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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// We used the "A Divide-And-Conquer Algorithm for the Bidiagonal SVD"
// research report written by Ming Gu and Stanley C.Eisenstat
// The code variable names correspond to the names they used in their
// report
//
// Copyright (C) 2013 Gauthier Brun <brun.gauthier@gmail.com>
// Copyright (C) 2013 Nicolas Carre <nicolas.carre@ensimag.fr>
// Copyright (C) 2013 Jean Ceccato <jean.ceccato@ensimag.fr>
// Copyright (C) 2013 Pierre Zoppitelli <pierre.zoppitelli@ensimag.fr>
// Copyright (C) 2013 Jitse Niesen <jitse@maths.leeds.ac.uk>
// Copyright (C) 2014-2017 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// 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_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 <iostream>
#endif
namespace Eigen {
#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
IOFormat bdcsvdfmt(8, 0, ", ", "\n", " [", "]");
#endif
template <typename MatrixType_, int Options>
class BDCSVD;
namespace internal {
template <typename MatrixType_, int Options>
struct traits<BDCSVD<MatrixType_, Options> > : svd_traits<MatrixType_, Options> {
typedef MatrixType_ MatrixType;
};
} // end namespace internal
/** \ingroup SVD_Module
*
*
* \class BDCSVD
*
* \brief class Bidiagonal Divide and Conquer SVD
*
* \tparam MatrixType_ the type of the matrix of which we are computing the SVD decomposition
*
* \tparam Options_ this optional parameter allows one to specify options for computing unitaries \a U and \a V.
* Possible values are #ComputeThinU, #ComputeThinV, #ComputeFullU, #ComputeFullV, and
* #DisableQRDecomposition. It is not possible to request both the thin and full version of \a U or
* \a V. By default, unitaries are not computed. BDCSVD uses R-Bidiagonalization to improve
* performance on tall and wide matrices. For backwards compatility, the option
* #DisableQRDecomposition can be used to disable this optimization.
*
* This class first reduces the input matrix to bi-diagonal form using class UpperBidiagonalization,
* and then performs a divide-and-conquer diagonalization. Small blocks are diagonalized using class JacobiSVD.
* You can control the switching size with the setSwitchSize() method, default is 16.
* For small matrice (<16), it is thus preferable to directly use JacobiSVD. For larger ones, BDCSVD is highly
* recommended and can several order of magnitude faster.
*
* \warning this algorithm is unlikely to provide accurate result when compiled with unsafe math optimizations.
* For instance, this concerns Intel's compiler (ICC), which performs such optimization by default unless
* you compile with the \c -fp-model \c precise option. Likewise, the \c -ffast-math option of GCC or clang will
* significantly degrade the accuracy.
*
* \sa class JacobiSVD
*/
template <typename MatrixType_, int Options_>
class BDCSVD : public SVDBase<BDCSVD<MatrixType_, Options_> > {
typedef SVDBase<BDCSVD> Base;
public:
using Base::cols;
using Base::computeU;
using Base::computeV;
using Base::diagSize;
using Base::rows;
typedef MatrixType_ MatrixType;
typedef typename Base::Scalar Scalar;
typedef typename Base::RealScalar RealScalar;
typedef typename NumTraits<RealScalar>::Literal Literal;
typedef typename Base::Index Index;
enum {
Options = Options_,
QRDecomposition = Options & internal::QRPreconditionerBits,
ComputationOptions = Options & internal::ComputationOptionsBits,
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, Dynamic, Dynamic, ColMajor> MatrixX;
typedef Matrix<RealScalar, Dynamic, Dynamic, ColMajor> MatrixXr;
typedef Matrix<RealScalar, Dynamic, 1> VectorType;
typedef Array<RealScalar, Dynamic, 1> ArrayXr;
typedef Array<Index, 1, Dynamic> ArrayXi;
typedef Ref<ArrayXr> ArrayRef;
typedef Ref<ArrayXi> IndicesRef;
/** \brief Default Constructor.
*
* The default constructor is useful in cases in which the user intends to
* perform decompositions via BDCSVD::compute(const MatrixType&).
*/
BDCSVD() : m_algoswap(16), m_isTranspose(false), m_compU(false), m_compV(false), m_numIters(0) {}
/** \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 BDCSVD()
*/
BDCSVD(Index rows, Index cols) : m_algoswap(16), m_numIters(0) {
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 and the \a computationOptions.
*
* 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 specification for computing Thin/Full unitaries U/V
* \sa BDCSVD()
*
* \deprecated Will be removed in the next major Eigen version. Options should
* be specified in the \a Options template parameter.
*/
EIGEN_DEPRECATED BDCSVD(Index rows, Index cols, unsigned int computationOptions) : m_algoswap(16), m_numIters(0) {
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 parameter.
