class Bidiagonal Divide and Conquer SVD More...

#include <src/eigen/Eigen/src/SVD/BDCSVD.h>

Inheritance diagram for Eigen::BDCSVD< _MatrixType >:

Collaboration diagram for Eigen::BDCSVD< _MatrixType >:

Public Types
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 _MatrixType	MatrixType

typedef MatrixType::Scalar	Scalar

typedef NumTraits< typenameMatrixType::Scalar >::Real	RealScalar

typedef NumTraits< RealScalar >::Literal	Literal

typedef Base::MatrixUType	MatrixUType

typedef Base::MatrixVType	MatrixVType

typedef 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

enum

typedef MatrixType::StorageIndex	StorageIndex

typedef Eigen::Index	Index

Public Member Functions
	BDCSVD ()
	Default Constructor.

	BDCSVD (Index rows, Index cols, unsigned int computationOptions=0)
	Default Constructor with memory preallocation.

	BDCSVD (const MatrixType &matrix, unsigned int computationOptions=0)
	Constructor performing the decomposition of given matrix.

	~BDCSVD ()

BDCSVD &	compute (const MatrixType &matrix, unsigned int computationOptions)
	Method performing the decomposition of given matrix using custom options.

BDCSVD &	compute (const MatrixType &matrix)
	Method performing the decomposition of given matrix using current options.

void	setSwitchSize (int s)

Index	rows () const

Index	cols () const

bool	computeU () const

bool	computeV () const

BDCSVD< _MatrixType > &	derived ()

const BDCSVD< _MatrixType > &	derived () const

const MatrixUType &	matrixU () const

const MatrixVType &	matrixV () const

const SingularValuesType &	singularValues () const

Index	nonzeroSingularValues () const

Index	rank () const

BDCSVD< _MatrixType > &	setThreshold (const RealScalar &threshold)

BDCSVD< _MatrixType > &	setThreshold (Default_t)

RealScalar	threshold () const

const Solve< BDCSVD< _MatrixType >, Rhs >	solve (const MatrixBase< Rhs > &b) const

EIGEN_DEVICE_FUNC void	_solve_impl (const RhsType &rhs, DstType &dst) const

void	_solve_impl (const RhsType &rhs, DstType &dst) const

Public Attributes
int	m_numIters

Static Protected Member Functions
static void	check_template_parameters ()

Protected Attributes
MatrixXr	m_naiveU

MatrixXr	m_naiveV

MatrixXr	m_computed

Index	m_nRec

ArrayXr	m_workspace

ArrayXi	m_workspaceI

int	m_algoswap

bool	m_isTranspose

bool	m_compU

bool	m_compV

SingularValuesType	m_singularValues

Index	m_diagSize

bool	m_computeFullU

bool	m_computeFullV

bool	m_computeThinU

bool	m_computeThinV

MatrixUType	m_matrixU

MatrixVType	m_matrixV

bool	m_isInitialized

Index	m_nonzeroSingularValues

bool	m_isAllocated

bool	m_usePrescribedThreshold

unsigned int	m_computationOptions

Index	m_rows

Index	m_cols

RealScalar	m_prescribedThreshold

Private Types
typedef SVDBase< BDCSVD >	Base

Private Member Functions
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 Private Member Functions
static RealScalar	secularEq (RealScalar x, const ArrayRef &col0, const ArrayRef &diag, const IndicesRef &perm, const ArrayRef &diagShifted, RealScalar shift)

Detailed Description

template<typename _MatrixType>
class Eigen::BDCSVD< _MatrixType >

class Bidiagonal Divide and Conquer SVD

Template Parameters

_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 -fp-model precise option. Likewise, the -ffast-math option of GCC or clang will significantly degrade the accuracy.

