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
 
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
#undef eigen_internal_assert
#define eigen_internal_assert(X) assert(X);
#endif
 
// IWYU pragma: private
#include "./InternalHeaderCheck.h"
 
#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
#include <iostream>
#endif
 
namespace Eigen {
 
#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
IOFormat bdcsvdfmt(8, 0, ", ", "\n", "  [", "]");
#endif
 
template <typename MatrixType_, int Options>
class BDCSVD;
 
namespace internal {
 
template <typename MatrixType_, int Options>
struct traits<BDCSVD<MatrixType_, Options> > : svd_traits<MatrixType_, Options> {
  typedef MatrixType_ MatrixType;
};
 
}  // end namespace internal
 
template <typename MatrixType_, int Options_>
class BDCSVD : public SVDBase<BDCSVD<MatrixType_, Options_> > {
  typedef SVDBase<BDCSVD> Base;
 
 public:
  using Base::cols;
  using Base::computeU;
  using Base::computeV;
  using Base::diagSize;
  using Base::rows;
 
  typedef MatrixType_ MatrixType;
  typedef typename Base::Scalar Scalar;
  typedef typename Base::RealScalar RealScalar;
  typedef typename NumTraits<RealScalar>::Literal Literal;
  typedef typename Base::Index Index;
  enum {
    Options = Options_,
    QRDecomposition = Options & internal::QRPreconditionerBits,
    ComputationOptions = Options & internal::ComputationOptionsBits,
    RowsAtCompileTime = Base::RowsAtCompileTime,
    ColsAtCompileTime = Base::ColsAtCompileTime,
    DiagSizeAtCompileTime = Base::DiagSizeAtCompileTime,
    MaxRowsAtCompileTime = Base::MaxRowsAtCompileTime,
    MaxColsAtCompileTime = Base::MaxColsAtCompileTime,
    MaxDiagSizeAtCompileTime = Base::MaxDiagSizeAtCompileTime,
    MatrixOptions = Base::MatrixOptions
  };
 
  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;
 
  BDCSVD() : m_algoswap(16), m_isTranspose(false), m_compU(false), m_compV(false), m_numIters(0) {}
 
  BDCSVD(Index rows, Index cols) : m_algoswap(16), m_numIters(0) {
    allocate(rows, cols, internal::get_computation_options(Options));
  }
 
  EIGEN_DEPRECATED BDCSVD(Index rows, Index cols, unsigned int computationOptions) : m_algoswap(16), m_numIters(0) {
    internal::check_svd_options_assertions<MatrixType, Options>(computationOptions, rows, cols);
    allocate(rows, cols, computationOptions);
  }
 
  BDCSVD(const MatrixType& matrix) : m_algoswap(16), m_numIters(0) {
    compute_impl(matrix, internal::get_computation_options(Options));
  }
 
  EIGEN_DEPRECATED BDCSVD(const MatrixType& matrix, unsigned int computationOptions) : m_algoswap(16), m_numIters(0) {
    internal::check_svd_options_assertions<MatrixType, Options>(computationOptions, matrix.rows(), matrix.cols());
    compute_impl(matrix, computationOptions);
  }
 
  ~BDCSVD() {}
 
  BDCSVD& compute(const MatrixType& matrix) { return compute_impl(matrix, m_computationOptions); }
 
  EIGEN_DEPRECATED BDCSVD& compute(const MatrixType& matrix, unsigned int computationOptions) {
    internal::check_svd_options_assertions<MatrixType, Options>(computationOptions, matrix.rows(), matrix.cols());
    return compute_impl(matrix, computationOptions);
  }
 
  void setSwitchSize(int s) {
    eigen_assert(s >= 3 && "BDCSVD the size of the algo switch has to be at least 3.");
    m_algoswap = s;
  }
 
