mirror of
https://gitlab.com/libeigen/eigen.git
synced 2026-04-10 11:34:33 +08:00
014c581a5b
we never assert on conditions that depend on the result of a computation!! also the assertion that rank>0 amounts to matrix!=0 which we have to leave under the responsibility of the user.
603 lines
23 KiB
C++
603 lines
23 KiB
C++
// This file is part of Eigen, a lightweight C++ template library
|
|
// for linear algebra.
|
|
//
|
|
// Copyright (C) 2006-2008 Benoit Jacob <jacob.benoit.1@gmail.com>
|
|
//
|
|
// Eigen is free software; you can redistribute it and/or
|
|
// modify it under the terms of the GNU Lesser General Public
|
|
// License as published by the Free Software Foundation; either
|
|
// version 3 of the License, or (at your option) any later version.
|
|
//
|
|
// Alternatively, you can redistribute it and/or
|
|
// modify it under the terms of the GNU General Public License as
|
|
// published by the Free Software Foundation; either version 2 of
|
|
// the License, or (at your option) any later version.
|
|
//
|
|
// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
|
|
// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
|
|
// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
|
|
// GNU General Public License for more details.
|
|
//
|
|
// You should have received a copy of the GNU Lesser General Public
|
|
// License and a copy of the GNU General Public License along with
|
|
// Eigen. If not, see <http://www.gnu.org/licenses/>.
|
|
|
|
#ifndef EIGEN_LU_H
|
|
#define EIGEN_LU_H
|
|
|
|
/** \ingroup LU_Module
|
|
*
|
|
* \class LU
|
|
*
|
|
* \brief LU decomposition of a matrix with complete pivoting, and related features
|
|
*
|
|
* \param MatrixType the type of the matrix of which we are computing the LU decomposition
|
|
*
|
|
* This class represents a LU decomposition of any matrix, with complete pivoting: the matrix A
|
|
* is decomposed as A = PLUQ where L is unit-lower-triangular, U is upper-triangular, and P and Q
|
|
* are permutation matrices. This is a rank-revealing LU decomposition. The eigenvalues (diagonal
|
|
* coefficients) of U are sorted in such a way that any zeros are at the end, so that the rank
|
|
* of A is the index of the first zero on the diagonal of U (with indices starting at 0) if any.
|
|
*
|
|
* This decomposition provides the generic approach to solving systems of linear equations, computing
|
|
* the rank, invertibility, inverse, kernel, and determinant.
|
|
*
|
|
* This LU decomposition is very stable and well tested with large matrices. Even exact rank computation
|
|
* works at sizes larger than 1000x1000. However there are use cases where the SVD decomposition is inherently
|
|
* more stable when dealing with numerically damaged input. For example, computing the kernel is more stable with
|
|
* SVD because the SVD can determine which singular values are negligible while LU has to work at the level of matrix
|
|
* coefficients that are less meaningful in this respect.
|
|
*
|
|
* The data of the LU decomposition can be directly accessed through the methods matrixLU(),
|
|
* permutationP(), permutationQ().
