orbital.math
Class LUDecomposition

java.lang.Object
  orbital.math.LUDecomposition

All Implemented Interfaces:

java.io.Serializable

public final class LUDecomposition
extends java.lang.Object
implements java.io.Serializable

LUDecomposition class, decomposing A into P∙A = L∙U. Solves linear equation systems.

Author:

André Platzer

See Also:

decompose(Matrix), NumericalAlgorithms, Serialized Form

Invariants:

!isInvertible() || getP().multiply(A).equals(getL().multiply(getU()))

Stereotype:

Structure, Wrapper

Note:

this class is more or less just a workaround for returning multiple values.

Constructor Summary

protected

LUDecomposition(Matrix A, Matrix P, boolean sign) Gaussian LU-decomposition implementation.

Method Summary

static LUDecomposition	decompose(Matrix M) Get the Gaussian LU-decomposition of a matrix.
Arithmetic	det() The determinant of A.
Matrix	getL() lower triangular matrix L with diagonal 1s.
Matrix	getP() permutation matrix.
Matrix	getU() upper triangular matrix U.
boolean	isInvertible() A is regular if and only if U is which depends upon whether there is a 0 on the diagonal.
boolean	isRegular() Deprecated. Since Orbital1.1 use isInvertible() instead.
int	linearRank() Rank of the matrix.
Vector	solve(Vector b) Solve linear equation system A∙b.

Methods inherited from class java.lang.Object

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

Constructor Detail

LUDecomposition

protected LUDecomposition(Matrix A,
                          Matrix P,
                          boolean sign)

Gaussian LU-decomposition implementation. Such that P.A = L.U

Preconditions:

A.isSquare()

Method Detail

decompose

public static LUDecomposition decompose(Matrix M)

Get the Gaussian LU-decomposition of a matrix. Such that P∙A = L∙U

Number of multiplications is 1/3*(n³-n)

Preconditions:

M.isSquare()

isInvertible

public boolean isInvertible()
                     throws java.lang.ArithmeticException

A is regular if and only if U is which depends upon whether there is a 0 on the diagonal.

Throws:

java.lang.ArithmeticException

See Also:

Matrix.isInvertible()

isRegular

public boolean isRegular()
                  throws java.lang.ArithmeticException

Deprecated. Since Orbital1.1 use isInvertible() instead.

Throws:

java.lang.ArithmeticException

linearRank

public int linearRank()

Rank of the matrix. i.e. the number of non-zero elements on the diagonal of U.

See Also:

Matrix.linearRank()

det

public Arithmetic det()

The determinant of A.

det A = (-1)^p*det U where p = sign P is the number of permutations in P. Since det(P)*det(A) = det(P∙A) = det(L∙U) = det(L)*det(U) = det(U).

See Also:

Matrix.det()

getL

public Matrix getL()

lower triangular matrix L with diagonal 1s.

Because of pivotising for numberical stability, this matrix only contains values with an absolute ≤1.

getU

public Matrix getU()

upper triangular matrix U.

getP

public Matrix getP()

permutation matrix.

solve

public Vector solve(Vector b)

Solve linear equation system A∙b.

Implementation solves L∙z = P∙b per forward-substitution, and then solves R∙x = z per backward-substitution.

Returns:

x such that A∙x = b.