Two-sided Jacobi SVD decomposition of a rectangular matrix. More...

#include <JacobiSVD.h>

Inheritance diagram for Eigen::JacobiSVD< _MatrixType, QRPreconditioner >:

Collaboration diagram for Eigen::JacobiSVD< _MatrixType, QRPreconditioner >:

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< typename MatrixType::Scalar >::Real	RealScalar
typedef Base::MatrixUType	MatrixUType
typedef Base::MatrixVType	MatrixVType
typedef Base::SingularValuesType	SingularValuesType
typedef internal::plain_row_type < MatrixType >::type	RowType
typedef internal::plain_col_type < MatrixType >::type	ColType
typedef Matrix< Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime, MatrixOptions, MaxDiagSizeAtCompileTime, MaxDiagSizeAtCompileTime >	WorkMatrixType
Public Member Functions
	JacobiSVD ()
	Default Constructor.
	JacobiSVD (Index p_rows, Index p_cols, unsigned int computationOptions=0)
	Default Constructor with memory preallocation.
	JacobiSVD (const MatrixType &matrix, unsigned int computationOptions=0)
	Constructor performing the decomposition of given matrix.
JacobiSVD &	compute (const MatrixType &matrix, unsigned int computationOptions)
	Method performing the decomposition of given matrix using custom options.
JacobiSVD &	compute (const MatrixType &matrix)
	Method performing the decomposition of given matrix using current options.
Protected Attributes
WorkMatrixType	m_workMatrix
internal::qr_preconditioner_impl < MatrixType, QRPreconditioner, internal::PreconditionIfMoreColsThanRows >	m_qr_precond_morecols
internal::qr_preconditioner_impl < MatrixType, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols >	m_qr_precond_morerows
MatrixType	m_scaledMatrix
Private Types
typedef SVDBase< JacobiSVD >	Base
Private Member Functions
void	allocate (Index rows, Index cols, unsigned int computationOptions)
Friends
struct	internal::svd_precondition_2x2_block_to_be_real
struct	internal::qr_preconditioner_impl

Detailed Description

template<typename _MatrixType, int QRPreconditioner>
class Eigen::JacobiSVD< _MatrixType, QRPreconditioner >

Two-sided Jacobi SVD decomposition of a rectangular matrix.

Template Parameters:

_MatrixType	the type of the matrix of which we are computing the SVD decomposition
QRPreconditioner	this optional parameter allows to specify the type of QR decomposition that will be used internally for the R-SVD step for non-square matrices. See discussion of possible values below.

SVD decomposition consists in decomposing any n-by-p matrix A as a product

$A = U S V^*$

where U is a n-by-n unitary, V is a p-by-p unitary, and S is a n-by-p real positive matrix which is zero outside of its main diagonal; the diagonal entries of S are known as the singular values of A and the columns of U and V are known as the left and right singular vectors of A respectively.

Singular values are always sorted in decreasing order.

This JacobiSVD decomposition computes only the singular values by default. If you want U or V, you need to ask for them explicitly.

You can ask for only thin U or V to be computed, meaning the following. In case of a rectangular n-by-p matrix, letting m be the smaller value among n and p, there are only m singular vectors; the remaining columns of U and V do not correspond to actual singular vectors. Asking for thin U or V means asking for only their m first columns to be formed. So U is then a n-by-m matrix, and V is then a p-by-m matrix. Notice that thin U and V are all you need for (least squares) solving.

Here's an example demonstrating basic usage:

Output:

This JacobiSVD class is a two-sided Jacobi R-SVD decomposition, ensuring optimal reliability and accuracy. The downside is that it's slower than bidiagonalizing SVD algorithms for large square matrices; however its complexity is still $ O(n^2p) $ where n is the smaller dimension and p is the greater dimension, meaning that it is still of the same order of complexity as the faster bidiagonalizing R-SVD algorithms. In particular, like any R-SVD, it takes advantage of non-squareness in that its complexity is only linear in the greater dimension.

If the input matrix has inf or nan coefficients, the result of the computation is undefined, but the computation is guaranteed to terminate in finite (and reasonable) time.

