com.imsl.math
Class SparseCholesky

java.lang.Object
  com.imsl.math.SparseCholesky

All Implemented Interfaces:

Serializable

public class SparseCholesky
extends Object
implements Serializable

Sparse Cholesky factorization of a matrix of type SparseMatrix.

Class SparseCholesky computes the Cholesky factorization of a sparse symmetric positive definite matrix A. This factorization can then be used to compute the solution of the linear system .

Typically, the solution of a large sparse positive definite system is done in 4 steps:

1. In step one, 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 . 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 , and the permutation .

Class SparseCholesky 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 symmetric 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

static class	SparseCholesky.NotSPDException The matrix is not symmetric, positive definite.
static class	SparseCholesky.NumericFactor The numeric Cholesky factorization of a matrix.
static class	SparseCholesky.SymbolicFactor The symbolic Cholesky factorization of a matrix.

Field Summary

static int	MULTIFRONTAL_METHOD Indicates the multifrontal method will be used for numeric factorization.
static int	STANDARD_METHOD Indicates the method of George/Liu (1981) will be used for numeric factorization.

Constructor Summary

SparseCholesky(SparseMatrix A)
Constructs the matrix structure for the Cholesky factorization of a sparse symmetric positive definite matrix of type SparseMatrix.

Method Summary

void	factorNumerically() Computes the numeric factorization of a sparse real symmetric positive definite matrix.
void	factorSymbolically() Computes the symbolic factorization of a sparse real symmetric 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.
SparseCholesky.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.
SparseCholesky.SymbolicFactor	getSymbolicFactor() Returns the symbolic Cholesky factor.
void	setNumericFactor(SparseCholesky.NumericFactor numericFactor) Sets the numeric Cholesky factor to use in solving of a sparse positive definite system of linear equations .
void	setNumericFactorizationMethod(int method) Defines the method used in the numerical factorization of the permuted input matrix.
void	setSymbolicFactor(SparseCholesky.SymbolicFactor symbolicFactor) Sets the symbolic Cholesky factor to use in solving a sparse positive definite system of linear equations .
double[]	solve(double[] b) Computes the solution of a sparse real symmetric positive definite system of linear equations .

Methods inherited from class java.lang.Object

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

Field Detail

MULTIFRONTAL_METHOD

public static final int MULTIFRONTAL_METHOD

Indicates the multifrontal method will be used for numeric factorization.

See Also:

Constant Field Values

STANDARD_METHOD

public static final int STANDARD_METHOD

Indicates the method of George/Liu (1981) will be used for numeric factorization.

See Also:

Constant Field Values

Constructor Detail

SparseCholesky

public SparseCholesky(SparseMatrix A)

Constructs the matrix structure for the Cholesky factorization of a sparse symmetric positive definite matrix of type SparseMatrix.

Parameters:

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

Method Detail

factorNumerically

public void factorNumerically()
                       throws SparseCholesky.NotSPDException

Computes the numeric factorization of a sparse real symmetric positive definite matrix.

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

1. First, matrix A is permuted to reduce fill-in, leading to a sparse symmetric positive definite matrix .
2. Then, matrix 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:

SparseCholesky.NotSPDException - is thrown if the input matrix is not symmetric, positive definite.

factorSymbolically

public void factorSymbolically()
                        throws SparseCholesky.NotSPDException

Computes the symbolic factorization of a sparse real symmetric positive definite matrix.

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

1. First, matrix A is permuted to reduce fill-in, leading to a sparse symmetric positive definite system .
2. Then, matrix is symbolically factored.

Throws:

SparseCholesky.NotSPDException - is thrown if the input matrix is not symmetric, positive definite.

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.

getNumericFactor

public SparseCholesky.NumericFactor getNumericFactor()

Returns the numeric Cholesky factor.

Returns:

a NumericFactor containing the numeric Cholesky factor.

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.

getSymbolicFactor

public SparseCholesky.SymbolicFactor getSymbolicFactor()

Returns the symbolic Cholesky factor.

Returns:

a SymbolicFactor containing the symbolic Cholesky factor.

setNumericFactor

public void setNumericFactor(SparseCholesky.NumericFactor numericFactor)

Sets the numeric Cholesky factor to use in solving of a sparse positive definite system of linear equations .

Parameters:

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

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.

setSymbolicFactor

public void setSymbolicFactor(SparseCholesky.SymbolicFactor symbolicFactor)

Sets the symbolic Cholesky factor to use in solving a sparse positive definite system of linear equations .

Parameters:

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

solve

public double[] solve(double[] b)
               throws SparseCholesky.NotSPDException

Computes the solution of a sparse real symmetric positive definite system of linear equations .

This method solves the linear system , where A is symmetric positive definite. The solution is obtained in several steps:

1. First, matrix A is permuted to reduce fill-in, leading to a sparse symmetric positive definite system .
2. Then, matrix is symbolically and numerically factored.
3. The final solution is obtained by solving the systems and .

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 for a given or precomputed symbolic or numeric factor.

Parameters:

b - a double vector of length equal to the order of matrix A representing the new right-hand side.

Returns:

a double vector of length equal to the order of matrix A representing the solution to the system of linear equations .

Throws:

SparseCholesky.NotSPDException - is thrown if the input matrix is not symmetric, positive definite.