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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2009-2010 Benoit Jacob <jacob.benoit.1@gmail.com>
// Copyright (C) 2013-2014 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// 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_JACOBISVD_H
#define EIGEN_JACOBISVD_H
// IWYU pragma: private
#include "./InternalHeaderCheck.h"
namespace Eigen {
namespace internal {
// forward declaration (needed by ICC)
// the empty body is required by MSVC
template <typename MatrixType, int Options, bool IsComplex = NumTraits<typename MatrixType::Scalar>::IsComplex>
struct svd_precondition_2x2_block_to_be_real {};
/*** QR preconditioners (R-SVD)
***
*** Their role is to reduce the problem of computing the SVD to the case of a square matrix.
*** This approach, known as R-SVD, is an optimization for rectangular-enough matrices, and is a requirement for
*** JacobiSVD which by itself is only able to work on square matrices.
***/
enum { PreconditionIfMoreColsThanRows, PreconditionIfMoreRowsThanCols };
template <typename MatrixType, int QRPreconditioner, int Case>
struct qr_preconditioner_should_do_anything {
enum {
a = MatrixType::RowsAtCompileTime != Dynamic && MatrixType::ColsAtCompileTime != Dynamic &&
MatrixType::ColsAtCompileTime <= MatrixType::RowsAtCompileTime,
b = MatrixType::RowsAtCompileTime != Dynamic && MatrixType::ColsAtCompileTime != Dynamic &&
MatrixType::RowsAtCompileTime <= MatrixType::ColsAtCompileTime,
ret = !((QRPreconditioner == NoQRPreconditioner) || (Case == PreconditionIfMoreColsThanRows && bool(a)) ||
(Case == PreconditionIfMoreRowsThanCols && bool(b)))
};
};
template <typename MatrixType, int Options, int QRPreconditioner, int Case,
bool DoAnything = qr_preconditioner_should_do_anything<MatrixType, QRPreconditioner, Case>::ret>
struct qr_preconditioner_impl {};
template <typename MatrixType, int Options, int QRPreconditioner, int Case>
class qr_preconditioner_impl<MatrixType, Options, QRPreconditioner, Case, false> {
public:
void allocate(const JacobiSVD<MatrixType, Options>&) {}
template <typename Xpr>
bool run(JacobiSVD<MatrixType, Options>&, const Xpr&) {
return false;
}
};
/*** preconditioner using FullPivHouseholderQR ***/
template <typename MatrixType, int Options>
class qr_preconditioner_impl<MatrixType, Options, FullPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols,
true> {
public:
typedef typename MatrixType::Scalar Scalar;
typedef JacobiSVD<MatrixType, Options> SVDType;
enum { WorkspaceSize = MatrixType::RowsAtCompileTime, MaxWorkspaceSize = MatrixType::MaxRowsAtCompileTime };
typedef Matrix<Scalar, 1, WorkspaceSize, RowMajor, 1, MaxWorkspaceSize> WorkspaceType;
void allocate(const SVDType& svd) {
if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols()) {
internal::destroy_at(&m_qr);
internal::construct_at(&m_qr, svd.rows(), svd.cols());
}
if (svd.m_computeFullU) m_workspace.resize(svd.rows());
}
template <typename Xpr>
bool run(SVDType& svd, const Xpr& matrix) {
if (matrix.rows() > matrix.cols()) {
m_qr.compute(matrix);
svd.m_workMatrix = m_qr.matrixQR().block(0, 0, matrix.cols(), matrix.cols()).template triangularView<Upper>();
if (svd.m_computeFullU) m_qr.matrixQ().evalTo(svd.m_matrixU, m_workspace);
if (svd.computeV()) svd.m_matrixV = m_qr.colsPermutation();
return true;
}
return false;
}
private:
typedef FullPivHouseholderQR<MatrixType> QRType;
QRType m_qr;
WorkspaceType m_workspace;
};
template <typename MatrixType, int Options>
class qr_preconditioner_impl<MatrixType, Options, FullPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows,
true> {
public:
typedef typename MatrixType::Scalar Scalar;
typedef JacobiSVD<MatrixType, Options> SVDType;
enum {
RowsAtCompileTime = MatrixType::RowsAtCompileTime,
ColsAtCompileTime = MatrixType::ColsAtCompileTime,
MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
