diff options
Diffstat (limited to 'extern/Eigen3/Eigen/src/SVD')
| -rw-r--r-- | extern/Eigen3/Eigen/src/SVD/BDCSVD.h | 1246 | ||||
| -rw-r--r-- | extern/Eigen3/Eigen/src/SVD/JacobiSVD.h | 392 | ||||
| -rw-r--r-- | extern/Eigen3/Eigen/src/SVD/JacobiSVD_LAPACKE.h | 91 | ||||
| -rw-r--r-- | extern/Eigen3/Eigen/src/SVD/SVDBase.h | 315 | ||||
| -rw-r--r-- | extern/Eigen3/Eigen/src/SVD/UpperBidiagonalization.h | 328 |
5 files changed, 2059 insertions, 313 deletions
diff --git a/extern/Eigen3/Eigen/src/SVD/BDCSVD.h b/extern/Eigen3/Eigen/src/SVD/BDCSVD.h new file mode 100644 index 00000000000..1134d66e7e9 --- /dev/null +++ b/extern/Eigen3/Eigen/src/SVD/BDCSVD.h @@ -0,0 +1,1246 @@ +// This file is part of Eigen, a lightweight C++ template library +// for linear algebra. +// +// We used the "A Divide-And-Conquer Algorithm for the Bidiagonal SVD" +// research report written by Ming Gu and Stanley C.Eisenstat +// The code variable names correspond to the names they used in their +// report +// +// Copyright (C) 2013 Gauthier Brun <brun.gauthier@gmail.com> +// Copyright (C) 2013 Nicolas Carre <nicolas.carre@ensimag.fr> +// Copyright (C) 2013 Jean Ceccato <jean.ceccato@ensimag.fr> +// Copyright (C) 2013 Pierre Zoppitelli <pierre.zoppitelli@ensimag.fr> +// Copyright (C) 2013 Jitse Niesen <jitse@maths.leeds.ac.uk> +// Copyright (C) 2014-2017 Gael Guennebaud <gael.guennebaud@inria.fr> +// +// Source Code Form is subject to the terms of the Mozilla +// Public License v. 2.0. If a copy of the MPL was not distributed +// with this file, You can obtain one at http://mozilla.org/MPL/2.0/. + +#ifndef EIGEN_BDCSVD_H +#define EIGEN_BDCSVD_H +// #define EIGEN_BDCSVD_DEBUG_VERBOSE +// #define EIGEN_BDCSVD_SANITY_CHECKS + +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); + // The following expression could be rewritten to involve only a single division, + // but this would make the expression more sensitive to overflow. + res += (col0(j) / (diagShifted(j) - mu)) * (col0(j) / (diag(j) + shift + mu)); + } + return res; + +} + +template <typename MatrixType> +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; + using std::sqrt; + + Index n = col0.size(); + Index actual_n = n; + // Note that here actual_n is computed based on col0(i)==0 instead of diag(i)==0 as above + // because 1) we have diag(i)==0 => col0(i)==0 and 2) if col0(i)==0, then diag(i) is already a singular value. + while(actual_n>1 && 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, + // recall that at this stage we assume that z[j]!=0 and all entries for which z[j]==0 have been put aside. + // This should be equivalent to using perm[] + Index l = k+1; + while(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) + { + // to avoid overflow, we must have mu > max(real_min, |z(k)|/sqrt(real_max)), + // the factor 2 is to be more conservative + leftShifted = numext::maxi<RealScalar>( (std::numeric_limits<RealScalar>::min)(), Literal(2) * abs(col0(k)) / sqrt((std::numeric_limits<RealScalar>::max)()) ); + + // check that we did it right: + eigen_internal_assert( (numext::isfinite)( (col0(k)/leftShifted)*(col0(k)/(diag(k)+shift+leftShifted)) ) ); + // I don't understand why the case k==0 would be special there: + // if (k == 0) rightShifted = right - left; else + rightShifted = (k==actual_n-1) ? right : ((right - left) * RealScalar(0.51)); // theoretically we can take 0.5, but let's be safe + } + else + { + leftShifted = -(right - left) * RealScalar(0.51); + if(k+1<n) + rightShifted = -numext::maxi<RealScalar>( (std::numeric_limits<RealScalar>::min)(), abs(col0(k+1)) / sqrt((std::numeric_limits<RealScalar>::max)()) ); + else + rightShifted = -(std::numeric_limits<RealScalar>::min)(); + } + + RealScalar fLeft = secularEq(leftShifted, col0, diag, perm, diagShifted, shift); + +#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 = numext::hypot(c,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 diff --git a/extern/Eigen3/Eigen/src/SVD/JacobiSVD.h b/extern/Eigen3/Eigen/src/SVD/JacobiSVD.h index 1b29774190a..43488b1e0c2 100644 --- a/extern/Eigen3/Eigen/src/SVD/JacobiSVD.h +++ b/extern/Eigen3/Eigen/src/SVD/JacobiSVD.h @@ -2,6 +2,7 @@ // for linear algebra. // // Copyright (C) 2009-2010 Benoit Jacob <jacob.benoit.1@gmail.com> +// Copyright (C) 2013-2014 Gael Guennebaud <gael.guennebaud@inria.fr> // // This Source Code Form is subject to the terms of the Mozilla // Public License v. 2.0. If a copy of the MPL was not distributed @@ -51,7 +52,6 @@ template<typename MatrixType, int QRPreconditioner, int Case> class qr_preconditioner_impl<MatrixType, QRPreconditioner, Case, false> { public: - typedef typename MatrixType::Index Index; void allocate(const JacobiSVD<MatrixType, QRPreconditioner>&) {} bool run(JacobiSVD<MatrixType, QRPreconditioner>&, const MatrixType&) { @@ -65,7 +65,6 @@ template<typename MatrixType> class qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true> { public: - typedef typename MatrixType::Index Index; typedef typename MatrixType::Scalar Scalar; enum { @@ -106,7 +105,6 @@ template<typename MatrixType> class qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true> { public: - typedef typename MatrixType::Index Index; typedef typename MatrixType::Scalar Scalar; enum { @@ -114,9 +112,11 @@ public: ColsAtCompileTime = MatrixType::ColsAtCompileTime, MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime, - Options = MatrixType::Options + TrOptions = RowsAtCompileTime==1 ? (MatrixType::Options & ~(RowMajor)) + : ColsAtCompileTime==1 ? (MatrixType::Options | RowMajor) + : MatrixType::Options }; - typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime> + typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, TrOptions, MaxColsAtCompileTime, MaxRowsAtCompileTime> TransposeTypeWithSameStorageOrder; void allocate(const JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd) @@ -156,8 +156,6 @@ template<typename MatrixType> class qr_preconditioner_impl<MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true> { public: - typedef typename MatrixType::Index Index; - void allocate(const JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd) { if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols()) @@ -197,7 +195,6 @@ template<typename MatrixType> class qr_preconditioner_impl<MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true> { public: - typedef typename MatrixType::Index Index; typedef typename MatrixType::Scalar Scalar; enum { @@ -205,10 +202,12 @@ public: ColsAtCompileTime = MatrixType::ColsAtCompileTime, MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime, - Options = MatrixType::Options + TrOptions = RowsAtCompileTime==1 ? (MatrixType::Options & ~(RowMajor)) + : ColsAtCompileTime==1 ? (MatrixType::Options | RowMajor) + : MatrixType::Options }; - typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime> + typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, TrOptions, MaxColsAtCompileTime, MaxRowsAtCompileTime> TransposeTypeWithSameStorageOrder; void allocate(const JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd) @@ -256,8 +255,6 @@ template<typename MatrixType> class qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true> { public: - typedef typename MatrixType::Index Index; - void allocate(const JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd) { if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols()) @@ -296,7 +293,6 @@ template<typename MatrixType> class qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true> { public: - typedef typename MatrixType::Index Index; typedef typename MatrixType::Scalar Scalar; enum { @@ -358,8 +354,8 @@ template<typename MatrixType, int QRPreconditioner> struct svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner, false> { typedef JacobiSVD<MatrixType, QRPreconditioner> SVD; - typedef typename SVD::Index Index; - static void run(typename SVD::WorkMatrixType&, SVD&, Index, Index) {} + typedef typename MatrixType::RealScalar RealScalar; + static bool run(typename SVD::WorkMatrixType&, SVD&, Index, Index, RealScalar&) { return true; } }; template<typename MatrixType, int QRPreconditioner> @@ -368,20 +364,30 @@ struct svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner, true> typedef JacobiSVD<MatrixType, QRPreconditioner> SVD; typedef typename MatrixType::Scalar Scalar; typedef typename MatrixType::RealScalar RealScalar; - typedef typename SVD::Index Index; - static void run(typename SVD::WorkMatrixType& work_matrix, SVD& svd, Index p, Index q) + static bool run(typename SVD::WorkMatrixType& work_matrix, SVD& svd, Index p, Index q, RealScalar& maxDiagEntry) { using std::sqrt; + using std::abs; Scalar z; JacobiRotation<Scalar> rot; RealScalar n = sqrt(numext::abs2(work_matrix.coeff(p,p)) + numext::abs2(work_matrix.coeff(q,p))); - + + const RealScalar considerAsZero = (std::numeric_limits<RealScalar>::min)(); + const RealScalar precision = NumTraits<Scalar>::epsilon(); + if(n==0) { - z = abs(work_matrix.coeff(p,q)) / work_matrix.coeff(p,q); - work_matrix.row(p) *= z; - if(svd.computeU()) svd.m_matrixU.col(p) *= conj(z); - if(work_matrix.coeff(q,q)!=Scalar(0)) + // make sure first column is zero + work_matrix.coeffRef(p,p) = work_matrix.coeffRef(q,p) = Scalar(0); + + if(abs(numext::imag(work_matrix.coeff(p,q)))>considerAsZero) + { + // work_matrix.coeff(p,q) can be zero if work_matrix.coeff(q,p) is not zero but small enough to underflow when computing n + z = abs(work_matrix.coeff(p,q)) / work_matrix.coeff(p,q); + work_matrix.row(p) *= z; + if(svd.computeU()) svd.m_matrixU.col(p) *= conj(z); + } + if(abs(numext::imag(work_matrix.coeff(q,q)))>considerAsZero) { z = abs(work_matrix.coeff(q,q)) / work_matrix.coeff(q,q); work_matrix.row(q) *= z; @@ -395,52 +401,33 @@ struct svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner, true> rot.s() = work_matrix.coeff(q,p) / n; work_matrix.applyOnTheLeft(p,q,rot); if(svd.computeU()) svd.m_matrixU.applyOnTheRight(p,q,rot.adjoint()); - if(work_matrix.coeff(p,q) != Scalar(0)) + if(abs(numext::imag(work_matrix.coeff(p,q)))>considerAsZero) { - Scalar z = abs(work_matrix.coeff(p,q)) / work_matrix.coeff(p,q); + z = abs(work_matrix.coeff(p,q)) / work_matrix.coeff(p,q); work_matrix.col(q) *= z; if(svd.computeV()) svd.m_matrixV.col(q) *= z; } - if(work_matrix.coeff(q,q) != Scalar(0)) + if(abs(numext::imag(work_matrix.coeff(q,q)))>considerAsZero) { z = abs(work_matrix.coeff(q,q)) / work_matrix.coeff(q,q); work_matrix.row(q) *= z; if(svd.computeU()) svd.m_matrixU.col(q) *= conj(z); } } + + // update largest diagonal entry + maxDiagEntry = numext::maxi<RealScalar>(maxDiagEntry,numext::maxi<RealScalar>(abs(work_matrix.coeff(p,p)), abs(work_matrix.coeff(q,q)))); + // and check whether the 2x2 block is already diagonal + RealScalar threshold = numext::maxi<RealScalar>(considerAsZero, precision * maxDiagEntry); + return abs(work_matrix.coeff(p,q))>threshold || abs(work_matrix.coeff(q,p)) > threshold; } }; -template<typename MatrixType, typename RealScalar, typename Index> -void real_2x2_jacobi_svd(const MatrixType& matrix, Index p, Index q, - JacobiRotation<RealScalar> *j_left, - JacobiRotation<RealScalar> *j_right) +template<typename _MatrixType, int QRPreconditioner> +struct traits<JacobiSVD<_MatrixType,QRPreconditioner> > { - using std::sqrt; - using std::abs; - Matrix<RealScalar,2,2> m; - m << numext::real(matrix.coeff(p,p)), numext::real(matrix.coeff(p,q)), - numext::real(matrix.coeff(q,p)), numext::real(matrix.coeff(q,q)); - JacobiRotation<RealScalar> rot1; - RealScalar t = m.coeff(0,0) + m.coeff(1,1); - RealScalar d = m.coeff(1,0) - m.coeff(0,1); - if(t == RealScalar(0)) - { - rot1.c() = RealScalar(0); - rot1.s() = d > RealScalar(0) ? RealScalar(1) : RealScalar(-1); - } - else - { - RealScalar t2d2 = numext::hypot(t,d); - rot1.c() = abs(t)/t2d2; - rot1.s() = d/t2d2; - if(t<RealScalar(0)) - rot1.s() = -rot1.s(); - } - m.applyOnTheLeft(0,1,rot1); - j_right->makeJacobi(m,0,1); - *j_left = rot1 * j_right->transpose(); -} + typedef _MatrixType MatrixType; +}; } // end namespace internal @@ -451,8 +438,8 @@ void real_2x2_jacobi_svd(const MatrixType& matrix, Index p, Index q, * * \brief Two-sided Jacobi SVD decomposition of a rectangular matrix * - * \param MatrixType the type of the matrix of which we are computing the SVD decomposition - * \param QRPreconditioner this optional parameter allows to specify the type of QR decomposition that will be used internally + * \tparam _MatrixType the type of the matrix of which we are computing the SVD decomposition + * \tparam QRPreconditioner this optional parameter allows to specify the type of QR decomposition that will be used internally * for the R-SVD step for non-square matrices. See discussion of possible values below. * * SVD decomposition consists in decomposing any n-by-p matrix \a A as a product @@ -498,13 +485,14 @@ void real_2x2_jacobi_svd(const MatrixType& matrix, Index p, Index q, * \sa MatrixBase::jacobiSvd() */ template<typename _MatrixType, int QRPreconditioner> class JacobiSVD + : public SVDBase<JacobiSVD<_MatrixType,QRPreconditioner> > { + typedef SVDBase<JacobiSVD> Base; public: typedef _MatrixType MatrixType; typedef typename MatrixType::Scalar Scalar; typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar; - typedef typename MatrixType::Index Index; enum { RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime, @@ -515,13 +503,10 @@ template<typename _MatrixType, int QRPreconditioner> class JacobiSVD MatrixOptions = MatrixType::Options }; - typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, - MatrixOptions, MaxRowsAtCompileTime, MaxRowsAtCompileTime> - MatrixUType; - typedef Matrix<Scalar, ColsAtCompileTime, ColsAtCompileTime, - MatrixOptions, MaxColsAtCompileTime, MaxColsAtCompileTime> - MatrixVType; - typedef typename internal::plain_diag_type<MatrixType, RealScalar>::type SingularValuesType; + typedef typename Base::MatrixUType MatrixUType; + typedef typename Base::MatrixVType MatrixVType; + typedef typename Base::SingularValuesType SingularValuesType; + typedef typename internal::plain_row_type<MatrixType>::type RowType; typedef typename internal::plain_col_type<MatrixType>::type ColType; typedef Matrix<Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime, @@ -534,11 +519,6 @@ template<typename _MatrixType, int QRPreconditioner> class JacobiSVD * perform decompositions via JacobiSVD::compute(const MatrixType&). */ JacobiSVD() - : m_isInitialized(false), - m_isAllocated(false), - m_usePrescribedThreshold(false), - m_computationOptions(0), - m_rows(-1), m_cols(-1), m_diagSize(0) {} @@ -549,11 +529,6 @@ template<typename _MatrixType, int QRPreconditioner> class JacobiSVD * \sa JacobiSVD() */ JacobiSVD(Index rows, Index cols, unsigned int computationOptions = 0) - : m_isInitialized(false), - m_isAllocated(false), - m_usePrescribedThreshold(false), - m_computationOptions(0), - m_rows(-1), m_cols(-1) { allocate(rows, cols, computationOptions); } @@ -568,12 +543,7 @@ template<typename _MatrixType, int