 private:
  BDCSVD& compute_impl(const MatrixType& matrix, 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);
  template <typename SVDType>
  void computeBaseCase(SVDType& svd, Index n, Index firstCol, Index firstRowW, Index firstColW, Index shift);
 
 protected:
  void allocate(Index rows, Index cols, unsigned int computationOptions);
  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, m_useQrDecomp;
  JacobiSVD<MatrixType, ComputationOptions> smallSvd;
  HouseholderQR<MatrixX> qrDecomp;
  internal::UpperBidiagonalization<MatrixX> bid;
  MatrixX copyWorkspace;
  MatrixX reducedTriangle;
 
  using Base::m_computationOptions;
  using Base::m_computeThinU;
  using Base::m_computeThinV;
  using Base::m_info;
  using Base::m_isInitialized;
  using Base::m_matrixU;
  using Base::m_matrixV;
  using Base::m_nonzeroSingularValues;
  using Base::m_singularValues;
 
 public:
  int m_numIters;
};  // end class BDCSVD
 
// Method to allocate and initialize matrix and attributes
template <typename MatrixType, int Options>
void BDCSVD<MatrixType, Options>::allocate(Index rows, Index cols, unsigned int computationOptions) {
  if (Base::allocate(rows, cols, computationOptions)) return;
 
  if (cols < m_algoswap)
    smallSvd.allocate(rows, cols, Options == 0 ? computationOptions : internal::get_computation_options(Options));
 
  m_computed = MatrixXr::Zero(diagSize() + 1, diagSize());
  m_compU = computeV();
  m_compV = computeU();
  m_isTranspose = (cols > rows);
  if (m_isTranspose) std::swap(m_compU, m_compV);
 
  // kMinAspectRatio is the crossover point that determines if we perform R-Bidiagonalization
  // or bidiagonalize the input matrix directly.
  // It is based off of LAPACK's dgesdd routine, which uses 11.0/6.0
  // we use a larger scalar to prevent a regression for relatively square matrices.
  constexpr Index kMinAspectRatio = 4;
  constexpr bool disableQrDecomp = static_cast<int>(QRDecomposition) == static_cast<int>(DisableQRDecomposition);
  m_useQrDecomp = !disableQrDecomp && ((rows / kMinAspectRatio > cols) || (cols / kMinAspectRatio > rows));
  if (m_useQrDecomp) {
    qrDecomp = HouseholderQR<MatrixX>((std::max)(rows, cols), (std::min)(rows, cols));
    reducedTriangle = MatrixX(diagSize(), diagSize());
  }
 
  copyWorkspace = MatrixX(m_isTranspose ? cols : rows, m_isTranspose ? rows : cols);
  bid = internal::UpperBidiagonalization<MatrixX>(m_useQrDecomp ? diagSize() : copyWorkspace.rows(),
                                                  m_useQrDecomp ? diagSize() : copyWorkspace.cols());
 
  if (m_compU)
    m_naiveU = MatrixXr::Zero(diagSize() + 1, diagSize() + 1);
  else
    m_naiveU = MatrixXr::Zero(2, diagSize() + 1);
 
  if (m_compV) m_naiveV = MatrixXr::Zero(diagSize(), diagSize());
 
  m_workspace.resize((diagSize() + 1) * (diagSize() + 1) * 3);
  m_workspaceI.resize(3 * diagSize());
}  // end allocate
 
template <typename MatrixType, int Options>
BDCSVD<MatrixType, Options>& BDCSVD<MatrixType, Options>::compute_impl(const MatrixType& matrix,
                                                                       unsigned int computationOptions) {
#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
  std::cout << "\n\n\n================================================================================================="
               "=====================\n\n\n";
#endif
  using std::abs;
 
  allocate(matrix.rows(), matrix.cols(), computationOptions);
 
