIFY(residual < tol);
+  }
+}
+
+// ---- Agreement with CPU HouseholderQR ---------------------------------------
+
+template <typename Scalar>
+void test_qr_vs_cpu(Index n, Index nrhs) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = Mat::Random(n, n);
+  Mat B = Mat::Random(n, nrhs);
+
+  GpuQR<Scalar> gpu_qr(A);
+  VERIFY_IS_EQUAL(gpu_qr.info(), Success);
+
+  Mat X_gpu = gpu_qr.solve(B);
+  Mat X_cpu = HouseholderQR<Mat>(A).solve(B);
+
+  RealScalar tol = RealScalar(100) * RealScalar(n) * NumTraits<Scalar>::epsilon();
+  VERIFY((X_gpu - X_cpu).norm() / X_cpu.norm() < tol);
+}
+
+// ---- Per-scalar driver ------------------------------------------------------
+
+template <typename Scalar>
+void test_scalar() {
+  CALL_SUBTEST(test_qr_solve_square<Scalar>(1, 1));
+  CALL_SUBTEST(test_qr_solve_square<Scalar>(64, 1));
+  CALL_SUBTEST(test_qr_solve_square<Scalar>(64, 4));
+  CALL_SUBTEST(test_qr_solve_square<Scalar>(256, 8));
+
+  CALL_SUBTEST(test_qr_solve_overdetermined<Scalar>(128, 64, 4));
+  CALL_SUBTEST(test_qr_solve_overdetermined<Scalar>(256, 128, 1));
+
+  CALL_SUBTEST(test_qr_solve_device<Scalar>(64, 4));
+  CALL_SUBTEST(test_qr_solve_overdetermined_device<Scalar>(128, 64, 4));
+  CALL_SUBTEST(test_qr_multiple_solves<Scalar>(64));
+  CALL_SUBTEST(test_qr_vs_cpu<Scalar>(64, 4));
+  CALL_SUBTEST(test_qr_vs_cpu<Scalar>(256, 8));
+}
+
+void test_qr_empty() {
+  GpuQR<double> qr(MatrixXd(0, 0));
+  VERIFY_IS_EQUAL(qr.info(), Success);
+  VERIFY_IS_EQUAL(qr.rows(), 0);
+  VERIFY_IS_EQUAL(qr.cols(), 0);
+}
+
+EIGEN_DECLARE_TEST(gpu_cusolver_qr) {
+  CALL_SUBTEST(test_scalar<float>());
+  CALL_SUBTEST(test_scalar<double>());
+  CALL_SUBTEST(test_scalar<std::complex<float>>());
+  CALL_SUBTEST(test_scalar<std::complex<double>>());
+  CALL_SUBTEST(test_qr_empty());
+}

test/gpu_cusolver_svd.cpp

@@ -0,0 +1,194 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2026 Rasmus Munk Larsen <rmlarsen@gmail.com>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+// Tests for GpuSVD: GPU SVD via cuSOLVER.
+
+#define EIGEN_USE_GPU
+#include "main.h"
+#include <Eigen/SVD>
+#include <Eigen/GPU>
+
+using namespace Eigen;
+
+// ---- SVD reconstruction: U * diag(S) * VT ≈ A ------------------------------
+
+template <typename Scalar, unsigned int Options>
+void test_svd_reconstruction(Index m, Index n) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = Mat::Random(m, n);
+  GpuSVD<Scalar> svd(A, Options);
+  VERIFY_IS_EQUAL(svd.info(), Success);
+
+  auto S = svd.singularValues();
+  Mat U = svd.matrixU();
+  Mat VT = svd.matrixVT();
+
+  const Index k = (std::min)(m, n);
+
+  // Reconstruct: A_hat = U[:,:k] * diag(S) * VT[:k,:].
+  Mat A_hat = U.leftCols(k) * S.asDiagonal() * VT.topRows(k);
+  RealScalar tol = RealScalar(5) * std::sqrt(static_cast<RealScalar>(k)) * NumTraits<Scalar>::epsilon() * A.norm();
+  VERIFY((A_hat - A).norm() < tol);
+
+  // Orthogonality: U^H * U ≈ I.
+  Mat UtU = U.adjoint() * U;
+  Mat I_u = Mat::Identity(U.cols(), U.cols());
+  VERIFY((UtU - I_u).norm() < tol);
+
+  // Orthogonality: VT * VT^H ≈ I.
