// 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) 2014-2016 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
namespace Eigen {
#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
IOFormat bdcsvdfmt(8, 0, ", ", "\n", " [", "]");
#endif
template<typename _MatrixType> class BDCSVD;
namespace internal {
template<typename _MatrixType>
struct traits<BDCSVD<_MatrixType> >
{
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
*
* 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 perfroms 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>
class BDCSVD : public SVDBase<BDCSVD<_MatrixType> >
{
typedef SVDBase<BDCSVD> Base;
public:
using Base::rows;
using Base::cols;
using Base::computeU;
using Base::computeV;
typedef _MatrixType MatrixType;
typedef typename MatrixType::Scalar Scalar;
typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
typedef typename NumTraits<RealScalar>::Literal Literal;
enum {
RowsAtCompileTime = MatrixType::RowsAtCompileTime,
ColsAtCompileTime = MatrixType::ColsAtCompileTime,
DiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(RowsAtCompileTime, ColsAtCompileTime),
MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
MaxDiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(MaxRowsAtCompileTime, MaxColsAtCompileTime),
MatrixOptions = MatrixType::Options
};
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_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.
* \sa BDCSVD()
*/
BDCSVD(Index rows, Index cols, unsigned int computationOptions = 0)
: m_algoswap(16), m_numIters(0)
{
allocate(rows, cols, computationOptions);
}
/** \brief Constructor performing the decomposition of given matrix.
*
* \param matrix the matrix to decompose
* \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed.
* By default, none is computed. This is a bit - field, the possible bits are #ComputeFullU, #ComputeThinU,
* #ComputeFullV, #ComputeThinV.
*
* Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not
* available with the (non - default) FullPivHouseholderQR preconditioner.
*/
BDCSVD(const MatrixType& matrix, unsigned int computationOptions = 0)
: m_algoswap(16), m_numIters(0)
{
compute(matrix, computationOptions);
}
~BDCSVD()
{
}
/** \brief Method performing the decomposition of given matrix using custom options.
*
* \param matrix the matrix to decompose
* \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed.
* By default, none is computed. This is a bit - field, the possible bits are #ComputeFullU, #ComputeThinU,
* #ComputeFullV, #ComputeThinV.
*
* Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not
* available with the (non - default) FullPivHouseholderQR preconditioner.
*/
BDCSVD& compute(const MatrixType& matrix, unsigned int computationOptions);
/** \brief Method performing the decomposition of given matrix using current options.
*
* \param matrix the matrix to decompose
*
* This method uses the current \a computationOptions, as already passed to the constructor or to compute(const MatrixType&, unsigned int).
*/
BDCSVD& compute(const MatrixType& matrix)
{
return compute(matrix, this->m_computationOptions);
}
void setSwitchSize(int s)
{
eigen_assert(s>3 && "BDCSVD the size of the algo switch has to be greater than 3");
m_algoswap = s;
}
private:
void allocate(Index rows, Index cols, 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);
protected:
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;
using Base::m_singularValues;
using Base::m_diagSize;
using Base::m_computeFullU;
using Base::m_computeFullV;
using Base::m_computeThinU;
using Base::m_computeThinV;
using Base::m_matrixU;
using Base::m_matrixV;
using Base::m_isInitialized;
using Base::m_nonzeroSingularValues;
public:
int m_numIters;
}; //end class BDCSVD
// Method to allocate and initialize matrix and attributes
template<typename MatrixType>
void BDCSVD<MatrixType>::allocate(Index rows, Index cols, unsigned int computationOptions)
{
m_isTranspose = (cols > rows);
if (Base::allocate(rows, cols, computationOptions))
return;
m_computed = MatrixXr::Zero(m_diagSize + 1, m_diagSize );
m_compU = computeV();
m_compV = computeU();
if (m_isTranspose)
std::swap(m_compU, m_compV);
if (m_compU) m_naiveU = MatrixXr::Zero(m_diagSize + 1, m_diagSize + 1 );
else m_naiveU = MatrixXr::Zero(2, m_diagSize + 1 );
if (m_compV) m_naiveV = MatrixXr::Zero(m_diagSize, m_diagSize);
m_workspace.resize((m_diagSize+1)*(m_diagSize+1)*3);
m_workspaceI.resize(3*m_diagSize);
}// end allocate
template<typename MatrixType>
BDCSVD<MatrixType>& BDCSVD<MatrixType>::compute(const MatrixType& matrix, unsigned int computationOptions)
{
#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
