com.imsl.math

Class SVD

java.lang.Object

com.imsl.math.SVD

public class SVD
extends Object

Singular Value Decomposition (SVD) of a rectangular matrix of type double.

SVD is based on the LINPACK routine SSVDC; see Dongarra et al. (1979).

Let n be the number of rows in A and let p be the number of columns in A. For any
n x p matrix A, there exists an n x n orthogonal matrix U and a p x p orthogonal matrix V such that

$$U^T A V = \left\{ \begin{array}{cl} \left[ \begin{array}{l} \Sigma \\ 0 \end{array} \right] & \mbox{if $n \ge p$} \\ \left[ \Sigma \,\, 0 \right] & \mbox{if $n \le p$} \end{array} \right.$$

where \(\Sigma = {\rm diag}(\sigma_1, \ldots, \sigma_m)\), and \(m = \min(n, p)\). The scalars \(\sigma_1 \geq \sigma_2 \geq \ldots \geq \sigma_m \geq 0\) are called the singular values of A. The columns of U are called the left singular vectors of A. The columns of V are called the right singular vectors of A.

The estimated rank of A is the number of \(\sigma_k\) that is larger than a tolerance \(\eta\). If \(\tau\) is the parameter tol in the program, then

$$\eta = \left\{ \begin{array}{cl} \tau \hfill & \mbox {if $\tau \gt 0$} \\ {\left| \tau \right|\left\| A \right\|_\infty } \hfill & \mbox {if $\tau \lt 0$} \end{array} \right.$$

The Moore-Penrose generalized inverse of the matrix is computed by partitioning the matrices U, V and \(\Sigma\) as \(U = (U_1,U_2)\), \(V = (V_1,V_2)\) and \(\Sigma_1 = {\rm diag}(\sigma_1,\ldots,\sigma_k)\) where the "1" matrices are k by k. The Moore-Penrose generalized inverse is \(V_1 \Sigma_1^{-1} U_1^T\).

See Also:

Example

Nested Class Summary

Nested Classes

Modifier and Type	Class and Description
static class	SVD.DidNotConvergeException The iteration did not converge

Constructor Summary

Constructors

Constructor and Description
SVD(double[][] a) Construct the singular value decomposition of a rectangular matrix with default tolerance.
SVD(double[][] a, double tol) Construct the singular value decomposition of a rectangular matrix with a given tolerance.

Method Summary

Modifier and Type	Method and Description
int	getInfo() Returns convergence information about S, U, and V.
double[][]	getInverse() Compute the Moore-Penrose generalized inverse.
int	getRank() Returns the rank of the matrix used to construct this instance.
double[]	getS() Returns the singular values.
double[][]	getU() Returns the left singular vectors.
double[][]	getV() Returns the right singular vectors.
double[][]	inverse() Deprecated. Method name does not adhere to the Java naming conventions. Use SVD.getInverse() instead.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

SVD

public SVD(double[][] a,
           double tol)
    throws SVD.DidNotConvergeException

Construct the singular value decomposition of a rectangular matrix with a given tolerance. If tol is positive, then a singular value is considered negligible if the singular value is less than or equal to tol. If tol is negative, then a singular value is considered negligible if the singular value is less than or equal to the absolute value of the product of tol and the infinity norm of the input matrix. In the latter case, the absolute value of tol generally contains an estimate of the level of the relative error in the data.

Parameters:

a - a double matrix for which the singular value decomposition is to be computed

tol - a double scalar containing the tolerance used to determine when a singular value is negligible

Throws:

IllegalArgumentException - is thrown when the row lengths of input matrix a are not equal (for example, the matrix edges are "jagged")

SVD.DidNotConvergeException - is thrown when the rank cannot be determined because convergence was not obtained for all singular values

SVD

public SVD(double[][] a)
    throws SVD.DidNotConvergeException

Construct the singular value decomposition of a rectangular matrix with default tolerance. The tolerance used is 2.2204460492503e-14. This tolerance is used to determine rank. A singular value is considered negligible if the singular value is less than or equal to this tolerance.

Parameters:

a - a double matrix for which the singular value decomposition is to be computed

Throws:

IllegalArgumentException - is thrown when the row lengths of input matrix a are not equal (i.e. the matrix edges are "jagged")

SVD.DidNotConvergeException

Method Detail

getRank

public int getRank()

Returns the rank of the matrix used to construct this instance.

Returns:

an int scalar containing the rank of the matrix used to construct this instance. The estimated rank of the input matrix is the number of singular values which are larger than a tolerance.

getU

public double[][] getU()

Returns the left singular vectors.

Returns:

a double matrix containing the left singular vectors

getV

public double[][] getV()

Returns the right singular vectors.

Returns:

a double matrix containing the right singular vectors

getS

public double[] getS()

Returns the singular values.

Returns:

a double array containing the singular values of the matrix

getInfo

public int getInfo()

Returns convergence information about S, U, and V.

Returns:

Convergence was obtained for the info, info+1, ..., min(nra,nca) singular values and their corresponding vectors. Here, nra and nca represent the number of rows and columns of the input matrix respectively.

inverse

@Deprecated
public double[][] inverse()

Deprecated. Method name does not adhere to the Java naming conventions. Use SVD.getInverse() instead.

Compute the Moore-Penrose generalized inverse of a real matrix.

Returns:

a double matrix containing the generalized inverse of the matrix used to construct this instance

getInverse

public double[][] getInverse()

Compute the Moore-Penrose generalized inverse.

Returns:

a double matrix containing the generalized inverse of the matrix used to construct this instance