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Diffstat (limited to 'runtimes/nn/depend/external/eigen/Eigen/src/SVD/BDCSVD.h')
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diff --git a/runtimes/nn/depend/external/eigen/Eigen/src/SVD/BDCSVD.h b/runtimes/nn/depend/external/eigen/Eigen/src/SVD/BDCSVD.h deleted file mode 100644 index d7a4271cb..000000000 --- a/runtimes/nn/depend/external/eigen/Eigen/src/SVD/BDCSVD.h +++ /dev/null @@ -1,1231 +0,0 @@ -// 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 |