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.

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 LES 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 LES 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.