  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) {
    smallSvd.compute(matrix);
    m_isInitialized = true;
    m_info = smallSvd.info();
    if (m_info == Success || m_info == NoConvergence) {
      if (computeU()) m_matrixU = smallSvd.matrixU();
      if (computeV()) m_matrixV = smallSvd.matrixV();
      m_singularValues = smallSvd.singularValues();
      m_nonzeroSingularValues = smallSvd.nonzeroSingularValues();
    }
    return *this;
  }
 
  //**** step 0 - Copy the input matrix and apply scaling to reduce over/under-flows
  RealScalar scale = matrix.cwiseAbs().template maxCoeff<PropagateNaN>();
  if (!(numext::isfinite)(scale)) {
    m_isInitialized = true;
    m_info = InvalidInput;
    return *this;
  }
 
  if (numext::is_exactly_zero(scale)) scale = Literal(1);
 
  if (m_isTranspose)
    copyWorkspace = matrix.adjoint() / scale;
  else
    copyWorkspace = matrix / scale;
 
  //**** step 1 - Bidiagonalization.
  // If the problem is sufficiently rectangular, we perform R-Bidiagonalization: compute A = Q(R/0)
  // and then bidiagonalize R. Otherwise, if the problem is relatively square, we
  // bidiagonalize the input matrix directly.
  if (m_useQrDecomp) {
    qrDecomp.compute(copyWorkspace);
    reducedTriangle = qrDecomp.matrixQR().topRows(diagSize());
    reducedTriangle.template triangularView<StrictlyLower>().setZero();
    bid.compute(reducedTriangle);
  } else {
    bid.compute(copyWorkspace);
  }
 
  //**** step 2 - Divide & Conquer
  m_naiveU.setZero();
  m_naiveV.setZero();
  // FIXME this line involves a temporary matrix
  m_computed.topRows(diagSize()) = bid.bidiagonal().toDenseMatrix().transpose();
  m_computed.template bottomRows<1>().setZero();
  divide(0, diagSize() - 1, 0, 0, 0);
  if (m_info != Success && m_info != NoConvergence) {
    m_isInitialized = true;
    return *this;
  }
 
  //**** step 3 - Copy singular values and vectors
  for (int i = 0; i < 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(diagSize() - i - 1).setZero();
      break;
    } else if (i == diagSize() - 1) {
      m_nonzeroSingularValues = i + 1;
      break;
    }
  }
 
  //**** step 4 - Finalize unitaries U and V
  if (m_isTranspose)
    copyUV(bid.householderV(), bid.householderU(), m_naiveV, m_naiveU);
  else
    copyUV(bid.householderU(), bid.householderV(), m_naiveU, m_naiveV);
 
  if (m_useQrDecomp) {
    if (m_isTranspose && computeV())
      m_matrixV.applyOnTheLeft(qrDecomp.householderQ());
    else if (!m_isTranspose && computeU())
      m_matrixU.applyOnTheLeft(qrDecomp.householderQ());
  }
 
  m_isInitialized = true;
  return *this;
}  // end compute
 
template <typename MatrixType, int Options>
template <typename HouseholderU, typename HouseholderV, typename NaiveU, typename NaiveV>
void BDCSVD<MatrixType, Options>::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 ? diagSize() : rows();
    m_matrixU = MatrixX::Identity(rows(), Ucols);
    m_matrixU.topLeftCorner(diagSize(), diagSize()) =
        naiveV.template cast<Scalar>().topLeftCorner(diagSize(), diagSize());
    // FIXME the following conditionals involve temporary buffers
    if (m_useQrDecomp)
      m_matrixU.topLeftCorner(householderU.cols(), diagSize()).applyOnTheLeft(householderU);
    else
      m_matrixU.applyOnTheLeft(householderU);
  }
  if (computeV()) {
    Index Vcols = m_computeThinV ? diagSize() : cols();
    m_matrixV = MatrixX::Identity(cols(), Vcols);
    m_matrixV.topLeftCorner(diagSize(), diagSize()) =
        naiveU.template cast<Scalar>().topLeftCorner(diagSize(), diagSize());
    // FIXME the following conditionals involve temporary buffers
    if (m_useQrDecomp)
      m_matrixV.topLeftCorner(householderV.cols(), diagSize()).applyOnTheLeft(householderV);
    else
      m_matrixV.applyOnTheLeft(householderV);
  }
}
 