The possible values for QRPreconditioner are:

ColPivHouseholderQRPreconditioner is the default. In practice it's very safe. It uses column-pivoting QR.
FullPivHouseholderQRPreconditioner, is the safest and slowest. It uses full-pivoting QR. Contrary to other QRs, it doesn't allow computing thin unitaries.
HouseholderQRPreconditioner is the fastest, and less safe and accurate than the pivoting variants. It uses non-pivoting QR. This is very similar in safety and accuracy to the bidiagonalization process used by bidiagonalizing SVD algorithms (since bidiagonalization is inherently non-pivoting). However the resulting SVD is still more reliable than bidiagonalizing SVDs because the Jacobi-based iterarive process is more reliable than the optimized bidiagonal SVD iterations.
NoQRPreconditioner allows not to use a QR preconditioner at all. This is useful if you know that you will only be computing JacobiSVD decompositions of square matrices. Non-square matrices require a QR preconditioner. Using this option will result in faster compilation and smaller executable code. It won't significantly speed up computation, since JacobiSVD is always checking if QR preconditioning is needed before applying it anyway.

See also:: MatrixBase::jacobiSvd()

Definition at line 498 of file JacobiSVD.h.

Member Typedef Documentation

template<typename _MatrixType, int QRPreconditioner>

typedef SVDBase<JacobiSVD> Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::Base [private]

Definition at line 501 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>

typedef internal::plain_col_type<MatrixType>::type Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::ColType

Definition at line 522 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>

typedef _MatrixType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::MatrixType

Reimplemented from Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >.

Definition at line 504 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>

typedef Base::MatrixUType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::MatrixUType

Reimplemented from Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >.

Definition at line 517 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>

typedef Base::MatrixVType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::MatrixVType

Reimplemented from Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >.

Definition at line 518 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>

typedef NumTraits<typename MatrixType::Scalar>::Real Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::RealScalar

Reimplemented from Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >.

Definition at line 506 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>

typedef internal::plain_row_type<MatrixType>::type Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::RowType

Definition at line 521 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>

typedef MatrixType::Scalar Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::Scalar

Reimplemented from Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >.

Definition at line 505 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>

typedef Base::SingularValuesType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::SingularValuesType

Reimplemented from Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >.

Definition at line 519 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>

typedef Matrix<Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime, MatrixOptions, MaxDiagSizeAtCompileTime, MaxDiagSizeAtCompileTime> Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::WorkMatrixType

Definition at line 525 of file JacobiSVD.h.

Member Enumeration Documentation

template<typename _MatrixType, int QRPreconditioner>

anonymous enum

Enumerator:

RowsAtCompileTime
ColsAtCompileTime
DiagSizeAtCompileTime
MaxRowsAtCompileTime
MaxColsAtCompileTime
MaxDiagSizeAtCompileTime
MatrixOptions

Definition at line 507 of file JacobiSVD.h.

         {
      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

template<typename _MatrixType, int QRPreconditioner>

Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::JacobiSVD ( ) [inline]

Default Constructor.

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

Definition at line 532 of file JacobiSVD.h.

{}

template<typename _MatrixType, int QRPreconditioner>

Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::JacobiSVD	(	Index	p_rows,
		Index	p_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:: JacobiSVD()

Definition at line 542 of file JacobiSVD.h.

    {
      allocate(p_rows, p_cols, computationOptions);
    }

template<typename _MatrixType, int QRPreconditioner>

Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::JacobiSVD	(	const MatrixType &	matrix,
		unsigned int	computationOptions = `0`
	)		`[inline, explicit]`

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.

Definition at line 557 of file JacobiSVD.h.

    {
      compute(matrix, computationOptions);
    }

Member Function Documentation

template<typename MatrixType , int QRPreconditioner>

void Eigen::JacobiSVD< MatrixType, QRPreconditioner >::allocate	(	Index	rows,
		Index	cols,
		unsigned int	computationOptions
	)		`[private]`

Reimplemented from Eigen::SVDBase< JacobiSVD< _MatrixType, QRPreconditioner > >.

Definition at line 624 of file JacobiSVD.h.