*
* \param matrix the matrix to decompose
*/
BDCSVD(const MatrixType& matrix) : m_algoswap(16), m_numIters(0) {
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 unitaries 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 specification for computing 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 BDCSVD(const MatrixType& matrix, unsigned int computationOptions) : m_algoswap(16), m_numIters(0) {
internal::check_svd_options_assertions<MatrixType, Options>(computationOptions, matrix.rows(), matrix.cols());
compute_impl(matrix, computationOptions);
}
~BDCSVD() {}
/** \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
*/
BDCSVD& 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 BDCSVD& compute(const MatrixType& matrix, unsigned int computationOptions) {
internal::check_svd_options_assertions<MatrixType, Options>(computationOptions, matrix.rows(), matrix.cols());
return compute_impl(matrix, computationOptions);
}
void setSwitchSize(int s) {
eigen_assert(s >= 3 && "BDCSVD the size of the algo switch has to be at least 3.");
m_algoswap = s;
}
private:
BDCSVD& compute_impl(const MatrixType& matrix, unsigned int computationOptions);
void divide(Index firstCol, Index lastCol, Index firstRowW, Index firstColW, Index shift);
void computeSVDofM(Index firstCol, Index n, MatrixXr& U, VectorType& singVals, MatrixXr& V);
void computeSingVals(const ArrayRef& col0, const ArrayRef& diag, const IndicesRef& perm, VectorType& singVals,
ArrayRef shifts, ArrayRef mus);
void perturbCol0(const ArrayRef& col0, const ArrayRef& diag, const IndicesRef& perm, const VectorType& singVals,
const ArrayRef& shifts, const ArrayRef& mus, ArrayRef zhat);
void computeSingVecs(const ArrayRef& zhat, const ArrayRef& diag, const IndicesRef& perm, const VectorType& singVals,
const ArrayRef& shifts, const ArrayRef& mus, MatrixXr& U, MatrixXr& V);
void deflation43(Index firstCol, Index shift, Index i, Index size);
void deflation44(Index firstColu, Index firstColm, Index firstRowW, Index firstColW, Index i, Index j, Index size);
void deflation(Index firstCol, Index lastCol, Index k, Index firstRowW, Index firstColW, Index shift);
template <typename HouseholderU, typename HouseholderV, typename NaiveU, typename NaiveV>
void copyUV(const HouseholderU& householderU, const HouseholderV& householderV, const NaiveU& naiveU,
const NaiveV& naivev);
void structured_update(Block<MatrixXr, Dynamic, Dynamic> A, const MatrixXr& B, Index n1);
static RealScalar secularEq(RealScalar x, const ArrayRef& col0, const ArrayRef& diag, const IndicesRef& perm,
const ArrayRef& diagShifted, RealScalar shift);
template <typename SVDType>
void computeBaseCase(SVDType& svd, Index n, Index firstCol, Index firstRowW, Index firstColW, Index shift);
protected:
void allocate(Index rows, Index cols, unsigned int computationOptions);
MatrixXr m_naiveU, m_naiveV;
MatrixXr m_computed;
Index m_nRec;
ArrayXr m_workspace;
ArrayXi m_workspaceI;
int m_algoswap;
bool m_isTranspose, m_compU, m_compV, m_useQrDecomp;
JacobiSVD<MatrixType, ComputationOptions> smallSvd;
HouseholderQR<MatrixX> qrDecomp;
internal::UpperBidiagonalization<MatrixX> bid;
MatrixX copyWorkspace;
MatrixX reducedTriangle;
using Base::m_computationOptions;
using Base::m_computeThinU;
using Base::m_computeThinV;
using Base::m_info;
using Base::m_isInitialized;
using Base::m_matrixU;
using Base::m_matrixV;
using Base::m_nonzeroSingularValues;
using Base::m_singularValues;
public:
int m_numIters;
}; // end class BDCSVD
// Method to allocate and initialize matrix and attributes
template <typename MatrixType, int Options>
void BDCSVD<MatrixType, Options>::allocate(Index rows, Index cols, unsigned int computationOptions) {
if (Base::allocate(rows, cols, computationOptions)) return;
if (cols < m_algoswap)
smallSvd.allocate(rows, cols, Options == 0 ? computationOptions : internal::get_computation_options(Options));
m_computed = MatrixXr::Zero(diagSize() + 1, diagSize());
m_compU = computeV();
m_compV = computeU();
m_isTranspose = (cols > rows);
if (m_isTranspose) std::swap(m_compU, m_compV);
// kMinAspectRatio is the crossover point that determines if we perform R-Bidiagonalization
// or bidiagonalize the input matrix directly.
// It is based off of LAPACK's dgesdd routine, which uses 11.0/6.0
// we use a larger scalar to prevent a regression for relatively square matrices.
constexpr Index kMinAspectRatio = 4;
constexpr bool disableQrDecomp = static_cast<int>(QRDecomposition) == static_cast<int>(DisableQRDecomposition);
m_useQrDecomp = !disableQrDecomp && ((rows / kMinAspectRatio > cols) || (cols / kMinAspectRatio > rows));
if (m_useQrDecomp) {
qrDecomp = HouseholderQR<MatrixX>((std::max)(rows, cols), (std::min)(rows, cols));
reducedTriangle = MatrixX(diagSize(), diagSize());
}
copyWorkspace = MatrixX(m_isTranspose ? cols : rows, m_isTranspose ? rows : cols);
bid = internal::UpperBidiagonalization<MatrixX>(m_useQrDecomp ? diagSize() : copyWorkspace.rows(),
m_useQrDecomp ? diagSize() : copyWorkspace.cols());
if (m_compU)
m_naiveU = MatrixXr::Zero(diagSize() + 1, diagSize() + 1);
else
m_naiveU = MatrixXr::Zero(2, diagSize() + 1);
if (m_compV) m_naiveV = MatrixXr::Zero(diagSize(), diagSize());
m_workspace.resize((diagSize() + 1) * (diagSize() + 1) * 3);
m_workspaceI.resize(3 * diagSize());
} // end allocate
template <typename MatrixType, int Options>
BDCSVD<MatrixType, Options>& BDCSVD<MatrixType, Options>::compute_impl(const MatrixType& matrix,
unsigned int computationOptions) {
#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
std::cout << "\n\n\n================================================================================================="
"=====================\n\n\n";
#endif
using std::abs;
allocate(matrix.rows(), matrix.cols(), computationOptions);
const RealScalar considerZero = (std::numeric_limits<RealScalar>::min)();
//**** step -1 - If the problem is too small, directly falls back to JacobiSVD and return
if (matrix.cols() < m_algoswap) {
smallSvd.compute(matrix);
m_isInitialized = true;
m_info = smallSvd.info();
if (m_info == Success || m_info == NoConvergence) {
if (computeU()) m_matrixU = smallSvd.matrixU();
if (computeV()) m_matrixV = smallSvd.matrixV();
m_singularValues = smallSvd.singularValues();
m_nonzeroSingularValues = smallSvd.nonzeroSingularValues();
}
return *this;
}
//**** step 0 - Copy the input matrix and apply scaling to reduce over/under-flows
RealScalar scale = matrix.cwiseAbs().template maxCoeff<PropagateNaN>();
if (!(numext::isfinite)(scale)) {
m_isInitialized = true;
m_info = InvalidInput;
return *this;
}
if (numext::is_exactly_zero(scale)) scale = Literal(1);
if (m_isTranspose)
copyWorkspace = matrix.adjoint() / scale;
else
copyWorkspace = matrix / scale;
//**** step 1 - Bidiagonalization.