std::cout << "\n\n\n======================================================================================================================\n\n\n";
#endif
allocate(matrix.rows(), matrix.cols(), computationOptions);
using std::abs;
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)
{
// FIXME this line involves temporaries
JacobiSVD<MatrixType> jsvd(matrix,computationOptions);
if(computeU()) m_matrixU = jsvd.matrixU();
if(computeV()) m_matrixV = jsvd.matrixV();
m_singularValues = jsvd.singularValues();
m_nonzeroSingularValues = jsvd.nonzeroSingularValues();
m_isInitialized = true;
return *this;
}
//**** step 0 - Copy the input matrix and apply scaling to reduce over/under-flows
RealScalar scale = matrix.cwiseAbs().maxCoeff();
if(scale==Literal(0)) scale = Literal(1);
MatrixX copy;
if (m_isTranspose) copy = matrix.adjoint()/scale;
else copy = matrix/scale;
//**** step 1 - Bidiagonalization
// FIXME this line involves temporaries
internal::UpperBidiagonalization<MatrixX> bid(copy);
//**** step 2 - Divide & Conquer
m_naiveU.setZero();
m_naiveV.setZero();
// FIXME this line involves a temporary matrix
m_computed.topRows(m_diagSize) = bid.bidiagonal().toDenseMatrix().transpose();
m_computed.template bottomRows<1>().setZero();
divide(0, m_diagSize - 1, 0, 0, 0);
//**** step 3 - Copy singular values and vectors
for (int i=0; i<m_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(m_diagSize - i - 1).setZero();
break;
}
else if (i == m_diagSize - 1)
{
m_nonzeroSingularValues = i + 1;
break;
}
}
#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
// std::cout << "m_naiveU\n" << m_naiveU << "\n\n";
// std::cout << "m_naiveV\n" << m_naiveV << "\n\n";
#endif
if(m_isTranspose) copyUV(bid.householderV(), bid.householderU(), m_naiveV, m_naiveU);
else copyUV(bid.householderU(), bid.householderV(), m_naiveU, m_naiveV);
m_isInitialized = true;
return *this;
}// end compute
template<typename MatrixType>
template<typename HouseholderU, typename HouseholderV, typename NaiveU, typename NaiveV>
void BDCSVD<MatrixType>::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 ? m_diagSize : householderU.cols();
m_matrixU = MatrixX::Identity(householderU.cols(), Ucols);
m_matrixU.topLeftCorner(m_diagSize, m_diagSize) = naiveV.template cast<Scalar>().topLeftCorner(m_diagSize, m_diagSize);
householderU.applyThisOnTheLeft(m_matrixU); // FIXME this line involves a temporary buffer
}
if (computeV())
{
Index Vcols = m_computeThinV ? m_diagSize : householderV.cols();
m_matrixV = MatrixX::Identity(householderV.cols(), Vcols);
m_matrixV.topLeftCorner(m_diagSize, m_diagSize) = naiveU.template cast<Scalar>().topLeftCorner(m_diagSize, m_diagSize);
householderV.applyThisOnTheLeft(m_matrixV); // FIXME this line involves a temporary buffer
}
}
/** \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>
void BDCSVD<MatrixType>::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;
}
}
// 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 firstRowW : 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>
void BDCSVD<MatrixType>::divide (Index firstCol, Index lastCol, Index firstRowW, Index firstColW, Index shift)
{
// requires rows = cols + 1;
using std::pow;
using std::sqrt;
using std::abs;
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 line involves temporaries
JacobiSVD<MatrixXr> b(m_computed.block(firstCol, firstCol, n + 1, n), ComputeFullU | (m_compV ? ComputeFullV : 0));
if (m_compU)
m_naiveU.block(firstCol, firstCol, n + 1, n + 1).real() = b.matrixU();
else
{
m_naiveU.row(0).segment(firstCol, n + 1).real() = b.matrixU().row(0);
m_naiveU.row(1).segment(firstCol, n + 1).real() = b.matrixU().row(n);
}
if (m_compV) m_naiveV.block(firstRowW, firstColW, n, n).real() = b.matrixV();
m_computed.block(firstCol + shift, firstCol + shift, n + 1, n).setZero();
m_computed.diagonal().segment(firstCol + shift, n) = b.singularValues().head(n);
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);
divide(firstCol, k - 1 + firstCol, firstRowW, firstColW + 1, shift + 1);
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
assert(m_naiveU.allFinite());
assert(m_naiveV.allFinite());
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
assert(m_naiveU.allFinite());
assert(m_naiveV.allFinite());
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";
assert((tmp1-tmp2).matrix().norm() < 1e-14*tmp2.matrix().norm());
// assert(count<681);
// 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
assert(UofSVD.allFinite());
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
assert(m_naiveU.allFinite());
assert(m_naiveV.allFinite());
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>
void BDCSVD<MatrixType>::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 && diag(actual_n-1)==Literal(0)) --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";
