smile.math.matrix

Class FloatMatrix.SVD

java.lang.Object

smile.math.matrix.FloatMatrix.SVD

All Implemented Interfaces:

java.io.Serializable

Enclosing class:

FloatMatrix

public static class FloatMatrix.SVD
extends java.lang.Object
implements java.io.Serializable

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 decomposition 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 pseudo-inverse, 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.

See Also:

Serialized Form

Field Summary

Fields

Modifier and Type	Field and Description
int	m The number of rows of matrix.
int	n The number of columns of matrix.
float[]	s The singular values in descending order.
FloatMatrix	U The left singular vectors U.
FloatMatrix	V The right singular vectors V.

Constructor Summary

Constructors

Constructor and Description
SVD(float[] s, FloatMatrix U, FloatMatrix V) Constructor.
SVD(int m, int n, float[] s) Constructor.

Method Summary

Modifier and Type	Method and Description
float	condition() Returns the L2 norm condition number, which is max(S) / min(S).
FloatMatrix	diag() Returns the diagonal matrix of singular values.
double	norm() Returns the L2 matrix norm.
int	nullity() Returns the dimension of null space.
FloatMatrix	nullspace() Returns the matrix which columns are the orthonormal basis for the null space.
FloatMatrix	pinv() Returns the pseudo inverse.
FloatMatrix	range() Returns the matrix which columns are the orthonormal basis for the range space.
int	rank() Returns the effective numerical matrix rank.
float[]	solve(float[] b) Solves the least squares min || B - A*X ||.

Methods inherited from class java.lang.Object

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

Field Detail

m

public final int m

The number of rows of matrix.

n

public final int n

The number of columns of matrix.

s

public final float[] s

The singular values in descending order.

U

public final FloatMatrix U

The left singular vectors U.

V

public final FloatMatrix V

The right singular vectors V.

Constructor Detail

SVD

public SVD(int m,
           int n,
           float[] s)

Constructor.

SVD

public SVD(float[] s,
           FloatMatrix U,
           FloatMatrix V)

Constructor.

Method Detail

diag

public FloatMatrix diag()

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 non-negligible singular values.

nullity

public int nullity()

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

condition

public float 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. The 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 FloatMatrix range()

Returns the matrix which columns are the orthonormal basis for the range space. Returns null if the rank is zero (if and only if zero matrix).

nullspace

public FloatMatrix nullspace()

Returns the matrix which columns are the orthonormal basis for the null space. Returns null if the matrix is of full rank.

pinv

public FloatMatrix pinv()

Returns the pseudo inverse.

solve

public float[] solve(float[] b)

Solves the least squares min || B - A*X ||.

Parameters:

b - the right hand side of overdetermined linear system.

Returns:

the solution vector beta that minimizes ||Y - X*beta||.

Throws:

java.lang.RuntimeException - if matrix is rank deficient.