// If the problem is sufficiently rectangular, we perform R-Bidiagonalization: compute A = Q(R/0)
// and then bidiagonalize R. Otherwise, if the problem is relatively square, we
// bidiagonalize the input matrix directly.
if (m_useQrDecomp) {
qrDecomp.compute(copyWorkspace);
reducedTriangle = qrDecomp.matrixQR().topRows(diagSize());
reducedTriangle.template triangularView<StrictlyLower>().setZero();
bid.compute(reducedTriangle);
} else {
bid.compute(copyWorkspace);
}
//**** step 2 - Divide & Conquer
m_naiveU.setZero();
m_naiveV.setZero();
// FIXME this line involves a temporary matrix
m_computed.topRows(diagSize()) = bid.bidiagonal().toDenseMatrix().transpose();
m_computed.template bottomRows<1>().setZero();
divide(0, diagSize() - 1, 0, 0, 0);
if (m_info != Success && m_info != NoConvergence) {
m_isInitialized = true;
return *this;
}
//**** step 3 - Copy singular values and vectors
for (int i = 0; i < diagSize(); i++) {
RealScalar a = abs(m_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;
}
}
//**** step 4 - Finalize unitaries U and V
if (m_isTranspose)
copyUV(bid.householderV(), bid.householderU(), m_naiveV, m_naiveU);
else
copyUV(bid.householderU(), bid.householderV(), m_naiveU, m_naiveV);
if (m_useQrDecomp) {
if (m_isTranspose && computeV())
m_matrixV.applyOnTheLeft(qrDecomp.householderQ());
else if (!m_isTranspose && computeU())
m_matrixU.applyOnTheLeft(qrDecomp.householderQ());
}
m_isInitialized = true;
return *this;
} // end compute
template <typename MatrixType, int Options>
template <typename HouseholderU, typename HouseholderV, typename NaiveU, typename NaiveV>
void BDCSVD<MatrixType, Options>::copyUV(const HouseholderU& householderU, const HouseholderV& householderV,
const NaiveU& naiveU, const NaiveV& naiveV) {
// Note exchange of U and V: m_matrixU is set from m_naiveV and vice versa
if (computeU()) {
Index Ucols = m_computeThinU ? diagSize() : rows();
m_matrixU = MatrixX::Identity(rows(), Ucols);
m_matrixU.topLeftCorner(diagSize(), diagSize()) =
naiveV.template cast<Scalar>().topLeftCorner(diagSize(), diagSize());
// FIXME the following conditionals involve temporary buffers
if (m_useQrDecomp)
m_matrixU.topLeftCorner(householderU.cols(), diagSize()).applyOnTheLeft(householderU);
else
m_matrixU.applyOnTheLeft(householderU);
}
if (computeV()) {
Index Vcols = m_computeThinV ? diagSize() : cols();
m_matrixV = MatrixX::Identity(cols(), Vcols);
m_matrixV.topLeftCorner(diagSize(), diagSize()) =
naiveU.template cast<Scalar>().topLeftCorner(diagSize(), diagSize());
// FIXME the following conditionals involve temporary buffers
if (m_useQrDecomp)
m_matrixV.topLeftCorner(householderV.cols(), diagSize()).applyOnTheLeft(householderV);
else
m_matrixV.applyOnTheLeft(householderV);
}
}
/** \internal
* Performs A = A * B exploiting the special structure of the matrix A. Splitting A as:
* A = [A1]
* [A2]
* such that A1.rows()==n1, then we assume that at least half of the columns of A1 and A2 are zeros.
* We can thus pack them prior to the the matrix product. However, this is only worth the effort if the matrix is large
* enough.
*/
template <typename MatrixType, int Options>
void BDCSVD<MatrixType, Options>::structured_update(Block<MatrixXr, Dynamic, Dynamic> A, const MatrixXr& B, Index n1) {
Index n = A.rows();
if (n > 100) {
// If the matrices are large enough, let's exploit the sparse structure of A by
// splitting it in half (wrt n1), and packing the non-zero columns.