|
|
*
|
|
* As an exemple, here is how the original matrix can be retrieved:
|
|
* \include class_LU.cpp
|
|
* Output: \verbinclude class_LU.out
|
|
*
|
|
* \sa MatrixBase::lu(), MatrixBase::determinant(), MatrixBase::inverse(), MatrixBase::computeInverse()
|
|
*/
|
|
template<typename MatrixType> class LU
|
|
{
|
|
public:
|
|
|
|
typedef typename MatrixType::Scalar Scalar;
|
|
typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
|
|
typedef Matrix<int, 1, MatrixType::ColsAtCompileTime> IntRowVectorType;
|
|
typedef Matrix<int, MatrixType::RowsAtCompileTime, 1> IntColVectorType;
|
|
typedef Matrix<Scalar, 1, MatrixType::ColsAtCompileTime> RowVectorType;
|
|
typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> ColVectorType;
|
|
|
|
enum { MaxSmallDimAtCompileTime = EIGEN_ENUM_MIN(
|
|
MatrixType::MaxColsAtCompileTime,
|
|
MatrixType::MaxRowsAtCompileTime)
|
|
};
|
|
|
|
typedef Matrix<typename MatrixType::Scalar,
|
|
MatrixType::ColsAtCompileTime, // the number of rows in the "kernel matrix" is the number of cols of the original matrix
|
|
// so that the product "matrix * kernel = zero" makes sense
|
|
Dynamic, // we don't know at compile-time the dimension of the kernel
|
|
MatrixType::Options,
|
|
MatrixType::MaxColsAtCompileTime, // see explanation for 2nd template parameter
|
|
MatrixType::MaxColsAtCompileTime // the kernel is a subspace of the domain space, whose dimension is the number
|
|
// of columns of the original matrix
|
|
> KernelResultType;
|
|
|
|
typedef Matrix<typename MatrixType::Scalar,
|
|
MatrixType::RowsAtCompileTime, // the image is a subspace of the destination space, whose dimension is the number
|
|
// of rows of the original matrix
|
|
Dynamic, // we don't know at compile time the dimension of the image (the rank)
|
|
MatrixType::Options,
|
|
MatrixType::MaxRowsAtCompileTime, // the image matrix will consist of columns from the original matrix,
|
|
MatrixType::MaxColsAtCompileTime // so it has the same number of rows and at most as many columns.
|
|
> ImageResultType;
|
|
|
|
/**
|
|
* \brief Default Constructor.
|
|
*
|
|
* The default constructor is useful in cases in which the user intends to
|
|
* perform decompositions via LU::compute(const MatrixType&).
|
|
*/
|
|
LU();
|
|
|
|
/** Constructor.
|
|
*
|
|
* \param matrix the matrix of which to compute the LU decomposition.
|
|
* It is required to be nonzero.
|
|
*/
|
|
LU(const MatrixType& matrix);
|
|
|
|
/** Computes the LU decomposition of the given matrix.
|
|
*
|
|
* \param matrix the matrix of which to compute the LU decomposition.
|
|
* It is required to be nonzero.
|
|
*/
|
|
void compute(const MatrixType& matrix);
|
|
|
|
/** \returns the LU decomposition matrix: the upper-triangular part is U, the
|
|
* unit-lower-triangular part is L (at least for square matrices; in the non-square
|
|
* case, special care is needed, see the documentation of class LU).
|
|
*
|
|
* \sa matrixL(), matrixU()
|
|
*/
|
|
inline const MatrixType& matrixLU() const
|
|
{
|
|
ei_assert(m_originalMatrix != 0 && "LU is not initialized.");
|
|
return m_lu;
|
|
}
|
|
|
|
/** \returns a vector of integers, whose size is the number of rows of the matrix being decomposed,
|
|
* representing the P permutation i.e. the permutation of the rows. For its precise meaning,
|
|
* see the examples given in the documentation of class LU.
|
|
*
|
|
* \sa permutationQ()
|
|
*/
|
|
inline const IntColVectorType& permutationP() const
|
|
{
|
|
ei_assert(m_originalMatrix != 0 && "LU is not initialized.");
|
|
return m_p;
|
|
}
|
|
|
|
/** \returns a vector of integers, whose size is the number of columns of the matrix being
|
|
* decomposed, representing the Q permutation i.e. the permutation of the columns.
|
|
* For its precise meaning, see the examples given in the documentation of class LU.
|
|
*
|
|
* \sa permutationP()
|
|
*/
|
|
inline const IntRowVectorType& permutationQ() const
|
|
{
|
|
ei_assert(m_originalMatrix != 0 && "LU is not initialized.");
|
|
return m_q;
|
|
}
|
|
|
|
/** Computes a basis of the kernel of the matrix, also called the null-space of the matrix.
|
|
*
|
|
* \note This method is only allowed on non-invertible matrices, as determined by
|
|
* isInvertible(). Calling it on an invertible matrix will make an assertion fail.
|
|
*
|
|
* \param result a pointer to the matrix in which to store the kernel. The columns of this
|
|
* matrix will be set to form a basis of the kernel (it will be resized
|
|
* if necessary).