MatrixOptions = traits<MatrixType>::Options
};
typedef typename internal::make_proper_matrix_type<Scalar, ColsAtCompileTime, RowsAtCompileTime, MatrixOptions,
MaxColsAtCompileTime, MaxRowsAtCompileTime>::type
TransposeTypeWithSameStorageOrder;
void allocate(const SVDType& svd) {
if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols()) {
internal::destroy_at(&m_qr);
internal::construct_at(&m_qr, svd.cols(), svd.rows());
}
if (svd.m_computeFullV) m_workspace.resize(svd.cols());
}
template <typename Xpr>
bool run(SVDType& svd, const Xpr& matrix) {
if (matrix.cols() > matrix.rows()) {
m_qr.compute(matrix.adjoint());
svd.m_workMatrix =
m_qr.matrixQR().block(0, 0, matrix.rows(), matrix.rows()).template triangularView<Upper>().adjoint();
if (svd.m_computeFullV) m_qr.matrixQ().evalTo(svd.m_matrixV, m_workspace);
if (svd.computeU()) svd.m_matrixU = m_qr.colsPermutation();
return true;
} else
return false;
}
private:
typedef FullPivHouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
QRType m_qr;
typename plain_row_type<MatrixType>::type m_workspace;
};
/*** preconditioner using ColPivHouseholderQR ***/
template <typename MatrixType, int Options>
class qr_preconditioner_impl<MatrixType, Options, ColPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols,
true> {
public:
typedef typename MatrixType::Scalar Scalar;
typedef JacobiSVD<MatrixType, Options> SVDType;
enum {
WorkspaceSize = internal::traits<SVDType>::MatrixUColsAtCompileTime,
MaxWorkspaceSize = internal::traits<SVDType>::MatrixUMaxColsAtCompileTime
};
typedef Matrix<Scalar, 1, WorkspaceSize, RowMajor, 1, MaxWorkspaceSize> WorkspaceType;
void allocate(const SVDType& svd) {
if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols()) {
internal::destroy_at(&m_qr);
internal::construct_at(&m_qr, svd.rows(), svd.cols());
}
if (svd.m_computeFullU)
m_workspace.resize(svd.rows());
else if (svd.m_computeThinU)
m_workspace.resize(svd.cols());
}
template <typename Xpr>
bool run(SVDType& svd, const Xpr& matrix) {
if (matrix.rows() > matrix.cols()) {
m_qr.compute(matrix);
svd.m_workMatrix = m_qr.matrixQR().block(0, 0, matrix.cols(), matrix.cols()).template triangularView<Upper>();
if (svd.m_computeFullU)
m_qr.householderQ().evalTo(svd.m_matrixU, m_workspace);
else if (svd.m_computeThinU) {
svd.m_matrixU.setIdentity(matrix.rows(), matrix.cols());
m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixU, m_workspace);
}
if (svd.computeV()) svd.m_matrixV = m_qr.colsPermutation();
return true;
}
return false;
}
private:
typedef ColPivHouseholderQR<MatrixType> QRType;
QRType m_qr;
WorkspaceType m_workspace;
};
template <typename MatrixType, int Options>
class qr_preconditioner_impl<MatrixType, Options, ColPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows,
true> {
public:
typedef typename MatrixType::Scalar Scalar;
typedef JacobiSVD<MatrixType, Options> SVDType;
enum {
RowsAtCompileTime = MatrixType::RowsAtCompileTime,
ColsAtCompileTime = MatrixType::ColsAtCompileTime,
MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
MatrixOptions = internal::traits<MatrixType>::Options,
WorkspaceSize = internal::traits<SVDType>::MatrixVColsAtCompileTime,
MaxWorkspaceSize = internal::traits<SVDType>::MatrixVMaxColsAtCompileTime
};
typedef Matrix<Scalar, WorkspaceSize, 1, ColMajor, MaxWorkspaceSize, 1> WorkspaceType;
typedef typename internal::make_proper_matrix_type<Scalar, ColsAtCompileTime, RowsAtCompileTime, MatrixOptions,
MaxColsAtCompileTime, MaxRowsAtCompileTime>::type
TransposeTypeWithSameStorageOrder;
void allocate(const SVDType& svd) {
if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols()) {
internal::destroy_at(&m_qr);
internal::construct_at(&m_qr, svd.cols(), svd.rows());
}
if (svd.m_computeFullV)
m_workspace.resize(svd.cols());
else if (svd.m_computeThinV)
m_workspace.resize(svd.rows());
}
template <typename Xpr>
bool run(SVDType& svd, const Xpr& matrix) {