Index n2 = n - n1;
Map<MatrixXr> A1(m_workspace.data(), n1, n);
Map<MatrixXr> A2(m_workspace.data() + n1 * n, n2, n);
Map<MatrixXr> B1(m_workspace.data() + n * n, n, n);
Map<MatrixXr> B2(m_workspace.data() + 2 * n * n, n, n);
Index k1 = 0, k2 = 0;
for (Index j = 0; j < n; ++j) {
if ((A.col(j).head(n1).array() != Literal(0)).any()) {
A1.col(k1) = A.col(j).head(n1);
B1.row(k1) = B.row(j);
++k1;
}
if ((A.col(j).tail(n2).array() != Literal(0)).any()) {
A2.col(k2) = A.col(j).tail(n2);
B2.row(k2) = B.row(j);
++k2;
}
}
A.topRows(n1).noalias() = A1.leftCols(k1) * B1.topRows(k1);
A.bottomRows(n2).noalias() = A2.leftCols(k2) * B2.topRows(k2);
} else {
Map<MatrixXr, Aligned> tmp(m_workspace.data(), n, n);
tmp.noalias() = A * B;
A = tmp;
}
}
template <typename MatrixType, int Options>
template <typename SVDType>
void BDCSVD<MatrixType, Options>::computeBaseCase(SVDType& svd, Index n, Index firstCol, Index firstRowW,
Index firstColW, Index shift) {
svd.compute(m_computed.block(firstCol, firstCol, n + 1, n));
m_info = svd.info();
if (m_info != Success && m_info != NoConvergence) return;
if (m_compU)
m_naiveU.block(firstCol, firstCol, n + 1, n + 1).real() = svd.matrixU();
else {
m_naiveU.row(0).segment(firstCol, n + 1).real() = svd.matrixU().row(0);
m_naiveU.row(1).segment(firstCol, n + 1).real() = svd.matrixU().row(n);
}
if (m_compV) m_naiveV.block(firstRowW, firstColW, n, n).real() = svd.matrixV();
m_computed.block(firstCol + shift, firstCol + shift, n + 1, n).setZero();
m_computed.diagonal().segment(firstCol + shift, n) = svd.singularValues().head(n);
}
// The divide algorithm is done "in place", we are always working on subsets of the same matrix. The divide methods
// takes as argument the place of the submatrix we are currently working on.
//@param firstCol : The Index of the first column of the submatrix of m_computed and for m_naiveU;
//@param lastCol : The Index of the last column of the submatrix of m_computed and for m_naiveU;
// lastCol + 1 - firstCol is the size of the submatrix.
//@param firstRowW : The Index of the first row of the matrix W that we are to change. (see the reference paper section
// 1 for more information on W)
//@param firstColW : Same as firstRowW with the column.
//@param shift : Each time one takes the left submatrix, one must add 1 to the shift. Why? Because! We actually want the
// last column of the U submatrix
// to become the first column (*coeff) and to shift all the other columns to the right. There are more details on the
// reference paper.
template <typename MatrixType, int Options>
void BDCSVD<MatrixType, Options>::divide(Index firstCol, Index lastCol, Index firstRowW, Index firstColW, Index shift) {
// requires rows = cols + 1;
using std::abs;
using std::pow;
using std::sqrt;
const Index n = lastCol - firstCol + 1;
const Index k = n / 2;
const RealScalar considerZero = (std::numeric_limits<RealScalar>::min)();
RealScalar alphaK;
RealScalar betaK;
RealScalar r0;
RealScalar lambda, phi, c0, s0;
VectorType l, f;
// We use the other algorithm which is more efficient for small
// matrices.
if (n < m_algoswap) {
// FIXME this block involves temporaries
if (m_compV) {
JacobiSVD<MatrixXr, ComputeFullU | ComputeFullV> baseSvd;
computeBaseCase(baseSvd, n, firstCol, firstRowW, firstColW, shift);
} else {
JacobiSVD<MatrixXr, ComputeFullU> baseSvd;
computeBaseCase(baseSvd, n, firstCol, firstRowW, firstColW, shift);
}
return;
}
// We use the divide and conquer algorithm
alphaK = m_computed(firstCol + k, firstCol + k);
betaK = m_computed(firstCol + k + 1, firstCol + k);
// The divide must be done in that order in order to have good results. Divide change the data inside the submatrices
// and the divide of the right submatrice reads one column of the left submatrice. That's why we need to treat the
// right submatrix before the left one.
divide(k + 1 + firstCol, lastCol, k + 1 + firstRowW, k + 1 + firstColW, shift);
if (m_info != Success && m_info != NoConvergence) return;
divide(firstCol, k - 1 + firstCol, firstRowW, firstColW + 1, shift + 1);
if (m_info != Success && m_info != NoConvergence) return;
if (m_compU) {
lambda = m_naiveU(firstCol + k, firstCol + k);
phi = m_naiveU(firstCol + k + 1, lastCol + 1);
} else {
lambda = m_naiveU(1, firstCol + k);
phi = m_naiveU(0, lastCol + 1);
}
r0 = sqrt((abs(alphaK * lambda) * abs(alphaK * lambda)) + abs(betaK * phi) * abs(betaK * phi));
if (m_compU) {
l = m_naiveU.row(firstCol + k).segment(firstCol, k);
f = m_naiveU.row(firstCol + k + 1).segment(firstCol + k + 1, n - k - 1);
} else {
l = m_naiveU.row(1).segment(firstCol, k);
f = m_naiveU.row(0).segment(firstCol + k + 1, n - k - 1);
}
if (m_compV) m_naiveV(firstRowW + k, firstColW) = Literal(1);
if (r0 < considerZero) {
c0 = Literal(1);
s0 = Literal(0);
} else {
c0 = alphaK * lambda / r0;
s0 = betaK * phi / r0;
}
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
eigen_internal_assert(m_naiveU.allFinite());
eigen_internal_assert(m_naiveV.allFinite());
eigen_internal_assert(m_computed.allFinite());
#endif
if (m_compU) {
MatrixXr q1(m_naiveU.col(firstCol + k).segment(firstCol, k + 1));
// we shiftW Q1 to the right
for (Index i = firstCol + k - 1; i >= firstCol; i--)
m_naiveU.col(i + 1).segment(firstCol, k + 1) = m_naiveU.col(i).segment(firstCol, k + 1);
// we shift q1 at the left with a factor c0
m_naiveU.col(firstCol).segment(firstCol, k + 1) = (q1 * c0);
// last column = q1 * - s0
m_naiveU.col(lastCol + 1).segment(firstCol, k + 1) = (q1 * (-s0));
// first column = q2 * s0
m_naiveU.col(firstCol).segment(firstCol + k + 1, n - k) =
m_naiveU.col(lastCol + 1).segment(firstCol + k + 1, n - k) * s0;
// q2 *= c0
m_naiveU.col(lastCol + 1).segment(firstCol + k + 1, n - k) *= c0;
} else {