{
  eigen_assert(p_rows >= 0 && p_cols >= 0);

  if (m_isAllocated &&
      p_rows == m_rows &&
      p_cols == m_cols &&
      computationOptions == m_computationOptions)
  {
    return;
  }

  m_rows = p_rows;
  m_cols = p_cols;
  m_isInitialized = false;
  m_isAllocated = true;
  m_computationOptions = computationOptions;
  m_computeFullU = (computationOptions & ComputeFullU) != 0;
  m_computeThinU = (computationOptions & ComputeThinU) != 0;
  m_computeFullV = (computationOptions & ComputeFullV) != 0;
  m_computeThinV = (computationOptions & ComputeThinV) != 0;
  eigen_assert(!(m_computeFullU && m_computeThinU) && "JacobiSVD: you can't ask for both full and thin U");
  eigen_assert(!(m_computeFullV && m_computeThinV) && "JacobiSVD: you can't ask for both full and thin V");
  eigen_assert(EIGEN_IMPLIES(m_computeThinU || m_computeThinV, MatrixType::ColsAtCompileTime==Dynamic) &&
              "JacobiSVD: thin U and V are only available when your matrix has a dynamic number of columns.");
  if (QRPreconditioner == FullPivHouseholderQRPreconditioner)
  {
      eigen_assert(!(m_computeThinU || m_computeThinV) &&
              "JacobiSVD: can't compute thin U or thin V with the FullPivHouseholderQR preconditioner. "
              "Use the ColPivHouseholderQR preconditioner instead.");
  }
  m_diagSize = (std::min)(m_rows, m_cols);
  m_singularValues.resize(m_diagSize);
  if(RowsAtCompileTime==Dynamic)
    m_matrixU.resize(m_rows, m_computeFullU ? m_rows
                            : m_computeThinU ? m_diagSize
                            : 0);
  if(ColsAtCompileTime==Dynamic)
    m_matrixV.resize(m_cols, m_computeFullV ? m_cols
                            : m_computeThinV ? m_diagSize
                            : 0);
  m_workMatrix.resize(m_diagSize, m_diagSize);
  
  if(m_cols>m_rows)   m_qr_precond_morecols.allocate(*this);
  if(m_rows>m_cols)   m_qr_precond_morerows.allocate(*this);
  if(m_rows!=m_cols)  m_scaledMatrix.resize(p_rows,p_cols);
}

template<typename MatrixType , int QRPreconditioner>

JacobiSVD< MatrixType, QRPreconditioner > & Eigen::JacobiSVD< MatrixType, QRPreconditioner >::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.

Definition at line 674 of file JacobiSVD.h.

{
  using std::abs;
  allocate(matrix.rows(), matrix.cols(), computationOptions);

  // currently we stop when we reach precision 2*epsilon as the last bit of precision can require an unreasonable number of iterations,
  // only worsening the precision of U and V as we accumulate more rotations
  const RealScalar precision = RealScalar(2) * NumTraits<Scalar>::epsilon();

  // limit for very small denormal numbers to be considered zero in order to avoid infinite loops (see bug 286)
  // FIXME What about considerering any denormal numbers as zero, using:
  // const RealScalar considerAsZero = (std::numeric_limits<RealScalar>::min)();
  const RealScalar considerAsZero = RealScalar(2) * std::numeric_limits<RealScalar>::denorm_min();

  // Scaling factor to reduce over/under-flows
  RealScalar scale = matrix.cwiseAbs().maxCoeff();
  if(scale==RealScalar(0)) scale = RealScalar(1);
  
  /*** step 1. The R-SVD step: we use a QR decomposition to reduce to the case of a square matrix */

  if(m_rows!=m_cols)
  {
    m_scaledMatrix = matrix / scale;
    m_qr_precond_morecols.run(*this, m_scaledMatrix);
    m_qr_precond_morerows.run(*this, m_scaledMatrix);
  }
  else
  {
    m_workMatrix = matrix.block(0,0,m_diagSize,m_diagSize) / scale;
    if(m_computeFullU) m_matrixU.setIdentity(m_rows,m_rows);
    if(m_computeThinU) m_matrixU.setIdentity(m_rows,m_diagSize);
    if(m_computeFullV) m_matrixV.setIdentity(m_cols,m_cols);
    if(m_computeThinV) m_matrixV.setIdentity(m_cols, m_diagSize);
  }

  /*** step 2. The main Jacobi SVD iteration. ***/

  bool finished = false;
  while(!finished)
  {
    finished = true;