+  Mat VtVh = VT * VT.adjoint();
+  Mat I_v = Mat::Identity(VT.rows(), VT.rows());
+  VERIFY((VtVh - I_v).norm() < tol);
+}
+
+// ---- Singular values match CPU BDCSVD ---------------------------------------
+
+template <typename Scalar>
+void test_svd_singular_values(Index m, Index n) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = Mat::Random(m, n);
+  GpuSVD<Scalar> svd(A, 0);  // values only
+  VERIFY_IS_EQUAL(svd.info(), Success);
+
+  auto S_gpu = svd.singularValues();
+  auto S_cpu = BDCSVD<Mat>(A, 0).singularValues();
+
+  RealScalar tol =
+      RealScalar(5) * std::sqrt(static_cast<RealScalar>((std::min)(m, n))) * NumTraits<Scalar>::epsilon() * S_cpu(0);
+  VERIFY((S_gpu - S_cpu).norm() < tol);
+}
+
+// ---- Solve: pseudoinverse ---------------------------------------------------
+
+template <typename Scalar>
+void test_svd_solve(Index m, Index n, Index nrhs) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = Mat::Random(m, n);
+  Mat B = Mat::Random(m, nrhs);
+
+  GpuSVD<Scalar> svd(A, ComputeThinU | ComputeThinV);
+  VERIFY_IS_EQUAL(svd.info(), Success);
+
+  Mat X = svd.solve(B);
+  VERIFY_IS_EQUAL(X.rows(), n);
+  VERIFY_IS_EQUAL(X.cols(), nrhs);
+
+  // Compare with CPU BDCSVD solve.
+  Mat X_cpu = BDCSVD<Mat>(A, ComputeThinU | ComputeThinV).solve(B);
+  RealScalar tol = RealScalar(100) * RealScalar((std::max)(m, n)) * NumTraits<Scalar>::epsilon();
+  VERIFY((X - X_cpu).norm() / X_cpu.norm() < tol);
+}
+
+// ---- Solve: truncated -------------------------------------------------------
+
+template <typename Scalar>
+void test_svd_solve_truncated(Index m, Index n) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = Mat::Random(m, n);
+  Mat B = Mat::Random(m, 1);
+  const Index k = (std::min)(m, n);
+  const Index trunc = k / 2;
+  eigen_assert(trunc > 0);
+
+  GpuSVD<Scalar> svd(A, ComputeThinU | ComputeThinV);
+  Mat X_trunc = svd.solve(B, trunc);
+
+  // Build CPU reference: truncated pseudoinverse.
+  auto cpu_svd = BDCSVD<Mat>(A, ComputeThinU | ComputeThinV);
+  auto S = cpu_svd.singularValues();
+  Mat U = cpu_svd.matrixU();
+  Mat V = cpu_svd.matrixV();
+
+  // D_ii = 1/S_i for i < trunc, 0 otherwise.
+  Matrix<RealScalar, Dynamic, 1> D = Matrix<RealScalar, Dynamic, 1>::Zero(k);
+  for (Index i = 0; i < trunc; ++i) D(i) = RealScalar(1) / S(i);
+  Mat X_ref = V * D.asDiagonal() * U.adjoint() * B;
+
+  RealScalar tol = RealScalar(100) * RealScalar(k) * NumTraits<Scalar>::epsilon();
+  VERIFY((X_trunc - X_ref).norm() / X_ref.norm() < tol);
+}
+
+// ---- Solve: Tikhonov regularized --------------------------------------------
+
+template <typename Scalar>
+void test_svd_solve_regularized(Index m, Index n) {
+  using Mat = Matrix<Scalar, Dynamic, Dynamic>;
+  using RealScalar = typename NumTraits<Scalar>::Real;
+
+  Mat A = Mat::Random(m, n);
+  Mat B = Mat::Random(m, 1);
+  RealScalar lambda = RealScalar(0.1);
+  const Index k = (std::min)(m, n);
+
+  GpuSVD<Scalar> svd(A, ComputeThinU | ComputeThinV);
+  Mat X_reg = svd.solve(B, lambda);
+
+  // CPU reference: D_ii = S_i / (S_i^2 + lambda^2).
