Please, help us to better know about our user community by answering the following short survey: https://forms.gle/wpyrxWi18ox9Z5ae9 Eigen  3.3.90 (git rev a8fdcae55d1f002966fc9b963597a404f30baa09) Eigen::JacobiSVD Class Reference

## Detailed Description

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;

Output:

Here is the matrix m:
0.68  0.597
-0.211  0.823
0.566 -0.605
Its singular values are:
1.19
0.899
Its left singular vectors are the columns of the thin U matrix:
0.388   0.866
0.712 -0.0634
-0.586   0.496
Its right singular vectors are the columns of the thin V matrix:
-0.183  0.983
0.983  0.183
Now consider this rhs vector:
1
0
0
A least-squares solution of m*x = rhs is:
0.888
0.496


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()

## Public Member Functions

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...

bool computeU () const

bool computeV () const

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...

Index rank () const

## ◆ JacobiSVD() [1/3]

 Eigen::JacobiSVD::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]

 Eigen::JacobiSVD::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]

 Eigen::JacobiSVD::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.

## ◆ compute() [1/2]

 JacobiSVD& Eigen::JacobiSVD::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]

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

## ◆ computeU()

 bool Eigen::SVDBase::computeU
inline
Returns
true if U (full or thin) is asked for in this SVD decomposition

## ◆ computeV()

 bool Eigen::SVDBase::computeV
inline
Returns
true if V (full or thin) is asked for in this SVD decomposition

## ◆ rank()

 Index Eigen::SVDBase::rank
inline
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&).

The documentation for this class was generated from the following files:
Eigen::ComputeThinU
@ ComputeThinU
Definition: Constants.h:394
Eigen::DenseBase::Random
static const RandomReturnType Random()
Definition: Random.h:113
Eigen::ComputeThinV
@ ComputeThinV
Definition: Constants.h:398