See also: class JacobiSVD

Member Typedef Documentation

◆ ArrayRef

template<typename _MatrixType >

typedef Ref<ArrayXr> Eigen::BDCSVD< _MatrixType >::ArrayRef

◆ ArrayXi

template<typename _MatrixType >

typedef Array<Index,1,Dynamic> Eigen::BDCSVD< _MatrixType >::ArrayXi

◆ ArrayXr

template<typename _MatrixType >

typedef Array<RealScalar, Dynamic, 1> Eigen::BDCSVD< _MatrixType >::ArrayXr

◆ Base

template<typename _MatrixType >

typedef SVDBase<BDCSVD> Eigen::BDCSVD< _MatrixType >::Base

private

◆ Index

typedef Eigen::Index Eigen::SVDBase< BDCSVD< _MatrixType > >::Index

inherited

◆ IndicesRef

template<typename _MatrixType >

typedef Ref<ArrayXi> Eigen::BDCSVD< _MatrixType >::IndicesRef

◆ Literal

template<typename _MatrixType >

typedef NumTraits<RealScalar>::Literal Eigen::BDCSVD< _MatrixType >::Literal

◆ MatrixType

template<typename _MatrixType >

typedef _MatrixType Eigen::BDCSVD< _MatrixType >::MatrixType

◆ MatrixUType

template<typename _MatrixType >

typedef Base::MatrixUType Eigen::BDCSVD< _MatrixType >::MatrixUType

◆ MatrixVType

template<typename _MatrixType >

typedef Base::MatrixVType Eigen::BDCSVD< _MatrixType >::MatrixVType

◆ MatrixX

template<typename _MatrixType >

typedef Matrix<Scalar, Dynamic, Dynamic, ColMajor> Eigen::BDCSVD< _MatrixType >::MatrixX

◆ MatrixXr

template<typename _MatrixType >

typedef Matrix<RealScalar, Dynamic, Dynamic, ColMajor> Eigen::BDCSVD< _MatrixType >::MatrixXr

◆ RealScalar

template<typename _MatrixType >

typedef NumTraits<typenameMatrixType::Scalar>::Real Eigen::BDCSVD< _MatrixType >::RealScalar

◆ Scalar

template<typename _MatrixType >

typedef MatrixType::Scalar Eigen::BDCSVD< _MatrixType >::Scalar

◆ SingularValuesType

template<typename _MatrixType >

typedef Base::SingularValuesType Eigen::BDCSVD< _MatrixType >::SingularValuesType

◆ StorageIndex

typedef MatrixType::StorageIndex Eigen::SVDBase< BDCSVD< _MatrixType > >::StorageIndex

inherited

◆ VectorType

template<typename _MatrixType >

typedef Matrix<RealScalar, Dynamic, 1> Eigen::BDCSVD< _MatrixType >::VectorType

Member Enumeration Documentation

◆ anonymous enum

anonymous enum

inherited

       {
    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
  };

◆ anonymous enum

template<typename _MatrixType >

anonymous enum

Enumerator
RowsAtCompileTime
ColsAtCompileTime
DiagSizeAtCompileTime
MaxRowsAtCompileTime
MaxColsAtCompileTime
MaxDiagSizeAtCompileTime
MatrixOptions

       {
    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
  };

Constructor & Destructor Documentation

◆ BDCSVD() [1/3]

template<typename _MatrixType >

Eigen::BDCSVD< _MatrixType >::BDCSVD ( )

inline

Default Constructor.

The default constructor is useful in cases in which the user intends to perform decompositions via BDCSVD::compute(const MatrixType&).

108 : m_algoswap(16), m_numIters(0)

109 {}

Eigen::BDCSVD::m_algoswap

int m_algoswap

Definition BDCSVD.h:194

Eigen::BDCSVD::m_numIters

int m_numIters

Definition BDCSVD.h:209

◆ BDCSVD() [2/3]

template<typename _MatrixType >

Eigen::BDCSVD< _MatrixType >::BDCSVD	(	Index	rows,
		Index	cols,
		unsigned int	computationOptions = `0`
	)

inline

Default Constructor with memory preallocation.

Like the default constructor but with preallocation of the internal data according to the specified problem size.

See also: BDCSVD()

    : m_algoswap(16), m_numIters(0)
  {
    allocate(rows, cols, computationOptions);
  }

References Eigen::BDCSVD< _MatrixType >::allocate(), Eigen::BDCSVD< _MatrixType >::cols(), and Eigen::BDCSVD< _MatrixType >::rows().

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◆ BDCSVD() [3/3]

template<typename _MatrixType >

Eigen::BDCSVD< _MatrixType >::BDCSVD	(	const MatrixType &	matrix,
		unsigned int	computationOptions = `0`
	)

inline

Constructor performing the decomposition of given matrix.

Parameters

matrix	the matrix to decompose
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.

    : m_algoswap(16), m_numIters(0)
  {
    compute(matrix, computationOptions);
  }

References Eigen::BDCSVD< _MatrixType >::compute().