|
|
*
|
|
* Example: \include LU_computeKernel.cpp
|
|
* Output: \verbinclude LU_computeKernel.out
|
|
*
|
|
* \sa kernel(), computeImage(), image()
|
|
*/
|
|
template<typename KernelMatrixType>
|
|
void computeKernel(KernelMatrixType *result) const;
|
|
|
|
/** Computes a basis of the image of the matrix, also called the column-space or range of he matrix.
|
|
*
|
|
* \note Calling this method on the zero matrix will make an assertion fail.
|
|
*
|
|
* \param result a pointer to the matrix in which to store the image. The columns of this
|
|
* matrix will be set to form a basis of the image (it will be resized
|
|
* if necessary).
|
|
*
|
|
* Example: \include LU_computeImage.cpp
|
|
* Output: \verbinclude LU_computeImage.out
|
|
*
|
|
* \sa image(), computeKernel(), kernel()
|
|
*/
|
|
template<typename ImageMatrixType>
|
|
void computeImage(ImageMatrixType *result) const;
|
|
|
|
/** \returns the kernel of the matrix, also called its null-space. The columns of the returned matrix
|
|
* will form a basis of the kernel.
|
|
*
|
|
* \note: this method is only allowed on non-invertible matrices, as determined by
|
|
* isInvertible(). Calling it on an invertible matrix will make an assertion fail.
|
|
*
|
|
* \note: this method returns a matrix by value, which induces some inefficiency.
|
|
* If you prefer to avoid this overhead, use computeKernel() instead.
|
|
*
|
|
* Example: \include LU_kernel.cpp
|
|
* Output: \verbinclude LU_kernel.out
|
|
*
|
|
* \sa computeKernel(), image()
|
|
*/
|
|
const KernelResultType kernel() const;
|
|
|
|
/** \returns the image of the matrix, also called its column-space. The columns of the returned matrix
|
|
* will form a basis of the kernel.
|
|
*
|
|
* \note: Calling this method on the zero matrix will make an assertion fail.
|
|
*
|
|
* \note: this method returns a matrix by value, which induces some inefficiency.
|
|
* If you prefer to avoid this overhead, use computeImage() instead.
|
|
*
|
|
* Example: \include LU_image.cpp
|
|
* Output: \verbinclude LU_image.out
|
|
*
|
|
* \sa computeImage(), kernel()
|
|
*/
|
|
const ImageResultType image() const;
|
|
|
|
/** This method finds a solution x to the equation Ax=b, where A is the matrix of which
|
|
* *this is the LU decomposition, if any exists.
|
|
*
|
|
* \param b the right-hand-side of the equation to solve. Can be a vector or a matrix,
|
|
* the only requirement in order for the equation to make sense is that
|
|
* b.rows()==A.rows(), where A is the matrix of which *this is the LU decomposition.
|
|
* \param result a pointer to the vector or matrix in which to store the solution, if any exists.
|
|
* Resized if necessary, so that result->rows()==A.cols() and result->cols()==b.cols().
|
|
* If no solution exists, *result is left with undefined coefficients.
|
|
*
|
|
* \returns true if any solution exists, false if no solution exists.
|
|
*
|
|
* \note If there exist more than one solution, this method will arbitrarily choose one.
|
|
* If you need a complete analysis of the space of solutions, take the one solution obtained
|
|
* by this method and add to it elements of the kernel, as determined by kernel().
|
|
*
|
|
* Example: \include LU_solve.cpp
|
|
* Output: \verbinclude LU_solve.out
|
|
*
|
|
* \sa TriangularView::solve(), kernel(), computeKernel(), inverse(), computeInverse()
|
|
*/
|
|
template<typename OtherDerived, typename ResultType>
|
|
bool solve(const MatrixBase<OtherDerived>& b, ResultType *result) const;
|
|
|
|
/** \returns the determinant of the matrix of which
|
|
* *this is the LU decomposition. It has only linear complexity
|
|
* (that is, O(n) where n is the dimension of the square matrix)
|
|
* as the LU decomposition has already been computed.
|
|
*
|
|
* \note This is only for square matrices.
