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diff --git a/Eigen/src/SVD/BDCSVD.h b/Eigen/src/SVD/BDCSVD.h new file mode 100644 index 000000000..25fca6f4d --- /dev/null +++ b/Eigen/src/SVD/BDCSVD.h @@ -0,0 +1,1230 @@ +// 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; + 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==RealScalar(0)) scale = RealScalar(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()!=0).any() ) + { + A1.col(k1) = A.col(j).head(n1); + B1.row(k1) = B.row(j); + ++k1; + } + if( (A.col(j).tail(n2).array()!=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) = 1; + if (r0<considerZero) + { + c0 = 1; + s0 = 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) = 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)==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 = 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)==0) --actual_n; + + for (Index k = 0; k < n; ++k) + { + if (col0(k) == 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) = 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)==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) / 2; + RealScalar fMid = secularEq(mid, col0, diag, perm, diag, 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 > 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>0; + while (fCur!=0 && abs(muCur - muPrev) > 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) / (1/muCur - 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 < 0 || muCur > right - left)) useBisection = true; + if (shift == right && (muCur < -(right - left) || muCur > 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 < 0); + + while (rightShifted - leftShifted > 2 * NumTraits<RealScalar>::epsilon() * numext::maxi<RealScalar>(abs(leftShifted), abs(rightShifted))) + { + RealScalar midShifted = (leftShifted + rightShifted) / 2; + fMid = secularEq(midShifted, col0, diag, perm, diagShifted, shift); + if (fLeft * fMid < 0) + { + rightShifted = midShifted; + } + else + { + leftShifted = midShifted; + fLeft = fMid; + } + } + + muCur = (leftShifted + rightShifted) / 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) == 0) // deflated + zhat(k) = 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) > 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) == 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) = 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) = -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 == 0) + { + m_computed(start+i, start+i) = 0; + return; + } + m_computed(start,start) = r; + m_computed(start+i, start) = 0; + m_computed(start+i, start+i) = 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==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) = 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 = 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) = 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 |