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◆ ~BDCSVD()

template<typename _MatrixType >

Eigen::BDCSVD< _MatrixType >::~BDCSVD ( )

inline

141 {

142 }

Member Function Documentation

◆ _solve_impl() [1/2]

EIGEN_DEVICE_FUNC void Eigen::SVDBase< BDCSVD< _MatrixType > >::_solve_impl	(	const RhsType &	rhs,
		DstType &	dst
	)		const

inherited

◆ _solve_impl() [2/2]

void Eigen::SVDBase< BDCSVD< _MatrixType > >::_solve_impl	(	const RhsType &	rhs,
		DstType &	dst
	)		const

inherited

{
  eigen_assert(rhs.rows() == rows());
 
  // A = U S V^*
  // So A^{-1} = V S^{-1} U^*
 
  Matrix<Scalar, Dynamic, RhsType::ColsAtCompileTime, 0, MatrixType::MaxRowsAtCompileTime, RhsType::MaxColsAtCompileTime> tmp;
  Index l_rank = rank();
  tmp.noalias() =  m_matrixU.leftCols(l_rank).adjoint() * rhs;
  tmp = m_singularValues.head(l_rank).asDiagonal().inverse() * tmp;
  dst = m_matrixV.leftCols(l_rank) * tmp;
}

◆ allocate()

template<typename MatrixType >

void Eigen::BDCSVD< MatrixType >::allocate	(	Index	rows,
		Index	cols,
		unsigned int	computationOptions
	)

private

{
  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

Referenced by Eigen::BDCSVD< _MatrixType >::BDCSVD().

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◆ check_template_parameters()

static void Eigen::SVDBase< BDCSVD< _MatrixType > >::check_template_parameters ( )

inlinestaticprotectedinherited

  {
    EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
  }

◆ cols()

template<typename _MatrixType >

Index Eigen::SVDBase< Derived >::cols ( ) const

inline

195{ return m_cols; }

Eigen::SVDBase< BDCSVD< _MatrixType > >::m_cols

Index m_cols

Definition SVDBase.h:238

Referenced by Eigen::BDCSVD< _MatrixType >::BDCSVD().

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◆ compute() [1/2]

template<typename _MatrixType >

BDCSVD & Eigen::BDCSVD< _MatrixType >::compute ( const MatrixType & matrix )

inline

Method performing the decomposition of given matrix using current options.

Parameters

matrix the matrix to decompose

This method uses the current computationOptions, as already passed to the constructor or to compute(const MatrixType&, unsigned int).

  {
    return compute(matrix, this->m_computationOptions);
  }

References Eigen::BDCSVD< _MatrixType >::compute(), and Eigen::SVDBase< BDCSVD< _MatrixType > >::m_computationOptions.

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◆ compute() [2/2]

template<typename MatrixType >

BDCSVD< MatrixType > & Eigen::BDCSVD< MatrixType >::compute	(	const MatrixType &	matrix,
		unsigned int	computationOptions
	)

Method performing the decomposition of given matrix using custom options.

Parameters

matrix	the matrix to decompose
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.

{
#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

References Eigen::internal::UpperBidiagonalization< _MatrixType >::bidiagonal(), Eigen::internal::UpperBidiagonalization< _MatrixType >::householderU(), Eigen::internal::UpperBidiagonalization< _MatrixType >::householderV(), Eigen::SVDBase< Derived >::matrixU(), Eigen::SVDBase< Derived >::matrixV(), Eigen::SVDBase< Derived >::nonzeroSingularValues(), scale(), Eigen::PlainObjectBase< Derived >::setZero(), Eigen::SVDBase< Derived >::singularValues(), and Eigen::internal::BandMatrixBase< Derived >::toDenseMatrix().

Referenced by Eigen::BDCSVD< _MatrixType >::BDCSVD(), and Eigen::BDCSVD< _MatrixType >::compute().

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◆ computeSingVals()

template<typename MatrixType >

void Eigen::BDCSVD< MatrixType >::computeSingVals	(	const ArrayRef &	col0,
		const ArrayRef &	diag,
		const IndicesRef &	perm,
		VectorType &	singVals,
		ArrayRef	shifts,
		ArrayRef	mus
	)

private

{
  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();
  }
}

References eigen_internal_assert, Eigen::numext::isfinite(), and sqrt().

