com.imsl.math
Class SuperLU

java.lang.Object
  com.imsl.math.SuperLU

All Implemented Interfaces:

Serializable

public class SuperLU
extends Object
implements Serializable

Computes the LU factorization of a general sparse matrix of type SparseMatrix by a column method and solves the real sparse linear system of equations .

Consider the sparse linear system of equations

Here, A is a general square, nonsingular, n by n sparse matrix, and x and b are vectors of length n. All entries in A, x and b are of type double.

Gaussian elimination, applied to the system above, can be shortly described as follows:
1. Compute a triangular factorization . Here, and are positive definite diagonal matrices to equilibrate the system and and are permutation matrices to ensure numerical stability and preserve sparsity. L is a unit lower triangular matrix and U is an upper triangular matrix.

2. Solve by evaluating

This is done efficiently by multiplying from right to left in the last expression: Scale the rows of b by . Multiplying means permuting the rows of . Multiplying means solving the triangular system of equations with matrix L by substitution. Similarly, multiplying means solving the triangular system with U.

Class SuperLU handles step 1 above in the solve method if it has not been computed prior to step 2. More precisely, before is solved the following steps are performed:

1. Equilibrate matrix A, i.e. compute diagonal matrices and so that is "better conditioned" than A, i.e. is less sensitive to perturbations in than is to perturbations in A.
2. Order the columns of to increase the sparsity of the computed L and U factors, i.e. replace by where is a column permutation matrix.
3. Compute the LU factorization of . For numerical stability, the rows of are eventually permuted through the factorization process by scaled partial pivoting, leading to the decomposition . The LU factorization is done by a left looking supernode-panel algorithm with 2-D blocking. See Demmel, Eisenstat, Gilbert et al. (1999) for further information on this technique.
4. Compute the reciprocal pivot growth factor

   

   where and denote the j-th column of matrices and U, respectively.
5. Estimate the reciprocal of the condition number of matrix .

Method solve uses this information to perform the following steps:

1. Solve the system using the computed triangular factors.
2. Iteratively refine the solution, again using the computed triangular factors. This is equivalent to Newton's method.
3. Compute forward and backward error bounds for the solution vector x.

Some of the steps mentioned above are optional. Their settings can be controlled by the set methods of class SuperLU.

Class SuperLU is based on the SuperLU code written by Demmel, Gilbert, Li et al. For more detailed explanations of the factorization and solve steps, see the SuperLU Users' Guide (1999).

Copyright (c) 2003, The Regents of the University of California, through Lawrence Berkeley National Laboratory (subject to receipt of any required approvals from U.S. Dept. of Energy)

All rights reserved.

Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:

(1) Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.

(2) Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.

(3) Neither the name of Lawrence Berkeley National Laboratory, U.S. Dept. of Energy nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission.

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Example, Serialized Form

Field Summary

static int	COLUMN_APPROXIMATE_MINIMUM_DEGREE For column ordering, use column approximate minimum degree ordering.
static int	COLUMN_SCALING Indicates that input matrix A was column scaled before factorization.
static int	FILL_FACTOR A performance tuning parameter which can be adjusted via method setPerformanceTuningParameters.
static int	MAXIMUM_SUPERNODE_SIZE A performance tuning parameter which can be adjusted via method setPerformanceTuningParameters.
static int	MINIMUM_COLUMN_DIMENSION A performance tuning parameter which can be adjusted via method setPerformanceTuningParameters.
static int	MINIMUM_DEGREE_AT_A For column ordering, use minimum degree ordering on the structure of .
static int	MINIMUM_DEGREE_AT_PLUS_A For column ordering, use minimum degree ordering on the structure of .
static int	MINIMUM_ROW_DIMENSION A performance tuning parameter which can be adjusted via method setPerformanceTuningParameters.
static int	NATURAL_ORDERING For column ordering, use the natural ordering.
static int	NO_SCALING Indicates that input matrix A was not equilibrated before factorization.
static int	PANEL_SIZE A performance tuning parameter which can be adjusted via method setPerformanceTuningParameters.
static int	RELAXATION_PARAMETER A performance tuning parameter which can be adjusted via method setPerformanceTuningParameters.
static int	ROW_AND_COLUMN_SCALING Indicates that input matrix A was row and column scaled before factorization.
static int	ROW_SCALING Indicates that input matrix A was row scaled before factorization.

Constructor Summary

SuperLU(SparseMatrix A)
Constructor for SuperLU.