RealScalar q1 = m_naiveU(0, firstCol + k);
// we shift Q1 to the right
for (Index i = firstCol + k - 1; i >= firstCol; i--) m_naiveU(0, i + 1) = m_naiveU(0, i);
// we shift q1 at the left with a factor c0
m_naiveU(0, firstCol) = (q1 * c0);
// last column = q1 * - s0
m_naiveU(0, lastCol + 1) = (q1 * (-s0));
// first column = q2 * s0
m_naiveU(1, firstCol) = m_naiveU(1, lastCol + 1) * s0;
// q2 *= c0
m_naiveU(1, lastCol + 1) *= c0;
m_naiveU.row(1).segment(firstCol + 1, k).setZero();
m_naiveU.row(0).segment(firstCol + k + 1, n - k - 1).setZero();
}
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
eigen_internal_assert(m_naiveU.allFinite());
eigen_internal_assert(m_naiveV.allFinite());
eigen_internal_assert(m_computed.allFinite());
#endif
m_computed(firstCol + shift, firstCol + shift) = r0;
m_computed.col(firstCol + shift).segment(firstCol + shift + 1, k) = alphaK * l.transpose().real();
m_computed.col(firstCol + shift).segment(firstCol + shift + k + 1, n - k - 1) = betaK * f.transpose().real();
#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);
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
eigen_internal_assert(UofSVD.allFinite());
eigen_internal_assert(VofSVD.allFinite());
#endif
if (m_compU)
structured_update(m_naiveU.block(firstCol, firstCol, n + 1, n + 1), UofSVD, (n + 2) / 2);
else {
Map<Matrix<RealScalar, 2, Dynamic>, Aligned> tmp(m_workspace.data(), 2, n + 1);
tmp.noalias() = m_naiveU.middleCols(firstCol, n + 1) * UofSVD;
m_naiveU.middleCols(firstCol, n + 1) = tmp;
}
if (m_compV) structured_update(m_naiveV.block(firstRowW, firstColW, n, n), VofSVD, (n + 1) / 2);
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
eigen_internal_assert(m_naiveU.allFinite());
eigen_internal_assert(m_naiveV.allFinite());
eigen_internal_assert(m_computed.allFinite());
#endif
m_computed.block(firstCol + shift, firstCol + shift, n, n).setZero();
m_computed.block(firstCol + shift, firstCol + shift, n, n).diagonal() = singVals;
} // end divide
// Compute SVD of m_computed.block(firstCol, firstCol, n + 1, n); this block only has non-zeros in
// the first column and on the diagonal and has undergone deflation, so diagonal is in increasing
// order except for possibly the (0,0) entry. The computed SVD is stored U, singVals and V, except
// that if m_compV is false, then V is not computed. Singular values are sorted in decreasing order.
//
// TODO Opportunities for optimization: better root finding algo, better stopping criterion, better
// handling of round-off errors, be consistent in ordering
// For instance, to solve the secular equation using FMM, see
// http://www.stat.uchicago.edu/~lekheng/courses/302/classics/greengard-rokhlin.pdf
template <typename MatrixType, int Options>
void BDCSVD<MatrixType, Options>::computeSVDofM(Index firstCol, Index n, MatrixXr& U, VectorType& singVals,
MatrixXr& V) {
const RealScalar considerZero = (std::numeric_limits<RealScalar>::min)();
using std::abs;
ArrayRef col0 = m_computed.col(firstCol).segment(firstCol, n);
m_workspace.head(n) = m_computed.block(firstCol, firstCol, n, n).diagonal();
ArrayRef diag = m_workspace.head(n);
diag(0) = Literal(0);
// Allocate space for singular values and vectors
singVals.resize(n);
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:
Index actual_n = n;
while (actual_n > 1 && numext::is_exactly_zero(diag(actual_n - 1))) {
--actual_n;
eigen_internal_assert(numext::is_exactly_zero(col0(actual_n)));
}
Index m = 0; // size of the deflated problem
for (Index k = 0; k < actual_n; ++k)
if (abs(col0(k)) > considerZero) m_workspaceI(m++) = k;
Map<ArrayXi> perm(m_workspaceI.data(), m);
Map<ArrayXr> shifts(m_workspace.data() + 1 * n, n);
Map<ArrayXr> mus(m_workspace.data() + 2 * n, n);
Map<ArrayXr> 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
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
eigen_internal_assert(singVals.allFinite());
eigen_internal_assert(mus.allFinite());
eigen_internal_assert(shifts.allFinite());
#endif
// Compute zhat
perturbCol0(col0, diag, perm, singVals, shifts, mus, zhat);
#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
std::cout << " zhat: " << zhat.transpose() << "\n";
#endif
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
eigen_internal_assert(zhat.allFinite());
#endif
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
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
eigen_internal_assert(m_naiveU.allFinite());
eigen_internal_assert(m_naiveV.allFinite());
eigen_internal_assert(m_computed.allFinite());
eigen_internal_assert(U.allFinite());
eigen_internal_assert(V.allFinite());
// eigen_internal_assert((U.transpose() * U - MatrixXr(MatrixXr::Identity(U.cols(),U.cols()))).norm() <
// 100*NumTraits<RealScalar>::epsilon() * n); eigen_internal_assert((V.transpose() * V -
// MatrixXr(MatrixXr::Identity(V.cols(),V.cols()))).norm() < 100*NumTraits<RealScalar>::epsilon() * n);
#endif
// 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) {
if (singVals(i) > singVals(i + 1)) {
using std::swap;
swap(singVals(i), singVals(i + 1));
U.col(i).swap(U.col(i + 1));
if (m_compV) V.col(i).swap(V.col(i + 1));
}
}
#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<MatrixXr> 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 <typename MatrixType, int Options>
typename BDCSVD<MatrixType, Options>::RealScalar BDCSVD<MatrixType, Options>::secularEq(
RealScalar mu, const ArrayRef& col0, const ArrayRef& diag, const IndicesRef& perm, const ArrayRef& diagShifted,
RealScalar shift) {
Index m = perm.size();
RealScalar res = Literal(1);
for (Index i = 0; i < m; ++i) {
Index j = perm(i);
// The following expression could be rewritten to involve only a single division,
// but this would make the expression more sensitive to overflow.
