com.numericalmethod.suanshu.matrix.doubles.factorization.gaussianelimination

Class GaussianElimination

java.lang.Object

com.numericalmethod.suanshu.matrix.doubles.factorization.gaussianelimination.GaussianElimination

All Implemented Interfaces:

java.io.Serializable

public class GaussianElimination
extends java.lang.Object
implements java.io.Serializable

The Gaussian elimination performs elementary row operations to reduce a matrix to the row echelon form. The two elementary row operations are: swapping rows, and adding multiples of a row to another row. This is equivalent to multiplying the original matrix with invertible matrices from the left. For a square matrix, this algorithm essentially computes an LU decomposition. We have,

 T * A == U
 

where T is the transformation matrix, U is in the row echelon form. And,

 P * A == L * U
 

where P is the permutation matrix, U is lower triangular, U is in the row echelon form.

See Also:

• Wikipedia: Gaussian elimination
• Wikipedia: Row echelon form

, Serialized Form

Constructor Summary

Constructors

Constructor and Description
GaussianElimination(Matrix A) Run the Gaussian elimination algorithm with partial pivoting.
GaussianElimination(Matrix A, boolean usePivoting, double epsilon) Run the Gaussian elimination algorithm.

Method Summary

Modifier and Type	Method and Description
Matrix	L() Get the lower triangular matrix L, such that P * A = L * U.
PermutationMatrix	P() Get the permutation matrix, P, such that P * A = L * U.
Matrix	T() Get the transformation matrix, T, such that T * A = U.
Matrix	U() Get the upper triangular matrix, U, such that T * A = U and P * A = L * U.

Methods inherited from class java.lang.Object

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

Constructor Detail

GaussianElimination

public GaussianElimination(Matrix A,
                           boolean usePivoting,
                           double epsilon)

Run the Gaussian elimination algorithm.

Parameters:

A - a matrix

usePivoting - true if to use partial pivoting, e.g., for numerical stability. In general, no pivoting means no row interchanges. It can be done only if Gaussian elimination never runs into zeros on the diagonal. Since division by zero is a fatal error we usually avoid no pivoting.

epsilon - a precision parameter: when a number |x| ≤ ε, it is considered 0

GaussianElimination

public GaussianElimination(Matrix A)

Run the Gaussian elimination algorithm with partial pivoting.

Parameters:

A - a matrix

Method Detail

T

public Matrix T()

Get the transformation matrix, T, such that T * A = U.

Returns:

T

U

public Matrix U()

Get the upper triangular matrix, U, such that T * A = U and P * A = L * U.

Returns:

U

L

public Matrix L()

Get the lower triangular matrix L, such that P * A = L * U.

Returns:

L

P

public PermutationMatrix P()

Get the permutation matrix, P, such that P * A = L * U.

Returns:

P