Squashed 'third_party/eigen/' changes from 61d72f6..cf794d3
Change-Id: I9b814151b01f49af6337a8605d0c42a3a1ed4c72
git-subtree-dir: third_party/eigen
git-subtree-split: cf794d3b741a6278df169e58461f8529f43bce5d
diff --git a/Eigen/src/SVD/BDCSVD.h b/Eigen/src/SVD/BDCSVD.h
new file mode 100644
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+++ b/Eigen/src/SVD/BDCSVD.h
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+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// We used the "A Divide-And-Conquer Algorithm for the Bidiagonal SVD"
+// research report written by Ming Gu and Stanley C.Eisenstat
+// The code variable names correspond to the names they used in their
+// report
+//
+// Copyright (C) 2013 Gauthier Brun <brun.gauthier@gmail.com>
+// Copyright (C) 2013 Nicolas Carre <nicolas.carre@ensimag.fr>
+// Copyright (C) 2013 Jean Ceccato <jean.ceccato@ensimag.fr>
+// Copyright (C) 2013 Pierre Zoppitelli <pierre.zoppitelli@ensimag.fr>
+// Copyright (C) 2013 Jitse Niesen <jitse@maths.leeds.ac.uk>
+// Copyright (C) 2014-2017 Gael Guennebaud <gael.guennebaud@inria.fr>
+//
+// Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+#ifndef EIGEN_BDCSVD_H
+#define EIGEN_BDCSVD_H
+// #define EIGEN_BDCSVD_DEBUG_VERBOSE
+// #define EIGEN_BDCSVD_SANITY_CHECKS
+
+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/Eigen/src/SVD/CMakeLists.txt b/Eigen/src/SVD/CMakeLists.txt
deleted file mode 100644
index 55efc44..0000000
--- a/Eigen/src/SVD/CMakeLists.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-FILE(GLOB Eigen_SVD_SRCS "*.h")
-
-INSTALL(FILES
- ${Eigen_SVD_SRCS}
- DESTINATION ${INCLUDE_INSTALL_DIR}/Eigen/src/SVD COMPONENT Devel
- )
diff --git a/Eigen/src/SVD/JacobiSVD.h b/Eigen/src/SVD/JacobiSVD.h
index 1b29774..43488b1 100644
--- a/Eigen/src/SVD/JacobiSVD.h
+++ b/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 @@
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 @@
class qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
{
public:
- typedef typename MatrixType::Index Index;
typedef typename MatrixType::Scalar Scalar;
enum
{
@@ -106,7 +105,6 @@
class qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
{
public:
- typedef typename MatrixType::Index Index;
typedef typename MatrixType::Scalar Scalar;
enum
{
@@ -114,9 +112,11 @@
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 @@
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 @@
class qr_preconditioner_impl<MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
{
public:
- typedef typename MatrixType::Index Index;
typedef typename MatrixType::Scalar Scalar;
enum
{
@@ -205,10 +202,12 @@
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 @@
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 @@
class qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
{
public:
- typedef typename MatrixType::Index Index;
typedef typename MatrixType::Scalar Scalar;
enum
{
@@ -358,8 +354,8 @@
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 @@
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 @@
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 @@
*
* \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 @@
* \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 @@
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 @@
* 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 @@
* \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 @@
* 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 @@
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 @@
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 @@
// 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 @@
}
/*** step 2. The main Jacobi SVD iteration. ***/
+ RealScalar maxDiagEntry = m_workMatrix.cwiseAbs().diagonal().maxCoeff();
bool finished = false;
while(!finished)
@@ -872,25 +710,27 @@
// 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);
+ // 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());
+ // 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);
+ 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 @@
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 @@
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/Eigen/src/SVD/JacobiSVD_LAPACKE.h b/Eigen/src/SVD/JacobiSVD_LAPACKE.h
new file mode 100644
index 0000000..ff0516f
--- /dev/null
+++ b/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/Eigen/src/SVD/JacobiSVD_MKL.h b/Eigen/src/SVD/JacobiSVD_MKL.h
deleted file mode 100644
index decda75..0000000
--- a/Eigen/src/SVD/JacobiSVD_MKL.h
+++ /dev/null
@@ -1,92 +0,0 @@
-/*
- 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 Intel(R) MKL
- * Singular Value Decomposition - SVD.