Method Summary

int	getColumnPermutationMethod() Returns the method that will be used to permute the columns of the input matrix.
double	getConditionNumber() Returns the estimate of the reciprocal condition number of the matrix A.
double	getDiagonalPivotThreshold() Returns the threshold used for a diagonal entry to be an acceptable pivot.
boolean	getEquilibrate() Returns the equilibration flag.
int	getEquilibrationMethod() Returns information on the type of equilibration used before matrix factorization.
double	getForwardErrorBound() Returns the estimated forward error bound for the solution vector.
boolean	getIterativeRefinement() Returns a value specifying whether iterative refinement is to be performed or not.
int	getPerformanceTuningParameters(int parameter) Returns a performance tuning parameter value.
boolean	getPivotGrowth() Returns the reciprocal pivot growth factor flag.
double	getReciprocalPivotGrowthFactor() Returns the reciprocal pivot growth factor.
double	getRelativeBackwardError() Returns the componentwise relative backward error of the solution vector.
boolean	getSymmetricMode() Returns the symmetric mode flag.
void	setColumnPermutationMethod(int colpermute) Specifies how to permute the columns of the input matrix.
void	setDiagonalPivotThreshold(double thresh) Specifies the threshold used for a diagonal entry to be an acceptable pivot.
void	setEquilibrate(boolean equilibrate) Determines if input matrix A should be equilibrated before factorization.
void	setIterativeRefinement(boolean refine) Specifies whether to perform iterative refinement.
void	setPerformanceTuningParameters(int parameter, int tunedValue) Sets performance tuning parameters.
void	setPivotGrowth(boolean growth) Specifies whether to compute the reciprocal pivot growth factor.
void	setSymmetricMode(boolean symmetric) Specifies whether to use the symmetric mode.
double[]	solve(double[] b) Computation of the solution vector for the system .
double[]	solveTranspose(double[] b) Computation of the solution vector for the system .

Methods inherited from class java.lang.Object

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

Field Detail

COLUMN_APPROXIMATE_MINIMUM_DEGREE

public static final int COLUMN_APPROXIMATE_MINIMUM_DEGREE

For column ordering, use column approximate minimum degree ordering.

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Constant Field Values

COLUMN_SCALING

public static final int COLUMN_SCALING

Indicates that input matrix A was column scaled before factorization. This is a return value for getEquilibrationMethod.

See Also:

Constant Field Values

FILL_FACTOR

public static final int FILL_FACTOR

A performance tuning parameter which can be adjusted via method setPerformanceTuningParameters.

See Also:

Constant Field Values

MAXIMUM_SUPERNODE_SIZE

public static final int MAXIMUM_SUPERNODE_SIZE

A performance tuning parameter which can be adjusted via method setPerformanceTuningParameters.

See Also:

Constant Field Values

MINIMUM_COLUMN_DIMENSION

public static final int MINIMUM_COLUMN_DIMENSION

A performance tuning parameter which can be adjusted via method setPerformanceTuningParameters.

See Also:

Constant Field Values

MINIMUM_DEGREE_AT_A

public static final int MINIMUM_DEGREE_AT_A

For column ordering, use minimum degree ordering on the structure of .

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Constant Field Values

MINIMUM_DEGREE_AT_PLUS_A

public static final int MINIMUM_DEGREE_AT_PLUS_A

For column ordering, use minimum degree ordering on the structure of .

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Constant Field Values

MINIMUM_ROW_DIMENSION

public static final int MINIMUM_ROW_DIMENSION

A performance tuning parameter which can be adjusted via method setPerformanceTuningParameters.

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Constant Field Values

NATURAL_ORDERING

public static final int NATURAL_ORDERING

For column ordering, use the natural ordering.

See Also:

Constant Field Values

NO_SCALING

public static final int NO_SCALING

Indicates that input matrix A was not equilibrated before factorization. This is a return value for getEquilibrationMethod.

See Also:

Constant Field Values

PANEL_SIZE

public static final int PANEL_SIZE

A performance tuning parameter which can be adjusted via method setPerformanceTuningParameters.

See Also:

Constant Field Values

RELAXATION_PARAMETER

public static final int RELAXATION_PARAMETER

A performance tuning parameter which can be adjusted via method setPerformanceTuningParameters.

See Also:

Constant Field Values

ROW_AND_COLUMN_SCALING

public static final int ROW_AND_COLUMN_SCALING

Indicates that input matrix A was row and column scaled before factorization. This is a return value for getEquilibrationMethod.

See Also:

Constant Field Values

ROW_SCALING

public static final int ROW_SCALING

Indicates that input matrix A was row scaled before factorization. This is a return value for getEquilibrationMethod.

See Also:

Constant Field Values

Constructor Detail

SuperLU

public SuperLU(SparseMatrix A)

Constructor for SuperLU.