res += (col0(j) / (diagShifted(j) - mu)) * (col0(j) / (diag(j) + shift + mu));
}
return res;
}
template <typename MatrixType, int Options>
void BDCSVD<MatrixType, Options>::computeSingVals(const ArrayRef& col0, const ArrayRef& diag, const IndicesRef& perm,
VectorType& singVals, ArrayRef shifts, ArrayRef mus) {
using std::abs;
using std::sqrt;
using std::swap;
Index n = col0.size();
Index actual_n = n;
// Note that here actual_n is computed based on col0(i)==0 instead of diag(i)==0 as above
// because 1) we have diag(i)==0 => col0(i)==0 and 2) if col0(i)==0, then diag(i) is already a singular value.
while (actual_n > 1 && numext::is_exactly_zero(col0(actual_n - 1))) --actual_n;
for (Index k = 0; k < n; ++k) {
if (numext::is_exactly_zero(col0(k)) || actual_n == 1) {
// if col0(k) == 0, then entry is deflated, so singular value is on diagonal
// if actual_n==1, then the deflated problem is already diagonalized
singVals(k) = k == 0 ? col0(0) : diag(k);
mus(k) = Literal(0);
shifts(k) = k == 0 ? col0(0) : diag(k);
continue;
}
// otherwise, use secular equation to find singular value
RealScalar left = diag(k);
RealScalar right; // was: = (k != actual_n-1) ? diag(k+1) : (diag(actual_n-1) + col0.matrix().norm());
if (k == actual_n - 1)
right = (diag(actual_n - 1) + col0.matrix().norm());
else {
// Skip deflated singular values,
// recall that at this stage we assume that z[j]!=0 and all entries for which z[j]==0 have been put aside.
// This should be equivalent to using perm[]
Index l = k + 1;
while (numext::is_exactly_zero(col0(l))) {
++l;
eigen_internal_assert(l < actual_n);
}
right = diag(l);
}
// 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
Map<ArrayXr> diagShifted(m_workspace.data() + 4 * n, n);
diagShifted = diag - shift;
if (k != actual_n - 1) {
// check that after the shift, f(mid) is still negative:
RealScalar midShifted = (right - left) / RealScalar(2);
// we can test exact equality here, because shift comes from `... ? left : right`
if (numext::equal_strict(shift, right)) midShifted = -midShifted;
RealScalar fMidShifted = secularEq(midShifted, col0, diag, perm, diagShifted, shift);
if (fMidShifted > 0) {
// fMid was erroneous, fix it:
shift = fMidShifted > Literal(0) ? left : right;
diagShifted = diag - shift;
}
}
// initial guess
RealScalar muPrev, muCur;
// we can test exact equality here, because shift comes from `... ? left : right`
if (numext::equal_strict(shift, left)) {
muPrev = (right - left) * RealScalar(0.1);
if (k == actual_n - 1)
muCur = right - left;
else
muCur = (right - left) * RealScalar(0.5);
} else {
muPrev = -(right - left) * RealScalar(0.1);
muCur = -(right - left) * RealScalar(0.5);
}
RealScalar fPrev = secularEq(muPrev, col0, diag, perm, diagShifted, shift);
RealScalar fCur = secularEq(muCur, col0, diag, perm, diagShifted, shift);
if (abs(fPrev) < abs(fCur)) {
swap(fPrev, fCur);
swap(muPrev, muCur);
}
// rational interpolation: fit a function of the form a / mu + b through the two previous
// iterates and use its zero to compute the next iterate
bool useBisection = fPrev * fCur > Literal(0);
while (!numext::is_exactly_zero(fCur) &&
abs(muCur - muPrev) >
Literal(8) * NumTraits<RealScalar>::epsilon() * numext::maxi<RealScalar>(abs(muCur), abs(muPrev)) &&
abs(fCur - fPrev) > NumTraits<RealScalar>::epsilon() && !useBisection) {
++m_numIters;
// Find a and b such that the function f(mu) = a / mu + b matches the current and previous samples.
RealScalar a = (fCur - fPrev) / (Literal(1) / muCur - Literal(1) / muPrev);
RealScalar b = fCur - a / muCur;
// And find mu such that f(mu)==0:
RealScalar muZero = -a / b;
RealScalar fZero = secularEq(muZero, col0, diag, perm, diagShifted, shift);
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
eigen_internal_assert((numext::isfinite)(fZero));
#endif
muPrev = muCur;
fPrev = fCur;
muCur = muZero;
fCur = fZero;
// we can test exact equality here, because shift comes from `... ? left : right`
if (numext::equal_strict(shift, left) && (muCur < Literal(0) || muCur > right - left)) useBisection = true;
if (numext::equal_strict(shift, right) && (muCur < -(right - left) || muCur > Literal(0))) useBisection = true;
if (abs(fCur) > abs(fPrev)) useBisection = true;
}
// 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)) {
// to avoid overflow, we must have mu > max(real_min, |z(k)|/sqrt(real_max)),
// the factor 2 is to be more conservative
leftShifted =
numext::maxi<RealScalar>((std::numeric_limits<RealScalar>::min)(),
Literal(2) * abs(col0(k)) / sqrt((std::numeric_limits<RealScalar>::max)()));
// check that we did it right:
eigen_internal_assert(
(numext::isfinite)((col0(k) / leftShifted) * (col0(k) / (diag(k) + shift + leftShifted))));
// I don't understand why the case k==0 would be special there:
// 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
} else {
leftShifted = -(right - left) * RealScalar(0.51);
if (k + 1 < n)
rightShifted = -numext::maxi<RealScalar>((std::numeric_limits<RealScalar>::min)(),
abs(col0(k + 1)) / sqrt((std::numeric_limits<RealScalar>::max)()));
else
rightShifted = -(std::numeric_limits<RealScalar>::min)();
}
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<RealScalar>::epsilon() *
numext::maxi<RealScalar>(abs(leftShifted), abs(rightShifted))) {
RealScalar midShifted = (leftShifted + rightShifted) / Literal(2);
fMid = secularEq(midShifted, col0, diag, perm, diagShifted, shift);
eigen_internal_assert((numext::isfinite)(fMid));
if (fLeft * fMid < Literal(0)) {
rightShifted = midShifted;
} else {
leftShifted = midShifted;
fLeft = fMid;
}
}
muCur = (leftShifted + rightShifted) / Literal(2);
} else {
// We have a problem as shifting on the left or right give either a positive or negative value
// at the middle of [left,right]...
