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

#include <JacobiSVD.h>

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 MatrixType::Index	Index

typedef Matrix< Scalar, RowsAtCompileTime, RowsAtCompileTime, MatrixOptions, MaxRowsAtCompileTime, MaxRowsAtCompileTime >	MatrixUType

typedef Matrix< Scalar, ColsAtCompileTime, ColsAtCompileTime, MatrixOptions, MaxColsAtCompileTime, MaxColsAtCompileTime >	MatrixVType

typedef internal::plain_diag_type< MatrixType, RealScalar >::type	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. More...

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

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

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

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

const MatrixUType &	matrixU () const

const MatrixVType &	matrixV () const

const SingularValuesType &	singularValues () const

bool	computeU () const

bool	computeV () const

template<typename Rhs >
const internal::solve_retval< JacobiSVD, Rhs >	solve (const MatrixBase< Rhs > &b) const

Index	nonzeroSingularValues () const

Index	rows () const

Index	cols () const

Protected Attributes
MatrixUType	m_matrixU

MatrixVType	m_matrixV

SingularValuesType	m_singularValues

WorkMatrixType	m_workMatrix

bool	m_isInitialized

bool	m_isAllocated

bool	m_computeFullU

bool	m_computeThinU

bool	m_computeFullV

bool	m_computeThinV

unsigned int	m_computationOptions

Index	m_nonzeroSingularValues

Index	m_rows

Index	m_cols

Index	m_diagSize

internal::qr_preconditioner_impl< MatrixType, QRPreconditioner, internal::PreconditionIfMoreColsThanRows >	m_qr_precond_morecols

internal::qr_preconditioner_impl< MatrixType, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols >	m_qr_precond_morerows

Friends
template<typename __MatrixType , int _QRPreconditioner, bool _IsComplex>
struct	internal::svd_precondition_2x2_block_to_be_real

template<typename __MatrixType , int _QRPreconditioner, int _Case, bool _DoAnything>
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.

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 224 of file ForwardDeclarations.h.

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 528 of file JacobiSVD.h.

       : m_isInitialized(false),
         m_isAllocated(false),
         m_computationOptions(0),
         m_rows(-1), m_cols(-1)
     {}

template<typename _MatrixType, int QRPreconditioner>

Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::JacobiSVD	(	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: JacobiSVD()

Definition at line 542 of file JacobiSVD.h.

       : m_isInitialized(false),
         m_isAllocated(false),
         m_computationOptions(0),
         m_rows(-1), m_cols(-1)
     {
       allocate(rows, cols, computationOptions);
     }

template<typename _MatrixType, int QRPreconditioner>

Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::JacobiSVD	(	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.

Definition at line 561 of file JacobiSVD.h.

       : m_isInitialized(false),
         m_isAllocated(false),
         m_computationOptions(0),
         m_rows(-1), m_cols(-1)
     {
       compute(matrix, computationOptions);
     }

Member Function Documentation

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 742 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)
   const RealScalar considerAsZero = RealScalar(2) * std::numeric_limits<RealScalar>::denorm_min();
 
   /*** step 1. The R-SVD step: we use a QR decomposition to reduce to the case of a square matrix */
 
   if(!m_qr_precond_morecols.run(*this, matrix) && !m_qr_precond_morerows.run(*this, matrix))
   {
     m_workMatrix = matrix.block(0,0,m_diagSize,m_diagSize);
     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.
         using std::max;
         RealScalar threshold = (max)(considerAsZero, precision * (max)(abs(m_workMatrix.coeff(p,p)),
                                                                        abs(m_workMatrix.coeff(q,q))));
         if((max)(abs(m_workMatrix.coeff(p,q)),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;
   }
 
   /*** 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 588 of file JacobiSVD.h.

     {
       return compute(matrix, m_computationOptions);
     }

template<typename _MatrixType, int QRPreconditioner>

bool Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::computeU ( ) const

inline

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

Definition at line 637 of file JacobiSVD.h.

637 { return m_computeFullU || m_computeThinU; }

template<typename _MatrixType, int QRPreconditioner>

bool Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::computeV ( ) const

inline

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

Definition at line 639 of file JacobiSVD.h.

639 { return m_computeFullV || m_computeThinV; }

template<typename _MatrixType, int QRPreconditioner>

const MatrixUType& Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::matrixU ( ) const

inline

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.

Definition at line 602 of file JacobiSVD.h.

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

template<typename _MatrixType, int QRPreconditioner>

const MatrixVType& Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::matrixV ( ) const

inline

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.

Definition at line 618 of file JacobiSVD.h.

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

template<typename _MatrixType, int QRPreconditioner>

Index Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::nonzeroSingularValues ( ) const

inline

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

Definition at line 660 of file JacobiSVD.h.

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

template<typename _MatrixType, int QRPreconditioner>

const SingularValuesType& Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::singularValues ( ) const

inline

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.

Definition at line 630 of file JacobiSVD.h.

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

template<typename _MatrixType, int QRPreconditioner>

template<typename Rhs >

const internal::solve_retval<JacobiSVD, Rhs> Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::solve ( const MatrixBase< Rhs > & b ) const

inline

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 $\Vert A x - b \Vert$ .

Definition at line 652 of file JacobiSVD.h.

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

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

C:/Users/Brig/Documents/ShapeWorksStudio/src/Surfworks/Eigen/src/Core/util/ForwardDeclarations.h
C:/Users/Brig/Documents/ShapeWorksStudio/src/Surfworks/Eigen/src/SVD/JacobiSVD.h

Public Types

Public Member Functions

Protected Attributes

Friends

Detailed Description

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

Constructor & Destructor Documentation

Member Function Documentation

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