|
|
*
|
|
* \note For fixed-size matrices of size up to 4, MatrixBase::determinant() offers
|
|
* optimized paths.
|
|
*
|
|
* \warning a determinant can be very big or small, so for matrices
|
|
* of large enough dimension, there is a risk of overflow/underflow.
|
|
*
|
|
* \sa MatrixBase::determinant()
|
|
*/
|
|
typename ei_traits<MatrixType>::Scalar determinant() const;
|
|
|
|
/** \returns the rank of the matrix of which *this is the LU decomposition.
|
|
*
|
|
* \note This is computed at the time of the construction of the LU decomposition. This
|
|
* method does not perform any further computation.
|
|
*/
|
|
inline int rank() const
|
|
{
|
|
ei_assert(m_originalMatrix != 0 && "LU is not initialized.");
|
|
return m_rank;
|
|
}
|
|
|
|
/** \returns the dimension of the kernel of the matrix of which *this is the LU decomposition.
|
|
*
|
|
* \note Since the rank is computed at the time of the construction of the LU decomposition, this
|
|
* method almost does not perform any further computation.
|
|
*/
|
|
inline int dimensionOfKernel() const
|
|
{
|
|
ei_assert(m_originalMatrix != 0 && "LU is not initialized.");
|
|
return m_lu.cols() - m_rank;
|
|
}
|
|
|
|
/** \returns true if the matrix of which *this is the LU decomposition represents an injective
|
|
* linear map, i.e. has trivial kernel; false otherwise.
|
|
*
|
|
* \note Since the rank is computed at the time of the construction of the LU decomposition, this
|
|
* method almost does not perform any further computation.
|
|
*/
|
|
inline bool isInjective() const
|
|
{
|
|
ei_assert(m_originalMatrix != 0 && "LU is not initialized.");
|
|
return m_rank == m_lu.cols();
|
|
}
|
|
|
|
/** \returns true if the matrix of which *this is the LU decomposition represents a surjective
|
|
* linear map; false otherwise.
|
|
*
|
|
* \note Since the rank is computed at the time of the construction of the LU decomposition, this
|
|
* method almost does not perform any further computation.
|
|
*/
|
|
inline bool isSurjective() const
|
|
{
|
|
ei_assert(m_originalMatrix != 0 && "LU is not initialized.");
|
|
return m_rank == m_lu.rows();
|
|
}
|
|
|
|
/** \returns true if the matrix of which *this is the LU decomposition is invertible.
|
|
*
|
|
* \note Since the rank is computed at the time of the construction of the LU decomposition, this
|
|
* method almost does not perform any further computation.
|
|
*/
|
|
inline bool isInvertible() const
|
|
{
|
|
ei_assert(m_originalMatrix != 0 && "LU is not initialized.");
|
|
return isInjective() && isSurjective();
|
|
}
|
|
|
|
/** Computes the inverse of the matrix of which *this is the LU decomposition.
|
|
*
|
|
* \param result a pointer to the matrix into which to store the inverse. Resized if needed.
|
|
*
|
|
* \note If this matrix is not invertible, *result is left with undefined coefficients.
|
|
* Use isInvertible() to first determine whether this matrix is invertible.
|
|
*
|
|
* \sa MatrixBase::computeInverse(), inverse()
|
|
*/
|
|
inline void computeInverse(MatrixType *result) const
|
|
{
|
|
ei_assert(m_originalMatrix != 0 && "LU is not initialized.");
|
|
solve(MatrixType::Identity(m_lu.rows(), m_lu.cols()), result);
|
|
}
|
|
|
|
/** \returns the inverse of the matrix of which *this is the LU decomposition.
|
|
*
|
|
* \note If this matrix is not invertible, the returned matrix has undefined coefficients.
|
|
* Use isInvertible() to first determine whether this matrix is invertible.