template <typename MatrixType, int Options>
void BDCSVD<MatrixType, Options>::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;
  }
}
 
template <typename MatrixType, int Options>
template <typename SVDType>
void BDCSVD<MatrixType, Options>::computeBaseCase(SVDType& svd, Index n, Index firstCol, Index firstRowW,
                                                  Index firstColW, Index shift) {
  svd.compute(m_computed.block(firstCol, firstCol, n + 1, n));
  m_info = svd.info();
  if (m_info != Success && m_info != NoConvergence) return;
  if (m_compU)
    m_naiveU.block(firstCol, firstCol, n + 1, n + 1).real() = svd.matrixU();
  else {
    m_naiveU.row(0).segment(firstCol, n + 1).real() = svd.matrixU().row(0);
    m_naiveU.row(1).segment(firstCol, n + 1).real() = svd.matrixU().row(n);
  }
  if (m_compV) m_naiveV.block(firstRowW, firstColW, n, n).real() = svd.matrixV();
  m_computed.block(firstCol + shift, firstCol + shift, n + 1, n).setZero();
  m_computed.diagonal().segment(firstCol + shift, n) = svd.singularValues().head(n);
}
 
// 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 firstColW : 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, int Options>
void BDCSVD<MatrixType, Options>::divide(Index firstCol, Index lastCol, Index firstRowW, Index firstColW, Index shift) {
  // requires rows = cols + 1;
  using std::abs;
  using std::pow;
  using std::sqrt;
  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 block involves temporaries
    if (m_compV) {
      JacobiSVD<MatrixXr, ComputeFullU | ComputeFullV> baseSvd;
      computeBaseCase(baseSvd, n, firstCol, firstRowW, firstColW, shift);
    } else {
      JacobiSVD<MatrixXr, ComputeFullU> baseSvd;
      computeBaseCase(baseSvd, n, firstCol, firstRowW, firstColW, shift);
    }
    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);
  if (m_info != Success && m_info != NoConvergence) return;
  divide(firstCol, k - 1 + firstCol, firstRowW, firstColW + 1, shift + 1);
  if (m_info != Success && m_info != NoConvergence) return;
 
  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
  eigen_internal_assert(m_naiveU.allFinite());
  eigen_internal_assert(m_naiveV.allFinite());
  eigen_internal_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
  eigen_internal_assert(m_naiveU.allFinite());
  eigen_internal_assert(m_naiveV.allFinite());
  eigen_internal_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";
  eigen_internal_assert((tmp1 - tmp2).matrix().norm() < 1e-14 * tmp2.matrix().norm());
//   eigen_internal_assert(count<681);
//   eigen_internal_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
  eigen_internal_assert(UofSVD.allFinite());
  eigen_internal_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
  eigen_internal_assert(m_naiveU.allFinite());
  eigen_internal_assert(m_naiveV.allFinite());
  eigen_internal_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, int Options>
void BDCSVD<MatrixType, Options>::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 && numext::is_exactly_zero(diag(actual_n - 1))) {
    --actual_n;
    eigen_internal_assert(numext::is_exactly_zero(col0(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";
 
  {
    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";
    eigen_internal_assert((((singVals.array() - (shifts + mus)) / singVals.array()).head(actual_n) >= 0).all());
    std::cout << "    check2 (>0)      : " << ((singVals.array() - diag) / singVals.array()).head(actual_n).transpose()
              << "\n\n";
    eigen_internal_assert((((singVals.array() - diag) / singVals.array()).head(actual_n) >= 0).all());
  }
#endif
 