if (matrix.cols() > matrix.rows()) {
m_qr.compute(matrix.adjoint());
svd.m_workMatrix =
m_qr.matrixQR().block(0, 0, matrix.rows(), matrix.rows()).template triangularView<Upper>().adjoint();
if (svd.m_computeFullV)
m_qr.householderQ().evalTo(svd.m_matrixV, m_workspace);
else if (svd.m_computeThinV) {
svd.m_matrixV.setIdentity(matrix.cols(), matrix.rows());
m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixV, m_workspace);
}
if (svd.computeU()) svd.m_matrixU = m_qr.colsPermutation();
return true;
} else
return false;
}
private:
typedef ColPivHouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
QRType m_qr;
WorkspaceType m_workspace;
};
/*** preconditioner using HouseholderQR ***/
template <typename MatrixType, int Options>
class qr_preconditioner_impl<MatrixType, Options, HouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true> {
public:
typedef typename MatrixType::Scalar Scalar;
typedef JacobiSVD<MatrixType, Options> SVDType;
enum {
WorkspaceSize = internal::traits<SVDType>::MatrixUColsAtCompileTime,
MaxWorkspaceSize = internal::traits<SVDType>::MatrixUMaxColsAtCompileTime
};
typedef Matrix<Scalar, 1, WorkspaceSize, RowMajor, 1, MaxWorkspaceSize> WorkspaceType;
void allocate(const SVDType& svd) {
if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols()) {
internal::destroy_at(&m_qr);
internal::construct_at(&m_qr, svd.rows(), svd.cols());
}
if (svd.m_computeFullU)
m_workspace.resize(svd.rows());
else if (svd.m_computeThinU)
m_workspace.resize(svd.cols());
}
template <typename Xpr>
bool run(SVDType& svd, const Xpr& matrix) {
if (matrix.rows() > matrix.cols()) {
m_qr.compute(matrix);
svd.m_workMatrix = m_qr.matrixQR().block(0, 0, matrix.cols(), matrix.cols()).template triangularView<Upper>();
if (svd.m_computeFullU)
m_qr.householderQ().evalTo(svd.m_matrixU, m_workspace);
else if (svd.m_computeThinU) {
svd.m_matrixU.setIdentity(matrix.rows(), matrix.cols());
m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixU, m_workspace);
}
if (svd.computeV()) svd.m_matrixV.setIdentity(matrix.cols(), matrix.cols());
return true;
}
return false;
}
private:
typedef HouseholderQR<MatrixType> QRType;
QRType m_qr;
WorkspaceType m_workspace;
};
template <typename MatrixType, int Options>
class qr_preconditioner_impl<MatrixType, Options, HouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true> {
public:
typedef typename MatrixType::Scalar Scalar;
typedef JacobiSVD<MatrixType, Options> SVDType;
enum {
RowsAtCompileTime = MatrixType::RowsAtCompileTime,
ColsAtCompileTime = MatrixType::ColsAtCompileTime,
MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
MatrixOptions = internal::traits<MatrixType>::Options,
WorkspaceSize = internal::traits<SVDType>::MatrixVColsAtCompileTime,
MaxWorkspaceSize = internal::traits<SVDType>::MatrixVMaxColsAtCompileTime
};
typedef Matrix<Scalar, WorkspaceSize, 1, ColMajor, MaxWorkspaceSize, 1> WorkspaceType;
typedef typename internal::make_proper_matrix_type<Scalar, ColsAtCompileTime, RowsAtCompileTime, MatrixOptions,
MaxColsAtCompileTime, MaxRowsAtCompileTime>::type
TransposeTypeWithSameStorageOrder;
void allocate(const SVDType& svd) {
if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols()) {
internal::destroy_at(&m_qr);
internal::construct_at(&m_qr, svd.cols(), svd.rows());
}
if (svd.m_computeFullV)
m_workspace.resize(svd.cols());
else if (svd.m_computeThinV)
m_workspace.resize(svd.rows());
}
template <typename Xpr>
bool run(SVDType& svd, const Xpr& matrix) {
if (matrix.cols() > matrix.rows()) {
m_qr.compute(matrix.adjoint());
svd.m_workMatrix =
m_qr.matrixQR().block(0, 0, matrix.rows(), matrix.rows()).template triangularView<Upper>().adjoint();
if (svd.m_computeFullV)
m_qr.householderQ().evalTo(svd.m_matrixV, m_workspace);
else if (svd.m_computeThinV) {
svd.m_matrixV.setIdentity(matrix.cols(), matrix.rows());