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◆ computeSingVecs()

template<typename MatrixType >

void Eigen::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
	)

private

{
  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);
}

References Eigen::PlainObjectBase< Derived >::setZero().

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◆ computeSVDofM()

template<typename MatrixType >

void Eigen::BDCSVD< MatrixType >::computeSVDofM	(	Index	firstCol,
		Index	n,
		MatrixXr &	U,
		VectorType &	singVals,
		MatrixXr &	V
	)

private

{
  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
}

References Eigen::PlainObjectBase< Derived >::cols(), head(), Eigen::PlainObjectBase< Derived >::resize(), Eigen::SVDBase< Derived >::singularValues(), and Eigen::PlainObjectBase< Derived >::swap().

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◆ computeU()

template<typename _MatrixType >

bool Eigen::SVDBase< Derived >::computeU ( ) const

inline

Returns: true if U (full or thin) is asked for in this SVD decomposition

190{ return m_computeFullU || m_computeThinU; }

Eigen::BDCSVD::m_computeThinU

bool m_computeThinU

Definition SVDBase.h:235

Eigen::BDCSVD::m_computeFullU

bool m_computeFullU

Definition SVDBase.h:235

◆ computeV()

template<typename _MatrixType >

bool Eigen::SVDBase< Derived >::computeV ( ) const

inline

Returns: true if V (full or thin) is asked for in this SVD decomposition

192{ return m_computeFullV || m_computeThinV; }

Eigen::BDCSVD::m_computeFullV

bool m_computeFullV

Definition SVDBase.h:236

Eigen::BDCSVD::m_computeThinV

bool m_computeThinV

Definition SVDBase.h:236

◆ copyUV()

template<typename MatrixType >

template<typename HouseholderU , typename HouseholderV , typename NaiveU , typename NaiveV >

void Eigen::BDCSVD< MatrixType >::copyUV	(	const HouseholderU &	householderU,
		const HouseholderV &	householderV,
		const NaiveU &	naiveU,
		const NaiveV &	naivev
	)

private

{
  // 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
  }
}

◆ deflation()

template<typename MatrixType >

void Eigen::BDCSVD< MatrixType >::deflation	(	Index	firstCol,
		Index	lastCol,
		Index	k,
		Index	firstRowW,
		Index	firstColW,
		Index	shift
	)

private

{
  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

References Eigen::Dynamic, and eigen_internal_assert.

◆ deflation43()

template<typename MatrixType >

void Eigen::BDCSVD< MatrixType >::deflation43	(	Index	firstCol,
		Index	shift,
		Index	i,
		Index	size
	)

private

{
  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

◆ deflation44()

template<typename MatrixType >

void Eigen::BDCSVD< MatrixType >::deflation44	(	Index	firstColu,
		Index	firstColm,
		Index	firstRowW,
		Index	firstColW,
		Index	i,
		Index	j,
		Index	size
	)

private

{
  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

References sqrt().

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◆ derived() [1/2]

BDCSVD< _MatrixType > & Eigen::SVDBase< BDCSVD< _MatrixType > >::derived ( )

inlineinherited

71{ return *static_cast<Derived*>(this); }

◆ derived() [2/2]

const BDCSVD< _MatrixType > & Eigen::SVDBase< BDCSVD< _MatrixType > >::derived ( ) const

inlineinherited

72{ return *static_cast<const Derived*>(this); }

◆ divide()

template<typename MatrixType >

void Eigen::BDCSVD< MatrixType >::divide	(	Index	firstCol,
		Index	lastCol,
		Index	firstRowW,
		Index	firstColW,
		Index	shift
	)

private

{
  // 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

References Eigen::Aligned, Eigen::ComputeFullU, Eigen::ComputeFullV, and sqrt().

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◆ matrixU()

const MatrixUType & Eigen::SVDBase< BDCSVD< _MatrixType > >::matrixU ( ) const

inlineinherited

Returns: the U matrix.

For the SVD decomposition of a n-by-p matrix, letting m be the minimum of n and p, the U matrix is n-by-n if you asked for ComputeFullU , and is n-by-m if you asked for ComputeThinU .

The m first columns of U are the left singular vectors of the matrix being decomposed.

This method asserts that you asked for U to be computed.

