com.numericalmethod.suanshu.matrix.doubles.operation
Class PseudoInverse

java.lang.Object
  com.numericalmethod.suanshu.matrix.doubles.matrixtype.MatrixMathImpl<T>
      com.numericalmethod.suanshu.matrix.doubles.matrixtype.MatrixStorageImpl<DenseMatrix>
          com.numericalmethod.suanshu.matrix.doubles.matrixtype.dense.DenseMatrix
              com.numericalmethod.suanshu.matrix.doubles.operation.PseudoInverse

All Implemented Interfaces:

DeepCopyable, AbelianGroup<Matrix>, Monoid<Matrix>, Ring<Matrix>, Matrix, MatrixAccessor, MatrixRing, Densifiable, MatrixDimension

public class PseudoInverse
extends DenseMatrix

The Moore–Penrose pseudoinverse of an m x n matrix A is A+. It is a generalization of the inverse matrix.

To compute the pseudoinverse of A, we use the Singular Value decomposition. Specifically,

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

It satisfies

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

For an invertible A, its pseudoinverse coincides with its inverse.

See Also:

SVD, Wikipedia: Moore–Penrose pseudoinverse

Field Summary

double

precision the precision used to truncate the negligible singular values

Constructor Summary

PseudoInverse(Matrix A)
Construct the Moore–Penrose pseudoinverse matrix of A.

PseudoInverse(Matrix A, double epsilon)
Construct the Moore–Penrose pseudoinverse matrix of A.

Method Summary

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

add, deepCopy, getColumn, getColumn, getMatrixData, getRow, getRow, getSample, minus, multiply, multiply, ONE, overwrite, scaled, t, toDense, ZERO

Methods inherited from class com.numericalmethod.suanshu.matrix.doubles.matrixtype.MatrixStorageImpl

equals, get, hashCode, set, setMatrixData

Methods inherited from class com.numericalmethod.suanshu.matrix.doubles.matrixtype.MatrixMathImpl

add, call, minus, multiply, nCols, nRows, opposite, setColumn, setColumn, setRow, setRow, toString

Methods inherited from class java.lang.Object

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

Field Detail

precision

public final double precision

the precision used to truncate the negligible singular values

Constructor Detail

PseudoInverse

public PseudoInverse(Matrix A,
                     double epsilon)

Construct the Moore–Penrose pseudoinverse matrix of A.

Parameters:

A - a m x n matrix

epsilon - a precision parameter: when a number |x| ≤ ε, it is considered 0. The threshold to truncate negligible singular values are 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 pseudoinverse matrix of A.

Parameters:

A - an m x n matrix