diff --git a/Eigen/src/SVD/BDCSVD.h b/Eigen/src/SVD/BDCSVD.h index ed219f0ac..a5e309cc3 100644 --- a/Eigen/src/SVD/BDCSVD.h +++ b/Eigen/src/SVD/BDCSVD.h @@ -19,21 +19,10 @@ #ifndef EIGEN_BDCSVD_H #define EIGEN_BDCSVD_H -// #define EIGEN_BDCSVD_DEBUG_VERBOSE -// #define EIGEN_BDCSVD_SANITY_CHECKS - -#ifdef EIGEN_BDCSVD_SANITY_CHECKS -#undef eigen_internal_assert -#define eigen_internal_assert(X) assert(X); -#endif // IWYU pragma: private #include "./InternalHeaderCheck.h" -#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE -#include -#endif - // Internal D&C implementation, templated only on RealScalar. #include "BDCSVDImpl.h" @@ -168,6 +157,20 @@ class BDCSVD : public SVDBase > { compute_impl(matrix, internal::get_computation_options(Options)); } + /** \brief Constructor performing the SVD of an upper bidiagonal matrix given its diagonal and superdiagonal. + * + * This skips the bidiagonalization step and directly runs the divide-and-conquer algorithm. + * The input vectors must be real-valued. For an n x n bidiagonal matrix, \a diagonal has n entries + * and \a superdiagonal has n-1 entries. + * + * \param diagonal the diagonal entries of the bidiagonal matrix + * \param superdiagonal the superdiagonal entries of the bidiagonal matrix + */ + template + BDCSVD(const MatrixBase& diagonal, const MatrixBase& superdiagonal) : m_numIters(0) { + compute_bidiagonal_impl(diagonal, superdiagonal, internal::get_computation_options(Options)); + } + /** \brief Constructor performing the decomposition of given matrix using specified options * for computing unitaries. * @@ -215,6 +218,20 @@ class BDCSVD : public SVDBase > { return compute_impl(matrix, computationOptions); } + /** \brief Compute the SVD of an upper bidiagonal matrix given its diagonal and superdiagonal. + * + * This skips the bidiagonalization step and directly runs the divide-and-conquer algorithm. + * The input vectors must be real-valued. For an n x n bidiagonal matrix, \a diagonal has n entries + * and \a superdiagonal has n-1 entries. + * + * \param diagonal the diagonal entries of the bidiagonal matrix + * \param superdiagonal the superdiagonal entries of the bidiagonal matrix + */ + template + BDCSVD& compute(const MatrixBase& diagonal, const MatrixBase& superdiagonal) { + return compute_bidiagonal_impl(diagonal, superdiagonal, m_computationOptions); + } + void setSwitchSize(int s) { eigen_assert(s >= 3 && "BDCSVD the size of the algo switch has to be at least 3."); m_impl.setAlgoSwap(s); @@ -223,6 +240,9 @@ class BDCSVD : public SVDBase > { private: template BDCSVD& compute_impl(const MatrixBase& matrix, unsigned int computationOptions); + template + BDCSVD& compute_bidiagonal_impl(const MatrixBase& diagonal, const MatrixBase& superdiagonal, + unsigned int computationOptions); template void copyUV(const HouseholderU& householderU, const HouseholderV& householderV, const NaiveU& naiveU, const NaiveV& naivev); @@ -292,10 +312,6 @@ EIGEN_DONT_INLINE BDCSVD& BDCSVD::comp EIGEN_STATIC_ASSERT((std::is_same::value), Input matrix must have the same Scalar type as the BDCSVD object.); -#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE - std::cout << "\n\n\n=================================================================================================" - "=====================\n\n\n"; -#endif using std::abs; allocate(matrix.rows(), matrix.cols(), computationOptions); @@ -419,6 +435,119 @@ EIGEN_DONT_INLINE void BDCSVD::copyUV(const HouseholderU& h } } +template +template +EIGEN_DONT_INLINE BDCSVD& BDCSVD::compute_bidiagonal_impl( + const MatrixBase& diagonal, const MatrixBase& superdiagonal, unsigned int computationOptions) { + EIGEN_STATIC_ASSERT(DerivedD::IsVectorAtCompileTime, THIS_METHOD_IS_ONLY_FOR_VECTORS); + EIGEN_STATIC_ASSERT(DerivedE::IsVectorAtCompileTime, THIS_METHOD_IS_ONLY_FOR_VECTORS); + EIGEN_STATIC_ASSERT((NumTraits::IsComplex == 0), + THIS_FUNCTION_IS_NOT_FOR_COMPLEX_VALUED_MATRICES); + EIGEN_STATIC_ASSERT((NumTraits::IsComplex == 0), + THIS_FUNCTION_IS_NOT_FOR_COMPLEX_VALUED_MATRICES); + + using std::abs; + const Index n = diagonal.size(); + eigen_assert((n == 0 || superdiagonal.size() == n - 1) && "superdiagonal must have size diagonal.size() - 1"); + + // For a bidiagonal matrix, rows == cols == n. + allocate(n, n, computationOptions); + + if (n == 0) { + m_isInitialized = true; + m_info = Success; + m_nonzeroSingularValues = 0; + return *this; + } + + // Check for non-finite inputs. + const RealScalar diagScale = diagonal.cwiseAbs().template maxCoeff(); + const RealScalar superdiagScale = n > 1 ? superdiagonal.cwiseAbs().template maxCoeff() : RealScalar(0); + RealScalar scale = numext::maxi(diagScale, superdiagScale); + if (!