QRPreconditioner> class JacobiSVD * 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. */ - JacobiSVD(const MatrixType& matrix, unsigned int computationOptions = 0) - : m_isInitialized(false), - m_isAllocated(false), - m_usePrescribedThreshold(false), - m_computationOptions(0), - m_rows(-1), m_cols(-1) + explicit JacobiSVD(const MatrixType& matrix, unsigned int computationOptions = 0) { compute(matrix, computationOptions); } @@ -601,164 +571,33 @@ template<typename _MatrixType, int QRPreconditioner> class JacobiSVD return compute(matrix, m_computationOptions); } - /** \returns the \a U matrix. - * - * For the SVD decomposition of a n-by-p matrix, letting \a m be the minimum of \a n and \a p, - * the U matrix is n-by-n if you asked for #ComputeFullU, and is n-by-m if you asked for #ComputeThinU. - * - * The \a m first columns of \a U are the left singular vectors of the matrix being decomposed. - * - * This method asserts that you asked for \a U to be computed. - */ - const MatrixUType& matrixU() const - { - eigen_assert(m_isInitialized && "JacobiSVD is not initialized."); - eigen_assert(computeU() && "This JacobiSVD decomposition didn't compute U. Did you ask for it?"); - return m_matrixU; - } - - /** \returns the \a V matrix. - * - * For the SVD decomposition of a n-by-p matrix, letting \a m be the minimum of \a n and \a p, - * the V matrix is p-by-p if you asked for #ComputeFullV, and is p-by-m if you asked for ComputeThinV. - * - * The \a m first columns of \a V are the right singular vectors of the matrix being decomposed. - * - * This method asserts that you asked for \a V to be computed. - */ - const MatrixVType& matrixV() const - { - eigen_assert(m_isInitialized && "JacobiSVD is not initialized."); - eigen_assert(computeV() && "This JacobiSVD decomposition didn't compute V. Did you ask for it?"); - return m_matrixV; - } - - /** \returns the vector of singular values. - * - * For the SVD decomposition of a n-by-p matrix, letting \a m be the minimum of \a n and \a p, the - * returned vector has size \a m. Singular values are always sorted in decreasing order. - */ - const SingularValuesType& singularValues() const - { - eigen_assert(m_isInitialized && "JacobiSVD is not initialized."); - return m_singularValues; - } - - /** \returns true if \a U (full or thin) is asked for in this SVD decomposition */ - inline bool computeU() const { return m_computeFullU || m_computeThinU; } - /** \returns true if \a V (full or thin) is asked for in this SVD decomposition */ - inline bool computeV() const { return m_computeFullV || m_computeThinV; } - - /** \returns a (least squares) solution of \f$ A x = b \f$ using the current SVD decomposition of A. - * - * \param b the right-hand-side of the equation to solve. - * - * \note Solving requires both U and V to be computed. Thin U and V are enough, there is no need for full U or V. - * - * \note SVD solving is implicitly least-squares. Thus, this method serves both purposes of exact solving and least-squares solving. - * In other words, the returned solution is guaranteed to minimize the Euclidean norm \f$ \Vert A x - b \Vert \f$. - */ - template<typename Rhs> - inline const internal::solve_retval<JacobiSVD, Rhs> - solve(const MatrixBase<Rhs>& b) const - { - eigen_assert(m_isInitialized && "JacobiSVD is not initialized."); - eigen_assert(computeU() && computeV() && "JacobiSVD::solve() requires both unitaries U and V to be computed (thin unitaries suffice)."); - return internal::solve_retval<JacobiSVD, Rhs>(*this, b.derived()); - } - - /** \returns the number of singular values that are not exactly 0 */ - Index nonzeroSingularValues() const - { - eigen_assert(m_isInitialized && "JacobiSVD is not initialized."); - return m_nonzeroSingularValues; - } - - /** \returns the rank of the matrix of which \c *this is the SVD. - * - * \note This method has to determine which singular values should be considered nonzero. - * For that, it uses the threshold value that you can control by calling - * setThreshold(const RealScalar&). - */ - inline Index rank() const - { - using std::abs; - eigen_assert(m_isInitialized && "JacobiSVD is not initialized."); - if(m_singularValues.size()==0) return 0; - RealScalar premultiplied_threshold = m_singularValues.coeff(0) * threshold(); - Index i = m_nonzeroSingularValues-1; - while(i>=0 && m_singularValues.coeff(i) < premultiplied_threshold) --i; - return i+1; - } - - /** Allows to prescribe a threshold to be used by certain methods, such as rank() and solve(), - * which need to determine when singular values are to be considered nonzero. - * This is not used for the SVD decomposition itself. - * - * When it needs to get the threshold value, Eigen calls threshold(). - * The default is \c NumTraits<Scalar>::epsilon() - * - * \param threshold The new value to use as the threshold. - * - * A singular value will be considered nonzero if its value is strictly greater than - * \f$ \vert singular value \vert \leqslant threshold \times \vert max singular value \vert \f$. - * - * If you want to come back to the default behavior, call setThreshold(Default_t) - */ - JacobiSVD& setThreshold(const RealScalar& threshold) - { - m_usePrescribedThreshold = true; - m_prescribedThreshold = threshold; - return *this; - } - - /** Allows to come back to the default behavior, letting Eigen use its default formula for - * determining the threshold. - * - * You should pass the special object Eigen::Default as parameter here. - * \code svd.setThreshold(Eigen::Default); \endcode - * - * See the documentation of setThreshold(const RealScalar&). - */ - JacobiSVD& setThreshold(Default_t) - { - m_usePrescribedThreshold = false; - return *this; - } - - /** Returns the threshold that will be used by certain methods such as rank(). - * - * See the documentation of setThreshold(const RealScalar&). - */ - RealScalar threshold() const - { - eigen_assert(m_isInitialized || m_usePrescribedThreshold); - return m_usePrescribedThreshold ? m_prescribedThreshold - : (std::max<Index>)(1,m_diagSize)*NumTraits<Scalar>::epsilon(); - } - - inline Index rows() const { return m_rows; } - inline Index cols() const { return m_cols; } + using Base::computeU; + using Base::computeV; + using Base::rows; + using Base::cols; + using Base::rank; private: void allocate(Index rows, Index cols, unsigned int computationOptions); - - static void check_template_parameters() - { - EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar); - } protected: - MatrixUType m_matrixU; - MatrixVType m_matrixV; - SingularValuesType m_singularValues; + using Base::m_matrixU; + using Base::m_matrixV; + using Base::m_singularValues; + using Base::m_isInitialized; + using Base::m_isAllocated; + using Base::m_usePrescribedThreshold; + using Base::m_computeFullU; + using Base::m_computeThinU; + using Base::m_computeFullV; + using Base::m_computeThinV; + using Base::m_computationOptions; + using Base::m_nonzeroSingularValues; + using Base::m_rows; + using Base::m_cols; + using Base::m_diagSize; + using Base::m_prescribedThreshold; WorkMatrixType m_workMatrix; - bool m_isInitialized, m_isAllocated, m_usePrescribedThreshold; - bool m_computeFullU, m_computeThinU; - bool m_computeFullV, m_computeThinV; - unsigned int m_computationOptions; - Index m_nonzeroSingularValues, m_rows, m_cols, m_diagSize; - RealScalar m_prescribedThreshold; template<typename __MatrixType, int _QRPreconditioner, bool _IsComplex> friend struct internal::svd_precondition_2x2_block_to_be_real; @@ -816,15 +655,13 @@ void JacobiSVD<MatrixType, QRPreconditioner>::allocate(Index rows, Index cols, u if(m_cols>m_rows) m_qr_precond_morecols.allocate(*this); if(m_rows>m_cols) m_qr_precond_morerows.allocate(*this); - if(m_cols!