{
Index actual_n = n;
while(actual_n>1 && abs(col0(actual_n-1))<considerZero) --actual_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";
std::cout << " check2 (>0) : " << ((singVals.array()-diag) / singVals.array()).head(actual_n).transpose() << "\n\n";
std::cout << " check3 (>0) : " << ((diag.segment(1,actual_n-1)-singVals.head(actual_n-1).array()) / singVals.head(actual_n-1).array()).transpose() << "\n\n\n";
std::cout << " check4 (>0) : " << ((singVals.segment(1,actual_n-1)-singVals.head(actual_n-1))).transpose() << "\n\n\n";
}
#endif
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
assert(singVals.allFinite());
assert(mus.allFinite());
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
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
assert(U.allFinite());
assert(V.allFinite());
assert((U.transpose() * U - MatrixXr(MatrixXr::Identity(U.cols(),U.cols()))).norm() < 1e-14 * n);
assert((V.transpose() * V - MatrixXr(MatrixXr::Identity(V.cols(),V.cols()))).norm() < 1e-14 * n);
assert(m_naiveU.allFinite());
assert(m_naiveV.allFinite());
assert(m_computed.allFinite());
#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));
}
}
// 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>
typename BDCSVD<MatrixType>::RealScalar BDCSVD<MatrixType>::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);
res += numext::abs2(col0(j)) / ((diagShifted(j) - mu) * (diag(j) + shift + mu));
}
return res;
}
template <typename MatrixType>
void BDCSVD<MatrixType>::computeSingVals(const ArrayRef& col0, const ArrayRef& diag, const IndicesRef &perm,
VectorType& singVals, ArrayRef shifts, ArrayRef mus)
{
using std::abs;
using std::swap;
Index n = col0.size();
Index actual_n = n;
while(actual_n>1 && col0(actual_n-1)==Literal(0)) --actual_n;
for (Index k = 0; k < n; ++k)
{
if (col0(k) == Literal(0) || 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
Index l = k+1;
while(col0(l)==Literal(0)) { ++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 << "\n";
std::cout << "fMid = " << fMid << " " << secularEq(mid-left, col0, diag, perm, diag-left, left) << " " << secularEq(mid-right, col0, diag, perm, diag-right, right) << "\n";
std::cout << " = " << secularEq(0.1*(left+right), col0, diag, perm, diag, 0)
<< " " << secularEq(0.2*(left+right), col0, diag, perm, diag, 0)
<< " " << secularEq(0.3*(left+right), col0, diag, perm, diag, 0)
<< " " << secularEq(0.4*(left+right), col0, diag, perm, diag, 0)
<< " " << secularEq(0.49*(left+right), col0, diag, perm, diag, 0)
<< " " << secularEq(0.5*(left+right), col0, diag, perm, diag, 0)
<< " " << secularEq(0.51*(left+right), col0, diag, perm, diag, 0)
<< " " << secularEq(0.6*(left+right), col0, diag, perm, diag, 0)
<< " " << secularEq(0.7*(left+right), col0, diag, perm, diag, 0)
<< " " << secularEq(0.8*(left+right), col0, diag, perm, diag, 0)
<< " " << secularEq(0.9*(left+right), 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;
// initial guess
RealScalar muPrev, muCur;
if (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 (fCur!=Literal(0) && 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);
muPrev = muCur;
fPrev = fCur;
muCur = muZero;
fCur = fZero;
if (shift == left && (muCur < Literal(0) || muCur > right - left)) useBisection = true;
if (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;
if (shift == left)
{
leftShifted = (std::numeric_limits<RealScalar>::min)();
// 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(0.6)); // theoretically we can take 0.5, but let's be safe
}
else
{
leftShifted = -(right - left) * RealScalar(0.6);
rightShifted = -(std::numeric_limits<RealScalar>::min)();
}
RealScalar fLeft = secularEq(leftShifted, col0, diag, perm, diagShifted, shift);
#if defined EIGEN_INTERNAL_DEBUGGING || defined EIGEN_BDCSVD_DEBUG_VERBOSE
RealScalar fRight = secularEq(rightShifted, col0, diag, perm, diagShifted, shift);
#endif
#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
if(!(fLeft * fRight<0))
{
std::cout << "fLeft: " << leftShifted << " - " << diagShifted.head(10).transpose() << "\n ; " << bool(left==shift) << " " << (left-shift) << "\n";
std::cout << k << " : " << fLeft << " * " << fRight << " == " << fLeft * fRight << " ; " << left << " - " << right << " -> " << leftShifted << " " << rightShifted << " shift=" << shift << "\n";
}
#endif
eigen_internal_assert(fLeft * fRight < 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);
if (fLeft * fMid < Literal(0))
{
rightShifted = midShifted;
}
else
{
leftShifted = midShifted;
fLeft = fMid;
}
}
muCur = (leftShifted + rightShifted) / Literal(2);
}
singVals[k] = shift + muCur;
shifts[k] = shift;
mus[k] = muCur;
// 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>