(numext::isfinite)(scale)) { + m_isInitialized = true; + m_info = InvalidInput; + return *this; + } + + const RealScalar considerZero = (std::numeric_limits::min)(); + if (numext::is_exactly_zero(scale)) scale = Literal(1); + + //**** Small problem: build dense bidiagonal and delegate to JacobiSVD. + if (n < m_impl.algoSwap()) { + // Build the dense upper bidiagonal matrix. + MatrixX B = MatrixX::Zero(n, n); + B.diagonal() = diagonal.template cast() / Scalar(scale); + if (n > 1) B.diagonal(1) = superdiagonal.template cast() / Scalar(scale); + smallSvd.compute(B); + m_isInitialized = true; + m_info = smallSvd.info(); + if (m_info == Success || m_info == NoConvergence) { + m_singularValues = smallSvd.singularValues() * scale; + m_nonzeroSingularValues = smallSvd.nonzeroSingularValues(); + if (computeU()) m_matrixU = smallSvd.matrixU(); + if (computeV()) m_matrixV = smallSvd.matrixV(); + } + return *this; + } + + //**** Fill m_computed with transposed bidiagonal format. + // D&C operates on B^T: m_computed(i,i) = d_i, m_computed(i+1,i) = e_i. + m_impl.naiveU().setZero(); + m_impl.naiveV().setZero(); + m_impl.computed().setZero(); + for (Index i = 0; i < n; ++i) { + m_impl.computed()(i, i) = RealScalar(diagonal.coeff(i)) / scale; + } + for (Index i = 0; i < n - 1; ++i) { + m_impl.computed()(i + 1, i) = RealScalar(superdiagonal.coeff(i)) / scale; + } + + m_isTranspose = false; + + //**** Run D&C. + m_impl.divide(0, n - 1, 0, 0, 0); + m_info = m_impl.info(); + m_numIters = m_impl.numIters(); + if (m_info != Success && m_info != NoConvergence) { + m_isInitialized = true; + return *this; + } + + //**** Extract singular values. + for (int i = 0; i < diagSize(); i++) { + RealScalar a = abs(m_impl.computed().coeff(i, i)); + m_singularValues.coeffRef(i) = a * scale; + if (a < considerZero) { + m_nonzeroSingularValues = i; + m_singularValues.tail(diagSize() - i - 1).setZero(); + break; + } else if (i == diagSize() - 1) { + m_nonzeroSingularValues = i + 1; + break; + } + } + + //**** Copy U and V directly (no Householder to apply). + // D&C computes B^T = naiveU * S * naiveV^T, so B = naiveV * S * naiveU^T. + // Thus U_of_B = naiveV, V_of_B = naiveU. + if (computeU()) { + Index Ucols = m_computeThinU ? diagSize() : rows(); + m_matrixU = MatrixX::Identity(rows(), Ucols); + m_matrixU.topLeftCorner(diagSize(), diagSize()) = + m_impl.naiveV().template cast().topLeftCorner(diagSize(), diagSize()); + } + if (computeV()) { + Index Vcols = m_computeThinV ? diagSize() : cols(); + m_matrixV = MatrixX::Identity(cols(), Vcols); + m_matrixV.topLeftCorner(diagSize(), diagSize()) = + m_impl.naiveU().template cast().topLeftCorner(diagSize(), diagSize()); + } + + m_isInitialized = true; + return *this; +} + /** \svd_module * * \return the singular value decomposition of \c *this computed by Divide & Conquer algorithm diff --git a/Eigen/src/SVD/BDCSVDImpl.h b/Eigen/src/SVD/BDCSVDImpl.h index 4d0e70f77..49267393c 100644 --- a/Eigen/src/SVD/BDCSVDImpl.h +++ b/Eigen/src/SVD/BDCSVDImpl.h @@ -25,18 +25,8 @@ namespace Eigen { -#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE -IOFormat bdcsvdfmt(8, 0, ", ", "\n", " [", "]"); -#endif - namespace internal { -#ifdef EIGEN_BDCSVD_SANITY_CHECKS -#define BDCSVD_SANITY_CHECK(expr) eigen_internal_assert(expr) -#else -#define BDCSVD_SANITY_CHECK(expr) -#endif - /** \internal * Implementation of the divide-and-conquer phase of BDCSVD. * @@ -256,10 +246,6 @@ void bdcsvd_impl::divide(Index firstCol, Index lastCol, Index first s0 = betaK * phi / r0; } - BDCSVD_SANITY_CHECK(m_naiveU.allFinite()); - BDCSVD_SANITY_CHECK(m_naiveV.allFinite()); - BDCSVD_SANITY_CHECK(m_computed.allFinite()); - if (m_compU) { MatrixXr q1(m_naiveU.col(firstCol + k).segment(firstCol, k + 1)); // we shiftW Q1 to the right @@ -290,39 +276,18 @@ void bdcsvd_impl::divide(Index firstCol, Index lastCol, Index first m_naiveU.row(0).segment(firstCol + k + 1, n - k - 1).setZero(); } - BDCSVD_SANITY_CHECK(m_naiveU.allFinite()); - BDCSVD_SANITY_CHECK(m_naiveV.allFinite()); - BDCSVD_SANITY_CHECK(m_computed.allFinite()); - m_computed(firstCol + shift, firstCol + shift) = r0; m_computed.col(firstCol + shift).segment(firstCol + shift + 1, k) = alphaK * l.transpose(); m_computed.col(firstCol + shift).segment(firstCol + shift + k + 1, n - k - 1) = betaK * f.transpose(); -#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE - ArrayXr tmp1 = (m_computed.block(firstCol + shift, firstCol + shift, n, n)).jacobiSvd().singularValues(); -#endif // Second part: try to deflate singular values in combined matrix deflation(firstCol, lastCol, k, firstRowW, firstColW, shift); -#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE - ArrayXr tmp2 = (m_computed.block(firstCol + shift, firstCol + shift, n, n)).jacobiSvd().singularValues(); - std::cout << "\n\nj1 = " << tmp1.transpose().format(bdcsvdfmt) << "\n"; - std::cout << "j2 = " << tmp2.transpose().format(bdcsvdfmt) << "\n\n"; - std::cout << "err: " << ((tmp1 - tmp2).abs() > 1e-12 * tmp2.abs()).transpose() << "\n"; - static int count = 0; - std::cout << "# " << ++count << "\n\n"; - eigen_internal_assert((tmp1 - tmp2).matrix().norm() < 1e-14 * tmp2.matrix().norm()); -// eigen_internal_assert(count<681); -// eigen_internal_assert(((tmp1-tmp2).abs()<1e-13*tmp2.abs()).all()); -#endif // Third part: compute SVD of combined matrix MatrixXr UofSVD, VofSVD; VectorType singVals; computeSVDofM(firstCol + shift, n, UofSVD, singVals, VofSVD); - BDCSVD_SANITY_CHECK(UofSVD.allFinite()); - BDCSVD_SANITY_CHECK(VofSVD.allFinite()); - if (m_compU) structured_update(m_naiveU.block(firstCol, firstCol, n + 1, n + 1), UofSVD, (n + 2) / 2); else { @@ -333,10 +298,6 @@ void bdcsvd_impl::divide(Index firstCol, Index lastCol, Index first if (m_compV) structured_update(m_naiveV.block(firstRowW, firstColW, n, n), VofSVD, (n + 1) / 2); - BDCSVD_SANITY_CHECK(m_naiveU.allFinite()); - BDCSVD_SANITY_CHECK(m_naiveV.allFinite()); - BDCSVD_SANITY_CHECK(m_computed.allFinite()); - m_computed.block(firstCol + shift, firstCol + shift, n, n).setZero(); m_computed.block(firstCol + shift, firstCol + shift, n, n).diagonal() = singVals; } // end divide @@ -364,10 +325,6 @@ void bdcsvd_impl::computeSVDofM(Index firstCol, Index n, MatrixXr& U.resize(n + 1, n + 1); if (m_compV) V.resize(n, n); -#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE - if (col0.hasNaN() || diag.hasNaN()) std::cout << "\n\nHAS NAN\n\n"; -#endif - // Many singular values might have been deflated, the zero ones have been moved to the end, // but others are interleaved and we must ignore them at this stage. // To this end, let's compute a permutation skipping them: @@ -385,63 +342,14 @@ void bdcsvd_impl::computeSVDofM(Index firstCol, Index n, MatrixXr& Map mus(m_workspace.data() + 2 * n, n); Map zhat(m_workspace.data() + 3 * n, n); -#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE - std::cout << "computeSVDofM using:\n"; - std::cout << " z: " << col0.transpose() << "\n"; - std::cout << " d: " << diag.transpose() << "\n"; -#endif - // Compute singVals, shifts, and mus computeSingVals(col0, diag, perm, singVals, shifts, mus); -#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE - std::cout << " j: " - << (m_computed.block(firstCol, firstCol, n, n)).jacobiSvd().singularValues().transpose().reverse() - << "\n\n"; - std::cout << " sing-val: " << singVals.transpose() << "\n"; - std::cout << " mu: " << mus.transpose() << "\n"; - std::cout << " shift: " << shifts.transpose() << "\n"; - - { - std::cout << "\n\n mus: " << mus.head(actual_n).transpose() << "\n\n"; - std::cout << " check1 (expect0) : " - << ((singVals.array() - (shifts + mus)) / singVals.array()).head(actual_n).transpose() << "\n\n"; - eigen_internal_assert((((singVals.array() - (shifts + mus)) / singVals.array()).head(actual_n) >= 0).all()); - std::cout << " check2 (>0) : " << ((singVals.array() - diag) / singVals.array()).head(actual_n).transpose() - << "\n\n"; - eigen_internal_assert((((singVals.array() - diag) / singVals.array()).head(actual_n) >= 0).all()); - } -#endif - - BDCSVD_SANITY_CHECK(singVals.allFinite()); - BDCSVD_SANITY_CHECK(mus.allFinite()); - BDCSVD_SANITY_CHECK(shifts.allFinite()); - // Compute zhat perturbCol0(col0, diag, perm, singVals, shifts, mus, zhat); -#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE - std::cout << " zhat: " << zhat.transpose() << "\n"; -#endif - - BDCSVD_SANITY_CHECK(zhat.allFinite()); computeSingVecs(zhat, diag, perm, singVals, shifts, mus, U, V); -#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE - std::cout << "U^T U: " << (U.transpose() * U - MatrixXr(MatrixXr::Identity(U.cols(), U.cols()))).norm() << "\n"; - std::cout << "V^T V: " << (V.transpose() * V - MatrixXr(MatrixXr::Identity(V.cols(), V.cols()))).norm() << "\n"; -#endif - - BDCSVD_SANITY_CHECK(m_naiveU.allFinite()); - BDCSVD_SANITY_CHECK(m_naiveV.allFinite()); - BDCSVD_SANITY_CHECK(m_computed.allFinite()); - BDCSVD_SANITY_CHECK(U.allFinite()); - BDCSVD_SANITY_CHECK(V.allFinite()); - // BDCSVD_SANITY_CHECK((U.transpose() * U - MatrixXr(MatrixXr::Identity(U.cols(),U.cols()))).norm() < - // 100*NumTraits::epsilon() * n); - // BDCSVD_SANITY_CHECK((V.transpose() * V - - // MatrixXr(MatrixXr::Identity(V.cols(),V.cols()))).norm() < 100*NumTraits::epsilon() * n); - // Because of deflation, the singular values might not be completely sorted. // Fortunately, reordering them is a O(n) problem for (Index i = 0; i < actual_n - 1; ++i) { @@ -453,28 +361,11 @@ void bdcsvd_impl::computeSVDofM(Index firstCol, Index n, MatrixXr& } } -#ifdef EIGEN_BDCSVD_SANITY_CHECKS - { - bool singular_values_sorted = - (((singVals.segment(1, actual_n - 1) - singVals.head(actual_n - 1))).array() >= 0).all(); - if (!singular_values_sorted) - std::cout << "Singular values are not sorted: " << singVals.segment(1, actual_n).transpose() << "\n"; - eigen_internal_assert(singular_values_sorted); - } -#endif - // Reverse order so that singular values in increased order // Because of deflation, the zeros singular-values are already at the end singVals.head(actual_n).reverseInPlace(); U.leftCols(actual_n).rowwise().reverseInPlace(); if (m_compV) V.leftCols(actual_n).rowwise().reverseInPlace(); - -#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE - JacobiSVD jsvd(m_computed.block(firstCol, firstCol, n, n)); - std::cout << " * j: " << jsvd.singularValues().transpose() << "\n\n"; - std::cout << " * sing-val: " << singVals.transpose() << "\n"; -// std::cout << " * err: " << ((jsvd.singularValues()-singVals)>1e-13*singVals.norm()).transpose() << "\n"; -#endif } template @@ -537,25 +428,6 @@ void bdcsvd_impl::computeSingVals(const ArrayRef& col0, const Array // first decide whether it's closer to the left end or the right end RealScalar mid = left + (right - left) / Literal(2); RealScalar fMid = secularEq(mid, col0, diag, perm, diag, Literal(0)); -#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE - std::cout << "right-left = " << right - left << "\n"; - // std::cout << "fMid = " << fMid << " " << secularEq(mid-left, col0, diag, perm, ArrayXr(diag-left), left) - // << " " << secularEq(mid-right, col0, diag, perm, ArrayXr(diag-right), right) << - // "\n"; - std::cout << " = " << secularEq(left + RealScalar(0.000001) * (right - left), col0, diag, perm, diag, 0) << " " - << secularEq(left + RealScalar(0.1) * (right - left), col0, diag, perm, diag, 0) << " " - << secularEq(left + RealScalar(0.2) * (right - left), col0, diag, perm, diag, 0) << " " - << secularEq(left + RealScalar(0.3) * (right - left), col0, diag, perm, diag, 0) << " " - << secularEq(left + RealScalar(0.4) * (right - left), col0, diag, perm, diag, 0) << " " - << secularEq(left + RealScalar(0.49) * (right - left), col0, diag, perm, diag, 0) << " " - << secularEq(left + RealScalar(0.5) * (right - left), col0, diag, perm, diag, 0) << " " - << secularEq(left + RealScalar(0.51) * (right - left), col0, diag, perm, diag, 0) << " " - << secularEq(left + RealScalar(0.6) * (right - left), col0, diag, perm, diag, 0) << " " - << secularEq(left + RealScalar(0.7) * (right - left), col0, diag, perm, diag, 0) << " " - << secularEq(left + RealScalar(0.8) * (right - left), col0, diag, perm, diag, 0) << " " - << secularEq(left + RealScalar(0.9) * (right - left), col0, diag, perm, diag, 0) << " " - << secularEq(left + RealScalar(0.999999) * (right - left), col0, diag, perm, diag, 0) << "\n"; -#endif RealScalar shift = (k == actual_n - 1 || fMid > Literal(0)) ? left : right; // measure everything relative to shift @@ -612,8 +484,6 @@ void bdcsvd_impl::computeSingVals(const ArrayRef& col0, const Array RealScalar muZero = -a / b; RealScalar fZero = secularEq(muZero, col0, diag, perm, diagShifted, shift); - BDCSVD_SANITY_CHECK((numext::isfinite)(fZero)); - muPrev = muCur; fPrev = fCur; muCur = muZero; @@ -627,9 +497,6 @@ void bdcsvd_impl::computeSingVals(const ArrayRef& col0, const Array // fall back on bisection method if rational interpolation did not work if (useBisection) { -#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE - std::cout << "useBisection for k = " << k << ", actual_n = " << actual_n << "\n"; -#endif RealScalar leftShifted, rightShifted; // we can test exact equality here, because