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