ComplexSuperLU (JMSL Numerical Library (jmsl) 2021.0.0 API)

java.lang.Object
- com.imsl.math.ComplexSuperLU

All Implemented Interfaces:

Serializable
```
public class ComplexSuperLU
extends Object
implements Serializable
```
Computes the LU factorization of a general sparse matrix of type ComplexSparseMatrix by a column method and solves a sparse linear system of equations $Ax=b$.
Consider the sparse linear system of equations $$ Ax=b.$$ 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 Complex.

Gaussian elimination, applied to the system above, can be shortly described as follows:
1. Compute a triangular factorization $P_rD_rAD_cP_c=LU$. Here, $D_r$ and $D_c$ are positive definite diagonal matrices to equilibrate the system and $P_r$ and $P_c$ 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 $Ax=b$ by evaluating $$x=A^{-1}b=D_c(P_c(U^{-1}(L^{-1}(P_r(D_rb))))).$$ This is done efficiently by multiplying from right to left in the last expression: Scale the rows of b by $D_r$. Multiplying $P_r(D_rb)$ means permuting the rows of $D_rb$. Multiplying $L^{-1}(P_rD_rb)$ means solving the triangular system of equations with matrix L by substitution. Similarly, multiplying $U^{-1}(L^{-1}(P_rD_rb))$ means solving the triangular system with U.
Class ComplexSuperLU handles step 1 above in the solve method if it is has not been computed prior to step 2. More precisely, before $Ax=b$ is solved the following steps are performed:
1. Equilibrate matrix A, i.e. compute diagonal matrices $D_r$ and $D_c$ so that $\hat{A}=D_rAD_c$ is "better conditioned" than A, i.e. $\hat{A}^{-1}$ is less sensitive to perturbations in $\hat{A}$ than $A^{-1}$ is to perturbations in A.
2. Order the columns of $\hat{A}$ to increase the sparsity of the computed L and U factors, i.e. replace $\hat{A}$ by $\hat{A}P_c$ where $P_c$ is a column permutation matrix.
3. Compute the LU factorization of $\hat{A}P_c$. For numerical stability, the rows of $\hat{A}P_c$ are eventually permuted through the factorization process by scaled partial pivoting, leading to the decomposition $\tilde{A}:=P_r\hat{A}P_c=LU$. 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 $$\max_{1 \le j \le n} \frac{\|\tilde{A}_j\|_\infty}{\|U_j\|_\infty},$$ where $\tilde{A}_j$ and $U_j$ denote the j-th column of matrices $\tilde{A}$ and U, respectively.
5. Estimate the reciprocal of the condition number of matrix $\tilde{A}$.
Method solve uses this information to perform the following steps:
1. Solve the system $Ax=b$ 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 ComplexSuperLU.

Class ComplexSuperLU 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.

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(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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See Also:

Example, Serialized Form

Field Summary

Fields
Modifier and Type	Field and Description
`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 $A^TA$.
`static int`	`MINIMUM_DEGREE_AT_PLUS_A` For column ordering, use minimum degree ordering on the structure of $A^T+A$.
`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

Constructors
Constructor and Description

ComplexSuperLU(ComplexSparseMatrix A)
Constructor for ComplexSuperLU.

Constructors
Constructor and Description
`ComplexSuperLU(ComplexSparseMatrix A)` Constructor for `ComplexSuperLU`.

Method Summary

All Methods Instance Methods Concrete Methods
Modifier and Type	Method and Description
`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 each 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)` Specifies 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.
`Complex[]`	`solve(Complex[] b)` Computation of the solution vector for the system $ Ax = b$.
`Complex[]`	`solveConjugateTranspose(Complex[] b)` Computation of the solution vector for the system $ A^Hx = b$.
`Complex[]`	`solveTranspose(Complex[] b)` Computation of the solution vector for the system $ A^Tx = b$.

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail
- NATURAL_ORDERING
```
public static final int NATURAL_ORDERING
```
  For column ordering, use the natural ordering.
  
  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 $A^TA$.
  
  See Also:
  
  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 $A^T+A$.
  
  See Also:
  
  Constant Field Values
- COLUMN_APPROXIMATE_MINIMUM_DEGREE
```
public static final int COLUMN_APPROXIMATE_MINIMUM_DEGREE
```
  For column ordering, use column approximate minimum degree ordering.
  