Parameters:

A - a SparseMatrix containing the sparse quadratic input matrix.

Method Detail

getColumnPermutationMethod

public int getColumnPermutationMethod()

Returns the method that will be used to permute the columns of the input matrix.

Returns:

an int scalar specifying how the columns of the input matrix are to be permuted for sparsity preservation.

return value	method
0 = NATURAL_ORDERING	natural ordering, that is , I the identity matrix.
1 = MINIMUM_DEGREE_AT_PLUS_A	minimum degree ordering on the structure of
2 = MINIMUM_DEGREE_AT_A	minimum degree ordering on the structure of
3 = COLUMN_APPROXIMATE_MINIMUM_DEGREE	column approximate minimum degree ordering

getConditionNumber

public double getConditionNumber()
                          throws SingularMatrixException

Returns the estimate of the reciprocal condition number of the matrix A.

Returns:

a double scalar containing the reciprocal condition number of the matrix A after equilibration and permutation of rows/columns (if done). If the return value is less than machine precision (in particular, if the return value = 0), the matrix is singular to working precision.

Throws:

SingularMatrixException

getDiagonalPivotThreshold

public double getDiagonalPivotThreshold()

Returns the threshold used for a diagonal entry to be an acceptable pivot.

Returns:

a double scalar specifying the threshold used for a diagonal entry to be an acceptable pivot.

getEquilibrate

public boolean getEquilibrate()

Returns the equilibration flag.

Returns:

a boolean specifying whether or not matrix A is equilibrated before factorization. if getEquilibrate returns true the system is equilibrated, if getEquilibrate returns false, no equilibration is performed.

getEquilibrationMethod

public int getEquilibrationMethod()

Returns information on the type of equilibration used before matrix factorization.

Returns:

an int value specifying the equilibration option used.

return value	Description
1 = NO_SCALING	No equilibration is performed.
2 = ROW_SCALING	Equilibration is performed with row scaling.
3 = COLUMN_SCALING	Equilibration is performed with column scaling.
4 = ROW_AND_COLUMN_SCALING	Equilibration is performed with row and column scaling.

getForwardErrorBound

public double getForwardErrorBound()

Returns the estimated forward error bound for the solution vector.

Returns:

a double containing the estimated forward error bound for the solution vector. The estimate is as reliable as the estimate for the reciprocal condition number, and is almost always a slight overestimate of the true error. If iterative refinement is not used, the return value = 1.0.

getIterativeRefinement

public boolean getIterativeRefinement()

Returns a value specifying whether iterative refinement is to be performed or not.

Returns:

a boolean scalar specifying whether iterative refinement is to be performed, true, or no iterative refinement is to be performed, false.

getPerformanceTuningParameters

public int getPerformanceTuningParameters(int parameter)

Returns a performance tuning parameter value.

Parameters:

parameter - an int scalar that specifies the parameter for which the value is to be returned.

parameter	return value description
PANEL_SIZE	The panel size.
RELAXATION_PARAMETER	The relaxation parameter to control supernode amalgamation.
MAXIMUM_SUPERNODE_SIZE	The maximum allowable size for a supernode.
MINIMUM_ROW_DIMENSIONM	The minimum row dimension to be used for 2D blocking.
MINIMUM_COLUMN_DIMENSION	The minimum column dimension to be used for 2D blocking.
FILL_FACTOR	The estimated fill factor for L and U, compared with A.

Returns:

an int specifying the current value used for the specified tuning parameter.

getPivotGrowth

public boolean getPivotGrowth()

Returns the reciprocal pivot growth factor flag.

Returns:

a boolean specifying whether to compute the reciprocal pivot growth factor. Returns true if the reciprocal pivot growth factor is to be computed.

getReciprocalPivotGrowthFactor

public double getReciprocalPivotGrowthFactor()
                                      throws SingularMatrixException

Returns the reciprocal pivot growth factor.

Returns:

a double scalar representing the reciprocal growth factor

If the returned value is much less than 1, the stability of the factorization could be poor.

Throws:

SingularMatrixException

getRelativeBackwardError

public double getRelativeBackwardError()

Returns the componentwise relative backward error of the solution vector.

Returns:

a double containing the componentwise relative backward error of the solution vector x. If iterative refinement is not used, the return value = 1.0.

getSymmetricMode

public boolean getSymmetricMode()

Returns the symmetric mode flag.

Returns:

a boolean scalar indicating if symmetric mode is to be used. Returns true if symmetric mode is to be used.

setColumnPermutationMethod

public void setColumnPermutationMethod(int colpermute)

Specifies how to permute the columns of the input matrix.