=m_cols) m_scaledMatrix.resize(rows,cols); + if(m_rows!=m_cols) m_scaledMatrix.resize(rows,cols); } template<typename MatrixType, int QRPreconditioner> JacobiSVD<MatrixType, QRPreconditioner>& JacobiSVD<MatrixType, QRPreconditioner>::compute(const MatrixType& matrix, unsigned int computationOptions) { - check_template_parameters(); - using std::abs; allocate(matrix.rows(), matrix.cols(), computationOptions); @@ -832,8 +669,8 @@ JacobiSVD<MatrixType, QRPreconditioner>::compute(const MatrixType& matrix, unsig // only worsening the precision of U and V as we accumulate more rotations const RealScalar precision = RealScalar(2) * NumTraits<Scalar>::epsilon(); - // limit for very small denormal numbers to be considered zero in order to avoid infinite loops (see bug 286) - const RealScalar considerAsZero = RealScalar(2) * std::numeric_limits<RealScalar>::denorm_min(); + // limit for denormal numbers to be considered zero in order to avoid infinite loops (see bug 286) + const RealScalar considerAsZero = (std::numeric_limits<RealScalar>::min)(); // Scaling factor to reduce over/under-flows RealScalar scale = matrix.cwiseAbs().maxCoeff(); @@ -857,6 +694,7 @@ JacobiSVD<MatrixType, QRPreconditioner>::compute(const MatrixType& matrix, unsig } /*** step 2. The main Jacobi SVD iteration. ***/ + RealScalar maxDiagEntry = m_workMatrix.cwiseAbs().diagonal().maxCoeff(); bool finished = false; while(!finished) @@ -872,25 +710,27 @@ JacobiSVD<MatrixType, QRPreconditioner>::compute(const MatrixType& matrix, unsig // if this 2x2 sub-matrix is not diagonal already... // notice that this comparison will evaluate to false if any NaN is involved, ensuring that NaN's don't // keep us iterating forever. Similarly, small denormal numbers are considered zero. - using std::max; - RealScalar threshold = (max)(considerAsZero, precision * (max)(abs(m_workMatrix.coeff(p,p)), - abs(m_workMatrix.coeff(q,q)))); - // We compare both values to threshold instead of calling max to be robust to NaN (See bug 791) + RealScalar threshold = numext::maxi<RealScalar>(considerAsZero, precision * maxDiagEntry); if(abs(m_workMatrix.coeff(p,q))>threshold || abs(m_workMatrix.coeff(q,p)) > threshold) { finished = false; - // perform SVD decomposition of 2x2 sub-matrix corresponding to indices p,q to make it diagonal - internal::svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner>::run(m_workMatrix, *this, p, q); - JacobiRotation<RealScalar> j_left, j_right; - internal::real_2x2_jacobi_svd(m_workMatrix, p, q, &j_left, &j_right); - - // accumulate resulting Jacobi rotations - m_workMatrix.applyOnTheLeft(p,q,j_left); - if(computeU()) m_matrixU.applyOnTheRight(p,q,j_left.transpose()); - - m_workMatrix.applyOnTheRight(p,q,j_right); - if(computeV()) m_matrixV.applyOnTheRight(p,q,j_right); + // the complex to real operation returns true if the updated 2x2 block is not already diagonal + if(internal::svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner>::run(m_workMatrix, *this, p, q, maxDiagEntry)) + { + JacobiRotation<RealScalar> j_left, j_right; + internal::real_2x2_jacobi_svd(m_workMatrix, p, q, &j_left, &j_right); + + // accumulate resulting Jacobi rotations + m_workMatrix.applyOnTheLeft(p,q,j_left); + if(computeU()) m_matrixU.applyOnTheRight(p,q,j_left.transpose()); + + m_workMatrix.applyOnTheRight(p,q,j_right); + if(computeV()) m_matrixV.applyOnTheRight(p,q,j_right); + + // keep track of the largest diagonal coefficient + maxDiagEntry = numext::maxi<RealScalar>(maxDiagEntry,numext::maxi<RealScalar>(abs(m_workMatrix.coeff(p,p)), abs(m_workMatrix.coeff(q,q)))); + } } } } @@ -900,10 +740,25 @@ JacobiSVD<MatrixType, QRPreconditioner>::compute(const MatrixType& matrix, unsig for(Index i = 0; i < m_diagSize; ++i) { - RealScalar a = abs(m_workMatrix.coeff(i,i)); - m_singularValues.coeffRef(i) = a; - if(computeU() && (a!=RealScalar(0))) m_matrixU.col(i) *= m_workMatrix.coeff(i,i)/a; + // For a complex matrix, some diagonal coefficients might note have been + // treated by svd_precondition_2x2_block_to_be_real, and the imaginary part + // of some diagonal entry might not be null. + if(NumTraits<Scalar>::IsComplex && abs(numext::imag(m_workMatrix.coeff(i,i)))>considerAsZero) + { + RealScalar a = abs(m_workMatrix.coeff(i,i)); + m_singularValues.coeffRef(i) = abs(a); + if(computeU()) m_matrixU.col(i) *= m_workMatrix.coeff(i,i)/a; + } + else + { + // m_workMatrix.coeff(i,i) is already real, no difficulty: + RealScalar a = numext::real(m_workMatrix.coeff(i,i)); + m_singularValues.coeffRef(i) = abs(a); + if(computeU() && (a<RealScalar(0))) m_matrixU.col(i) = -m_matrixU.col(i); + } } + + m_singularValues *= scale; /*** step 4. Sort singular values in descending order and compute the number of nonzero singular values ***/ @@ -925,38 +780,11 @@ JacobiSVD<MatrixType, QRPreconditioner>::compute(const MatrixType& matrix, unsig if(computeV()) m_matrixV.col(pos).swap(m_matrixV.col(i)); } } - - m_singularValues *= scale; m_isInitialized = true; return *this; } -namespace internal { -template<typename _MatrixType, int QRPreconditioner, typename Rhs> -struct solve_retval<JacobiSVD<_MatrixType, QRPreconditioner>, Rhs> - : solve_retval_base<JacobiSVD<_MatrixType, QRPreconditioner>, Rhs> -{ - typedef JacobiSVD<_MatrixType, QRPreconditioner> JacobiSVDType; - EIGEN_MAKE_SOLVE_HELPERS(JacobiSVDType,Rhs) - - template<typename Dest> void evalTo(Dest& dst) const - { - eigen_assert(rhs().rows() == dec().rows()); - - // A = U S V^* - // So A^{-1} = V S^{-1} U^* - - Matrix<Scalar, Dynamic, Rhs::ColsAtCompileTime, 0, _MatrixType::MaxRowsAtCompileTime, Rhs::MaxColsAtCompileTime> tmp; - Index rank = dec().rank(); - - tmp.noalias() = dec().matrixU().leftCols(rank).adjoint() * rhs(); - tmp = dec().singularValues().head(rank).asDiagonal().inverse() * tmp; - dst = dec().matrixV().leftCols(rank) * tmp; - } -}; -} // end namespace internal - /** \svd_module * * \return the singular value decomposition of \c *this computed by two-sided diff --git a/extern/Eigen3/Eigen/src/SVD/JacobiSVD_LAPACKE.h b/extern/Eigen3/Eigen/src/SVD/JacobiSVD_LAPACKE.h new file mode 100644 index 00000000000..ff0516f6116 --- /dev/null +++ b/extern/Eigen3/Eigen/src/SVD/JacobiSVD_LAPACKE.h @@ -0,0 +1,91 @@ +/* + Copyright (c) 2011, Intel Corporation. All rights reserved. + + Redistribution and use in source and binary forms, with or without modification, + are permitted provided that the following conditions are met: + + * Redistributions of source code must retain the above copyright notice, this + list of conditions and the following disclaimer. + * Redistributions in binary form must reproduce the above copyright notice, + this list of conditions and the following disclaimer in the documentation + and/or other materials provided with the distribution. + * Neither the name of Intel Corporation nor the names of its contributors may + be used to endorse or promote products derived from this software without + specific prior written permission. + + THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND + ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED + WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE + DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR + ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES + (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; + LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON + ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT + (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS + SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + + ******************************************************************************** + * Content : Eigen bindings to LAPACKe + * Singular Value Decomposition - SVD. + ******************************************************************************** +*/ + +#ifndef EIGEN_JACOBISVD_LAPACKE_H +#define EIGEN_JACOBISVD_LAPACKE_H + +namespace Eigen { + +/** \internal Specialization for the data types supported by LAPACKe */ + +#define EIGEN_LAPACKE_SVD(EIGTYPE, LAPACKE_TYPE, LAPACKE_RTYPE, LAPACKE_PREFIX, EIGCOLROW, LAPACKE_COLROW) \ +template<> inline \ +JacobiSVD<Matrix<EIGTYPE, Dynamic, Dynamic, EIGCOLROW, Dynamic, Dynamic>, ColPivHouseholderQRPreconditioner>& \ +JacobiSVD<Matrix<EIGTYPE, Dynamic, Dynamic, EIGCOLROW, Dynamic, Dynamic>, ColPivHouseholderQRPreconditioner>::compute(const Matrix<EIGTYPE, Dynamic, Dynamic, EIGCOLROW, Dynamic, Dynamic>& matrix, unsigned int computationOptions) \ +{ \ + typedef Matrix<EIGTYPE, Dynamic, Dynamic, EIGCOLROW, Dynamic, Dynamic> MatrixType; \ + /*typedef MatrixType::Scalar Scalar;*/ \ + /*typedef MatrixType::RealScalar RealScalar;*/ \ + allocate(matrix.rows(), matrix.cols(), computationOptions); \ +\ + /*const RealScalar precision = RealScalar(2) * NumTraits<Scalar>::epsilon();*/ \ + m_nonzeroSingularValues = m_diagSize; \ +\ + lapack_int lda = internal::convert_index<lapack_int>(matrix.outerStride()), ldu, ldvt; \ + lapack_int matrix_order = LAPACKE_COLROW; \ + char jobu, jobvt; \ + LAPACKE_TYPE *u, *vt, dummy; \ + jobu = (m_computeFullU) ? 'A' : (m_computeThinU) ? 'S' : 'N'; \ + jobvt = (m_computeFullV) ? 'A' : (m_computeThinV) ? 'S' : 'N'; \ + if (computeU()) { \ + ldu = internal::convert_index<lapack_int>(m_matrixU.outerStride()); \ + u = (LAPACKE_TYPE*)m_matrixU.data(); \ + } else { ldu=1; u=&dummy; }\ + MatrixType localV; \ + lapack_int vt_rows = (m_computeFullV) ? internal::convert_index<lapack_int>(m_cols) : (m_computeThinV) ? internal::convert_index<lapack_int>(m_diagSize) : 1; \ + if (computeV()) { \ + localV.resize(vt_rows, m_cols); \ + ldvt = internal::convert_index<lapack_int>(localV.outerStride()); \ + vt = (LAPACKE_TYPE*)localV.data(); \ + } else { ldvt=1; vt=&dummy; }\ + Matrix<LAPACKE_RTYPE, Dynamic, Dynamic> superb; superb.resize(m_diagSize, 1); \ + MatrixType m_temp; m_temp = matrix; \ + LAPACKE_##LAPACKE_PREFIX##gesvd( matrix_order, jobu, jobvt, internal::convert_index<lapack_int>(m_rows), internal::convert_index<lapack_int>(m_cols), (LAPACKE_TYPE*)m_temp.data(), lda, (LAPACKE_RTYPE*)m_singularValues.data(), u, ldu, vt, ldvt, superb.data()); \ + if (computeV()) m_matrixV = localV.adjoint(); \ + /* for(int i=0;i<m_diagSize;i++) if (m_singularValues.coeffRef(i) < precision) { m_nonzeroSingularValues--; m_singularValues.coeffRef(i)=RealScalar(0);}*/ \ + m_isInitialized = true; \ + return *this; \ +} + +EIGEN_LAPACKE_SVD(double, double, double, d, ColMajor, LAPACK_COL_MAJOR) +EIGEN_LAPACKE_SVD(float, float, float , s, ColMajor, LAPACK_COL_MAJOR) +EIGEN_LAPACKE_SVD(dcomplex, lapack_complex_double, double, z, ColMajor, LAPACK_COL_MAJOR) +EIGEN_LAPACKE_SVD(scomplex, lapack_complex_float, float , c, ColMajor, LAPACK_COL_MAJOR) + +EIGEN_LAPACKE_SVD(double, double, double, d, RowMajor, LAPACK_ROW_MAJOR) +EIGEN_LAPACKE_SVD(float, float, float , s, RowMajor, LAPACK_ROW_MAJOR) +EIGEN_LAPACKE_SVD(dcomplex, lapack_complex_double, double, z, RowMajor, LAPACK_ROW_MAJOR) +EIGEN_LAPACKE_SVD(scomplex, lapack_complex_float, float , c, RowMajor, LAPACK_ROW_MAJOR) + +} // end namespace Eigen + +#endif // EIGEN_JACOBISVD_LAPACKE_H diff --git a/extern/Eigen3/Eigen/src/SVD/SVDBase.h b/extern/Eigen3/Eigen/src/SVD/SVDBase.h new file mode 100644 index 00000000000..3d1ef373ea7 --- /dev/null +++ b/extern/Eigen3/Eigen/src/SVD/SVDBase.h @@ -0,0 +1,315 @@ +// This file is part of Eigen, a lightweight C++ template library +// for linear algebra. +// +// Copyright (C) 2009-2010 Benoit Jacob <jacob.benoit.1@gmail.com> +// Copyright (C) 2014 Gael Guennebaud <gael.guennebaud@inria.fr> +// +// 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> +// +// This Source Code Form is subject to the terms of the Mozilla +// Public License v. 2.0. If a copy of the MPL was not distributed +// with this file, You can obtain one at http://mozilla.org/MPL/2.0/. + +#ifndef EIGEN_SVDBASE_H +#define EIGEN_SVDBASE_H + +namespace Eigen { +/** \ingroup SVD_Module + * + * + * \class SVDBase + * + * \brief Base class of SVD algorithms + * + * \tparam Derived the type of the actual SVD decomposition + * + * SVD decomposition consists in decomposing any n-by-p matrix \a A as a product + * \f[ A = U S V^* \f] + * where \a U is a n-by-n unitary, \a V is a p-by-p unitary, and \a S is a n-by-p real positive matrix which is zero outside of its main diagonal; + * the diagonal entries of S are known as the \em singular \em values of \a A and the columns of \a U and \a V are known as the left + * and right \em singular \em vectors of \a A respectively. + * + * Singular values are always sorted in decreasing order. + * + * + * You can ask for only \em thin \a U or \a V to be computed, meaning the following. In case of a rectangular n-by-p matrix, letting \a m be the + * smaller value among \a n and \a p, there are only \a m singular vectors; the remaining columns of \a U and \a V do not correspond to actual + * singular vectors. Asking for \em thin \a U or \a V means asking for only their \a m first columns to be formed. So \a U is then a n-by-m matrix, + * and \a V is then a p-by-m matrix. Notice that thin \a U and \a V are all you need for (least squares) solving. + * + * If the input matrix has inf or nan coefficients, the result of the computation is undefined, but the computation is guaranteed to + * terminate in finite (and reasonable) time. + * \sa class BDCSVD, class JacobiSVD + */ +template<typename Derived> +class SVDBase +{ + +public: + typedef typename internal::traits<Derived>::MatrixType MatrixType; + typedef typename MatrixType::Scalar Scalar; + typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar; + typedef typename MatrixType::StorageIndex StorageIndex; + typedef Eigen::Index Index; ///< \deprecated since Eigen 3.3 + 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 Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, MatrixOptions, MaxRowsAtCompileTime, MaxRowsAtCompileTime> MatrixUType; + typedef Matrix<Scalar, ColsAtCompileTime, ColsAtCompileTime, MatrixOptions, MaxColsAtCompileTime, MaxColsAtCompileTime> MatrixVType; + typedef typename internal::plain_diag_type<MatrixType, RealScalar>::type SingularValuesType; + + Derived& derived() { return *static_cast<Derived*>(this); } + const Derived& derived() const { return *static_cast<const Derived*>(this); } + + /** \returns the \a U matrix. + * + * For the SVD decomposition of a n-by-p matrix, letting \a m be the minimum of \a n and \a p, + * the U matrix is n-by-n if you asked for \link Eigen::ComputeFullU ComputeFullU \endlink, and is n-by-m if you asked for \link Eigen::ComputeThinU ComputeThinU \endlink. + * + * The \a m first columns of \a U are the left singular vectors of the matrix being decomposed. + * + * This method asserts that you asked for \a U to be computed. + */ + const MatrixUType& matrixU() const + { + eigen_assert(m_isInitialized && "SVD is not initialized."); + eigen_assert(computeU() && "This SVD decomposition didn't compute U. Did you ask for it?"); + return m_matrixU; + } + + /** \returns the \a V matrix. + * + * For the SVD decomposition of a n-by-p matrix, letting \a m be the minimum of \a n and \a p, + * the V matrix is p-by-p if you asked for \link Eigen::ComputeFullV ComputeFullV \endlink, and is p-by-m if you asked for \link Eigen::ComputeThinV ComputeThinV \endlink. + * + * The \a m first columns of \a V are the right singular vectors of the matrix being decomposed. + * + * This method asserts that you asked for \a V to be computed. + */ + const MatrixVType& matrixV() const + { + eigen_assert(m_isInitialized && "SVD is not initialized."); + eigen_assert(computeV() && "This SVD decomposition didn't compute V. Did you ask for it?"); + return m_matrixV; + } + + /** \returns the vector of singular values. + * + * For the SVD decomposition of a n-by-p matrix, letting \a m be the minimum of \a n and \a p, the + * returned vector has size \a m. Singular values are always sorted in decreasing order. + */ + const SingularValuesType& singularValues() const + { + eigen_assert(m_isInitialized && "SVD is not initialized."); + return m_singularValues; + } + + /** \returns the number of singular values that are not exactly 0 */ + Index nonzeroSingularValues() const + { + eigen_assert(m_isInitialized && "SVD is not initialized."); + return m_nonzeroSingularValues; + } + + /** \returns the rank of the matrix of which \c *this is the SVD. + * + * \note This method has to determine which singular values should be considered nonzero. + * For that, it uses the threshold value that you can control by calling + * setThreshold(const RealScalar&). + */ + inline Index rank() const + { + using std::abs; + eigen_assert(m_isInitialized && "JacobiSVD is not initialized."); + if(m_singularValues.size()==0) return 0; + RealScalar premultiplied_threshold = numext::maxi<RealScalar>(m_singularValues.coeff(0) * threshold(), (std::numeric_limits<RealScalar>::min)()); + Index i = m_nonzeroSingularValues-1; + while(i>=0 && m_singularValues.coeff(i) < premultiplied_threshold) --i; + return i+1; + } + + /** Allows to prescribe a threshold to be used by certain methods, such as rank() and solve(), + * which need to determine when singular values are to be considered nonzero. + * This is not used for the SVD decomposition itself. + * + * When it needs to get the threshold value, Eigen calls threshold(). + * The default is \c NumTraits<Scalar>::epsilon() + * + * \param threshold The new value to use as the threshold. + * + * A singular value will be considered nonzero if its value is strictly greater than + * \f$ \vert singular value \vert \leqslant threshold \times \vert max singular value \vert \f$. + * + * If you want to come back to the default behavior, call setThreshold(Default_t) + */ + Derived& setThreshold(const RealScalar& threshold) + { + m_usePrescribedThreshold = true; + m_prescribedThreshold = threshold; + return derived(); + } + + /** Allows to come back to the default behavior, letting Eigen use its default formula for + * determining the threshold. + * + * You should pass the special object Eigen::Default as parameter here. + * \code svd.setThreshold(Eigen::Default); \endcode + * + * See the documentation of setThreshold(const RealScalar&). + */ + Derived& setThreshold(Default_t) + { + m_usePrescribedThreshold = false; + return derived(); + } + + /** Returns the threshold that will be used by certain methods such as rank(). + * + * See the documentation of setThreshold(const RealScalar&). + */ + RealScalar threshold() const + { + eigen_assert(m_isInitialized || m_usePrescribedThreshold); + // this temporary is needed to workaround a MSVC issue + Index diagSize = (std::max<Index>)(1,m_diagSize); + return m_usePrescribedThreshold ? m_prescribedThreshold + : diagSize*NumTraits<Scalar>::epsilon(); + } + + /** \returns true if \a U (full or thin) is asked for in this SVD decomposition */ + inline bool computeU() const { return m_computeFullU || m_computeThinU; } + /** \returns true if \a V (full or thin) is asked for in this SVD decomposition */ + inline bool computeV() const { return m_computeFullV || m_computeThinV; } + + inline Index rows() const { return m_rows; } + inline Index cols() const { return m_cols; } + + /** \returns a (least squares) solution of \f$ A x = b \f$ using the current SVD decomposition of A. + * + * \param b the right-hand-side of the equation to solve. + * + * \note Solving requires both U and V to be computed. Thin U and V are enough, there is no need for full U or V. + * + * \note SVD solving is implicitly least-squares. Thus, this method serves both purposes of exact solving and least-squares solving. + * In other words, the returned solution is guaranteed to minimize the Euclidean norm \f$ \Vert A x - b \Vert \f$. + */ + template<typename Rhs> + inline const Solve<Derived, Rhs> + solve(const MatrixBase<Rhs>& b) const + { + eigen_assert(m_isInitialized && "SVD is not initialized."); + eigen_assert(computeU() && computeV() && "SVD::solve() requires both unitaries U and V to be computed (thin unitaries suffice)."); + return Solve<Derived, Rhs>(derived(), b.derived()); + } + + #ifndef EIGEN_PARSED_BY_DOXYGEN + template<typename RhsType, typename DstType> + EIGEN_DEVICE_FUNC + void _solve_impl(const RhsType &rhs, DstType &dst) const; + #endif + +protected: + + static void check_template_parameters() + { + EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar); + } + + // return true if already allocated + bool allocate(Index rows, Index cols, unsigned int computationOptions) ; + + MatrixUType m_matrixU; + MatrixVType m_matrixV; + SingularValuesType m_singularValues; + bool m_isInitialized, m_isAllocated, m_usePrescribedThreshold; + bool m_computeFullU, m_computeThinU; + bool m_computeFullV, m_computeThinV; + unsigned int m_computationOptions; + Index m_nonzeroSingularValues, m_rows, m_cols, m_diagSize; + RealScalar m_prescribedThreshold; + + /** \brief Default Constructor. + * + * Default constructor of SVDBase + */ + SVDBase() + : m_isInitialized(false), + m_isAllocated(false), + m_usePrescribedThreshold(false), + m_computationOptions(0), + m_rows(-1), m_cols(-1), m_diagSize(0) + { + check_template_parameters(); + } + + +}; + +#ifndef EIGEN_PARSED_BY_DOXYGEN +template<typename Derived> +template<typename RhsType, typename DstType> +void SVDBase<Derived>::_solve_impl(const RhsType &rhs, DstType &dst) const +{ + eigen_assert(rhs.rows() == rows()); + + // A = U S V^* + // So A^{-1} = V S^{-1} U^* + + Matrix<Scalar, Dynamic, RhsType::ColsAtCompileTime, 0, MatrixType::MaxRowsAtCompileTime, RhsType::MaxColsAtCompileTime> tmp; + Index l_rank = rank(); + tmp.noalias() = m_matrixU.leftCols(l_rank).adjoint() * rhs; + tmp = m_singularValues.head(l_rank).asDiagonal().inverse() * tmp; + dst = m_matrixV.leftCols(l_rank) * tmp; +} +#endif + +template<typename MatrixType> +bool SVDBase<MatrixType>::allocate(Index rows, Index cols, unsigned int computationOptions) +{ + eigen_assert(rows >= 0 && cols >= 0); + + if (m_isAllocated && + rows == m_rows && + cols == m_cols && + computationOptions == m_computationOptions) + { + return true; + } + + m_rows = rows; + m_cols = cols; + m_isInitialized = false; + m_isAllocated = true; + m_computationOptions = computationOptions; + m_computeFullU = (computationOptions & ComputeFullU) != 0; + m_computeThinU = (computationOptions & ComputeThinU) != 0; + m_computeFullV = (computationOptions & ComputeFullV) != 0; + m_computeThinV = (computationOptions & ComputeThinV) != 0; + eigen_assert(!