com.imsl.math

Class ComplexSparseCholesky

java.lang.Object

com.imsl.math.ComplexSparseCholesky

All Implemented Interfaces:

Serializable

public class ComplexSparseCholesky
extends Object
implements Serializable

Sparse Cholesky factorization of a matrix of type ComplexSparseMatrix.

Class ComplexSparseCholesky computes the Cholesky factorization of a sparse Hermitian positive definite matrix A. This factorization can then be used to compute the solution of the linear system \(Ax = b\).

Typically, the solution of a large sparse positive definite system \(Ax = b\) is done in four steps.

1. In the first step, an ordering algorithm is used to preserve sparsity in the Cholesky factor L of matrix A during the numerical factorization process. The new order can be described by a permutation matrix P.
2. Step two consists of setting up the data structure for the Cholesky factor L, where \(PAP^T=LL^T\). This step is called the symbolic factorization phase of the computation. During symbolic factorization, only the sparsity pattern of sparse matrix A, i.e., the locations of the nonzero entries of matrix A are needed but not any of the elements themselves.
3. In step 3, the numerical factorization phase, the Cholesky factorization is done numerically.
4. Step 4 is the solution phase. Here, the numerical solution, x, to the original system is obtained by solving the two triangular systems \(Ly_1=Pb\), \(L^Ty_2=y_1\) and the permutation \(x=P^Ty_2\).

Class ComplexSparseCholesky realizes all four steps by algorithms described in George and Liu (1981). Especially, step one, is a realization of a minimum degree ordering algorithm. The numerical factorization in its standard form is based on a sparse compressed storage scheme. Alternatively, a multifrontal method can be used. The multifrontal method requires more storage but will be faster than the standard method in certain cases. The multifrontal method is based on the routines in Liu (1987). For a detailed description of this method, see Liu (1990), also Duff and Reid(1983, 1984), Ashcraft (1987) et al. (1987), and Liu (1986, 1989, 1992). The numerical factorization method can be specified by using the setNumericFactorizationMethod.

The solve method will compute the symbolic and numeric factorizations if they have not already been computed or supplied by the user through the factorSymbolically , factorNumerically, setNumericFactor, or setSymbolicFactor methods. These factorizations are retained for later use by the solve method when different right-hand sides are to be solved.

There is a special situation where computations can be simplified. If an application generates different sparse Hermitian positive definite coefficient matrices that all have the same sparsity pattern, then by using methods getSymbolicFactor and setSymbolicFactor the symbolic factorization needs only be computed once.

See Also:

Example, Serialized Form

Nested Class Summary

Nested Classes

Modifier and Type	Class and Description
static class	ComplexSparseCholesky.NotSPDException The matrix is not Hermitian, positive definite.
static class	ComplexSparseCholesky.NumericFactor Data structures and functions for the numeric Cholesky factor.
static class	ComplexSparseCholesky.SymbolicFactor Data structures and functions for the symbolic Cholesky factor.

Field Summary

Fields

Modifier and Type	Field and Description
static int	MULTIFRONTAL_METHOD Indicates the multifrontal method will be used for numeric factorization.
static int	STANDARD_METHOD Indicates that the method of George/Liu (1981) is used for numeric factorization.

Constructor Summary

Constructors

Constructor and Description
ComplexSparseCholesky(ComplexSparseMatrix A) Constructs the matrix structure for the Cholesky factorization of a sparse Hermitian positive definite matrix of type ComplexSparseMatrix.

Method Summary

Modifier and Type	Method and Description
void	factorNumerically() Computes the numeric factorization of a sparse Hermitian positive definite matrix.
void	factorSymbolically() Computes the symbolic factorization of a sparse Hermitian positive definite matrix.
double	getLargestDiagonalElement() Returns the largest diagonal element of the Cholesky factor.
long	getNumberOfNonzeros() Returns the number of nonzeros in the Cholesky factor.
ComplexSparseCholesky.NumericFactor	getNumericFactor() Returns the numeric Cholesky factor.
int	getNumericFactorizationMethod() Returns the method used in the numerical factorization of the permuted input matrix.
double	getSmallestDiagonalElement() Returns the smallest diagonal element of the Cholesky factor.
ComplexSparseCholesky.SymbolicFactor	getSymbolicFactor() Returns the symbolic Cholesky factor.
void	setNumericFactor(ComplexSparseCholesky.NumericFactor numericFactor) Sets the numeric Cholesky factor to use in solving a sparse complex Hermitian positive definite system of linear equations \(Ax=b\).
void	setNumericFactorizationMethod(int method) Defines the method used in the numerical factorization of the permuted input matrix.
void	setSymbolicFactor(ComplexSparseCholesky.SymbolicFactor symbolicFactor) Sets the symbolic Cholesky factor to use in solving a sparse complex Hermitian positive definite system of linear equations \(Ax=b\).
Complex[]	solve(Complex[] b) Computes the solution of a sparse Hermitian positive definite system of linear equations \(Ax=b\).

Methods inherited from class java.lang.Object

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

Field Detail

STANDARD_METHOD

public static final int STANDARD_METHOD

Indicates that the method of George/Liu (1981) is used for numeric factorization.

See Also:

Constant Field Values

MULTIFRONTAL_METHOD

public static final int MULTIFRONTAL_METHOD

Indicates the multifrontal method will be used for numeric factorization.

See Also:

Constant Field Values

Constructor Detail

ComplexSparseCholesky

public ComplexSparseCholesky(ComplexSparseMatrix A)

Constructs the matrix structure for the Cholesky factorization of a sparse Hermitian positive definite matrix of type ComplexSparseMatrix.