Parameters:

colpermute - an int scalar specifying how to permute the columns of the input matrix for sparsity preservation.

colpermute	method
NATURAL_ORDERING	natural ordering, that is , I the identity matrix.
MINIMUM_DEGREE_AT_PLUS_A	minimum degree ordering on the structure of
MINIMUM_DEGREE_AT_A	minimum degree ordering on the structure of
COLUMN_APPROXIMATE_MINIMUM_DEGREE	column approximate minimum degree ordering

If this method is not called, colpermute is set to SuperLU.COLUMN_APPROXIMATE_MINIMUM_DEGREE.

Throws:

IllegalArgumentException - is thrown when colpermute is not one of the above values.

setDiagonalPivotThreshold

public void setDiagonalPivotThreshold(double thresh)

Specifies the threshold used for a diagonal entry to be an acceptable pivot.

Parameters:

thresh - a double scalar specifying the threshold used for a diagonal entry to be an acceptable pivot.
Default: thresh=1.0, i.e. classical partial pivoting.

Throws:

IllegalArgumentException - is thrown when thresh is not in the interval .

setEquilibrate

public void setEquilibrate(boolean equilibrate)

Determines if input matrix A should be equilibrated before factorization.

Parameters:

equilibrate - a boolean determining if matrix A should be equilibrated before the factorization.

equilibrate	action
false	do not equilibrate
true	equilibrate

If this method is not called, equilibrate is set to true.

setIterativeRefinement

public void setIterativeRefinement(boolean refine)

Specifies whether to perform iterative refinement.

Parameters:

refine - a boolean specifying whether to use iterative refinement, refine = true, or no iterative refinement, refine = false.
Default: refine = false.

setPerformanceTuningParameters

public void setPerformanceTuningParameters(int parameter,
                                           int tunedValue)

Sets performance tuning parameters.

Parameters:

parameter - an int scalar that specifies the parameter to be tuned.

tunedValue - an int scalar that specifies the value to be used for the specified tuning parameter.

parameter	Description	Default
PANEL_SIZE	The panel size.	10
RELAXATION_PARAMETER	The relaxation parameter to control supernode amalgamation.	5
MAXIMUM_SUPERNODE_SIZE	The maximum allowable size for a supernode.	100
MINIMUM_ROW_DIMENSION	The minimum row dimension to be used for 2D blocking.	200
MINIMUM_COLUMN_DIMENSION	The minimum column dimension to be used for 2D blocking.	40
FILL_FACTOR	The estimated fill factor for L and U, compared with A.	20

Throws:

IllegalArgumentException - is thrown when a) parameter is not in the interval or b) tunedValue is not greater than zero.

setPivotGrowth

public void setPivotGrowth(boolean growth)

Specifies whether to compute the reciprocal pivot growth factor.

Parameters:

growth - a boolean specifying whether to compute the reciprocal pivot growth factor.

growth	action
false	don't compute growth factor
true	compute growth factor

Default: growth = false.

setSymmetricMode

public void setSymmetricMode(boolean symmetric)

Specifies whether to use the symmetric mode.

Parameters:

symmetric - a boolean indicating if symmetric mode is to be used. This mode should be applied if the input matrix A is diagonally dominant or nearly so. The user should then define a small diagonal pivot threshold (e.g. 0.0 or 0.01) by method setDiagonalPivotThreshold and choose an ()-based column permutation algorithm (e.g. column permutation method SuperLU.MINIMUM_DEGREE_AT_PLUS_A).

symmetric	action
false	symmetric mode is not used
true	symmetric mode is used

Default: symmetric=false.

solve

public double[] solve(double[] b)
               throws SingularMatrixException

Computation of the solution vector for the system .

Parameters:

b - a double vector of length n, n the order of input matrix A, containing the right hand side.

Returns:

a double vector containing the solution to the system . Optionally, the solution is improved by iterative refinement, if setIterativeRefinement is set to true. Method solve internally first factorizes matrix A (step 1 of the introduction) if the factorization has not been done before.

Throws:

SingularMatrixException

solveTranspose

public double[] solveTranspose(double[] b)
                        throws SingularMatrixException

Computation of the solution vector for the system .

Parameters:

b - a double vector of length n, n the order of input matrix A, containing the right hand side.

Returns:

a double vector containing the solution to the system . Optionally, the solution is improved by iterative refinement, if setIterativeRefinement is set to true . Method solveTranspose internally first factorizes matrix A (step 1 of the introduction) if the factorization has not been done before.

Throws:

SingularMatrixException