m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixV, m_workspace);
}
if (svd.computeU()) svd.m_matrixU.setIdentity(matrix.rows(), matrix.rows());
return true;
} else
return false;
}
private:
typedef HouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
QRType m_qr;
WorkspaceType m_workspace;
};
/*** 2x2 SVD implementation
***
*** JacobiSVD consists in performing a series of 2x2 SVD subproblems
***/
template <typename MatrixType, int Options>
struct svd_precondition_2x2_block_to_be_real<MatrixType, Options, false> {
typedef JacobiSVD<MatrixType, Options> SVD;
typedef typename MatrixType::RealScalar RealScalar;
static bool run(typename SVD::WorkMatrixType&, SVD&, Index, Index, RealScalar&) { return true; }
};
template <typename MatrixType, int Options>
struct svd_precondition_2x2_block_to_be_real<MatrixType, Options, true> {
typedef JacobiSVD<MatrixType, Options> SVD;
typedef typename MatrixType::Scalar Scalar;
typedef typename MatrixType::RealScalar RealScalar;
static bool run(typename SVD::WorkMatrixType& work_matrix, SVD& svd, Index p, Index q, RealScalar& maxDiagEntry) {
using std::abs;
using std::sqrt;
Scalar z;
JacobiRotation<Scalar> rot;
RealScalar n = sqrt(numext::abs2(work_matrix.coeff(p, p)) + numext::abs2(work_matrix.coeff(q, p)));
const RealScalar considerAsZero = (std::numeric_limits<RealScalar>::min)();
const RealScalar precision = NumTraits<Scalar>::epsilon();
if (numext::is_exactly_zero(n)) {
// make sure first column is zero
work_matrix.coeffRef(p, p) = work_matrix.coeffRef(q, p) = Scalar(0);
if (abs(numext::imag(work_matrix.coeff(p, q))) > considerAsZero) {
// work_matrix.coeff(p,q) can be zero if work_matrix.coeff(q,p) is not zero but small enough to underflow when
// computing n
z = abs(work_matrix.coeff(p, q)) / work_matrix.coeff(p, q);
work_matrix.row(p) *= z;
if (svd.computeU()) svd.m_matrixU.col(p) *= conj(z);
}
if (abs(numext::imag(work_matrix.coeff(q, q))) > considerAsZero) {
z = abs(work_matrix.coeff(q, q)) / work_matrix.coeff(q, q);
work_matrix.row(q) *= z;
if (svd.computeU()) svd.m_matrixU.col(q) *= conj(z);
}
// otherwise the second row is already zero, so we have nothing to do.
} else {
rot.c() = conj(work_matrix.coeff(p, p)) / n;
rot.s() = work_matrix.coeff(q, p) / n;
work_matrix.applyOnTheLeft(p, q, rot);
if (svd.computeU()) svd.m_matrixU.applyOnTheRight(p, q, rot.adjoint());
if (abs(numext::imag(work_matrix.coeff(p, q))) > considerAsZero) {
z = abs(work_matrix.coeff(p, q)) / work_matrix.coeff(p, q);
work_matrix.col(q) *= z;
if (svd.computeV()) svd.m_matrixV.col(q) *= z;
}
if (abs(numext::imag(work_matrix.coeff(q, q))) > considerAsZero) {
z = abs(work_matrix.coeff(q, q)) / work_matrix.coeff(q, q);
work_matrix.row(q) *= z;
if (svd.computeU()) svd.m_matrixU.col(q) *= conj(z);
}
}
// update largest diagonal entry
maxDiagEntry = numext::maxi<RealScalar>(
maxDiagEntry, numext::maxi<RealScalar>(abs(work_matrix.coeff(p, p)), abs(work_matrix.coeff(q, q))));
// and check whether the 2x2 block is already diagonal
RealScalar threshold = numext::maxi<RealScalar>(considerAsZero, precision * maxDiagEntry);
return abs(work_matrix.coeff(p, q)) > threshold || abs(work_matrix.coeff(q, p)) > threshold;
}
};
template <typename MatrixType_, int Options>
struct traits<JacobiSVD<MatrixType_, Options> > : svd_traits<MatrixType_, Options> {
typedef MatrixType_ MatrixType;
};
} // end namespace internal
/** \ingroup SVD_Module
*
*
* \class JacobiSVD
*
* \brief Two-sided Jacobi SVD decomposition of a rectangular matrix
*
* \tparam MatrixType_ the type of the matrix of which we are computing the SVD decomposition
* \tparam Options this optional parameter allows one to specify the type of QR decomposition that will be used
* internally for the R-SVD step for non-square matrices. Additionally, it allows one to specify whether to compute thin
* or full unitaries \a U and \a V. See discussion of possible values below.