    // do a sweep: for all index pairs (p,q), perform SVD of the corresponding 2x2 sub-matrix

    for(Index p = 1; p < m_diagSize; ++p)
    {
      for(Index q = 0; q < p; ++q)
      {
        // if this 2x2 sub-matrix is not diagonal already...
        // notice that this comparison will evaluate to false if any NaN is involved, ensuring that NaN's don't
        // keep us iterating forever. Similarly, small denormal numbers are considered zero.
        RealScalar threshold = numext::maxi<RealScalar>(considerAsZero,
                   precision * numext::maxi<RealScalar>(abs(m_workMatrix.coeff(p,p)),
                                                        abs(m_workMatrix.coeff(q,q))));
        // We compare both values to threshold instead of calling max to be robust to NaN (See bug 791)
        if(abs(m_workMatrix.coeff(p,q))>threshold || abs(m_workMatrix.coeff(q,p)) > threshold)
        {
          finished = false;

          // perform SVD decomposition of 2x2 sub-matrix corresponding to indices p,q to make it diagonal
          internal::svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner>::run(m_workMatrix, *this, p, q);
          JacobiRotation<RealScalar> j_left, j_right;
          internal::real_2x2_jacobi_svd(m_workMatrix, p, q, &j_left, &j_right);

          // accumulate resulting Jacobi rotations
          m_workMatrix.applyOnTheLeft(p,q,j_left);
          if(computeU()) m_matrixU.applyOnTheRight(p,q,j_left.transpose());

          m_workMatrix.applyOnTheRight(p,q,j_right);
          if(computeV()) m_matrixV.applyOnTheRight(p,q,j_right);
        }
      }
    }
  }

  /*** step 3. The work matrix is now diagonal, so ensure it's positive so its diagonal entries are the singular values ***/

  for(Index i = 0; i < m_diagSize; ++i)
  {
    RealScalar a = abs(m_workMatrix.coeff(i,i));
    m_singularValues.coeffRef(i) = a;
    if(computeU() && (a!=RealScalar(0))) m_matrixU.col(i) *= m_workMatrix.coeff(i,i)/a;
  }
  
  m_singularValues *= scale;

  /*** step 4. Sort singular values in descending order and compute the number of nonzero singular values ***/

  m_nonzeroSingularValues = m_diagSize;
  for(Index i = 0; i < m_diagSize; i++)
  {
    Index pos;
    RealScalar maxRemainingSingularValue = m_singularValues.tail(m_diagSize-i).maxCoeff(&pos);
    if(maxRemainingSingularValue == RealScalar(0))
    {
      m_nonzeroSingularValues = i;
      break;
    }
    if(pos)
    {
      pos += i;
      std::swap(m_singularValues.coeffRef(i), m_singularValues.coeffRef(pos));
      if(computeU()) m_matrixU.col(pos).swap(m_matrixU.col(i));
      if(computeV()) m_matrixV.col(pos).swap(m_matrixV.col(i));
    }
  }

  m_isInitialized = true;
  return *this;
}

template<typename _MatrixType, int QRPreconditioner>

JacobiSVD& Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::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).

Definition at line 580 of file JacobiSVD.h.

    {
      return compute(matrix, m_computationOptions);
    }

Friends And Related Function Documentation

template<typename _MatrixType, int QRPreconditioner>

friend struct internal::qr_preconditioner_impl [friend]

Definition at line 616 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>

friend struct internal::svd_precondition_2x2_block_to_be_real [friend]

Definition at line 614 of file JacobiSVD.h.

Member Data Documentation

template<typename _MatrixType, int QRPreconditioner>

internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreColsThanRows> Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_qr_precond_morecols [protected]

Definition at line 618 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>

internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols> Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_qr_precond_morerows [protected]

Definition at line 619 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>

MatrixType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_scaledMatrix [protected]

Definition at line 620 of file JacobiSVD.h.

template<typename _MatrixType, int QRPreconditioner>

WorkMatrixType Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::m_workMatrix [protected]

Definition at line 611 of file JacobiSVD.h.

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

JacobiSVD.h

Public Types

Public Member Functions

Protected Attributes

Private Types

Private Member Functions

Friends

Detailed Description

template<typename _MatrixType, int QRPreconditioner> class Eigen::JacobiSVD< _MatrixType, QRPreconditioner >

Member Typedef Documentation

Member Enumeration Documentation

Constructor & Destructor Documentation

Member Function Documentation

Friends And Related Function Documentation

Member Data Documentation

template<typename _MatrixType, int QRPreconditioner>
class Eigen::JacobiSVD< _MatrixType, QRPreconditioner >