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
  eigen_internal_assert(singVals.allFinite());
  eigen_internal_assert(mus.allFinite());
  eigen_internal_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
  eigen_internal_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
  eigen_internal_assert(m_naiveU.allFinite());
  eigen_internal_assert(m_naiveV.allFinite());
  eigen_internal_assert(m_computed.allFinite());
  eigen_internal_assert(U.allFinite());
  eigen_internal_assert(V.allFinite());
//   eigen_internal_assert((U.transpose() * U - MatrixXr(MatrixXr::Identity(U.cols(),U.cols()))).norm() <
//   100*NumTraits<RealScalar>::epsilon() * n); eigen_internal_assert((V.transpose() * V -
//   MatrixXr(MatrixXr::Identity(V.cols(),V.cols()))).norm() < 100*NumTraits<RealScalar>::epsilon() * n);
#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));
    }
  }
 
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
  {
    bool singular_values_sorted =
        (((singVals.segment(1, actual_n - 1) - singVals.head(actual_n - 1))).array() >= 0).all();
    if (!singular_values_sorted)
      std::cout << "Singular values are not sorted: " << singVals.segment(1, actual_n).transpose() << "\n";
    eigen_internal_assert(singular_values_sorted);
  }
#endif
 
  // 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, int Options>
typename BDCSVD<MatrixType, Options>::RealScalar BDCSVD<MatrixType, Options>::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, int Options>
void BDCSVD<MatrixType, Options>::computeSingVals(const ArrayRef& col0, const ArrayRef& diag, const IndicesRef& perm,
                                                  VectorType& singVals, ArrayRef shifts, ArrayRef mus) {
  using std::abs;
  using std::sqrt;
  using std::swap;
 
  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 && numext::is_exactly_zero(col0(actual_n - 1))) --actual_n;
 
  for (Index k = 0; k < n; ++k) {
    if (numext::is_exactly_zero(col0(k)) || 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 (numext::is_exactly_zero(col0(l))) {
        ++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 = " << right - left << "\n";
    //     std::cout << "fMid = " << fMid << " " << secularEq(mid-left, col0, diag, perm, ArrayXr(diag-left), left)
    //                            << " " << secularEq(mid-right, col0, diag, perm, ArrayXr(diag-right), right)   <<
    //                            "\n";
    std::cout << "     = " << secularEq(left + RealScalar(0.000001) * (right - left), col0, diag, perm, diag, 0) << " "
              << secularEq(left + RealScalar(0.1) * (right - left), col0, diag, perm, diag, 0) << " "
              << secularEq(left + RealScalar(0.2) * (right - left), col0, diag, perm, diag, 0) << " "
              << secularEq(left + RealScalar(0.3) * (right - left), col0, diag, perm, diag, 0) << " "
              << secularEq(left + RealScalar(0.4) * (right - left), col0, diag, perm, diag, 0) << " "
              << secularEq(left + RealScalar(0.49) * (right - left), col0, diag, perm, diag, 0) << " "
              << secularEq(left + RealScalar(0.5) * (right - left), col0, diag, perm, diag, 0) << " "
              << secularEq(left + RealScalar(0.51) * (right - left), col0, diag, perm, diag, 0) << " "
              << secularEq(left + RealScalar(0.6) * (right - left), col0, diag, perm, diag, 0) << " "
              << secularEq(left + RealScalar(0.7) * (right - left), col0, diag, perm, diag, 0) << " "
              << secularEq(left + RealScalar(0.8) * (right - left), col0, diag, perm, diag, 0) << " "
              << secularEq(left + RealScalar(0.9) * (right - left), col0, diag, perm, diag, 0) << " "
              << secularEq(left + RealScalar(0.999999) * (right - left), 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;
 
    if (k != actual_n - 1) {
      // check that after the shift, f(mid) is still negative:
      RealScalar midShifted = (right - left) / RealScalar(2);
      // we can test exact equality here, because shift comes from `... ? left : right`
      if (numext::equal_strict(shift, right)) midShifted = -midShifted;
      RealScalar fMidShifted = secularEq(midShifted, col0, diag, perm, diagShifted, shift);
      if (fMidShifted > 0) {
        // fMid was erroneous, fix it:
        shift = fMidShifted > Literal(0) ? left : right;
        diagShifted = diag - shift;
      }
    }
 