*
* SVD decomposition consists in decomposing any n-by-p matrix \a A as a product
* \f[ A = U S V^* \f]
* where \a U is a n-by-n unitary, \a V is a p-by-p unitary, and \a S is a n-by-p real positive matrix which is zero
* outside of its main diagonal; the diagonal entries of S are known as the \em singular \em values of \a A and the
* columns of \a U and \a V are known as the left and right \em singular \em vectors of \a A respectively.
*
* Singular values are always sorted in decreasing order.
*
* This JacobiSVD decomposition computes only the singular values by default. If you want \a U or \a V, you need to ask
* for them explicitly.
*
* You can ask for only \em thin \a U or \a V to be computed, meaning the following. In case of a rectangular n-by-p
* matrix, letting \a m be the smaller value among \a n and \a p, there are only \a m singular vectors; the remaining
* columns of \a U and \a V do not correspond to actual singular vectors. Asking for \em thin \a U or \a V means asking
* for only their \a m first columns to be formed. So \a U is then a n-by-m matrix, and \a V is then a p-by-m matrix.
* Notice that thin \a U and \a V are all you need for (least squares) solving.
*
* Here's an example demonstrating basic usage:
* \include JacobiSVD_basic.cpp
* Output: \verbinclude JacobiSVD_basic.out
*
* This JacobiSVD class is a two-sided Jacobi R-SVD decomposition, ensuring optimal reliability and accuracy. The
* downside is that it's slower than bidiagonalizing SVD algorithms for large square matrices; however its complexity is
* still \f$ O(n^2p) \f$ where \a n is the smaller dimension and \a p is the greater dimension, meaning that it is still
* of the same order of complexity as the faster bidiagonalizing R-SVD algorithms. In particular, like any R-SVD, it
* takes advantage of non-squareness in that its complexity is only linear in the greater dimension.
*
* If the input matrix has inf or nan coefficients, the result of the computation is undefined, but the computation is
* guaranteed to terminate in finite (and reasonable) time.
*
* The possible QR preconditioners that can be set with Options template parameter are:
* \li ColPivHouseholderQRPreconditioner is the default. In practice it's very safe. It uses column-pivoting QR.
* \li FullPivHouseholderQRPreconditioner, is the safest and slowest. It uses full-pivoting QR.
* Contrary to other QRs, it doesn't allow computing thin unitaries.
* \li HouseholderQRPreconditioner is the fastest, and less safe and accurate than the pivoting variants. It uses
* non-pivoting QR. This is very similar in safety and accuracy to the bidiagonalization process used by bidiagonalizing
* SVD algorithms (since bidiagonalization is inherently non-pivoting). However the resulting SVD is still more reliable
* than bidiagonalizing SVDs because the Jacobi-based iterarive process is more reliable than the optimized bidiagonal
* SVD iterations. \li NoQRPreconditioner allows not to use a QR preconditioner at all. This is useful if you know that
* you will only be computing JacobiSVD decompositions of square matrices. Non-square matrices require a QR
* preconditioner. Using this option will result in faster compilation and smaller executable code. It won't
* significantly speed up computation, since JacobiSVD is always checking if QR preconditioning is needed before
* applying it anyway.
*
* One may also use the Options template parameter to specify how the unitaries should be computed. The options are
* #ComputeThinU, #ComputeThinV, #ComputeFullU, #ComputeFullV. It is not possible to request both the thin and full
* versions of a unitary. By default, unitaries will not be computed.