  {
    eigen_assert(m_isInitialized && "SVD is not initialized.");
    eigen_assert(computeU() && "This SVD decomposition didn't compute U. Did you ask for it?");
    return m_matrixU;
  }

◆ matrixV()

const MatrixVType & Eigen::SVDBase< BDCSVD< _MatrixType > >::matrixV ( ) const

inlineinherited

Returns: the V matrix.

For the SVD decomposition of a n-by-p matrix, letting m be the minimum of n and p, the V matrix is p-by-p if you asked for ComputeFullV , and is p-by-m if you asked for ComputeThinV .

The m first columns of V are the right singular vectors of the matrix being decomposed.

This method asserts that you asked for V to be computed.

  {
    eigen_assert(m_isInitialized && "SVD is not initialized.");
    eigen_assert(computeV() && "This SVD decomposition didn't compute V. Did you ask for it?");
    return m_matrixV;
  }

◆ nonzeroSingularValues()

Index Eigen::SVDBase< BDCSVD< _MatrixType > >::nonzeroSingularValues ( ) const

inlineinherited

Returns: the number of singular values that are not exactly 0

  {
    eigen_assert(m_isInitialized && "SVD is not initialized.");
    return m_nonzeroSingularValues;
  }

◆ perturbCol0()

template<typename MatrixType >

void Eigen::BDCSVD< MatrixType >::perturbCol0	(	const ArrayRef &	col0,
		const ArrayRef &	diag,
		const IndicesRef &	perm,
		const VectorType &	singVals,
		const ArrayRef &	shifts,
		const ArrayRef &	mus,
		ArrayRef	zhat
	)

private

{
  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;
    }
  }
}

References sqrt().

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◆ rank()

Index Eigen::SVDBase< BDCSVD< _MatrixType > >::rank ( ) const

inlineinherited

Returns: the rank of the matrix of which *this is the SVD.

Note: This method has to determine which singular values should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

  {
    using std::abs;
    eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
    if(m_singularValues.size()==0) return 0;
    RealScalar premultiplied_threshold = numext::maxi<RealScalar>(m_singularValues.coeff(0) * threshold(), (std::numeric_limits<RealScalar>::min)());
    Index i = m_nonzeroSingularValues-1;
    while(i>=0 && m_singularValues.coeff(i) < premultiplied_threshold) --i;
    return i+1;
  }

◆ rows()

template<typename _MatrixType >

Index Eigen::SVDBase< Derived >::rows ( ) const

inline

194{ return m_rows; }

Eigen::SVDBase< BDCSVD< _MatrixType > >::m_rows

Index m_rows

Definition SVDBase.h:238

Referenced by Eigen::BDCSVD< _MatrixType >::BDCSVD().

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◆ secularEq()

template<typename MatrixType >

BDCSVD< MatrixType >::RealScalar Eigen::BDCSVD< MatrixType >::secularEq	(	RealScalar	x,
		const ArrayRef &	col0,
		const ArrayRef &	diag,
		const IndicesRef &	perm,
		const ArrayRef &	diagShifted,
		RealScalar	shift
	)

staticprivate

{
  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;
 
}

◆ setSwitchSize()

template<typename _MatrixType >

void Eigen::BDCSVD< _MatrixType >::setSwitchSize ( int s )

inline

  {
    eigen_assert(s>3 && "BDCSVD the size of the algo switch has to be greater than 3");
    m_algoswap = s;
  }

References eigen_assert, and Eigen::BDCSVD< _MatrixType >::m_algoswap.

◆ setThreshold() [1/2]

BDCSVD< _MatrixType > & Eigen::SVDBase< BDCSVD< _MatrixType > >::setThreshold ( const RealScalar & threshold )

inlineinherited

Allows to prescribe a threshold to be used by certain methods, such as rank() and solve(), which need to determine when singular values are to be considered nonzero. This is not used for the SVD decomposition itself.

When it needs to get the threshold value, Eigen calls threshold(). The default is NumTraits<Scalar>::epsilon()

Parameters

threshold The new value to use as the threshold.

A singular value will be considered nonzero if its value is strictly greater than $\vert singular value \vert \leqslant threshold \times \vert max singular value \vert$ .

If you want to come back to the default behavior, call setThreshold(Default_t)

  {
    m_usePrescribedThreshold = true;
    m_prescribedThreshold = threshold;
    return derived();
  }

◆ setThreshold() [2/2]

BDCSVD< _MatrixType > & Eigen::SVDBase< BDCSVD< _MatrixType > >::setThreshold ( Default_t )

inlineinherited

Allows to come back to the default behavior, letting Eigen use its default formula for determining the threshold.