|
|
*
|
|
* \sa computeInverse(), MatrixBase::inverse()
|
|
*/
|
|
inline MatrixType inverse() const
|
|
{
|
|
MatrixType result;
|
|
computeInverse(&result);
|
|
return result;
|
|
}
|
|
|
|
protected:
|
|
const MatrixType* m_originalMatrix;
|
|
MatrixType m_lu;
|
|
IntColVectorType m_p;
|
|
IntRowVectorType m_q;
|
|
int m_det_pq;
|
|
int m_rank;
|
|
RealScalar m_precision;
|
|
};
|
|
|
|
template<typename MatrixType>
|
|
LU<MatrixType>::LU()
|
|
: m_originalMatrix(0),
|
|
m_lu(),
|
|
m_p(),
|
|
m_q(),
|
|
m_det_pq(0),
|
|
m_rank(-1),
|
|
m_precision(precision<RealScalar>())
|
|
{
|
|
}
|
|
|
|
template<typename MatrixType>
|
|
LU<MatrixType>::LU(const MatrixType& matrix)
|
|
: m_originalMatrix(0),
|
|
m_lu(),
|
|
m_p(),
|
|
m_q(),
|
|
m_det_pq(0),
|
|
m_rank(-1),
|
|
m_precision(precision<RealScalar>())
|
|
{
|
|
compute(matrix);
|
|
}
|
|
|
|
template<typename MatrixType>
|
|
void LU<MatrixType>::compute(const MatrixType& matrix)
|
|
{
|
|
m_originalMatrix = &matrix;
|
|
m_lu = matrix;
|
|
m_p.resize(matrix.rows());
|
|
m_q.resize(matrix.cols());
|
|
|
|
const int size = matrix.diagonalSize();
|
|
const int rows = matrix.rows();
|
|
const int cols = matrix.cols();
|
|
|
|
// this formula comes from experimenting (see "LU precision tuning" thread on the list)
|
|
// and turns out to be identical to Higham's formula used already in LDLt.
|
|
m_precision = machine_epsilon<Scalar>() * size;
|
|
|
|
IntColVectorType rows_transpositions(matrix.rows());
|
|
IntRowVectorType cols_transpositions(matrix.cols());
|
|
int number_of_transpositions = 0;
|
|
|
|
RealScalar biggest = RealScalar(0);
|
|
m_rank = size;
|
|
for(int k = 0; k < size; ++k)
|
|
{
|
|
int row_of_biggest_in_corner, col_of_biggest_in_corner;
|
|
RealScalar biggest_in_corner;
|
|
|
|
biggest_in_corner = m_lu.corner(Eigen::BottomRight, rows-k, cols-k)
|
|
.cwise().abs()
|
|
.maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner);
|
|
row_of_biggest_in_corner += k;
|
|
col_of_biggest_in_corner += k;
|
|
if(k==0) biggest = biggest_in_corner;
|
|
|
|
// if the corner is negligible, then we have less than full rank, and we can finish early
|
|
if(ei_isMuchSmallerThan(biggest_in_corner, biggest, m_precision))
|
|
{
|
|
m_rank = k;
|
|
for(int i = k; i < size; i++)
|
|
{
|
|
rows_transpositions.coeffRef(i) = i;
|
|
cols_transpositions.coeffRef(i) = i;
|
|
}
|
|
break;
|
|
}
|
|
|
|
rows_transpositions.coeffRef(k) = row_of_biggest_in_corner;
|
|
cols_transpositions.coeffRef(k) = col_of_biggest_in_corner;
|
|
if(k != row_of_biggest_in_corner) {
|
|
m_lu.row(k).swap(m_lu.row(row_of_biggest_in_corner));
|
|
++number_of_transpositions;
|
|
}
|
|
if(k != col_of_biggest_in_corner) {
|
|
m_lu.col(k).swap(m_lu.col(col_of_biggest_in_corner));
|
|
++number_of_transpositions;
|
|
}
|
|
if(k<rows-1)
|
|
m_lu.col(k).end(rows-k-1) /= m_lu.coeff(k,k);
|
|
if(k<size-1) {
|
|
/* I know it's tempting to replace this for loop by a single matrix product. But actually there's no reason why it
|
|
* should be faster because it's just an exterior vector product; and in practice this gives much slower code with
|
|
* GCC 4.2-4.4 (this is weird, would be interesting to investigate). On the other hand, it would be worth having a variant
|
|
* for row-major matrices, traversing in the other direction for better performance, with a meta selector to compile only
|
|
* one path
|
|
*/
|
|
for(int col = k + 1; col < cols; ++col)
|
|
m_lu.col(col).end(rows-k-1) -= m_lu.col(k).end(rows-k-1) * m_lu.coeff(k,col);