+  auto cpu_svd = BDCSVD<Mat>(A, ComputeThinU | ComputeThinV);
+  auto S = cpu_svd.singularValues();
+  Mat U = cpu_svd.matrixU();
+  Mat V = cpu_svd.matrixV();
+
+  Matrix<RealScalar, Dynamic, 1> D(k);
+  for (Index i = 0; i < k; ++i) D(i) = S(i) / (S(i) * S(i) + lambda * lambda);
+  Mat X_ref = V * D.asDiagonal() * U.adjoint() * B;
+
+  RealScalar tol = RealScalar(100) * RealScalar(k) * NumTraits<Scalar>::epsilon();
+  VERIFY((X_reg - X_ref).norm() / X_ref.norm() < tol);
+}
+
+// ---- Empty matrix -----------------------------------------------------------
+
+void test_svd_empty() {
+  GpuSVD<double> svd(MatrixXd(0, 0), 0);
+  VERIFY_IS_EQUAL(svd.info(), Success);
+  VERIFY_IS_EQUAL(svd.rows(), 0);
+  VERIFY_IS_EQUAL(svd.cols(), 0);
+}
+
+// ---- Per-scalar driver ------------------------------------------------------
+
+template <typename Scalar>
+void test_scalar() {
+  // Reconstruction + orthogonality (thin and full, identical test logic).
+  CALL_SUBTEST((test_svd_reconstruction<Scalar, ComputeThinU | ComputeThinV>(64, 64)));
+  CALL_SUBTEST((test_svd_reconstruction<Scalar, ComputeThinU | ComputeThinV>(128, 64)));
+  CALL_SUBTEST((test_svd_reconstruction<Scalar, ComputeThinU | ComputeThinV>(64, 128)));  // wide (m < n)
+  CALL_SUBTEST((test_svd_reconstruction<Scalar, ComputeFullU | ComputeFullV>(64, 64)));
+  CALL_SUBTEST((test_svd_reconstruction<Scalar, ComputeFullU | ComputeFullV>(128, 64)));
+
+  // Singular values.
+  CALL_SUBTEST(test_svd_singular_values<Scalar>(64, 64));
+  CALL_SUBTEST(test_svd_singular_values<Scalar>(128, 64));
+
+  // Solve.
+  CALL_SUBTEST(test_svd_solve<Scalar>(64, 64, 4));
+  CALL_SUBTEST(test_svd_solve<Scalar>(128, 64, 4));
+  CALL_SUBTEST(test_svd_solve<Scalar>(64, 128, 4));  // wide (m < n)
+
+  // Truncated and regularized solve.
+  CALL_SUBTEST(test_svd_solve_truncated<Scalar>(64, 64));
+  CALL_SUBTEST(test_svd_solve_regularized<Scalar>(64, 64));
+}
+
+EIGEN_DECLARE_TEST(gpu_cusolver_svd) {
+  CALL_SUBTEST(test_scalar<float>());
+  CALL_SUBTEST(test_scalar<double>());
+  CALL_SUBTEST(test_scalar<std::complex<float>>());
+  CALL_SUBTEST(test_scalar<std::complex<double>>());
+  CALL_SUBTEST(test_svd_empty());
+}

test/gpu_device_matrix.cpp

@@ -0,0 +1,245 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2026 Rasmus Munk Larsen <rmlarsen@gmail.com>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+// Tests for DeviceMatrix and HostTransfer: typed RAII GPU memory wrapper.
+// No cuSOLVER dependency — only CUDA runtime.