Parameters:

A - The ComplexSparseMatrix Hermitian positive definite matrix to be factored. Only the lower triangular part of the input matrix is used.

Method Detail

solve

public Complex[] solve(Complex[] b)
                throws ComplexSparseCholesky.NotSPDException

Computes the solution of a sparse Hermitian positive definite system of linear equations \(Ax=b\).

This method solves the linear system \(Ax=b\), where A is Hermitian positive definite. The solution is obtained in several steps:

1. First, matrix A is permuted to reduce fill-in, leading to a sparse Hermitian positive definite system \(PAP^T=Pb\).
2. Then, matrix \(PAP^T\) is symbolically and numerically factored.
3. The final solution is obtained by solving the systems \(Ly_1=Pb, L^Ty_2=y_1\) and \(x=P^Ty_2\).

By default this method implements all of the above steps. The factorizations are retained for later use by subsequent solves. By choosing appropriate methods within this class, the computation can be reduced to the solution of the system \(Ax=b\) for a given or precomputed symbolic or numeric factor.

Parameters:

b - A Complex vector of length equal to the order of A containing the right-hand side.

Returns:

a Complex vector of length equal to the order of matrix A containing the solution of the system \(Ax=b\).

Throws:

ComplexSparseCholesky.NotSPDException - is thrown if the input matrix is not Hermitian, positive definite.

factorSymbolically

public void factorSymbolically()
                        throws ComplexSparseCholesky.NotSPDException

Computes the symbolic factorization of a sparse Hermitian positive definite matrix.

This method symbolically factors the instance of the constructed matrix A, where A is of type ComplexSparseMatrix and is Hermitian positive definite. The factorization is obtained in several steps:

1. First, matrix A is permuted to reduce fill-in, leading to a sparse Hermitian positive definite matrix \(PAP^T\).
2. Then, matrix \(PAP^T\) is symbolically factored.

Throws:

ComplexSparseCholesky.NotSPDException - is thrown if the input matrix is not Hermitian, positive definite.

factorNumerically

public void factorNumerically()
                       throws ComplexSparseCholesky.NotSPDException

Computes the numeric factorization of a sparse Hermitian positive definite matrix.

This method numerically factors the instance of the constructed matrix A, where A is of type ComplexSparseMatrix and is Hermitian positive definite. The factorization is obtained in several steps:

1. First, matrix A is permuted to reduce fill-in, leading to a sparse Hermitian positive definite matrix \(PAP^T\).
2. Then, matrix \(PAP^T\) is symbolically and numerically factored.

Note that the symbolic factorization is not done if the symbolic factor has been supplied by the user through the setSymbolicFactor method.

Throws:

ComplexSparseCholesky.NotSPDException - is thrown if the input matrix is not Hermitian, positive definite.

setSymbolicFactor

public void setSymbolicFactor(ComplexSparseCholesky.SymbolicFactor symbolicFactor)

Sets the symbolic Cholesky factor to use in solving a sparse complex Hermitian positive definite system of linear equations \(Ax=b\).

Parameters:

symbolicFactor - a SymbolicFactor containing the symbolic Cholesky factor. By default the symbolic factorization is computed.

getSymbolicFactor

public ComplexSparseCholesky.SymbolicFactor getSymbolicFactor()

Returns the symbolic Cholesky factor.

Returns:

a SymbolicFactor containing the symbolic Cholesky factor.

setNumericFactor

public void setNumericFactor(ComplexSparseCholesky.NumericFactor numericFactor)

Sets the numeric Cholesky factor to use in solving a sparse complex Hermitian positive definite system of linear equations \(Ax=b\).

Parameters:

numericFactor - a NumericFactor object containing the numeric Cholesky factor. By default the numeric factorization is computed.

getNumericFactor

public ComplexSparseCholesky.NumericFactor getNumericFactor()

Returns the numeric Cholesky factor.

Returns:

a NumericFactor containing the numeric Cholesky factor.

setNumericFactorizationMethod

public void setNumericFactorizationMethod(int method)

Defines the method used in the numerical factorization of the permuted input matrix.

Parameters:

method - an int value specifying the method to choose:

Method Name	Description
STANDARD_METHOD	standard method as described by George/Liu (1981). This is the default.
MULTIFRONTAL_METHOD	multifrontal method

Throws:

IllegalArgumentException - This exception is thrown when the value for method is not STANDARD_METHOD or MULTIFRONTAL_METHOD.

getNumericFactorizationMethod

public int getNumericFactorizationMethod()

Returns the method used in the numerical factorization of the permuted input matrix.

Returns:

an int value equal to STANDARD_METHOD = 0 or MULTIFRONTAL_METHOD = 1 representing the method used in the numeric factorization of the permuted input matrix. See setNumericFactorizationMethod for more details.

getSmallestDiagonalElement

public double getSmallestDiagonalElement()

Returns the smallest diagonal element of the Cholesky factor.

Returns:

a double value specifying the smallest diagonal element of the Cholesky factor. Use of this method is only sensible if a numeric factorization of the input matrix was done beforehand.

getLargestDiagonalElement

public double getLargestDiagonalElement()

Returns the largest diagonal element of the Cholesky factor.

Returns:

a double value specifying the largest diagonal element of the Cholesky factor. Use of this method is only sensible if a numeric factorization of the input matrix was done beforehand.

getNumberOfNonzeros

public long getNumberOfNonzeros()

Returns the number of nonzeros in the Cholesky factor.

Returns:

a long value specifying the number of nonzeros (including the diagonal) of the Cholesky factor.