Eigen  3.4.90 (git rev a4098ac676528a83cfb73d4d26ce1b42ec05f47c)
Eigen::JacobiSVD< MatrixType_, QRPreconditioner > Class Template Reference

## 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:

MatrixXf m = MatrixXf::Random(3,2);
cout << "Here is the matrix m:" << endl << m << endl;
JacobiSVD<MatrixXf> svd(m, ComputeThinU | ComputeThinV);
cout << "Its singular values are:" << endl << svd.singularValues() << endl;
cout << "Its left singular vectors are the columns of the thin U matrix:" << endl << svd.matrixU() << endl;
cout << "Its right singular vectors are the columns of the thin V matrix:" << endl << svd.matrixV() << endl;
Vector3f rhs(1, 0, 0);
cout << "Now consider this rhs vector:" << endl << rhs << endl;
cout << "A least-squares solution of m*x = rhs is:" << endl << svd.solve(rhs) << endl;
static const RandomReturnType Random()
Definition: Random.h:115
@ ComputeThinV
Definition: Constants.h:401
@ ComputeThinU
Definition: Constants.h:397

Output:

Here is the matrix m:
-1 -0.0827
-0.737  0.0655
0.511  -0.562
Its singular values are:
1.36
0.534
Its left singular vectors are the columns of the thin U matrix:
-0.716  -0.46
-0.543 -0.106
0.439 -0.881
Its right singular vectors are the columns of the thin V matrix:
0.986  0.164
-0.164  0.986
Now consider this rhs vector:
1
0
0
A least-squares solution of m*x = rhs is:
-0.661
-0.764

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.
MatrixBase::jacobiSvd()
Inheritance diagram for Eigen::JacobiSVD< MatrixType_, QRPreconditioner >:

## Public Member Functions

EIGEN_CONSTEXPR Index cols () const EIGEN_NOEXCEPT

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

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

JacobiSVD ()
Default Constructor. More...

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

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

EIGEN_CONSTEXPR Index rows () const EIGEN_NOEXCEPT

Public Member Functions inherited from Eigen::SVDBase< JacobiSVD< MatrixType_, QRPreconditioner > >
bool computeU () const

bool computeV () const

ComputationInfo info () const
Reports whether previous computation was successful. More...

const MatrixUTypematrixU () const

const MatrixVTypematrixV () const

Index nonzeroSingularValues () const

Index rank () const

JacobiSVD< MatrixType_, QRPreconditioner > & setThreshold (const RealScalar &threshold)

JacobiSVD< MatrixType_, QRPreconditioner > & setThreshold (Default_t)

const SingularValuesType & singularValues () const

const Solve< JacobiSVD< MatrixType_, QRPreconditioner >, Rhs > solve (const MatrixBase< Rhs > &b) const

RealScalar threshold () const

Public Member Functions inherited from Eigen::SolverBase< Derived >

Derived & derived ()

const Derived & derived () const

template<typename Rhs >
const Solve< Derived, Rhs > solve (const MatrixBase< Rhs > &b) const

SolverBase ()

ConstTransposeReturnType transpose () const

Public Member Functions inherited from Eigen::EigenBase< Derived >
EIGEN_CONSTEXPR Index cols () const EIGEN_NOEXCEPT

Derived & derived ()

const Derived & derived () const

EIGEN_CONSTEXPR Index rows () const EIGEN_NOEXCEPT

EIGEN_CONSTEXPR Index size () const EIGEN_NOEXCEPT

Public Types inherited from Eigen::SVDBase< JacobiSVD< MatrixType_, QRPreconditioner > >
typedef Eigen::Index Index

Public Types inherited from Eigen::EigenBase< Derived >
typedef Eigen::Index Index
The interface type of indices. More...

Protected Member Functions inherited from Eigen::SVDBase< JacobiSVD< MatrixType_, QRPreconditioner > >
SVDBase ()
Default Constructor. More...

## ◆ JacobiSVD() [1/3]

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&).

## ◆ JacobiSVD() [2/3]

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.

JacobiSVD()

## ◆ JacobiSVD() [3/3]

template<typename MatrixType_ , int QRPreconditioner>
 Eigen::JacobiSVD< MatrixType_, QRPreconditioner >::JacobiSVD ( const MatrixType & matrix, unsigned int computationOptions = 0 )
inlineexplicit

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.

## ◆ cols()

template<typename MatrixType_ , int QRPreconditioner>
 EIGEN_CONSTEXPR Index Eigen::EigenBase< Derived >::cols ( void ) const
inline
Returns
the number of columns.
rows(), ColsAtCompileTime

## ◆ compute() [1/2]

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).

## ◆ compute() [2/2]

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.

## ◆ rows()

template<typename MatrixType_ , int QRPreconditioner>
 EIGEN_CONSTEXPR Index Eigen::EigenBase< Derived >::rows ( void ) const
inline
Returns
the number of rows.