- ********************************************************************************
-*/
-
-#ifndef EIGEN_JACOBISVD_MKL_H
-#define EIGEN_JACOBISVD_MKL_H
-
-#include "Eigen/src/Core/util/MKL_support.h"
-
-namespace Eigen {
-
-/** \internal Specialization for the data types supported by MKL */
-
-#define EIGEN_MKL_SVD(EIGTYPE, MKLTYPE, MKLRTYPE, MKLPREFIX, EIGCOLROW, MKLCOLROW) \
-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 = matrix.outerStride(), ldu, ldvt; \
- lapack_int matrix_order = MKLCOLROW; \
- char jobu, jobvt; \
- MKLTYPE *u, *vt, dummy; \
- jobu = (m_computeFullU) ? 'A' : (m_computeThinU) ? 'S' : 'N'; \
- jobvt = (m_computeFullV) ? 'A' : (m_computeThinV) ? 'S' : 'N'; \
- if (computeU()) { \
- ldu = m_matrixU.outerStride(); \
- u = (MKLTYPE*)m_matrixU.data(); \
- } else { ldu=1; u=&dummy; }\
- MatrixType localV; \
- ldvt = (m_computeFullV) ? m_cols : (m_computeThinV) ? m_diagSize : 1; \
- if (computeV()) { \
- localV.resize(ldvt, m_cols); \
- vt = (MKLTYPE*)localV.data(); \
- } else { ldvt=1; vt=&dummy; }\
- Matrix<MKLRTYPE, Dynamic, Dynamic> superb; superb.resize(m_diagSize, 1); \
- MatrixType m_temp; m_temp = matrix; \
- LAPACKE_##MKLPREFIX##gesvd( matrix_order, jobu, jobvt, m_rows, m_cols, (MKLTYPE*)m_temp.data(), lda, (MKLRTYPE*)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_MKL_SVD(double, double, double, d, ColMajor, LAPACK_COL_MAJOR)
-EIGEN_MKL_SVD(float, float, float , s, ColMajor, LAPACK_COL_MAJOR)
-EIGEN_MKL_SVD(dcomplex, MKL_Complex16, double, z, ColMajor, LAPACK_COL_MAJOR)
-EIGEN_MKL_SVD(scomplex, MKL_Complex8, float , c, ColMajor, LAPACK_COL_MAJOR)
-
-EIGEN_MKL_SVD(double, double, double, d, RowMajor, LAPACK_ROW_MAJOR)
-EIGEN_MKL_SVD(float, float, float , s, RowMajor, LAPACK_ROW_MAJOR)
-EIGEN_MKL_SVD(dcomplex, MKL_Complex16, double, z, RowMajor, LAPACK_ROW_MAJOR)
-EIGEN_MKL_SVD(scomplex, MKL_Complex8, float , c, RowMajor, LAPACK_ROW_MAJOR)
-
-} // end namespace Eigen
-
-#endif // EIGEN_JACOBISVD_MKL_H
diff --git a/Eigen/src/SVD/SVDBase.h b/Eigen/src/SVD/SVDBase.h
new file mode 100644
index 0000000..3d1ef37
--- /dev/null
+++ b/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/Eigen/src/SVD/UpperBidiagonalization.h b/Eigen/src/SVD/UpperBidiagonalization.h
index 587de37..11ac847 100644
--- a/Eigen/src/SVD/UpperBidiagonalization.h
+++ b/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 @@
};
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 @@
*/
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 @@
}
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 @@
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 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());
+
+ // 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);
+
+ // 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;
}