(m_computeFullU && m_computeThinU) && "SVDBase: you can't ask for both full and thin U"); + eigen_assert(!(m_computeFullV && m_computeThinV) && "SVDBase: you can't ask for both full and thin V"); + eigen_assert(EIGEN_IMPLIES(m_computeThinU || m_computeThinV, MatrixType::ColsAtCompileTime==Dynamic) && + "SVDBase: thin U and V are only available when your matrix has a dynamic number of columns."); + + m_diagSize = (std::min)(m_rows, m_cols); + m_singularValues.resize(m_diagSize); + if(RowsAtCompileTime==Dynamic) + m_matrixU.resize(m_rows, m_computeFullU ? m_rows : m_computeThinU ? m_diagSize : 0); + if(ColsAtCompileTime==Dynamic) + m_matrixV.resize(m_cols, m_computeFullV ? m_cols : m_computeThinV ? m_diagSize : 0); + + return false; +} + +}// end namespace + +#endif // EIGEN_SVDBASE_H diff --git a/extern/Eigen3/Eigen/src/SVD/UpperBidiagonalization.h b/extern/Eigen3/Eigen/src/SVD/UpperBidiagonalization.h index 587de37a5ad..11ac847e1db 100644 --- a/extern/Eigen3/Eigen/src/SVD/UpperBidiagonalization.h +++ b/extern/Eigen3/Eigen/src/SVD/UpperBidiagonalization.h @@ -2,6 +2,7 @@ // for linear algebra. // // Copyright (C) 2010 Benoit Jacob <jacob.benoit.1@gmail.com> +// Copyright (C) 2013-2014 Gael Guennebaud <gael.guennebaud@inria.fr> // // This Source Code Form is subject to the terms of the Mozilla // Public License v. 2.0. If a copy of the MPL was not distributed @@ -28,15 +29,15 @@ template<typename _MatrixType> class UpperBidiagonalization }; typedef typename MatrixType::Scalar Scalar; typedef typename MatrixType::RealScalar RealScalar; - typedef typename MatrixType::Index Index; + typedef Eigen::Index Index; ///< \deprecated since Eigen 3.3 typedef Matrix<Scalar, 1, ColsAtCompileTime> RowVectorType; typedef Matrix<Scalar, RowsAtCompileTime, 1> ColVectorType; - typedef BandMatrix<RealScalar, ColsAtCompileTime, ColsAtCompileTime, 1, 0> BidiagonalType; + typedef BandMatrix<RealScalar, ColsAtCompileTime, ColsAtCompileTime, 1, 0, RowMajor> BidiagonalType; typedef Matrix<Scalar, ColsAtCompileTime, 1> DiagVectorType; typedef Matrix<Scalar, ColsAtCompileTimeMinusOne, 1> SuperDiagVectorType; typedef HouseholderSequence< const MatrixType, - CwiseUnaryOp<internal::scalar_conjugate_op<Scalar>, const Diagonal<const MatrixType,0> > + const typename internal::remove_all<typename Diagonal<const MatrixType,0>::ConjugateReturnType>::type > HouseholderUSequenceType; typedef HouseholderSequence< const typename internal::remove_all<typename MatrixType::ConjugateReturnType>::type, @@ -52,7 +53,7 @@ template<typename _MatrixType> class UpperBidiagonalization */ UpperBidiagonalization() : m_householder(), m_bidiagonal(), m_isInitialized(false) {} - UpperBidiagonalization(const MatrixType& matrix) + explicit UpperBidiagonalization(const MatrixType& matrix) : m_householder(matrix.rows(), matrix.cols()), m_bidiagonal(matrix.cols(), matrix.cols()), m_isInitialized(false) @@ -61,6 +62,7 @@ template<typename _MatrixType> class UpperBidiagonalization } UpperBidiagonalization& compute(const MatrixType& matrix); + UpperBidiagonalization& computeUnblocked(const MatrixType& matrix); const MatrixType& householder() const { return m_householder; } const BidiagonalType& bidiagonal() const { return m_bidiagonal; } @@ -85,45 +87,309 @@ template<typename _MatrixType> class UpperBidiagonalization bool m_isInitialized; }; -template<typename _MatrixType> -UpperBidiagonalization<_MatrixType>& UpperBidiagonalization<_MatrixType>::compute(const _MatrixType& matrix) +// Standard upper bidiagonalization without fancy optimizations +// This version should be faster for small matrix size +template<typename MatrixType> +void upperbidiagonalization_inplace_unblocked(MatrixType& mat, + typename MatrixType::RealScalar *diagonal, + typename MatrixType::RealScalar *upper_diagonal, + typename MatrixType::Scalar* tempData = 0) { - Index rows = matrix.rows(); - Index cols = matrix.cols(); - - eigen_assert(rows >= cols && "UpperBidiagonalization is only for matrices satisfying rows>=cols."); - - m_householder = matrix; + typedef typename MatrixType::Scalar Scalar; - ColVectorType temp(rows); + Index rows = mat.rows(); + Index cols = mat.cols(); + + typedef Matrix<Scalar,Dynamic,1,ColMajor,MatrixType::MaxRowsAtCompileTime,1> TempType; + TempType tempVector; + if(tempData==0) + { + tempVector.resize(rows); + tempData = tempVector.data(); + } for (Index k = 0; /* breaks at k==cols-1 below */ ; ++k) { Index remainingRows = rows - k; Index remainingCols = cols - k - 1; - // construct left householder transform in-place in m_householder - m_householder.col(k).tail(remainingRows) - .makeHouseholderInPlace(m_householder.coeffRef(k,k), - m_bidiagonal.template diagonal<0>().coeffRef(k)); - // apply householder transform to remaining part of m_householder on the left - m_householder.bottomRightCorner(remainingRows, remainingCols) - .applyHouseholderOnTheLeft(m_householder.col(k).tail(remainingRows-1), - m_householder.coeff(k,k), - temp.data()); + // construct left householder transform in-place in A + mat.col(k).tail(remainingRows) + .makeHouseholderInPlace(mat.coeffRef(k,k), diagonal[k]); + // apply householder transform to remaining part of A on the left + mat.bottomRightCorner(remainingRows, remainingCols) + .applyHouseholderOnTheLeft(mat.col(k).tail(remainingRows-1), mat.coeff(k,k), tempData); if(k == cols-1) break; + + // construct right householder transform in-place in mat + mat.row(k).tail(remainingCols) + .makeHouseholderInPlace(mat.coeffRef(k,k+1), upper_diagonal[k]); + // apply householder transform to remaining part of mat on the left + mat.bottomRightCorner(remainingRows-1, remainingCols) + .applyHouseholderOnTheRight(mat.row(k).tail(remainingCols-1).transpose(), mat.coeff(k,k+1), tempData); + } +} + +/** \internal + * Helper routine for the block reduction to upper bidiagonal form. + * + * Let's partition the matrix A: + * + * | A00 A01 | + * A = | | + * | A10 A11 | + * + * This function reduces to bidiagonal form the left \c rows x \a blockSize vertical panel [A00/A10] + * and the \a blockSize x \c cols horizontal panel [A00 A01] of the matrix \a A. The bottom-right block A11 + * is updated using matrix-matrix products: + * A22 -= V * Y^T - X * U^T + * where V and U contains the left and right Householder vectors. U and V are stored in A10, and A01 + * respectively, and the update matrices X and Y are computed during the reduction. + * + */ +template<typename MatrixType> +void upperbidiagonalization_blocked_helper(MatrixType& A, + typename MatrixType::RealScalar *diagonal, + typename MatrixType::RealScalar *upper_diagonal, + Index bs, + Ref<Matrix<typename MatrixType::Scalar, Dynamic, Dynamic, + traits<MatrixType>::Flags & RowMajorBit> > X, + Ref<Matrix<typename MatrixType::Scalar, Dynamic, Dynamic, + traits<MatrixType>::Flags & RowMajorBit> > Y) +{ + typedef typename MatrixType::Scalar Scalar; + typedef typename MatrixType::RealScalar RealScalar; + typedef typename NumTraits<RealScalar>::Literal Literal; + enum { StorageOrder = traits<MatrixType>::Flags & RowMajorBit }; + typedef InnerStride<int(StorageOrder) == int(ColMajor) ? 