*
* You can set the QRPreconditioner and unitary options together: JacobiSVD<MatrixType,
* ColPivHouseholderQRPreconditioner | ComputeThinU | ComputeFullV>
*
* \sa MatrixBase::jacobiSvd()
*/
template <typename MatrixType_, int Options_>
class JacobiSVD : public SVDBase<JacobiSVD<MatrixType_, Options_> > {
typedef SVDBase<JacobiSVD> Base;
public:
typedef MatrixType_ MatrixType;
typedef typename Base::Scalar Scalar;
typedef typename Base::RealScalar RealScalar;
enum : int {
Options = Options_,
QRPreconditioner = internal::get_qr_preconditioner(Options),
RowsAtCompileTime = Base::RowsAtCompileTime,
ColsAtCompileTime = Base::ColsAtCompileTime,
DiagSizeAtCompileTime = Base::DiagSizeAtCompileTime,
MaxRowsAtCompileTime = Base::MaxRowsAtCompileTime,
MaxColsAtCompileTime = Base::MaxColsAtCompileTime,
MaxDiagSizeAtCompileTime = Base::MaxDiagSizeAtCompileTime,
MatrixOptions = Base::MatrixOptions
};
typedef typename Base::MatrixUType MatrixUType;
typedef typename Base::MatrixVType MatrixVType;
typedef typename Base::SingularValuesType SingularValuesType;
typedef Matrix<Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime, MatrixOptions, MaxDiagSizeAtCompileTime,
MaxDiagSizeAtCompileTime>
WorkMatrixType;
/** \brief Default Constructor.
*
* The default constructor is useful in cases in which the user intends to
* perform decompositions via JacobiSVD::compute(const MatrixType&).
*/
JacobiSVD() {}
/** \brief Default Constructor with memory preallocation
*
* Like the default constructor but with preallocation of the internal data
* according to the specified problem size and \a Options template parameter.
*
* \sa JacobiSVD()
*/
JacobiSVD(Index rows, Index cols) { allocate(rows, cols, internal::get_computation_options(Options)); }
/** \brief Default Constructor with memory preallocation
*
* Like the default constructor but with preallocation of the internal data
* according to the specified problem size.
*
* One \b cannot request unitaries using both the \a Options template parameter
* and the constructor. If possible, prefer using the \a Options template parameter.
*
* \param computationOptions specify whether to compute Thin/Full unitaries U/V
* \sa JacobiSVD()
*
* \deprecated Will be removed in the next major Eigen version. Options should
* be specified in the \a Options template parameter.
*/
EIGEN_DEPRECATED JacobiSVD(Index rows, Index cols, unsigned int computationOptions) {
internal::check_svd_options_assertions<MatrixType, Options>(computationOptions, rows, cols);
allocate(rows, cols, computationOptions);
}
/** \brief Constructor performing the decomposition of given matrix, using the custom options specified
* with the \a Options template paramter.
*
* \param matrix the matrix to decompose
*/
explicit JacobiSVD(const MatrixType& matrix) { compute_impl(matrix, internal::get_computation_options(Options)); }
/** \brief Constructor performing the decomposition of given matrix using specified options
* for computing unitaries.
*
* One \b cannot request unitiaries using both the \a Options template parameter
* and the constructor. If possible, prefer using the \a Options template parameter.
*
* \param matrix the matrix to decompose
* \param computationOptions specify whether to compute Thin/Full unitaries U/V
*
* \deprecated Will be removed in the next major Eigen version. Options should
* be specified in the \a Options template parameter.
*/
// EIGEN_DEPRECATED // TODO(cantonios): re-enable after fixing a few 3p libraries that error on deprecation warnings.
JacobiSVD(const MatrixType& matrix, unsigned int computationOptions) {
internal::check_svd_options_assertions<MatrixType, Options>(computationOptions, matrix.rows(), matrix.cols());
compute_impl(matrix, computationOptions);
}
/** \brief Method performing the decomposition of given matrix. Computes Thin/Full unitaries U/V if specified
* using the \a Options template parameter or the class constructor.
*
* \param matrix the matrix to decompose
*/
JacobiSVD& compute(const MatrixType& matrix) { return compute_impl(matrix, m_computationOptions); }
/** \brief Method performing the decomposition of given matrix, as specified by
* the `computationOptions` parameter.
*
* \param matrix the matrix to decompose
* \param computationOptions specify whether to compute Thin/Full unitaries U/V
*
* \deprecated Will be removed in the next major Eigen version. Options should
* be specified in the \a Options template parameter.