    // initial guess
    RealScalar muPrev, muCur;
    // we can test exact equality here, because shift comes from `... ? left : right`
    if (numext::equal_strict(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 (!numext::is_exactly_zero(fCur) &&
           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);
 
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
      eigen_internal_assert((numext::isfinite)(fZero));
#endif
 
      muPrev = muCur;
      fPrev = fCur;
      muCur = muZero;
      fCur = fZero;
 
      // we can test exact equality here, because shift comes from `... ? left : right`
      if (numext::equal_strict(shift, left) && (muCur < Literal(0) || muCur > right - left)) useBisection = true;
      if (numext::equal_strict(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;
      // we can test exact equality here, because shift comes from `... ? left : right`
      if (numext::equal_strict(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);
      eigen_internal_assert(fLeft < Literal(0));
 
#if defined EIGEN_BDCSVD_DEBUG_VERBOSE || defined EIGEN_BDCSVD_SANITY_CHECKS || defined EIGEN_INTERNAL_DEBUGGING
      RealScalar fRight = secularEq(rightShifted, col0, diag, perm, diagShifted, shift);
#endif
 
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
      if (!(numext::isfinite)(fLeft))
        std::cout << "f(" << leftShifted << ") =" << fLeft << " ; " << left << " " << shift << " " << right << "\n";
      eigen_internal_assert((numext::isfinite)(fLeft));
 
      if (!(numext::isfinite)(fRight))
        std::cout << "f(" << rightShifted << ") =" << fRight << " ; " << left << " " << shift << " " << right << "\n";
        // eigen_internal_assert((numext::isfinite)(fRight));
#endif
 
#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
      if (!(fLeft * fRight < 0)) {
        std::cout << "f(leftShifted) using  leftShifted=" << leftShifted
                  << " ;  diagShifted(1:10):" << diagShifted.head(10).transpose() << "\n ; "
                  << "left==shift=" << bool(left == shift) << " ; left-shift = " << (left - shift) << "\n";
        std::cout << "k=" << k << ", " << fLeft << " * " << fRight << " == " << fLeft * fRight << "  ;  "
                  << "[" << left << " .. " << right << "] -> [" << leftShifted << " " << rightShifted
                  << "], shift=" << shift << " ,  f(right)=" << secularEq(0, col0, diag, perm, diagShifted, shift)
                  << " == " << secularEq(right, col0, diag, perm, diag, 0) << " == " << fRight << "\n";
      }
#endif
      eigen_internal_assert(fLeft * fRight < Literal(0));
 
      if (fLeft < 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);
          eigen_internal_assert((numext::isfinite)(fMid));
 
          if (fLeft * fMid < Literal(0)) {
            rightShifted = midShifted;
          } else {
            leftShifted = midShifted;
            fLeft = fMid;
          }
        }
        muCur = (leftShifted + rightShifted) / Literal(2);
      } else {
        // We have a problem as shifting on the left or right give either a positive or negative value
        // at the middle of [left,right]...
        // Instead of abbording or entering an infinite loop,
        // let's just use the middle as the estimated zero-crossing:
        muCur = (right - left) * RealScalar(0.5);
        // we can test exact equality here, because shift comes from `... ? left : right`
        if (numext::equal_strict(shift, right)) muCur = -muCur;
      }
    }
 
    singVals[k] = shift + muCur;
    shifts[k] = shift;
    mus[k] = muCur;
 