+
+#define EIGEN_USE_GPU
+#include "main.h"
+#include <Eigen/GPU>
+
+using namespace Eigen;
+
+// ---- Default construction ---------------------------------------------------
+
+void test_default_construct() {
+  DeviceMatrix<double> dm;
+  VERIFY(dm.empty());
+  VERIFY_IS_EQUAL(dm.rows(), 0);
+  VERIFY_IS_EQUAL(dm.cols(), 0);
+  VERIFY(dm.data() == nullptr);
+  VERIFY_IS_EQUAL(dm.sizeInBytes(), size_t(0));
+}
+
+// ---- Allocate uninitialized -------------------------------------------------
+
+template <typename Scalar>
+void test_allocate(Index rows, Index cols) {
+  DeviceMatrix<Scalar> dm(rows, cols);
+  VERIFY(!dm.empty());
+  VERIFY_IS_EQUAL(dm.rows(), rows);
+  VERIFY_IS_EQUAL(dm.cols(), cols);
+  VERIFY(dm.data() != nullptr);
+  VERIFY_IS_EQUAL(dm.sizeInBytes(), size_t(rows) * size_t(cols) * sizeof(Scalar));
+}
+
+// ---- fromHost / toHost roundtrip (synchronous) ------------------------------
+
+template <typename Scalar>
+void test_roundtrip(Index rows, Index cols) {
+  using MatrixType = Matrix<Scalar, Dynamic, Dynamic>;
+  MatrixType host = MatrixType::Random(rows, cols);
+
+  auto dm = DeviceMatrix<Scalar>::fromHost(host);
+  VERIFY_IS_EQUAL(dm.rows(), rows);
+  VERIFY_IS_EQUAL(dm.cols(), cols);
+  VERIFY(!dm.empty());
+
+  MatrixType result = dm.toHost();
+  VERIFY_IS_EQUAL(result.rows(), rows);
+  VERIFY_IS_EQUAL(result.cols(), cols);
+  VERIFY_IS_APPROX(result, host);
+}
+
+// ---- fromHostAsync / toHostAsync roundtrip -----------------------------------
+
+template <typename Scalar>
+void test_roundtrip_async(Index rows, Index cols) {
+  using MatrixType = Matrix<Scalar, Dynamic, Dynamic>;
+  MatrixType host = MatrixType::Random(rows, cols);
+
+  cudaStream_t stream;
+  EIGEN_CUDA_RUNTIME_CHECK(cudaStreamCreate(&stream));
+
+  // Async upload from raw pointer.
+  auto dm = DeviceMatrix<Scalar>::fromHostAsync(host.data(), rows, cols, stream);
+  VERIFY_IS_EQUAL(dm.rows(), rows);
+  VERIFY_IS_EQUAL(dm.cols(), cols);
+
+  // Async download via HostTransfer future.
+  auto transfer = dm.toHostAsync(stream);
+
+  // get() blocks and returns the matrix.
+  MatrixType result = transfer.get();
+  VERIFY_IS_APPROX(result, host);
+
+  EIGEN_CUDA_RUNTIME_CHECK(cudaStreamDestroy(stream));
+}
+
+// ---- HostTransfer::ready() and idempotent get() -----------------------------
+
+void test_host_transfer_ready() {
+  using MatrixType = Matrix<double, Dynamic, Dynamic>;
+  MatrixType host = MatrixType::Random(100, 100);
+
+  auto dm = DeviceMatrix<double>::fromHost(host);
+  auto transfer = dm.toHostAsync();
+
+  // After get(), ready() must return true.
+  MatrixType result = transfer.get();
+  VERIFY(transfer.ready());
+  VERIFY_IS_APPROX(result, host);
+
+  // get() is idempotent.
+  MatrixType& result2 = transfer.get();
+  VERIFY_IS_APPROX(result2, host);
+}
+
+// ---- HostTransfer move ------------------------------------------------------
+
+void test_host_transfer_move() {
+  using MatrixType = Matrix<double, Dynamic, Dynamic>;
+  MatrixType host = MatrixType::Random(50, 50);
+
+  auto dm = DeviceMatrix<double>::fromHost(host);
+  auto transfer = dm.toHostAsync();
+
+  HostTransfer<double> moved(std::move(transfer));
+  MatrixType result = moved.get();
+  VERIFY_IS_APPROX(result, host);
+}
+
+// ---- clone() produces independent copy --------------------------------------
+
+template <typename Scalar>
+void test_clone(Index rows, Index cols) {
+  using MatrixType = Matrix<Scalar, Dynamic, Dynamic>;
+  MatrixType host = MatrixType::Random(rows, cols);
+
+  auto dm = DeviceMatrix<Scalar>::fromHost(host);
+  auto cloned = dm.clone();
+
+  // Overwrite original with different data.
+  MatrixType other = MatrixType::Random(rows, cols);
+  dm = DeviceMatrix<Scalar>::fromHost(other);
+
+  // Clone still holds the original data.
+  MatrixType clone_result = cloned.toHost();
+  VERIFY_IS_APPROX(clone_result, host);
+
+  // Original holds the new data.
