smile.math.matrix

Class SingularValueDecomposition

java.lang.Object

smile.math.matrix.SingularValueDecomposition

public class SingularValueDecomposition
extends java.lang.Object

Singular Value Decomposition.

For an m-by-n matrix A with m ≥ n, the singular value decomposition is an m-by-n orthogonal matrix U, an n-by-n diagonal matrix Σ, and an n-by-n orthogonal matrix V so that A = U*Σ*V'.

For m < n, only the first m columns of V are computed and Σ is m-by-m.

The singular values, σ_k = Σ_kk, are ordered so that σ₀ ≥ σ₁ ≥ ... ≥ σ_n-1.

The singular value decompostion always exists. The matrix condition number and the effective numerical rank can be computed from this decomposition.

SVD is a very powerful technique for dealing with sets of equations or matrices that are either singular or else numerically very close to singular. In many cases where Gaussian elimination and LU decomposition fail to give satisfactory results, SVD will diagnose precisely what the problem is. SVD is also the method of choice for solving most linear least squares problems.

Applications which employ the SVD include computing the pseudoinverse, least squares fitting of data, matrix approximation, and determining the rank, range and null space of a matrix. The SVD is also applied extensively to the study of linear inverse problems, and is useful in the analysis of regularization methods such as that of Tikhonov. It is widely used in statistics where it is related to principal component analysis. Yet another usage is latent semantic indexing in natural language text processing.

Constructor Summary

Constructors

Constructor and Description
SingularValueDecomposition(DenseMatrix A) Returns the singular value decomposition.
SingularValueDecomposition(double[][] A) Constructor.

Method Summary

Modifier and Type	Method and Description
double	condition() Returns the L2 norm condition number, which is max(S) / min(S).
DenseMatrix	getS() Returns the diagonal matrix of singular values
double[]	getSingularValues() Returns the one-dimensional array of singular values, ordered by from largest to smallest.
DenseMatrix	getU() Returns the left singular vectors
DenseMatrix	getV() Returns the right singular vectors
double	norm() Returns the L2 matrix norm.
int	nullity() Returns the dimension of null space.
DenseMatrix	nullspace() Returns a matrix of which columns give an orthonormal basis for the null space.
DenseMatrix	range() Returns a matrix of which columns give an orthonormal basis for the range space.
int	rank() Returns the effective numerical matrix rank.
void	solve(double[][] B, double[][] X) Solve A * X = B using the pseudoinverse of A as obtained by SVD.
void	solve(double[] b, double[] x) Solve A * x = b using the pseudoinverse of A as obtained by SVD.
CholeskyDecomposition	toCholesky() Returns the Cholesky decomposition of A'A.

Methods inherited from class java.lang.Object

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

Constructor Detail

SingularValueDecomposition

public SingularValueDecomposition(double[][] A)

Constructor. The decomposition will be stored in a new create matrix. The input matrix will not be modified.

Parameters:

A - input matrix

SingularValueDecomposition

public SingularValueDecomposition(DenseMatrix A)

Returns the singular value decomposition. Note that the input matrix A will hold U on output.

Parameters:

A - rectangular matrix. Row number should be equal to or larger than column number for current implementation.

Method Detail

getU

public DenseMatrix getU()

Returns the left singular vectors

getV

public DenseMatrix getV()

Returns the right singular vectors

getSingularValues

public double[] getSingularValues()

Returns the one-dimensional array of singular values, ordered by from largest to smallest.

getS

public DenseMatrix getS()

Returns the diagonal matrix of singular values

norm

public double norm()

Returns the L2 matrix norm. The largest singular value.

rank

public int rank()

Returns the effective numerical matrix rank. The number of nonnegligible singular values.

nullity

public int nullity()

Returns the dimension of null space. The number of negligible singular values.

condition

public double condition()

Returns the L₂ norm condition number, which is max(S) / min(S). A system of equations is considered to be well-conditioned if a small change in the coefficient matrix or a small change in the right hand side results in a small change in the solution vector. Otherwise, it is called ill-conditioned. Condition number is defined as the product of the norm of A and the norm of A^-1. If we use the usual L₂ norm on vectors and the associated matrix norm, then the condition number is the ratio of the largest singular value of matrix A to the smallest. Condition number depends on the underlying norm. However, regardless of the norm, it is always greater or equal to 1. If it is close to one, the matrix is well conditioned. If the condition number is large, then the matrix is said to be ill-conditioned. A matrix that is not invertible has the condition number equal to infinity.

range

public DenseMatrix range()

Returns a matrix of which columns give an orthonormal basis for the range space.

nullspace

public DenseMatrix nullspace()

Returns a matrix of which columns give an orthonormal basis for the null space.

toCholesky

public CholeskyDecomposition toCholesky()

Returns the Cholesky decomposition of A'A.

solve

public void solve(double[] b,
                  double[] x)

Solve A * x = b using the pseudoinverse of A as obtained by SVD.

solve

public void solve(double[][] B,
                  double[][] X)

Solve A * X = B using the pseudoinverse of A as obtained by SVD.