#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
    if (k + 1 < n)
      std::cout << "found " << singVals[k] << " == " << shift << " + " << muCur << " from " << diag(k) << " .. "
                << diag(k + 1) << "\n";
#endif
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
    eigen_internal_assert(k == 0 || singVals[k] >= singVals[k - 1]);
    eigen_internal_assert(singVals[k] >= diag(k));
#endif
 
    // 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, int Options>
void BDCSVD<MatrixType, Options>::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 lastIdx = perm(m - 1);
  // The offset permits to skip deflated entries while computing zhat
  for (Index k = 0; k < n; ++k) {
    if (numext::is_exactly_zero(col0(k)))  // deflated
      zhat(k) = Literal(0);
    else {
      // see equation (3.6)
      RealScalar dk = diag(k);
      RealScalar prod = (singVals(lastIdx) + dk) * (mus(lastIdx) + (shifts(lastIdx) - dk));
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
      if (prod < 0) {
        std::cout << "k = " << k << " ;  z(k)=" << col0(k) << ", diag(k)=" << dk << "\n";
        std::cout << "prod = "
                  << "(" << singVals(lastIdx) << " + " << dk << ") * (" << mus(lastIdx) << " + (" << shifts(lastIdx)
                  << " - " << dk << "))"
                  << "\n";
        std::cout << "     = " << singVals(lastIdx) + dk << " * " << mus(lastIdx) + (shifts(lastIdx) - dk) << "\n";
      }
      eigen_internal_assert(prod >= 0);
#endif
 
      for (Index l = 0; l < m; ++l) {
        Index i = perm(l);
        if (i != k) {
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
          if (i >= k && (l == 0 || l - 1 >= m)) {
            std::cout << "Error in perturbCol0\n";
            std::cout << "  " << k << "/" << n << " " << l << "/" << m << " " << i << "/" << n << " ; " << col0(k)
                      << " " << diag(k) << " "
                      << "\n";
            std::cout << "  " << diag(i) << "\n";
            Index j = (i < k /*|| l==0*/) ? i : perm(l - 1);
            std::cout << "  "
                      << "j=" << j << "\n";
          }
#endif
          Index j = i < k ? i : l > 0 ? perm(l - 1) : i;
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
          if (!(dk != Literal(0) || diag(i) != Literal(0))) {
            std::cout << "k=" << k << ", i=" << i << ", l=" << l << ", perm.size()=" << perm.size() << "\n";
          }
          eigen_internal_assert(dk != Literal(0) || diag(i) != Literal(0));
#endif
          prod *= ((singVals(j) + dk) / ((diag(i) + dk))) * ((mus(j) + (shifts(j) - dk)) / ((diag(i) - dk)));
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
          eigen_internal_assert(prod >= 0);
#endif
#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
          if (i != k &&
              numext::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(lastIdx) + dk) << " * "
                << mus(lastIdx) + shifts(lastIdx) << " - " << dk << "\n";
#endif
      RealScalar tmp = sqrt(prod);
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
      eigen_internal_assert((numext::isfinite)(tmp));
#endif
      zhat(k) = col0(k) > Literal(0) ? RealScalar(tmp) : RealScalar(-tmp);
    }
  }
}
 
// compute singular vectors
template <typename MatrixType, int Options>
void BDCSVD<MatrixType, Options>::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 (numext::is_exactly_zero(zhat(k))) {
      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, and set di = 0.
template <typename MatrixType, int Options>
void BDCSVD<MatrixType, Options>::deflation43(Index firstCol, Index shift, Index i, Index size) {
  using std::abs;
  using std::pow;
  using std::sqrt;
  Index start = firstCol + shift;
  RealScalar c = m_computed(start, start);
  RealScalar s = m_computed(start + i, start);
  RealScalar r = numext::hypot(c, s);
  if (numext::is_exactly_zero(r)) {
    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 zi = 0, and dj = di.
template <typename MatrixType, int Options>
void BDCSVD<MatrixType, Options>::deflation44(Index firstColu, Index firstColm, Index firstRowW, Index firstColW,
                                              Index i, Index j, Index size) {
  using std::abs;
  using std::conj;
  using std::pow;
  using std::sqrt;
 