*/
EIGEN_DEPRECATED JacobiSVD& compute(const MatrixType& matrix, unsigned int computationOptions) {
internal::check_svd_options_assertions<MatrixType, Options>(m_computationOptions, matrix.rows(), matrix.cols());
return compute_impl(matrix, computationOptions);
}
using Base::cols;
using Base::computeU;
using Base::computeV;
using Base::diagSize;
using Base::rank;
using Base::rows;
private:
void allocate(Index rows_, Index cols_, unsigned int computationOptions) {
if (Base::allocate(rows_, cols_, computationOptions)) return;
eigen_assert(!(ShouldComputeThinU && int(QRPreconditioner) == int(FullPivHouseholderQRPreconditioner)) &&
!(ShouldComputeThinU && int(QRPreconditioner) == int(FullPivHouseholderQRPreconditioner)) &&
"JacobiSVD: can't compute thin U or thin V with the FullPivHouseholderQR preconditioner. "
"Use the ColPivHouseholderQR preconditioner instead.");
m_workMatrix.resize(diagSize(), diagSize());
if (cols() > rows()) m_qr_precond_morecols.allocate(*this);
if (rows() > cols()) m_qr_precond_morerows.allocate(*this);
}
JacobiSVD& compute_impl(const MatrixType& matrix, unsigned int computationOptions);
protected:
using Base::m_computationOptions;
using Base::m_computeFullU;
using Base::m_computeFullV;
using Base::m_computeThinU;
using Base::m_computeThinV;
using Base::m_info;
using Base::m_isAllocated;
using Base::m_isInitialized;
using Base::m_matrixU;
using Base::m_matrixV;
using Base::m_nonzeroSingularValues;
using Base::m_prescribedThreshold;
using Base::m_singularValues;
using Base::m_usePrescribedThreshold;
using Base::ShouldComputeThinU;
using Base::ShouldComputeThinV;
EIGEN_STATIC_ASSERT(!(ShouldComputeThinU && int(QRPreconditioner) == int(FullPivHouseholderQRPreconditioner)) &&
!(ShouldComputeThinU && int(QRPreconditioner) == int(FullPivHouseholderQRPreconditioner)),
"JacobiSVD: can't compute thin U or thin V with the FullPivHouseholderQR preconditioner. "
"Use the ColPivHouseholderQR preconditioner instead.")
template <typename MatrixType__, int Options__, bool IsComplex_>
friend struct internal::svd_precondition_2x2_block_to_be_real;
template <typename MatrixType__, int Options__, int QRPreconditioner_, int Case_, bool DoAnything_>
friend struct internal::qr_preconditioner_impl;
internal::qr_preconditioner_impl<MatrixType, Options, QRPreconditioner, internal::PreconditionIfMoreColsThanRows>
m_qr_precond_morecols;
internal::qr_preconditioner_impl<MatrixType, Options, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols>
m_qr_precond_morerows;
WorkMatrixType m_workMatrix;
};
template <typename MatrixType, int Options>
JacobiSVD<MatrixType, Options>& JacobiSVD<MatrixType, Options>::compute_impl(const MatrixType& matrix,
unsigned int computationOptions) {
using std::abs;
allocate(matrix.rows(), matrix.cols(), computationOptions);
// currently we stop when we reach precision 2*epsilon as the last bit of precision can require an unreasonable number
// of iterations, only worsening the precision of U and V as we accumulate more rotations
const RealScalar precision = RealScalar(2) * NumTraits<Scalar>::epsilon();
// limit for denormal numbers to be considered zero in order to avoid infinite loops (see bug 286)
const RealScalar considerAsZero = (std::numeric_limits<RealScalar>::min)();
// Scaling factor to reduce over/under-flows
RealScalar scale = matrix.cwiseAbs().template maxCoeff<PropagateNaN>();
if (!(numext::isfinite)(scale)) {
m_isInitialized = true;
m_info = InvalidInput;
m_nonzeroSingularValues = 0;
return *this;
}
if (numext::is_exactly_zero(scale)) scale = RealScalar(1);
/*** step 1. The R-SVD step: we use a QR decomposition to reduce to the case of a square matrix */
if (rows() != cols()) {
m_qr_precond_morecols.run(*this, matrix / scale);
m_qr_precond_morerows.run(*this, matrix / scale);
} else {
m_workMatrix =
matrix.template topLeftCorner<DiagSizeAtCompileTime, DiagSizeAtCompileTime>(diagSize(), diagSize()) / scale;
if (m_computeFullU) m_matrixU.setIdentity(rows(), rows());
if (m_computeThinU) m_matrixU.setIdentity(rows(), diagSize());
if (m_computeFullV) m_matrixV.setIdentity(cols(), cols());
if (m_computeThinV) m_matrixV.setIdentity(cols(), diagSize());
}
/*** step 2. The main Jacobi SVD iteration. ***/
RealScalar maxDiagEntry = m_workMatrix.cwiseAbs().diagonal().maxCoeff();
bool finished = false;
while (!finished) {
finished = true;
// do a sweep: for all index pairs (p,q), perform SVD of the corresponding 2x2 sub-matrix
for (Index p = 1; p < diagSize(); ++p) {
for (Index q = 0; q < p; ++q) {
// if this 2x2 sub-matrix is not diagonal already...
