com.numericalmethod.suanshu.algebra.linear.matrix.doubles.operation

Class PseudoInverse

java.lang.Object

com.numericalmethod.suanshu.algebra.linear.matrix.doubles.matrixtype.dense.DenseMatrix

com.numericalmethod.suanshu.algebra.linear.matrix.doubles.operation.PseudoInverse

All Implemented Interfaces:

Matrix, MatrixAccess, MatrixRing, MatrixTable, Densifiable, AbelianGroup<Matrix>, Monoid<Matrix>, Ring<Matrix>, Table, DeepCopyable

public class PseudoInverse
extends DenseMatrix

The Moore-Penrose pseudo-inverse of an m x n matrix A is A⁺. It is a generalization of the inverse matrix. This implementation uses the Singular Value decomposition to compute the pseudo-inverse. Specifically,

 A = U * D * V'
 A+ = V * D+ * U'
 

The properties are

• A⁺ has the dimension of n x m.
• A * A⁺ * A = A
• A⁺ * A * A⁺ = A⁺
• (A * A⁺)' = A * A⁺
• (A⁺ * A)' = A⁺ * A

When A is invertible, its pseudo-inverse coincides with its inverse.

See Also:

SVD, Wikipedia: Moore-Penrose pseudo-inverse

Constructor Summary

Constructors

Constructor and Description
PseudoInverse(Matrix A) Construct the Moore-Penrose pseudo-inverse matrix of A.
PseudoInverse(Matrix A, double epsilon) Construct the Moore-Penrose pseudo-inverse matrix of a matrix.

Method Summary

Methods inherited from class com.numericalmethod.suanshu.algebra.linear.matrix.doubles.matrixtype.dense.DenseMatrix

add, deepCopy, equals, get, getColumn, getColumn, getRow, getRow, hashCode, minus, multiply, multiply, nCols, nRows, ONE, opposite, scaled, set, setColumn, setRow, t, toDense, toString, ZERO

Methods inherited from class java.lang.Object

clone, finalize, getClass, notify, notifyAll, wait, wait, wait

Constructor Detail

PseudoInverse

public PseudoInverse(Matrix A,
                     double epsilon)

Construct the Moore-Penrose pseudo-inverse matrix of a matrix.

Parameters:

A - an m x n matrix

epsilon - a precision parameter: when a number |x| ≤ ε, it is considered 0. This threshold truncates negligible singular values less than the smaller of ε and (t = machine ε * max(m,n) * max(D)).

See Also:

Wikipedia: The general case and the SVD method

PseudoInverse

public PseudoInverse(Matrix A)

Construct the Moore-Penrose pseudo-inverse matrix of A.

Parameters:

A - an m x n matrix