Package com.imsl.math

Class ComplexLU

java.lang.Object

com.imsl.math.ComplexLU

All Implemented Interfaces:

Serializable, Cloneable

public class ComplexLU
extends Object
implements Serializable, Cloneable

LU factorization of a matrix of type Complex.

ComplexLU performs an LU factorization of a complex general coefficient matrix. ComplexLU's method condition estimates the reciprocal of the \(L_1\) condition number of the matrix. The LU factorization is done using scaled partial pivoting. Scaled partial pivoting differs from partial pivoting in that the pivoting strategy is the same as if each row were scaled to have the same infinity norm.

The \(L_1\) condition number of the matrix A is defined to be \(\kappa \left( A \right) = \left\| A \right\|_1 \left\| {A ^{-1}} \right\|_1\). Since it is expensive to compute \(\left\| {A^{-1}} \right\|_1\), the condition number is only estimated. The estimation algorithm is the same as used by LINPACK and is described by Cline et al. (1979).

Note that A is not retained for use by other methods of this class, only the factorization of A is retained. Thus, A is a required parameter to the condition method.

An estimated condition number greater than \(1/\epsilon\) (where \(\epsilon\) is machine precision) indicates that very small changes in A can cause very large changes in the solution x. Iterative refinement can sometimes find the solution to such a system. If there is conern about the input matrix being ill-conditioned, the user of this class should check the condition number of the input matrix using the condition method before using one of the other class methods.

ComplexLU fails if U, the upper triangular part of the factorization, has a zero diagonal element. This can occur only if A either is singular or is very close to a singular matrix.

The solve method can be used to solve systems of equations. The method determinant can be called to compute the determinant of the coefficient matrix.

ComplexLU is based on the LINPACK routine CGECO; see Dongarra et al. (1979). CGECO uses unscaled partial pivoting.

See Also:

• Example
• Serialized Form

Field Summary

Fields

Modifier and Type	Field	Description
protected Complex[][]	factor	This is an n by n Complex matrix containing the LU factorization of the matrix A.
protected int[]	ipvt	Vector of length n containing the pivot sequence for the factorization.

Constructor Summary

Constructors

Constructor	Description
ComplexLU(Complex[][] a)	Creates the LU factorization of a square matrix of type Complex.

Method Summary

Modifier and Type	Method	Description
double	condition(Complex[][] a)	Return an estimate of the reciprocal of the \(L_1\) condition number.
Complex	determinant()	Return the determinant of the matrix used to construct this instance.
Complex[][]	getL()	Returns the lower triangular portion of the LU factorization of A.
Complex[][]	getPermutationMatrix()	Returns the permutation matrix which results from the LU factorization of A.
Complex[][]	getU()	Returns the unit upper triangular portion of the LU factorization of A.
Complex[][]	inverse()	Returns the inverse of the matrix used to construct this instance.
Complex[]	solve(Complex[] b)	Return the solution x of the linear system Ax = b using the LU factorization of A.
static Complex[]	solve(Complex[][] a, Complex[] b)	Solve Ax = b for x using the LU factorization of A.
Complex[]	solveTranspose(Complex[] b)	Return the solution x of the linear system \(A^T x = b\).

Methods inherited from class java.lang.Object

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

Field Details

factor

protected Complex[][] factor

This is an n by n Complex matrix containing the LU factorization of the matrix A.

ipvt

protected int[] ipvt

Vector of length n containing the pivot sequence for the factorization.

Constructor Details

ComplexLU

public ComplexLU(Complex[][] a)
          throws SingularMatrixException

Creates the LU factorization of a square matrix of type Complex.

Parameters:

a - Complex square matrix to be factored

Throws:

IllegalArgumentException - is thrown when the row lengths of input matrix are not equal (for example, the matrix edges are "jagged".)

SingularMatrixException - is thrown when the input matrix is singular.

Method Details

getL

public Complex[][] getL()

Returns the lower triangular portion of the LU factorization of A.

Scaled partial pivoting is used to achieve the LU factorization. The resulting factorization is such that \(AP = LU\), where A is the input matrix a, P is the permutation matrix returned by getPermutationMatrix, L is the lower triangular matrix returned by getL, and U is the unit upper triangular matrix returned by getU.

Returns:

a Complex matrix containing L, the lower triangular portion of the LU factorization of A.

getU

public Complex[][] getU()

Returns the unit upper triangular portion of the LU factorization of A.

Scaled partial pivoting is used to achieve the LU factorization. The resulting factorization is such that \(AP = LU\), where A is the input matrix a, P is the permutation matrix returned by getPermutationMatrix, L is the lower triangular matrix returned by getL, and U is the unit upper triangular matrix returned by getU.

Returns:

a Complex matrix containing U, the unit upper triangular portion of the LU factorization of A.

getPermutationMatrix

public Complex[][] getPermutationMatrix()

Returns the permutation matrix which results from the LU factorization of A.

Scaled partial pivoting is used to achieve the LU factorization. The resulting factorization is such that \(AP = LU\), where A is the input matrix a, P is the permutation matrix returned by getPermutationMatrix, L is the lower triangular matrix returned by getL, and U is the unit upper triangular matrix returned by getU.

Returns:

A Complex matrix containing the permuted identity matrix as a result of the LU factorization of A.

solve

public Complex[] solve(Complex[] b)

Return the solution x of the linear system Ax = b using the LU factorization of A.

Parameters:

b - Complex array containing the right-hand side of the linear system

Returns:

Complex array containing the solution to the linear system of equations

solveTranspose

public Complex[] solveTranspose(Complex[] b)

Return the solution x of the linear system \(A^T x = b\).

Parameters:

b - Complex array containing the right-hand side of the linear system

Returns:

Complex array containing the solution to the linear system of equations

determinant

public Complex determinant()

Return the determinant of the matrix used to construct this instance.

Returns:

a Complex scalar containing the determinant of the matrix used to construct this instance

solve

public static Complex[] solve(Complex[][] a,
 Complex[] b)
                       throws SingularMatrixException

Solve Ax = b for x using the LU factorization of A.

Parameters:

a - a Complex square matrix

b - a Complex column vector

Returns:

a Complex column vector containing the solution to the linear system of equations.

Throws:

IllegalArgumentException - This exception is thrown when (1) the lengths of the rows of the input matrix are not uniform, and (2) the number of rows in the input matrix is not equal to the number of elements in x.

SingularMatrixException - is thrown when the matrix is singular.

inverse

public Complex[][] inverse()

Returns the inverse of the matrix used to construct this instance.

Returns:

a Complex matrix containing the inverse of the matrix used to construct this object.

condition

public double condition(Complex[][] a)

Return an estimate of the reciprocal of the \(L_1\) condition number.

Parameters:

a - a Complex matrix

Returns:

a double scalar value representing the estimate of the reciprocal of the \(L_1\) condition number of the matrix A, where A is the parameter a