shift comes from `... ? left : right` if (numext::equal_strict(shift, left)) { @@ -642,8 +509,6 @@ void bdcsvd_impl::computeSingVals(const ArrayRef& col0, const Array // check that we did it right: eigen_internal_assert( (numext::isfinite)((col0(k) / leftShifted) * (col0(k) / (diag(k) + shift + leftShifted)))); - // It is unclear why k==0 would need special handling here: - // if (k == 0) rightShifted = right - left; else rightShifted = (k == actual_n - 1) ? right : ((right - left) * RealScalar(0.51)); // theoretically we can take 0.5, but let's be safe @@ -658,33 +523,6 @@ void bdcsvd_impl::computeSingVals(const ArrayRef& col0, const Array RealScalar fLeft = secularEq(leftShifted, col0, diag, perm, diagShifted, shift); eigen_internal_assert(fLeft < Literal(0)); -#if defined EIGEN_BDCSVD_DEBUG_VERBOSE || defined EIGEN_BDCSVD_SANITY_CHECKS || defined EIGEN_INTERNAL_DEBUGGING - RealScalar fRight = secularEq(rightShifted, col0, diag, perm, diagShifted, shift); -#endif - -#ifdef EIGEN_BDCSVD_SANITY_CHECKS - if (!(numext::isfinite)(fLeft)) - std::cout << "f(" << leftShifted << ") =" << fLeft << " ; " << left << " " << shift << " " << right << "\n"; - eigen_internal_assert((numext::isfinite)(fLeft)); - - if (!(numext::isfinite)(fRight)) - std::cout << "f(" << rightShifted << ") =" << fRight << " ; " << left << " " << shift << " " << right << "\n"; - // eigen_internal_assert((numext::isfinite)(fRight)); -#endif - -#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE - if (!(fLeft * fRight < 0)) { - std::cout << "f(leftShifted) using leftShifted=" << leftShifted - << " ; diagShifted(1:10):" << diagShifted.head(10).transpose() << "\n ; " - << "left==shift=" << bool(left == shift) << " ; left-shift = " << (left - shift) << "\n"; - std::cout << "k=" << k << ", " << fLeft << " * " << fRight << " == " << fLeft * fRight << " ; " - << "[" << left << " .. " << right << "] -> [" << leftShifted << " " << rightShifted - << "], shift=" << shift << " , f(right)=" << secularEq(0, col0, diag, perm, diagShifted, shift) - << " == " << secularEq(right, col0, diag, perm, diag, 0) << " == " << fRight << "\n"; - } -#endif - eigen_internal_assert(fLeft * fRight < Literal(0)); - if (fLeft < Literal(0)) { while (rightShifted - leftShifted > Literal(2) * NumTraits::epsilon() * numext::maxi(abs(leftShifted), abs(rightShifted))) { @@ -714,20 +552,6 @@ void bdcsvd_impl::computeSingVals(const ArrayRef& col0, const Array singVals[k] = shift + muCur; shifts[k] = shift; mus[k] = muCur; - -#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE - if (k + 1 < n) - std::cout << "found " << singVals[k] << " == " << shift << " + " << muCur << " from " << diag(k) << " .. " - << diag(k + 1) << "\n"; -#endif - BDCSVD_SANITY_CHECK(k == 0 || singVals[k] >= singVals[k - 1]); - BDCSVD_SANITY_CHECK(singVals[k] >= diag(k)); - - // perturb singular value slightly if it equals diagonal entry to avoid division by zero later - // (deflation is supposed to avoid this from happening) - // - this does no seem to be necessary anymore - - // if (singVals[k] == left) singVals[k] *= 1 + NumTraits::epsilon(); - // if (singVals[k] == right) singVals[k] *= 1 - NumTraits::epsilon(); } } @@ -752,59 +576,15 @@ void bdcsvd_impl::perturbCol0(const ArrayRef& col0, const ArrayRef& // see equation (3.6) RealScalar dk = diag(k); RealScalar prod = (singVals(lastIdx) + dk) * (mus(lastIdx) + (shifts(lastIdx) - dk)); -#ifdef EIGEN_BDCSVD_SANITY_CHECKS - if (prod < 0) { - std::cout << "k = " << k << " ; z(k)=" << col0(k) << ", diag(k)=" << dk << "\n"; - std::cout << "prod = " - << "(" << singVals(lastIdx) << " + " << dk << ") * (" << mus(lastIdx) << " + (" << shifts(lastIdx) - << " - " << dk << "))" - << "\n"; - std::cout << " = " << singVals(lastIdx) + dk << " * " << mus(lastIdx) + (shifts(lastIdx) - dk) << "\n"; - } - eigen_internal_assert(prod >= 0); -#endif for (Index l = 0; l < m; ++l) { Index i = perm(l); if (i != k) { -#ifdef EIGEN_BDCSVD_SANITY_CHECKS - if (i >= k && (l == 0 || l - 1 >= m)) { - std::cout << "Error in perturbCol0\n"; - std::cout << " " << k << "/" << n << " " << l << "/" << m << " " << i << "/" << n << " ; " << col0(k) - << " " << diag(k) << " " - << "\n"; - std::cout << " " << diag(i) << "\n"; - Index j = (i < k /*|| l==0*/) ? i : perm(l - 1); - std::cout << " " - << "j=" << j << "\n"; - } -#endif Index j = i < k ? i : l > 0 ? perm(l - 1) : i; -#ifdef EIGEN_BDCSVD_SANITY_CHECKS - if (!