|
|
}
|
|
}
|
|
|
|
for(int k = 0; k < matrix.rows(); ++k) m_p.coeffRef(k) = k;
|
|
for(int k = size-1; k >= 0; --k)
|
|
std::swap(m_p.coeffRef(k), m_p.coeffRef(rows_transpositions.coeff(k)));
|
|
|
|
for(int k = 0; k < matrix.cols(); ++k) m_q.coeffRef(k) = k;
|
|
for(int k = 0; k < size; ++k)
|
|
std::swap(m_q.coeffRef(k), m_q.coeffRef(cols_transpositions.coeff(k)));
|
|
|
|
m_det_pq = (number_of_transpositions%2) ? -1 : 1;
|
|
}
|
|
|
|
template<typename MatrixType>
|
|
typename ei_traits<MatrixType>::Scalar LU<MatrixType>::determinant() const
|
|
{
|
|
ei_assert(m_originalMatrix != 0 && "LU is not initialized.");
|
|
return Scalar(m_det_pq) * m_lu.diagonal().prod();
|
|
}
|
|
|
|
template<typename MatrixType>
|
|
template<typename KernelMatrixType>
|
|
void LU<MatrixType>::computeKernel(KernelMatrixType *result) const
|
|
{
|
|
ei_assert(m_originalMatrix != 0 && "LU is not initialized.");
|
|
const int dimker = dimensionOfKernel(), cols = m_lu.cols();
|
|
result->resize(cols, dimker);
|
|
|
|
/* Let us use the following lemma:
|
|
*
|
|
* Lemma: If the matrix A has the LU decomposition PAQ = LU,
|
|
* then Ker A = Q(Ker U).
|
|
*
|
|
* Proof: trivial: just keep in mind that P, Q, L are invertible.
|
|
*/
|
|
|
|
/* Thus, all we need to do is to compute Ker U, and then apply Q.
|
|
*
|
|
* U is upper triangular, with eigenvalues sorted so that any zeros appear at the end.
|
|
* Thus, the diagonal of U ends with exactly
|
|
* m_dimKer zero's. Let us use that to construct m_dimKer linearly
|
|
* independent vectors in Ker U.
|
|
*/
|
|
|
|
Matrix<Scalar, Dynamic, Dynamic, MatrixType::Options,
|
|
MatrixType::MaxColsAtCompileTime, MatrixType::MaxColsAtCompileTime>
|
|
y(-m_lu.corner(TopRight, m_rank, dimker));
|
|
|
|
m_lu.corner(TopLeft, m_rank, m_rank)
|
|
.template triangularView<UpperTriangular>().solveInPlace(y);
|
|
|
|
for(int i = 0; i < m_rank; ++i) result->row(m_q.coeff(i)) = y.row(i);
|
|
for(int i = m_rank; i < cols; ++i) result->row(m_q.coeff(i)).setZero();
|
|
for(int k = 0; k < dimker; ++k) result->coeffRef(m_q.coeff(m_rank+k), k) = Scalar(1);
|
|
}
|
|
|
|
template<typename MatrixType>
|
|
const typename LU<MatrixType>::KernelResultType
|
|
LU<MatrixType>::kernel() const
|
|
{
|
|
ei_assert(m_originalMatrix != 0 && "LU is not initialized.");
|
|
KernelResultType result(m_lu.cols(), dimensionOfKernel());
|
|
computeKernel(&result);
|
|
return result;
|
|
}
|
|
|
|
template<typename MatrixType>
|
|
template<typename ImageMatrixType>
|
|
void LU<MatrixType>::computeImage(ImageMatrixType *result) const
|
|
{
|
|
ei_assert(m_originalMatrix != 0 && "LU is not initialized.");
|
|
result->resize(m_originalMatrix->rows(), m_rank);
|
|
for(int i = 0; i < m_rank; ++i)
|
|
result->col(i) = m_originalMatrix->col(m_q.coeff(i));
|
|
}
|
|
|
|
template<typename MatrixType>
|
|
const typename LU<MatrixType>::ImageResultType
|
|
LU<MatrixType>::image() const
|
|
{
|
|
ei_assert(m_originalMatrix != 0 && "LU is not initialized.");
|
|
ImageResultType result(m_originalMatrix->rows(), m_rank);
|
|
computeImage(&result);
|
|
return result;
|
|
}
|
|
|
|
template<typename MatrixType>
|
|
template<typename OtherDerived, typename ResultType>
|
|
bool LU<MatrixType>::solve(
|
|
const MatrixBase<OtherDerived>& b,
|
|
ResultType *result
|
|
) const
|
|
{
|
|
ei_assert(m_originalMatrix != 0 && "LU is not initialized.");
|
|
|
|
/* The decomposition PAQ = LU can be rewritten as A = P^{-1} L U Q^{-1}.