+  MatrixType dm_result = dm.toHost();
+  VERIFY_IS_APPROX(dm_result, other);
+}
+
+// ---- Move construct ---------------------------------------------------------
+
+template <typename Scalar>
+void test_move_construct(Index rows, Index cols) {
+  using MatrixType = Matrix<Scalar, Dynamic, Dynamic>;
+  MatrixType host = MatrixType::Random(rows, cols);
+
+  auto dm = DeviceMatrix<Scalar>::fromHost(host);
+  DeviceMatrix<Scalar> moved(std::move(dm));
+
+  VERIFY(dm.empty());
+  VERIFY(dm.data() == nullptr);
+
+  VERIFY_IS_EQUAL(moved.rows(), rows);
+  VERIFY_IS_EQUAL(moved.cols(), cols);
+  MatrixType result = moved.toHost();
+  VERIFY_IS_APPROX(result, host);
+}
+
+// ---- Move assign ------------------------------------------------------------
+
+template <typename Scalar>
+void test_move_assign(Index rows, Index cols) {
+  using MatrixType = Matrix<Scalar, Dynamic, Dynamic>;
+  MatrixType host = MatrixType::Random(rows, cols);
+
+  auto dm = DeviceMatrix<Scalar>::fromHost(host);
+  DeviceMatrix<Scalar> dest;
+  dest = std::move(dm);
+
+  VERIFY(dm.empty());
+  VERIFY_IS_EQUAL(dest.rows(), rows);
+  MatrixType result = dest.toHost();
+  VERIFY_IS_APPROX(result, host);
+}
+
+// ---- resize() ---------------------------------------------------------------
+
+void test_resize() {
+  DeviceMatrix<double> dm(10, 20);
+  VERIFY_IS_EQUAL(dm.rows(), 10);
+  VERIFY_IS_EQUAL(dm.cols(), 20);
+
+  dm.resize(50, 30);
+  VERIFY_IS_EQUAL(dm.rows(), 50);
+  VERIFY_IS_EQUAL(dm.cols(), 30);
+  VERIFY(dm.data() != nullptr);
+
+  // Resize to same dimensions is a no-op.
+  double* ptr_before = dm.data();
+  dm.resize(50, 30);
+  VERIFY(dm.data() == ptr_before);
+}
+
+// ---- Empty / 0x0 matrix -----------------------------------------------------
+
+void test_empty() {
+  using MatrixType = Matrix<double, Dynamic, Dynamic>;
+  MatrixType empty_mat(0, 0);
+
+  auto dm = DeviceMatrix<double>::fromHost(empty_mat);
+  VERIFY(dm.empty());
+  VERIFY_IS_EQUAL(dm.rows(), 0);
+  VERIFY_IS_EQUAL(dm.cols(), 0);
+
+  MatrixType result = dm.toHost();
+  VERIFY_IS_EQUAL(result.rows(), 0);
+  VERIFY_IS_EQUAL(result.cols(), 0);
+}
+
+// ---- Per-scalar driver ------------------------------------------------------
+
+template <typename Scalar>
+void test_scalar() {
+  // Square.
+  CALL_SUBTEST(test_roundtrip<Scalar>(1, 1));
+  CALL_SUBTEST(test_roundtrip<Scalar>(64, 64));
+  CALL_SUBTEST(test_roundtrip<Scalar>(256, 256));
+
+  // Rectangular.
+  CALL_SUBTEST(test_roundtrip<Scalar>(100, 7));
+  CALL_SUBTEST(test_roundtrip<Scalar>(7, 100));
+
+  // Async roundtrip.
+  CALL_SUBTEST(test_roundtrip_async<Scalar>(64, 64));
+  CALL_SUBTEST(test_roundtrip_async<Scalar>(100, 7));
+
+  CALL_SUBTEST(test_clone<Scalar>(64, 64));
+  CALL_SUBTEST(test_move_construct<Scalar>(64, 64));
+  CALL_SUBTEST(test_move_assign<Scalar>(64, 64));
+}
+
+EIGEN_DECLARE_TEST(gpu_device_matrix) {
+  CALL_SUBTEST(test_default_construct());
+  CALL_SUBTEST(test_empty());
+  CALL_SUBTEST(test_resize());
+  CALL_SUBTEST(test_host_transfer_ready());
+  CALL_SUBTEST(test_host_transfer_move());
+  CALL_SUBTEST((test_allocate<float>(100, 50)));
+  CALL_SUBTEST((test_allocate<double>(100, 50)));
+  CALL_SUBTEST(test_scalar<float>());
+  CALL_SUBTEST(test_scalar<double>());
+  CALL_SUBTEST(test_scalar<std::complex<float>>());
+  CALL_SUBTEST(test_scalar<std::complex<double>>());
+}

test/gpu_library_example.cu

@@ -0,0 +1,110 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2026 Rasmus Munk Larsen <rmlarsen@gmail.com>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+// Smoke test for GPU library test infrastructure.