// Instead of abbording or entering an infinite loop,
// let's just use the middle as the estimated zero-crossing:
muCur = (right - left) * RealScalar(0.5);
// we can test exact equality here, because shift comes from `... ? left : right`
if (numext::equal_strict(shift, right)) muCur = -muCur;
}
}
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
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
eigen_internal_assert(k == 0 || singVals[k] >= singVals[k - 1]);
eigen_internal_assert(singVals[k] >= diag(k));
#endif
// 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<RealScalar>::epsilon();
// if (singVals[k] == right) singVals[k] *= 1 - NumTraits<RealScalar>::epsilon();
}
}
// zhat is perturbation of col0 for which singular vectors can be computed stably (see Section 3.1)
template <typename MatrixType, int Options>
void BDCSVD<MatrixType, Options>::perturbCol0(const ArrayRef& col0, const ArrayRef& diag, const IndicesRef& perm,
const VectorType& singVals, const ArrayRef& shifts, const ArrayRef& mus,
ArrayRef zhat) {
using std::sqrt;
Index n = col0.size();
Index m = perm.size();
if (m == 0) {
zhat.setZero();
return;
}
Index lastIdx = perm(m - 1);
// The offset permits to skip deflated entries while computing zhat
for (Index k = 0; k < n; ++k) {
if (numext::is_exactly_zero(col0(k))) // deflated
zhat(k) = Literal(0);
else {
// 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)));
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
eigen_internal_assert(prod >= 0);
#endif
#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);
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
eigen_internal_assert((numext::isfinite)(tmp));
#endif
zhat(k) = col0(k) > Literal(0) ? RealScalar(tmp) : RealScalar(-tmp);
}
}
}
// compute singular vectors
template <typename MatrixType, int Options>
void BDCSVD<MatrixType, Options>::computeSingVecs(const ArrayRef& zhat, const ArrayRef& diag, const IndicesRef& perm,
const VectorType& singVals, const ArrayRef& shifts,
const ArrayRef& mus, MatrixXr& U, MatrixXr& V) {
Index n = zhat.size();
Index m = perm.size();
for (Index k = 0; k < n; ++k) {
if (numext::is_exactly_zero(zhat(k))) {
U.col(k) = VectorType::Unit(n + 1, k);
if (m_compV) V.col(k) = VectorType::Unit(n, k);
} else {
U.col(k).setZero();
for (Index l = 0; l < m; ++l) {
Index i = perm(l);
U(i, k) = zhat(i) / (((diag(i) - shifts(k)) - mus(k))) / ((diag(i) + singVals[k]));
}
U(n, k) = Literal(0);
U.col(k).normalize();
if (m_compV) {
V.col(k).setZero();
for (Index l = 1; l < m; ++l) {
Index i = perm(l);
V(i, k) = diag(i) * zhat(i) / (((diag(i) - shifts(k)) - mus(k))) / ((diag(i) + singVals[k]));
}
V(0, k) = Literal(-1);
V.col(k).normalize();
}
}
}
U.col(n) = VectorType::Unit(n + 1, n);
}
// page 12_13
// i >= 1, di almost null and zi non null.
// We use a rotation to zero out zi applied to the left of M, and set di = 0.
template <typename MatrixType, int Options>
void BDCSVD<MatrixType, Options>::deflation43(Index firstCol, Index shift, Index i, Index size) {
using std::abs;
using std::pow;
using std::sqrt;
Index start = firstCol + shift;
RealScalar c = m_computed(start, start);
RealScalar s = m_computed(start + i, start);
RealScalar r = numext::hypot(c, s);
if (numext::is_exactly_zero(r)) {
m_computed(start + i, start + i) = Literal(0);
return;
}
m_computed(start, start) = r;
m_computed(start + i, start) = Literal(0);
m_computed(start + i, start + i) = Literal(0);
JacobiRotation<RealScalar> J(c / r, -s / r);
if (m_compU)
m_naiveU.middleRows(firstCol, size + 1).applyOnTheRight(firstCol, firstCol + i, J);
else
m_naiveU.applyOnTheRight(firstCol, firstCol + i, J);
} // end deflation 43
// page 13
// i,j >= 1, i > j, and |di - dj| < epsilon * norm2(M)
// We apply two rotations to have zi = 0, and dj = di.