1 : Dynamic> ColInnerStride; + typedef InnerStride<int(StorageOrder) == int(ColMajor) ? Dynamic : 1> RowInnerStride; + typedef Ref<Matrix<Scalar, Dynamic, 1>, 0, ColInnerStride> SubColumnType; + typedef Ref<Matrix<Scalar, 1, Dynamic>, 0, RowInnerStride> SubRowType; + typedef Ref<Matrix<Scalar, Dynamic, Dynamic, StorageOrder > > SubMatType; + + Index brows = A.rows(); + Index bcols = A.cols(); + + Scalar tau_u, tau_u_prev(0), tau_v; + + for(Index k = 0; k < bs; ++k) + { + Index remainingRows = brows - k; + Index remainingCols = bcols - k - 1; + + SubMatType X_k1( X.block(k,0, remainingRows,k) ); + SubMatType V_k1( A.block(k,0, remainingRows,k) ); + + // 1 - update the k-th column of A + SubColumnType v_k = A.col(k).tail(remainingRows); + v_k -= V_k1 * Y.row(k).head(k).adjoint(); + if(k) v_k -= X_k1 * A.col(k).head(k); + + // 2 - construct left Householder transform in-place + v_k.makeHouseholderInPlace(tau_v, diagonal[k]); + + if(k+1<bcols) + { + SubMatType Y_k ( Y.block(k+1,0, remainingCols, k+1) ); + SubMatType U_k1 ( A.block(0,k+1, k,remainingCols) ); + + // this eases the application of Householder transforAions + // A(k,k) will store tau_v later + A(k,k) = Scalar(1); + + // 3 - Compute y_k^T = tau_v * ( A^T*v_k - Y_k-1*V_k-1^T*v_k - U_k-1*X_k-1^T*v_k ) + { + SubColumnType y_k( Y.col(k).tail(remainingCols) ); + + // let's use the begining of column k of Y as a temporary vector + SubColumnType tmp( Y.col(k).head(k) ); + y_k.noalias() = A.block(k,k+1, remainingRows,remainingCols).adjoint() * v_k; // bottleneck + tmp.noalias() = V_k1.adjoint() * v_k; + y_k.noalias() -= Y_k.leftCols(k) * tmp; + tmp.noalias() = X_k1.adjoint() * v_k; + y_k.noalias() -= U_k1.adjoint() * tmp; + y_k *= numext::conj(tau_v); + } + + // 4 - update k-th row of A (it will become u_k) + SubRowType u_k( A.row(k).tail(remainingCols) ); + u_k = u_k.conjugate(); + { + u_k -= Y_k * A.row(k).head(k+1).adjoint(); + if(k) u_k -= U_k1.adjoint() * X.row(k).head(k).adjoint(); + } + + // 5 - construct right Householder transform in-place + u_k.makeHouseholderInPlace(tau_u, upper_diagonal[k]); + + // this eases the application of Householder transformations + // A(k,k+1) will store tau_u later + A(k,k+1) = Scalar(1); + + // 6 - Compute x_k = tau_u * ( A*u_k - X_k-1*U_k-1^T*u_k - V_k*Y_k^T*u_k ) + { + SubColumnType x_k ( X.col(k).tail(remainingRows-1) ); + + // let's use the begining of column k of X as a temporary vectors + // note that tmp0 and tmp1 overlaps + SubColumnType tmp0 ( X.col(k).head(k) ), + tmp1 ( X.col(k).head(k+1) ); + + x_k.noalias() = A.block(k+1,k+1, remainingRows-1,remainingCols) * u_k.transpose(); // bottleneck + tmp0.noalias() = U_k1 * u_k.transpose(); + x_k.noalias() -= X_k1.bottomRows(remainingRows-1) * tmp0; + tmp1.noalias() = Y_k.adjoint() * u_k.transpose(); + x_k.noalias() -= A.block(k+1,0, remainingRows-1,k+1) * tmp1; + x_k *= numext::conj(tau_u); + tau_u = numext::conj(tau_u); + u_k = u_k.conjugate(); + } + + if(k>0) A.coeffRef(k-1,k) = tau_u_prev; + tau_u_prev = tau_u; + } + else + A.coeffRef(k-1,k) = tau_u_prev; + + A.coeffRef(k,k) = tau_v; + } + + if(bs<bcols) + A.coeffRef(bs-1,bs) = tau_u_prev; + + // update A22 + if(bcols>bs && brows>bs) + { + SubMatType A11( A.bottomRightCorner(brows-bs,bcols-bs) ); + SubMatType A10( A.block(bs,0, brows-bs,bs) ); + SubMatType A01( A.block(0,bs, bs,bcols-bs) ); + Scalar tmp = A01(bs-1,0); + A01(bs-1,0) = Literal(1); + A11.noalias() -= A10 * Y.topLeftCorner(bcols,bs).bottomRows(bcols-bs).adjoint(); + A11.noalias() -= X.topLeftCorner(brows,bs).bottomRows(brows-bs) * A01; + A01(bs-1,0) = tmp; + } +} + +/** \internal + * + * Implementation of a block-bidiagonal reduction. + * It is based on the following paper: + * The Design of a Parallel Dense Linear Algebra Software Library: Reduction to Hessenberg, Tridiagonal, and Bidiagonal Form. + * by Jaeyoung Choi, Jack J. Dongarra, David W. Walker. (1995) + * section 3.3 + */ +template<typename MatrixType, typename BidiagType> +void upperbidiagonalization_inplace_blocked(MatrixType& A, BidiagType& bidiagonal, + Index maxBlockSize=32, + typename MatrixType::Scalar* /*tempData*/ = 0) +{ + typedef typename MatrixType::Scalar Scalar; + typedef Block<MatrixType,Dynamic,Dynamic> BlockType; + + Index rows = A.rows(); + Index cols = A.cols(); + Index size = (std::min)(rows, cols); + + // X and Y are work space + enum { StorageOrder = traits<MatrixType>::Flags & RowMajorBit }; + Matrix<Scalar, + MatrixType::RowsAtCompileTime, + Dynamic, + StorageOrder, + MatrixType::MaxRowsAtCompileTime> X(rows,maxBlockSize); + Matrix<Scalar, + MatrixType::ColsAtCompileTime, + Dynamic, + StorageOrder, + MatrixType::MaxColsAtCompileTime> Y(cols,maxBlockSize); + Index blockSize = (std::min)(maxBlockSize,size); + + Index k = 0; + for(k = 0; k < size; k += blockSize) + { + Index bs = (std::min)(size-k,blockSize); // actual size of the block + Index brows = rows - k; // rows of the block + Index bcols = cols - k; // columns of the block + + // partition the matrix A: + // + // | A00 A01 A02 | + // | | + // A = | A10 A11 A12 | + // | | + // | A20 A21 A22 | + // + // where A11 is a bs x bs diagonal block, + // and let: + // | A11 A12 | + // B = | | + // | A21 A22 | + + BlockType B = A.block(k,k,brows,bcols); - // construct right householder transform in-place in m_householder - m_householder.row(k).tail(remainingCols) - .makeHouseholderInPlace(m_householder.coeffRef(k,k+1), - m_bidiagonal.template diagonal<1>().coeffRef(k)); - // apply householder transform to remaining part of m_householder on the left - m_householder.bottomRightCorner(remainingRows-1, remainingCols) - .applyHouseholderOnTheRight(m_householder.row(k).tail(remainingCols-1).transpose(), - m_householder.coeff(k,k+1), - temp.data()); + // This stage performs the bidiagonalization of A11, A21, A12, and updating of A22. + // Finally, the algorithm continue on the updated A22. + // + // However, if B is too small, or A22 empty, then let's use an unblocked strategy + if(k+bs==cols || bcols<48) // somewhat arbitrary threshold + { + upperbidiagonalization_inplace_unblocked(B, + &(bidiagonal.template diagonal<0>().coeffRef(k)), + &(bidiagonal.template diagonal<1>().coeffRef(k)), + X.data() + ); + break; // We're done + } + else + { + upperbidiagonalization_blocked_helper<BlockType>( B, + &(bidiagonal.template diagonal<0>().coeffRef(k)), + &(bidiagonal.template diagonal<1>().coeffRef(k)), + bs, + X.topLeftCorner(brows,bs), + Y.topLeftCorner(bcols,bs) + ); + } } +} + +template<typename _MatrixType> +UpperBidiagonalization<_MatrixType>& UpperBidiagonalization<_MatrixType>::computeUnblocked(const _MatrixType& matrix) +{ + Index rows = matrix.rows(); + Index cols = matrix.cols(); + EIGEN_ONLY_USED_FOR_DEBUG(cols); + + eigen_assert(rows >= cols && "UpperBidiagonalization is only for Arices satisfying rows>=cols."); + + m_householder = matrix; + + ColVectorType temp(rows); + + upperbidiagonalization_inplace_unblocked(m_householder, + &(m_bidiagonal.template diagonal<0>().coeffRef(0)), + &(m_bidiagonal.template diagonal<1>().coeffRef(0)), + temp.data()); + + m_isInitialized = true; + return *this; +} + +template<typename _MatrixType> +UpperBidiagonalization<_MatrixType>& UpperBidiagonalization<_MatrixType>::compute(const _MatrixType& matrix) +{ + Index rows = matrix.rows(); + Index cols = matrix.cols(); + EIGEN_ONLY_USED_FOR_DEBUG(rows); + EIGEN_ONLY_USED_FOR_DEBUG(cols); + + eigen_assert(rows >= cols && "UpperBidiagonalization is only for Arices satisfying rows>=cols."); + + m_householder = matrix; + upperbidiagonalization_inplace_blocked(m_householder, m_bidiagonal); + m_isInitialized = true; return *this; } |