  RealScalar s = m_computed(firstColm + i, firstColm);
  RealScalar c = m_computed(firstColm + j, firstColm);
  RealScalar r = numext::hypot(c, 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 (numext::is_exactly_zero(r)) {
    m_computed(firstColm + j, firstColm + j) = m_computed(firstColm + i, firstColm + i);
    return;
  }
  c /= r;
  s /= r;
  m_computed(firstColm + j, firstColm) = r;
  m_computed(firstColm + j, firstColm + j) = m_computed(firstColm + i, firstColm + i);
  m_computed(firstColm + i, firstColm) = Literal(0);
 
  JacobiRotation<RealScalar> J(c, -s);
  if (m_compU)
    m_naiveU.middleRows(firstColu, size + 1).applyOnTheRight(firstColu + j, firstColu + i, J);
  else
    m_naiveU.applyOnTheRight(firstColu + j, firstColu + i, J);
  if (m_compV) m_naiveV.middleRows(firstRowW, size).applyOnTheRight(firstColW + j, firstColW + i, J);
}  // end deflation 44
 
// acts on block from (firstCol+shift, firstCol+shift) to (lastCol+shift, lastCol+shift) [inclusive]
template <typename MatrixType, int Options>
void BDCSVD<MatrixType, Options>::deflation(Index firstCol, Index lastCol, Index k, Index firstRowW, Index firstColW,
                                            Index shift) {
  using std::abs;
  using std::sqrt;
  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
  eigen_internal_assert(m_naiveU.allFinite());
  eigen_internal_assert(m_naiveV.allFinite());
  eigen_internal_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
  eigen_internal_assert(m_naiveU.allFinite());
  eigen_internal_assert(m_naiveV.allFinite());
  eigen_internal_assert(m_computed.allFinite());
#endif
#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
  std::cout << "to be sorted: " << diag.transpose() << "\n\n";
  std::cout << "            : " << col0.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.
    const bool total_deflation = (col0.tail(length - 1).array().abs() < 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 (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 (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;
    // Find last non-deflated entry.
    while (i > 0 && (diag(i) < considerZero || abs(col0(i)) < considerZero)) --i;
 
    for (; i > 1; --i)
      if ((diag(i) - diag(i - 1)) < epsilon_strict) {
#ifdef EIGEN_BDCSVD_DEBUG_VERBOSE
        std::cout << "deflation 4.4 with i = " << i << " because " << diag(i) << " - " << diag(i - 1)
                  << " == " << (diag(i) - diag(i - 1)) << " < " << epsilon_strict << "\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, i - 1, length);
      }
  }
 
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
  for (Index j = 2; j < length; ++j) eigen_internal_assert(diag(j - 1) <= diag(j) || abs(diag(j)) < considerZero);
#endif
 
#ifdef EIGEN_BDCSVD_SANITY_CHECKS
  eigen_internal_assert(m_naiveU.allFinite());
  eigen_internal_assert(m_naiveV.allFinite());
  eigen_internal_assert(m_computed.allFinite());
#endif
}  // end deflation
 
template <typename Derived>
template <int Options>
BDCSVD<typename MatrixBase<Derived>::PlainObject, Options> MatrixBase<Derived>::bdcSvd() const {
  return BDCSVD<PlainObject, Options>(*this);
}
 
template <typename Derived>
template <int Options>
BDCSVD<typename MatrixBase<Derived>::PlainObject, Options> MatrixBase<Derived>::bdcSvd(
    unsigned int computationOptions) const {
  return BDCSVD<PlainObject, Options>(*this, computationOptions);
}
 
}  // end namespace Eigen
 
#endif