template <typename MatrixType, int Options>
void BDCSVD<MatrixType, Options>::deflation44(Index firstColu, Index firstColm, Index firstRowW, Index firstColW,
Index i, Index j, Index size) {
using std::abs;
using std::conj;
using std::pow;
using std::sqrt;
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;
}
c /= r;
s /= r;
m_computed(firstColm + j, firstColm) = r;
m_computed(firstColm + j, firstColm + j) = m_computed(firstColm + i, firstColm + i);
m_computed(firstColm + i, firstColm) = Literal(0);
JacobiRotation<RealScalar> J(c, -s);
if (m_compU)
m_naiveU.middleRows(firstColu, size + 1).applyOnTheRight(firstColu + j, firstColu + i, J);
else
m_naiveU.applyOnTheRight(firstColu + j, firstColu + i, J);
if (m_compV) m_naiveV.middleRows(firstRowW, size).applyOnTheRight(firstColW + j, firstColW + i, J);
} // end deflation 44
// acts on block from (firstCol+shift, firstCol+shift) to (lastCol+shift, lastCol+shift) [inclusive]
template <typename MatrixType, int Options>
void BDCSVD<MatrixType, Options>::deflation(Index firstCol, Index lastCol, Index k, Index firstRowW, Index firstColW,
Index shift) {
using std::abs;
using std::sqrt;
const Index length = lastCol + 1 - firstCol;
Block<MatrixXr, Dynamic, 1> col0(m_computed, firstCol + shift, firstCol + shift, length, 1);
Diagonal<MatrixXr> fulldiag(m_computed);
VectorBlock<Diagonal<MatrixXr>, Dynamic> diag(fulldiag, firstCol + shift, length);
const RealScalar considerZero = (std::numeric_limits<RealScalar>::min)();
RealScalar maxDiag = diag.tail((std::max)(Index(1), length - 1)).cwiseAbs().maxCoeff();
RealScalar epsilon_strict = numext::maxi<RealScalar>(considerZero, NumTraits<RealScalar>::epsilon() * maxDiag);
RealScalar epsilon_coarse =
Literal(8) * NumTraits<RealScalar>::epsilon() * numext::maxi<RealScalar>(col0.cwiseAbs().maxCoeff(), maxDiag);
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
eigen_internal_assert(m_naiveU.allFinite());
eigen_internal_assert(m_naiveV.allFinite());
eigen_internal_assert(m_computed.allFinite());
#endif
#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);
}
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
eigen_internal_assert(m_naiveU.allFinite());
eigen_internal_assert(m_naiveV.allFinite());
eigen_internal_assert(m_computed.allFinite());
#endif
#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.
const bool total_deflation = (col0.tail(length - 1).array().abs() < considerZero).all();
// Sort the diagonal entries, since diag(1:k-1) and diag(k:length) are already sorted, let's do a sorted merge.
// First, compute the respective permutation.
Index* permutation = m_workspaceI.data();
{
permutation[0] = 0;
Index p = 1;
// Move deflated diagonal entries at the end.
for (Index i = 1; i < length; ++i)
if (diag(i) < considerZero) permutation[p++] = i;
Index i = 1, j = k + 1;
for (; p < length; ++p) {
if (i > k)
permutation[p] = j++;
else if (j >= length)
permutation[p] = i++;
else if (diag(i) < diag(j))
permutation[p] = j++;
else
permutation[p] = i++;
}
}
// If we have a total deflation, then we have to insert diag(0) at the right place
if (total_deflation) {
for (Index i = 1; i < length; ++i) {
Index pi = permutation[i];
if (diag(pi) < considerZero || diag(0) < diag(pi))
permutation[i - 1] = permutation[i];
else {
permutation[i - 1] = 0;
break;
}
}
}
// Current index of each col, and current column of each index
Index* realInd = m_workspaceI.data() + length;
Index* realCol = m_workspaceI.data() + 2 * length;
for (int pos = 0; pos < length; pos++) {
realCol[pos] = pos;
realInd[pos] = pos;
}
for (Index i = total_deflation ? 0 : 1; i < length; i++) {
const Index pi = permutation[length - (total_deflation ? i + 1 : i)];
const Index J = realCol[pi];
using std::swap;
// swap diagonal and first column entries:
swap(diag(i), diag(J));
if (i != 0 && J != 0) swap(col0(i), col0(J));
// change columns
if (m_compU)
m_naiveU.col(firstCol + i)
.segment(firstCol, length + 1)
.swap(m_naiveU.col(firstCol + J).segment(firstCol, length + 1));
else
m_naiveU.col(firstCol + i).segment(0, 2).swap(m_naiveU.col(firstCol + J).segment(0, 2));
if (m_compV)
m_naiveV.col(firstColW + i)
.segment(firstRowW, length)
.swap(m_naiveV.col(firstColW + J).segment(firstRowW, length));
// update real pos
const Index realI = realInd[i];
realCol[realI] = J;
realCol[pi] = i;
realInd[J] = realI;
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
{
Index i = length - 1;
// Find last non-deflated entry.
while (i > 0 && (diag(i) < considerZero || abs(col0(i)) < considerZero)) --i;
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
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
eigen_internal_assert(m_naiveU.allFinite());
eigen_internal_assert(m_naiveV.allFinite());
eigen_internal_assert(m_computed.allFinite());
#endif
} // end deflation
/** \svd_module
*
* \return the singular value decomposition of \c *this computed by Divide & Conquer algorithm
*
* \sa class BDCSVD
*/
template <typename Derived>
template <int Options>
BDCSVD<typename MatrixBase<Derived>::PlainObject, Options> MatrixBase<Derived>::bdcSvd() const {
return BDCSVD<PlainObject, Options>(*this);
}
/** \svd_module
*
* \return the singular value decomposition of \c *this computed by Divide & Conquer algorithm
*
* \sa class BDCSVD
*/
template <typename Derived>
template <int Options>
BDCSVD<typename MatrixBase<Derived>::PlainObject, Options> MatrixBase<Derived>::bdcSvd(
unsigned int computationOptions) const {
return BDCSVD<PlainObject, Options>(*this, computationOptions);
}
} // end namespace Eigen
#endif