(dk != Literal(0) || diag(i) != Literal(0))) { - std::cout << "k=" << k << ", i=" << i << ", l=" << l << ", perm.size()=" << perm.size() << "\n"; - } - eigen_internal_assert(dk != Literal(0) || diag(i) != Literal(0)); -#endif prod *= ((singVals(j) + dk) / ((diag(i) + dk))) * ((mus(j) + (shifts(j) - dk)) / ((diag(i) - dk))); - BDCSVD_SANITY_CHECK(prod >= 0); -#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE - if (i != k && - numext::abs(((singVals(j) + dk) * (mus(j) + (shifts(j) - dk))) / ((diag(i) + dk) * (diag(i) - dk)) - 1) > - 0.9) - std::cout << " " - << ((singVals(j) + dk) * (mus(j) + (shifts(j) - dk))) / ((diag(i) + dk) * (diag(i) - dk)) - << " == (" << (singVals(j) + dk) << " * " << (mus(j) + (shifts(j) - dk)) << ") / (" - << (diag(i) + dk) << " * " << (diag(i) - dk) << ")\n"; -#endif } } -#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE - std::cout << "zhat(" << k << ") = sqrt( " << prod << ") ; " << (singVals(lastIdx) + dk) << " * " - << mus(lastIdx) + shifts(lastIdx) << " - " << dk << "\n"; -#endif RealScalar tmp = sqrt(prod); - BDCSVD_SANITY_CHECK((numext::isfinite)(tmp)); zhat(k) = col0(k) > Literal(0) ? RealScalar(tmp) : RealScalar(-tmp); } } @@ -883,14 +663,6 @@ void bdcsvd_impl::deflation44(Index firstColu, Index firstColm, Ind RealScalar s = m_computed(firstColm + i, firstColm); RealScalar c = m_computed(firstColm + j, firstColm); RealScalar r = numext::hypot(c, s); -#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE - std::cout << "deflation 4.4: " << i << "," << j << " -> " << c << " " << s << " " << r << " ; " - << m_computed(firstColm + i - 1, firstColm) << " " << m_computed(firstColm + i, firstColm) << " " - << m_computed(firstColm + i + 1, firstColm) << " " << m_computed(firstColm + i + 2, firstColm) << "\n"; - std::cout << m_computed(firstColm + i - 1, firstColm + i - 1) << " " << m_computed(firstColm + i, firstColm + i) - << " " << m_computed(firstColm + i + 1, firstColm + i + 1) << " " - << m_computed(firstColm + i + 2, firstColm + i + 2) << "\n"; -#endif if (numext::is_exactly_zero(r)) { m_computed(firstColm + j, firstColm + j) = m_computed(firstColm + i, firstColm + i); return; @@ -927,51 +699,23 @@ void bdcsvd_impl::deflation(Index firstCol, Index lastCol, Index k, RealScalar epsilon_coarse = Literal(8) * NumTraits::epsilon() * numext::maxi(col0.cwiseAbs().maxCoeff(), maxDiag); - BDCSVD_SANITY_CHECK(m_naiveU.allFinite()); - BDCSVD_SANITY_CHECK(m_naiveV.allFinite()); - BDCSVD_SANITY_CHECK(m_computed.allFinite()); - -#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE - std::cout << "\ndeflate:" << diag.head(k + 1).transpose() << " | " - << diag.segment(k + 1, length - k - 1).transpose() << "\n"; -#endif - // condition 4.1 if (diag(0) < epsilon_coarse) { -#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE - std::cout << "deflation 4.1, because " << diag(0) << " < " << epsilon_coarse << "\n"; -#endif diag(0) = epsilon_coarse; } // condition 4.2 for (Index i = 1; i < length; ++i) if (abs(col0(i)) < epsilon_strict) { -#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE - std::cout << "deflation 4.2, set z(" << i << ") to zero because " << abs(col0(i)) << " < " << epsilon_strict - << " (diag(" << i << ")=" << diag(i) << ")\n"; -#endif col0(i) = Literal(0); } // condition 4.3 for (Index i = 1; i < length; i++) if (diag(i) < epsilon_coarse) { -#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE - std::cout << "deflation 4.3, cancel z(" << i << ")=" << col0(i) << " because diag(" << i << ")=" << diag(i) - << " < " << epsilon_coarse << "\n"; -#endif deflation43(firstCol, shift, i, length); } - BDCSVD_SANITY_CHECK(m_naiveU.allFinite()); - BDCSVD_SANITY_CHECK(m_naiveV.allFinite()); - BDCSVD_SANITY_CHECK(m_computed.allFinite()); - -#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE - std::cout << "to be sorted: " << diag.transpose() << "\n\n"; - std::cout << " : " << col0.transpose() << "\n\n"; -#endif { // Check for total deflation: // If we have a total deflation, then we have to consider col0(0)==diag(0) as a singular value during sorting. @@ -1052,10 +796,6 @@ void bdcsvd_impl::deflation(Index firstCol, Index lastCol, Index k, realInd[i] = pi; } } -#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE - std::cout << "sorted: " << diag.transpose().format(bdcsvdfmt) << "\n"; - std::cout << " : " << col0.transpose() << "\n\n"; -#endif // condition 4.4 { @@ -1065,23 +805,12 @@ void bdcsvd_impl::deflation(Index firstCol, Index lastCol, Index k, for (; i > 1; --i) if ((diag(i) - diag(i - 1)) < epsilon_strict) { -#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE - std::cout << "deflation 4.4 with i = " << i << " because " << diag(i) << " - " << diag(i - 1) - << " == " << (diag(i) - diag(i - 1)) << " < " << epsilon_strict << "\n"; -#endif eigen_internal_assert(abs(diag(i) - diag(i - 1)) < epsilon_coarse && " diagonal entries are not properly sorted"); deflation44(firstCol, firstCol + shift, firstRowW, firstColW, i, i - 1, length); } } -#ifdef EIGEN_BDCSVD_SANITY_CHECKS - for (Index j = 2; j < length; ++j) eigen_internal_assert(diag(j - 1) <= diag(j) || abs(diag(j)) < considerZero); -#endif - - BDCSVD_SANITY_CHECK(m_naiveU.allFinite()); - BDCSVD_SANITY_CHECK(m_naiveV.allFinite()); - BDCSVD_SANITY_CHECK(m_computed.allFinite()); } // end deflation } // end namespace internal diff --git a/test/bdcsvd.cpp b/test/bdcsvd.cpp index b1e02318d..de0fa7d21 100644 --- a/test/bdcsvd.cpp +++ b/test/bdcsvd.cpp @@ -85,6 +85,212 @@ void bdcsvd_check_convergence(const MatrixType& input) { VERIFY_IS_APPROX(input, D); } +// Verify SVD of bidiagonal matrix given as diagonal + superdiagonal vectors. +template +void verify_bidiagonal_svd(const Matrix& diag, + const Matrix& superdiag) { + typedef Matrix MatrixXr; + typedef Matrix VectorXr; + const Index n = diag.size(); + + BDCSVD bdcsvd(diag, superdiag); + VERIFY(bdcsvd.info() == Success); + + const VectorXr& sv = bdcsvd.singularValues(); + + // Singular values must be non-negative. + for (Index i = 0; i < sv.size(); ++i) { + VERIFY(sv(i) >= RealScalar(0)); + } + + // Singular values must be sorted descending. + for (Index i = 1; i < sv.size(); ++i) { + VERIFY(sv(i - 1) >= sv(i)); + } + + // Orthogonality of U and V. + VERIFY_IS_APPROX(bdcsvd.matrixU().transpose() * bdcsvd.matrixU(), MatrixXr::Identity(n, n)); + VERIFY_IS_APPROX(bdcsvd.matrixV().transpose() * bdcsvd.matrixV(), MatrixXr::Identity(n, n)); + + // Reconstruction: U * S * V^T should equal the original bidiagonal. + MatrixXr B = MatrixXr::Zero(n, n); + B.diagonal() = diag; + if (n > 1) B.diagonal(1) = superdiag; + MatrixXr recon = bdcsvd.matrixU() * sv.asDiagonal() * bdcsvd.matrixV().transpose(); + VERIFY_IS_APPROX(recon, B); + + // Cross-validate singular values against JacobiSVD. + JacobiSVD jacobi(B); + VERIFY_IS_APPROX(sv, jacobi.singularValues()); +} + +// Verify that bidiagonal API and matrix API produce matching singular values. +template +void verify_bidiagonal_vs_matrix_svd(const Matrix& diag, + const Matrix& superdiag) { + typedef Matrix MatrixXr; + const Index n = diag.size(); + + // Build dense bidiagonal matrix. + MatrixXr B = MatrixXr::Zero(n, n); + B.diagonal() = diag; + if (n > 1) B.diagonal(1) = superdiag; + + BDCSVD bidiag_svd(diag, superdiag); + BDCSVD matrix_svd(B); + + VERIFY(bidiag_svd.info() == Success); + VERIFY(matrix_svd.info() == Success); + VERIFY_IS_APPROX(bidiag_svd.singularValues(), matrix_svd.singularValues()); +} + +template +void bdcsvd_bidiagonal_hard_cases() { + using std::abs; + using std::cos; + using std::pow; + using std::sin; + typedef Matrix VectorXr; + + Eigen::internal::set_is_malloc_allowed(true); + + const RealScalar eps = NumTraits::epsilon(); + + // Test sizes: cover n=1, very small, below/above algoSwap (16), and larger. + const int sizes[] = {1, 2, 3, 5, 10, 16, 20, 50, 100}; + const int numSizes = sizeof(sizes) / sizeof(sizes[0]); + + for (int si = 0; si < numSizes; ++si) { + const Index n = sizes[si]; + VectorXr diag(n), superdiag(n > 1 ? n - 1 : 0); + + // 1. Identity: d=[1,...,1], e=[0,...,0] + diag.setOnes(); + superdiag.setZero(); + verify_bidiagonal_svd(diag, superdiag); + verify_bidiagonal_vs_matrix_svd(diag, superdiag); + + // 2. Zero: d=[0,...,0], e=[0,...,0] + diag.setZero(); + superdiag.setZero(); + verify_bidiagonal_svd(diag, superdiag); + + // 3. Scalar (only meaningful for n=1, but runs for all) + if (n == 1) { + diag(0) = RealScalar(3.14); + verify_bidiagonal_svd(diag, superdiag); + } + + // 4. Golub-Kahan: d=[1,...,1], e=[1,...,1] + diag.setOnes(); + if (n > 1) superdiag.setOnes(); + verify_bidiagonal_svd(diag, superdiag); + + // 5. Kahan matrix: d_i = s^(i-1), e_i = -c*s^(i-1) + // Clamp exponents so condition number stays bounded by 1/eps. + { + const RealScalar theta = RealScalar(0.3); + const RealScalar s = sin(theta); + const RealScalar c = cos(theta); + using std::log; + const RealScalar maxPower = -log(eps) / (-log(s)); + for (Index i = 0; i < n; ++i) diag(i) = pow(s, numext::mini(RealScalar(i), maxPower)); + for (Index i = 0; i < n - 1; ++i) superdiag(i) = -c * pow(s, numext::mini(RealScalar(i), maxPower)); + verify_bidiagonal_svd(diag, superdiag); + } + + // 6. Geometric decay diagonal: d_i = 0.5^i, e=[0,...,0] + // Clamp so condition number stays bounded by 1/eps. + { + using std::log; + const RealScalar base = RealScalar(0.5); + const RealScalar maxPower = -log(eps) / (-log(base)); + for (Index i = 0; i < n; ++i) diag(i) = pow(base, numext::mini(RealScalar(i), maxPower)); + superdiag.setZero(); + verify_bidiagonal_svd(diag, superdiag); + } + + // 7. Geometric decay superdiagonal: d=[1,...,1], e_i = 0.5^i + diag.setOnes(); + for (Index i = 0; i < n - 1; ++i) superdiag(i) = pow(RealScalar(0.5), RealScalar(i)); + verify_bidiagonal_svd(diag, superdiag); + + // 8. Clustered at 1: d_i = 1 + i*eps, e=[0,...,0] + for (Index i = 0; i < n; ++i) diag(i) = RealScalar(1) + RealScalar(i) * eps; + superdiag.setZero(); + verify_bidiagonal_svd(diag, superdiag); + + // 9. Two clusters: half ≈ 1, half ≈ eps + for (Index i = 0; i < n; ++i) diag(i) = (i < n / 2) ? RealScalar(1) : eps; + superdiag.setZero(); + verify_bidiagonal_svd(diag, superdiag); + + // 10. Single tiny singular value: d=[1,...,1,eps], e=[eps^2,...] + diag.setOnes(); + diag(n - 1) = eps; + for (Index i = 0; i < n - 1; ++i) superdiag(i) = eps * eps; + verify_bidiagonal_svd(diag, superdiag); + + // 11. Graded: d_i = 10^(-i), e_i = 10^(-i) + for (Index i = 0; i < n; ++i) diag(i) = pow(RealScalar(10), -RealScalar(i)); + for (Index i = 0; i < n - 1; ++i) superdiag(i) = pow(RealScalar(10), -RealScalar(i)); + verify_bidiagonal_svd(diag, superdiag); + + // 12. Nearly diagonal: random diag, eps * random superdiag + diag = VectorXr::Random(n).cwiseAbs() + VectorXr::Constant(n, RealScalar(0.1)); + for (Index i = 0; i < n - 1; ++i) superdiag(i) = eps * (RealScalar(0.5) + abs(internal::random())); + verify_bidiagonal_svd(diag, superdiag); + + // 13. All equal: d=[c,...,c], e=[c,...,c] + diag.setConstant(RealScalar(2.5)); + if (n > 1) superdiag.setConstant(RealScalar(2.5)); + verify_bidiagonal_svd(diag, superdiag); + + // 14. Wilkinson: d_i = |n/2 - i|, e=[1,...,1] + for (Index i = 0; i < n; ++i) diag(i) = abs(RealScalar(n / 2) - RealScalar(i)); + if (n > 1) superdiag.setOnes(); + verify_bidiagonal_svd(diag, superdiag); + + // 15. Overflow/underflow: alternating big/tiny diagonal, tiny/big superdiagonal + { + const RealScalar big = (std::numeric_limits::max)() / RealScalar(1000); + const RealScalar tiny = (std::numeric_limits::min)() * RealScalar(1000); + for (Index i = 0; i < n; ++i) diag(i) = (i % 2 == 0) ? big : tiny; + for (Index i = 0; i < n - 1; ++i) superdiag(i) = (i % 2 == 0) ? tiny : big; + verify_bidiagonal_svd(diag, superdiag); + } + + // 16. Prescribed condition number: d_i = kappa^(-i/(n-1)), e_i = eps * random + if (n > 1) { + const RealScalar kappa = RealScalar(1) / eps; + for (Index i = 0; i < n; ++i) diag(i) = pow(kappa, -RealScalar(i) / RealScalar(n - 1)); + for (Index i = 0; i < n - 1; ++i) superdiag(i) = eps * abs(internal::random()); + verify_bidiagonal_svd(diag, superdiag); + } + + // 17. Rank-deficient: d=[1,..,0,..,0,..,1], e=[0,...,0] + for (Index i = 0; i < n; ++i) diag(i) = (i < n / 3 || i >= 2 * n / 3) ? RealScalar(1) : RealScalar(0); + superdiag.setZero(); + verify_bidiagonal_svd(diag, superdiag); + + // 18. Arrowhead stress: d_i = linspace(1, n), e_i = 1/(i+1) + for (Index i = 0; i < n; ++i) diag(i) = RealScalar(1) + RealScalar(i); + for (Index i = 0; i < n - 1; ++i) superdiag(i) = RealScalar(1) / RealScalar(i + 1); + verify_bidiagonal_svd(diag, superdiag); + + // 19. Repeated singular values: d=[1,2,3,1,2,3,...], e=[0,...,0] + for (Index i = 0; i < n; ++i) diag(i) = RealScalar((i % 3) + 1); + superdiag.setZero(); + verify_bidiagonal_svd(diag, superdiag); + + // 20. Glued identity: d=[1,...,1], e=0 except e[n/2-1]=eps + diag.setOnes(); + superdiag.setZero(); + if (n > 2) superdiag(n / 2 - 1) = eps; + verify_bidiagonal_svd(diag, superdiag); + } +} + EIGEN_DECLARE_TEST(bdcsvd) { CALL_SUBTEST_1((bdcsvd_verify_assert())); CALL_SUBTEST_2((bdcsvd_verify_assert())); @@ -174,4 +380,8 @@ EIGEN_DECLARE_TEST(bdcsvd) { // Convergence for large constant matrix (https://gitlab.com/libeigen/eigen/-/issues/2491) CALL_SUBTEST_49(bdcsvd_check_convergence(MatrixXf::Constant(500, 500, 1))); + + // Bidiagonal SVD hard test cases + CALL_SUBTEST_50((bdcsvd_bidiagonal_hard_cases())); + CALL_SUBTEST_51((bdcsvd_bidiagonal_hard_cases())); }