void BDCSVD<MatrixType>::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 last = perm(m-1);
// The offset permits to skip deflated entries while computing zhat
for (Index k = 0; k < n; ++k)
{
if (col0(k) == Literal(0)) // deflated
zhat(k) = Literal(0);
else
{
// see equation (3.6)
RealScalar dk = diag(k);
RealScalar prod = (singVals(last) + dk) * (mus(last) + (shifts(last) - dk));
for(Index l = 0; l<m; ++l)
{
Index i = perm(l);
if(i!=k)
{
Index j = i<k ? i : perm(l-1);
prod *= ((singVals(j)+dk) / ((diag(i)+dk))) * ((mus(j)+(shifts(j)-dk)) / ((diag(i)-dk)));
#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
if(i!=k && std::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(last) + dk) << " * " << mus(last) + shifts(last) << " - " << dk << "\n";
#endif
RealScalar tmp = sqrt(prod);
zhat(k) = col0(k) > Literal(0) ? tmp : -tmp;
}
}
}
// compute singular vectors
template <typename MatrixType>
void BDCSVD<MatrixType>::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 (zhat(k) == Literal(0))
{
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
template <typename MatrixType>
void BDCSVD<MatrixType>::deflation43(Index firstCol, Index shift, Index i, Index size)
{
using std::abs;
using std::sqrt;
using std::pow;
Index start = firstCol + shift;
RealScalar c = m_computed(start, start);
RealScalar s = m_computed(start+i, start);
RealScalar r = sqrt(numext::abs2(c) + numext::abs2(s));
if (r == Literal(0))
{
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 zj = 0;
// TODO deflation44 is still broken and not properly tested
template <typename MatrixType>
void BDCSVD<MatrixType>::deflation44(Index firstColu , Index firstColm, Index firstRowW, Index firstColW, Index i, Index j, Index size)
{
using std::abs;
using std::sqrt;
using std::conj;
using std::pow;
RealScalar c = m_computed(firstColm+i, firstColm);
RealScalar s = m_computed(firstColm+j, firstColm);
RealScalar r = sqrt(numext::abs2(c) + numext::abs2(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 (r==Literal(0))
{
m_computed(firstColm + i, firstColm + i) = m_computed(firstColm + j, firstColm + j);
return;
}
c/=r;
s/=r;
m_computed(firstColm + i, firstColm) = r;
m_computed(firstColm + j, firstColm + j) = m_computed(firstColm + i, firstColm + i);
m_computed(firstColm + j, firstColm) = Literal(0);
JacobiRotation<RealScalar> J(c,-s);
if (m_compU) m_naiveU.middleRows(firstColu, size+1).applyOnTheRight(firstColu + i, firstColu + j, J);
else m_naiveU.applyOnTheRight(firstColu+i, firstColu+j, J);
if (m_compV) m_naiveV.middleRows(firstRowW, size).applyOnTheRight(firstColW + i, firstColW + j, J);
}// end deflation 44
// acts on block from (firstCol+shift, firstCol+shift) to (lastCol+shift, lastCol+shift) [inclusive]
template <typename MatrixType>
void BDCSVD<MatrixType>::deflation(Index firstCol, Index lastCol, Index k, Index firstRowW, Index firstColW, Index shift)
{
using std::sqrt;
using std::abs;
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
assert(m_naiveU.allFinite());
assert(m_naiveV.allFinite());
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
assert(m_naiveU.allFinite());
assert(m_naiveV.allFinite());
assert(m_computed.allFinite());
#endif
#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
std::cout << "to be sorted: " << diag.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
bool total_deflation = (col0.tail(length-1).array()<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(abs(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(abs(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;
while(i>0 && (abs(diag(i))<considerZero || abs(col0(i))<considerZero)) --i;
for(; i>1;--i)
if( (diag(i) - diag(i-1)) < NumTraits<RealScalar>::epsilon()*maxDiag )
{
#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
std::cout << "deflation 4.4 with i = " << i << " because " << (diag(i) - diag(i-1)) << " < " << NumTraits<RealScalar>::epsilon()*diag(i) << "\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-1, i, length);
}
}
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
for(Index j=2;j<length;++j)
assert(diag(j-1)<=diag(j) || abs(diag(j))<considerZero);
#endif
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
assert(m_naiveU.allFinite());
assert(m_naiveV.allFinite());
assert(m_computed.allFinite());
#endif
}//end deflation
#ifndef __CUDACC__
/** \svd_module
*
* \return the singular value decomposition of \c *this computed by Divide & Conquer algorithm
*
* \sa class BDCSVD
*/
template<typename Derived>
BDCSVD<typename MatrixBase<Derived>::PlainObject>
MatrixBase<Derived>::bdcSvd(unsigned int computationOptions) const
{
return BDCSVD<PlainObject>(*this, computationOptions);
}
#endif
} // end namespace Eigen
#endif