You should pass the special object Eigen::Default as parameter here.

svd.setThreshold(Eigen::Default);

Eigen::Default

@ Default

Definition Constants.h:352

See the documentation of setThreshold(const RealScalar&).

  {
    m_usePrescribedThreshold = false;
    return derived();
  }

◆ singularValues()

const SingularValuesType & Eigen::SVDBase< BDCSVD< _MatrixType > >::singularValues ( ) const

inlineinherited

Returns: the vector of singular values.

For the SVD decomposition of a n-by-p matrix, letting m be the minimum of n and p, the returned vector has size m. Singular values are always sorted in decreasing order.

  {
    eigen_assert(m_isInitialized && "SVD is not initialized.");
    return m_singularValues;
  }

◆ solve()

const Solve< BDCSVD< _MatrixType > , Rhs > Eigen::SVDBase< BDCSVD< _MatrixType > >::solve ( const MatrixBase< Rhs > & b ) const

inlineinherited

Returns: a (least squares) solution of using the current SVD decomposition of A.

Parameters

b	the right-hand-side of the equation to solve.

Note: Solving requires both U and V to be computed. Thin U and V are enough, there is no need for full U or V.; SVD solving is implicitly least-squares. Thus, this method serves both purposes of exact solving and least-squares solving. In other words, the returned solution is guaranteed to minimize the Euclidean norm $\Vert A x - b \Vert$ .

  {
    eigen_assert(m_isInitialized && "SVD is not initialized.");
    eigen_assert(computeU() && computeV() && "SVD::solve() requires both unitaries U and V to be computed (thin unitaries suffice).");
    return Solve<Derived, Rhs>(derived(), b.derived());
  }

◆ structured_update()

template<typename MatrixType >

void Eigen::BDCSVD< MatrixType >::structured_update	(	Block< MatrixXr, Dynamic, Dynamic >	A,
		const MatrixXr &	B,
		Index	n1
	)

private

{
  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;
  }
}

◆ threshold()

RealScalar Eigen::SVDBase< BDCSVD< _MatrixType > >::threshold ( ) const

inlineinherited

Returns the threshold that will be used by certain methods such as rank().

See the documentation of setThreshold(const RealScalar&).

  {
    eigen_assert(m_isInitialized || m_usePrescribedThreshold);
    // this temporary is needed to workaround a MSVC issue
    Index diagSize = (std::max<Index>)(1,m_diagSize);
    return m_usePrescribedThreshold ? m_prescribedThreshold
                                    : diagSize*NumTraits<Scalar>::epsilon();
  }

Member Data Documentation

◆ m_algoswap

template<typename _MatrixType >

int Eigen::BDCSVD< _MatrixType >::m_algoswap

protected

Referenced by Eigen::BDCSVD< _MatrixType >::setSwitchSize().