// notice that this comparison will evaluate to false if any NaN is involved, ensuring that NaN's don't
// keep us iterating forever. Similarly, small denormal numbers are considered zero.
RealScalar threshold = numext::maxi<RealScalar>(considerAsZero, precision * maxDiagEntry);
if (abs(m_workMatrix.coeff(p, q)) > threshold || abs(m_workMatrix.coeff(q, p)) > threshold) {
finished = false;
// perform SVD decomposition of 2x2 sub-matrix corresponding to indices p,q to make it diagonal
// the complex to real operation returns true if the updated 2x2 block is not already diagonal
if (internal::svd_precondition_2x2_block_to_be_real<MatrixType, Options>::run(m_workMatrix, *this, p, q,
maxDiagEntry)) {
JacobiRotation<RealScalar> j_left, j_right;
internal::real_2x2_jacobi_svd(m_workMatrix, p, q, &j_left, &j_right);
// accumulate resulting Jacobi rotations
m_workMatrix.applyOnTheLeft(p, q, j_left);
if (computeU()) m_matrixU.applyOnTheRight(p, q, j_left.transpose());
m_workMatrix.applyOnTheRight(p, q, j_right);
if (computeV()) m_matrixV.applyOnTheRight(p, q, j_right);
// keep track of the largest diagonal coefficient
maxDiagEntry = numext::maxi<RealScalar>(
maxDiagEntry, numext::maxi<RealScalar>(abs(m_workMatrix.coeff(p, p)), abs(m_workMatrix.coeff(q, q))));
}
}
}
}
}
/*** step 3. The work matrix is now diagonal, so ensure it's positive so its diagonal entries are the singular values
* ***/
for (Index i = 0; i < diagSize(); ++i) {
// For a complex matrix, some diagonal coefficients might note have been
// treated by svd_precondition_2x2_block_to_be_real, and the imaginary part
// of some diagonal entry might not be null.
if (NumTraits<Scalar>::IsComplex && abs(numext::imag(m_workMatrix.coeff(i, i))) > considerAsZero) {
RealScalar a = abs(m_workMatrix.coeff(i, i));
m_singularValues.coeffRef(i) = abs(a);
if (computeU()) m_matrixU.col(i) *= m_workMatrix.coeff(i, i) / a;
} else {
// m_workMatrix.coeff(i,i) is already real, no difficulty:
RealScalar a = numext::real(m_workMatrix.coeff(i, i));
m_singularValues.coeffRef(i) = abs(a);
if (computeU() && (a < RealScalar(0))) m_matrixU.col(i) = -m_matrixU.col(i);
}
}
m_singularValues *= scale;
/*** step 4. Sort singular values in descending order and compute the number of nonzero singular values ***/
m_nonzeroSingularValues = diagSize();
for (Index i = 0; i < diagSize(); i++) {
Index pos;
RealScalar maxRemainingSingularValue = m_singularValues.tail(diagSize() - i).maxCoeff(&pos);
if (numext::is_exactly_zero(maxRemainingSingularValue)) {
m_nonzeroSingularValues = i;
break;
}
if (pos) {
pos += i;
std::swap(m_singularValues.coeffRef(i), m_singularValues.coeffRef(pos));
if (computeU()) m_matrixU.col(pos).swap(m_matrixU.col(i));
if (computeV()) m_matrixV.col(pos).swap(m_matrixV.col(i));
}
}
m_isInitialized = true;
return *this;
}
/** \svd_module
*
* \return the singular value decomposition of \c *this computed by two-sided
* Jacobi transformations.
*
* \sa class JacobiSVD
*/
template <typename Derived>
template <int Options>
JacobiSVD<typename MatrixBase<Derived>::PlainObject, Options> MatrixBase<Derived>::jacobiSvd() const {
return JacobiSVD<PlainObject, Options>(*this);
}
template <typename Derived>
template <int Options>
JacobiSVD<typename MatrixBase<Derived>::PlainObject, Options> MatrixBase<Derived>::jacobiSvd(
unsigned int computationOptions) const {
return JacobiSVD<PlainObject, Options>(*this, computationOptions);
}
} // end namespace Eigen
#endif // EIGEN_JACOBISVD_H