+// Verifies GpuContext, GpuBuffer, and host<->device matrix transfers
+// without requiring any NVIDIA library (cuBLAS, cuSOLVER, etc.).
+
+#define EIGEN_USE_GPU
+#include "main.h"
+#include "gpu_context.h"
+#include "gpu_library_test_helper.h"
+
+using namespace Eigen;
+using namespace Eigen::test;
+
+// Test that GpuContext initializes, reports valid device info, and owns a cuSOLVER handle.
+void test_gpu_context() {
+  GpuContext ctx;
+  VERIFY(ctx.device() >= 0);
+  VERIFY(ctx.deviceProperties().major >= 7);  // sm_70 minimum
+  VERIFY(ctx.stream != nullptr);
+  VERIFY(ctx.cusolver != nullptr);
+  std::cout << "  GPU: " << ctx.deviceProperties().name << " (sm_" << ctx.deviceProperties().major
+            << ctx.deviceProperties().minor << ")\n";
+}
+
+// Test dense matrix roundtrip: host -> device -> host.
+template <typename MatrixType>
+void test_dense_roundtrip() {
+  GpuContext ctx;
+  const Index rows = 64;
+  const Index cols = 32;
+
+  MatrixType A = MatrixType::Random(rows, cols);
+  auto buf = gpu_copy_to_device(ctx.stream, A);
+  VERIFY(buf.data != nullptr);
+  VERIFY(buf.size == rows * cols);
+
+  MatrixType B(rows, cols);
+  B.setZero();
+  gpu_copy_to_host(ctx.stream, buf, B);
+  ctx.synchronize();
+
+  VERIFY_IS_EQUAL(A, B);
+}
+
+// Test GpuBuffer RAII: move semantics, async zero-init.
+void test_gpu_buffer() {
+  GpuContext ctx;
+
+  GpuBuffer<float> a(128);
+  VERIFY(a.data != nullptr);
+  VERIFY(a.size == 128);
+
+  // Move construction.
+  GpuBuffer<float> b(std::move(a));
+  VERIFY(a.data == nullptr);
+  VERIFY(b.data != nullptr);
+  VERIFY(b.size == 128);
+
+  // Move assignment.
+  GpuBuffer<float> c;
+  c = std::move(b);
+  VERIFY(b.data == nullptr);
+  VERIFY(c.data != nullptr);
+
+  // setZeroAsync.
+  c.setZeroAsync(ctx.stream);
+  ctx.synchronize();
+
+  std::vector<float> host(128);
+  GPU_CHECK(cudaMemcpy(host.data(), c.data, 128 * sizeof(float), cudaMemcpyDeviceToHost));
+  for (int i = 0; i < 128; ++i) {
+    VERIFY_IS_EQUAL(host[i], 0.0f);
+  }
+}
+
+// Test with vectors (1D).
+template <typename Scalar>
+void test_vector_roundtrip() {
+  GpuContext ctx;
+  const Index n = 256;
+  Matrix<Scalar, Dynamic, 1> v = Matrix<Scalar, Dynamic, 1>::Random(n);
+  auto buf = gpu_copy_to_device(ctx.stream, v);
+
+  Matrix<Scalar, Dynamic, 1> w(n);
+  w.setZero();
+  gpu_copy_to_host(ctx.stream, buf, w);
+  ctx.synchronize();
+
+  VERIFY_IS_EQUAL(v, w);
+}
+
+EIGEN_DECLARE_TEST(gpu_library_example) {
+  CALL_SUBTEST(test_gpu_context());
+  CALL_SUBTEST(test_gpu_buffer());
+  CALL_SUBTEST(test_dense_roundtrip<MatrixXf>());
+  CALL_SUBTEST(test_dense_roundtrip<MatrixXd>());
+  CALL_SUBTEST((test_dense_roundtrip<Matrix<float, Dynamic, Dynamic, RowMajor>>()));
+  CALL_SUBTEST((test_dense_roundtrip<Matrix<double, Dynamic, Dynamic, RowMajor>>()));
+  CALL_SUBTEST(test_vector_roundtrip<float>());
+  CALL_SUBTEST(test_vector_roundtrip<double>());
+  CALL_SUBTEST(test_vector_roundtrip<std::complex<float>>());
+}

test/gpu_library_test_helper.h

@@ -0,0 +1,90 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2026 Rasmus Munk Larsen <rmlarsen@gmail.com>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+#ifndef EIGEN_TEST_GPU_LIBRARY_TEST_HELPER_H
+#define EIGEN_TEST_GPU_LIBRARY_TEST_HELPER_H
+
+// Helpers for GPU tests that call NVIDIA library APIs (cuBLAS, cuSOLVER, etc.)