|
|
* So we proceed as follows:
|
|
* Step 1: compute c = Pb.
|
|
* Step 2: replace c by the solution x to Lx = c. Exists because L is invertible.
|
|
* Step 3: replace c by the solution x to Ux = c. Check if a solution really exists.
|
|
* Step 4: result = Qc;
|
|
*/
|
|
|
|
const int rows = m_lu.rows(), cols = m_lu.cols();
|
|
ei_assert(b.rows() == rows);
|
|
const int smalldim = std::min(rows, cols);
|
|
|
|
typename OtherDerived::PlainMatrixType c(b.rows(), b.cols());
|
|
|
|
// Step 1
|
|
for(int i = 0; i < rows; ++i) c.row(m_p.coeff(i)) = b.row(i);
|
|
|
|
// Step 2
|
|
m_lu.corner(Eigen::TopLeft,smalldim,smalldim).template triangularView<UnitLowerTriangular>()
|
|
.solveInPlace(c.corner(Eigen::TopLeft, smalldim, c.cols()));
|
|
if(rows>cols)
|
|
{
|
|
c.corner(Eigen::BottomLeft, rows-cols, c.cols())
|
|
-= m_lu.corner(Eigen::BottomLeft, rows-cols, cols) * c.corner(Eigen::TopLeft, cols, c.cols());
|
|
}
|
|
|
|
// Step 3
|
|
if(!isSurjective())
|
|
{
|
|
// is c is in the image of U ?
|
|
RealScalar biggest_in_c = c.corner(TopLeft, m_rank, c.cols()).cwise().abs().maxCoeff();
|
|
for(int col = 0; col < c.cols(); ++col)
|
|
for(int row = m_rank; row < c.rows(); ++row)
|
|
if(!ei_isMuchSmallerThan(c.coeff(row,col), biggest_in_c, m_precision))
|
|
return false;
|
|
}
|
|
m_lu.corner(TopLeft, m_rank, m_rank)
|
|
.template triangularView<UpperTriangular>()
|
|
.solveInPlace(c.corner(TopLeft, m_rank, c.cols()));
|
|
|
|
// Step 4
|
|
result->resize(m_lu.cols(), b.cols());
|
|
for(int i = 0; i < m_rank; ++i) result->row(m_q.coeff(i)) = c.row(i);
|
|
for(int i = m_rank; i < m_lu.cols(); ++i) result->row(m_q.coeff(i)).setZero();
|
|
return true;
|
|
}
|
|
|
|
/** \lu_module
|
|
*
|
|
* \return the LU decomposition of \c *this.
|
|
*
|
|
* \sa class LU
|
|
*/
|
|
template<typename Derived>
|
|
inline const LU<typename MatrixBase<Derived>::PlainMatrixType>
|
|
MatrixBase<Derived>::lu() const
|
|
{
|
|
return LU<PlainMatrixType>(eval());
|
|
}
|
|
|
|
#endif // EIGEN_LU_H
|