◆ m_cols

Index Eigen::SVDBase< BDCSVD< _MatrixType > >::m_cols

protectedinherited

◆ m_compU

template<typename _MatrixType >

bool Eigen::BDCSVD< _MatrixType >::m_compU

protected

◆ m_computationOptions

unsigned int Eigen::SVDBase< BDCSVD< _MatrixType > >::m_computationOptions

protectedinherited

◆ m_computed

template<typename _MatrixType >

MatrixXr Eigen::BDCSVD< _MatrixType >::m_computed

protected

◆ m_computeFullU

template<typename _MatrixType >

bool Eigen::SVDBase< Derived >::m_computeFullU

protected

◆ m_computeFullV

template<typename _MatrixType >

bool Eigen::SVDBase< Derived >::m_computeFullV

protected

◆ m_computeThinU

template<typename _MatrixType >

bool Eigen::SVDBase< Derived >::m_computeThinU

protected

◆ m_computeThinV

template<typename _MatrixType >

bool Eigen::SVDBase< Derived >::m_computeThinV

protected

◆ m_compV

template<typename _MatrixType >

bool Eigen::BDCSVD< _MatrixType >::m_compV

protected

◆ m_diagSize

template<typename _MatrixType >

Index Eigen::SVDBase< Derived >::m_diagSize

protected

◆ m_isAllocated

bool Eigen::SVDBase< BDCSVD< _MatrixType > >::m_isAllocated

protectedinherited

◆ m_isInitialized

template<typename _MatrixType >

bool Eigen::SVDBase< Derived >::m_isInitialized

protected

◆ m_isTranspose

template<typename _MatrixType >

bool Eigen::BDCSVD< _MatrixType >::m_isTranspose

protected

◆ m_matrixU

template<typename _MatrixType >

MatrixUType Eigen::SVDBase< Derived >::m_matrixU

protected

◆ m_matrixV

template<typename _MatrixType >

MatrixVType Eigen::SVDBase< Derived >::m_matrixV

protected

◆ m_naiveU

template<typename _MatrixType >

MatrixXr Eigen::BDCSVD< _MatrixType >::m_naiveU

protected

◆ m_naiveV

template<typename _MatrixType >

MatrixXr Eigen::BDCSVD< _MatrixType >::m_naiveV

protected

◆ m_nonzeroSingularValues

template<typename _MatrixType >

Index Eigen::SVDBase< Derived >::m_nonzeroSingularValues

protected

◆ m_nRec

template<typename _MatrixType >

Index Eigen::BDCSVD< _MatrixType >::m_nRec

protected

◆ m_numIters

template<typename _MatrixType >

int Eigen::BDCSVD< _MatrixType >::m_numIters

◆ m_prescribedThreshold

RealScalar Eigen::SVDBase< BDCSVD< _MatrixType > >::m_prescribedThreshold

protectedinherited

◆ m_rows

Index Eigen::SVDBase< BDCSVD< _MatrixType > >::m_rows

protectedinherited

◆ m_singularValues

template<typename _MatrixType >

SingularValuesType Eigen::SVDBase< Derived >::m_singularValues

protected

◆ m_usePrescribedThreshold

bool Eigen::SVDBase< BDCSVD< _MatrixType > >::m_usePrescribedThreshold

protectedinherited

◆ m_workspace

template<typename _MatrixType >

ArrayXr Eigen::BDCSVD< _MatrixType >::m_workspace

protected

◆ m_workspaceI

template<typename _MatrixType >

ArrayXi Eigen::BDCSVD< _MatrixType >::m_workspaceI

protected

The documentation for this class was generated from the following files:

src/eigen/Eigen/src/Core/util/ForwardDeclarations.h
src/eigen/Eigen/src/SVD/BDCSVD.h

Public Types

Public Member Functions

Public Attributes

Static Protected Member Functions

Protected Attributes

Private Types

Private Member Functions

Static Private Member Functions

Detailed Description

Member Typedef Documentation

◆ ArrayRef

◆ ArrayXi

◆ ArrayXr

◆ Base

◆ Index

◆ IndicesRef

◆ Literal

◆ MatrixType

◆ MatrixUType

◆ MatrixVType

◆ MatrixX

◆ MatrixXr

◆ RealScalar

◆ Scalar

◆ SingularValuesType

◆ StorageIndex

◆ VectorType

Member Enumeration Documentation

◆ anonymous enum

◆ anonymous enum

Constructor & Destructor Documentation

◆ BDCSVD() [1/3]

◆ BDCSVD() [2/3]

◆ BDCSVD() [3/3]

◆ ~BDCSVD()

Member Function Documentation

◆ _solve_impl() [1/2]

◆ _solve_impl() [2/2]

◆ allocate()

◆ check_template_parameters()

◆ cols()

◆ compute() [1/2]

◆ compute() [2/2]

◆ computeSingVals()

◆ computeSingVecs()

◆ computeSVDofM()

◆ computeU()

◆ computeV()

◆ copyUV()

◆ deflation()

◆ deflation43()

◆ deflation44()

◆ derived() [1/2]

◆ derived() [2/2]

◆ divide()

◆ matrixU()

◆ matrixV()

◆ nonzeroSingularValues()

◆ perturbCol0()

◆ rank()

◆ rows()

◆ secularEq()

◆ setSwitchSize()

◆ setThreshold() [1/2]

◆ setThreshold() [2/2]

◆ singularValues()

◆ solve()

◆ structured_update()

◆ threshold()

Member Data Documentation

◆ m_algoswap

◆ m_cols

◆ m_compU

◆ m_computationOptions

◆ m_computed

◆ m_computeFullU

◆ m_computeFullV

◆ m_computeThinU

◆ m_computeThinV

◆ m_compV