+// from the host side. Provides RAII GPU memory management and async matrix transfer.
+//
+// This is separate from gpu_common.h (element-parallel device kernels) and
+// gpu_test_helper.h (serialization-based device kernels). Those patterns run
+// user functors inside GPU kernels. This helper is for host-orchestrated tests
+// that call library APIs which launch their own kernels internally.
+//
+// All transfers use an explicit stream and cudaMemcpyAsync. Callers must
+// synchronize (ctx.synchronize() or cudaStreamSynchronize) before reading
+// results back on the host.
+
+#include "gpu_test_helper.h"
+
+namespace Eigen {
+namespace test {
+
+// RAII wrapper for GPU device memory. Prevents leaks when VERIFY macros abort.
+template <typename Scalar>
+struct GpuBuffer {
+  Scalar* data = nullptr;
+  Index size = 0;
+
+  GpuBuffer() = default;
+
+  explicit GpuBuffer(Index n) : size(n) { GPU_CHECK(gpuMalloc(reinterpret_cast<void**>(&data), n * sizeof(Scalar))); }
+
+  ~GpuBuffer() {
+    if (data) GPU_CHECK(gpuFree(data));
+  }
+
+  // Move-only.
+  GpuBuffer(GpuBuffer&& other) noexcept : data(other.data), size(other.size) {
+    other.data = nullptr;
+    other.size = 0;
+  }
+  GpuBuffer& operator=(GpuBuffer&& other) noexcept {
+    if (this != &other) {
+      if (data) GPU_CHECK(gpuFree(data));
+      data = other.data;
+      size = other.size;
+      other.data = nullptr;
+      other.size = 0;
+    }
+    return *this;
+  }
+
+  GpuBuffer(const GpuBuffer&) = delete;
+  GpuBuffer& operator=(const GpuBuffer&) = delete;
+
+  // Async zero the buffer on the given stream.
+  void setZeroAsync(cudaStream_t stream) { GPU_CHECK(cudaMemsetAsync(data, 0, size * sizeof(Scalar), stream)); }
+};
+
+// Copy a dense Eigen matrix to a new GPU buffer, async on the given stream.
+// Caller must synchronize before the host matrix is freed or modified.
+template <typename Derived>
+GpuBuffer<typename Derived::Scalar> gpu_copy_to_device(cudaStream_t stream, const MatrixBase<Derived>& host_mat) {
+  using Scalar = typename Derived::Scalar;
+  const auto& mat = host_mat.derived();
+  GpuBuffer<Scalar> buf(mat.size());
+  GPU_CHECK(cudaMemcpyAsync(buf.data, mat.data(), mat.size() * sizeof(Scalar), cudaMemcpyHostToDevice, stream));
+  return buf;
+}
+
+// Copy GPU buffer contents back to a dense Eigen matrix, async on the given stream.
+// Caller must synchronize before reading from host_mat.
+template <typename Scalar, typename Derived>
+void gpu_copy_to_host(cudaStream_t stream, const GpuBuffer<Scalar>& buf, MatrixBase<Derived>& host_mat) {
+  auto& mat = host_mat.derived();
+  eigen_assert(buf.size == mat.size());
+  GPU_CHECK(cudaMemcpyAsync(mat.data(), buf.data, mat.size() * sizeof(Scalar), cudaMemcpyDeviceToHost, stream));
+}
+
+}  // namespace test
+}  // namespace Eigen
+
+#endif  // EIGEN_TEST_GPU_LIBRARY_TEST_HELPER_H