  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
- 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_ROW_DIMENSION
```
public static final int MINIMUM_ROW_DIMENSION
```
  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
- FILL_FACTOR
```
public static final int FILL_FACTOR
```
  A performance tuning parameter which can be adjusted via method setPerformanceTuningParameters.
  
  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
- 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
- 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
- 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

Constructor Detail
- ComplexSuperLU
```
public ComplexSuperLU(ComplexSparseMatrix A)
```
  Constructor for ComplexSuperLU.
  
  Parameters:
  
  A - a ComplexSparseMatrix containing the sparse quadratic input matrix.

Method Detail

solve
```
public Complex[] solve(Complex[] b)
                throws SingularMatrixException
```
Computation of the solution vector for the system $ Ax = b$.

Parameters:

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

Returns:

a Complex vector containing the solution to the system $ Ax = b$. 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 Complex[] solveTranspose(Complex[] b)
                         throws SingularMatrixException
```
Computation of the solution vector for the system $ A^Tx = b$.

Parameters:

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

Returns:

a Complex vector containing the solution to the system $ A^Tx = b$. 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

solveConjugateTranspose
```
public Complex[] solveConjugateTranspose(Complex[] b)
                                  throws SingularMatrixException
```
Computation of the solution vector for the system $ A^Hx = b$.

Parameters:

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

Returns:

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

Throws:

SingularMatrixException

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	option 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.

setEquilibrate

public void setEquilibrate(boolean equilibrate)

Specifies 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

Default: equilibrate = true.

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 $P_c=I$, I the identity matrix
`MINIMUM_DEGREE_AT_PLUS_A`	minimum degree ordering on the structure of $A^T+A$
`MINIMUM_DEGREE_AT_A`	minimum degree ordering on the structure of $A^TA$
`COLUMN_APPROXIMATE_MINIMUM_DEGREE`	column approximate minimum degree ordering

Default: colpermute = SuperLU.COLUMN_APPROXIMATE_MINIMUM_DEGREE.

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 $P_c=I$, I the identity matrix
1 = `MINIMUM_DEGREE_AT_PLUS_A`	minimum degree ordering on the structure of $A^T+A$
2 = `MINIMUM_DEGREE_AT_A`	minimum degree ordering on the structure of $A^TA$
3 = `COLUMN_APPROXIMATE_MINIMUM_DEGREE`	column approximate minimum degree ordering

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 ($A^T+A$)-based column permutation algorithm (e.g. column permutation method ComplexSuperLU.MINIMUM_DEGREE_AT_PLUS_A).

symmetric action

false symmetric mode is not used

true symmetric mode is used

Default: symmetric=false.

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

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.

setIterativeRefinement
```
public void setIterativeRefinement(boolean refine)
```
Specifies whether to perform iterative refinement.

Parameters:

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

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.

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 if thresh is not in the interval $[0.0,1.0]$.

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.

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.

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

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 $$\max_{1 \le j \le n} \frac{\|\tilde{A}_j\|_\infty}{\|U_j\|_\infty}.$$ If the returned value is much less than 1, the stability of the $LU$ factorization could be poor.

Throws:

SingularMatrixException

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 reciprocal condition number is less than machine precision, in particular if it is equal to 0, the matrix is singular to working precision.

Throws:

SingularMatrixException

getForwardErrorBound
```
public double getForwardErrorBound()
```
Returns the estimated forward error bound for each 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.

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 setIterativeRefinement is not set to true, then getRelativeBackwardError returns 1.0.

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.

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 $[1,\ldots,6]$ or b) tunedValue is not greater than zero.

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_DIMENSION`	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.

`colpermute`	method
`NATURAL_ORDERING`	natural ordering, that is \(P_c=I\), I the identity matrix
`MINIMUM_DEGREE_AT_PLUS_A`	minimum degree ordering on the structure of \(A^T+A\)
`MINIMUM_DEGREE_AT_A`	minimum degree ordering on the structure of \(A^TA\)
`COLUMN_APPROXIMATE_MINIMUM_DEGREE`	column approximate minimum degree ordering

Class ComplexSuperLU

Field Summary

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Field Detail

NATURAL_ORDERING

MINIMUM_DEGREE_AT_A

MINIMUM_DEGREE_AT_PLUS_A

COLUMN_APPROXIMATE_MINIMUM_DEGREE

PANEL_SIZE

RELAXATION_PARAMETER

MAXIMUM_SUPERNODE_SIZE

MINIMUM_ROW_DIMENSION

MINIMUM_COLUMN_DIMENSION

FILL_FACTOR

NO_SCALING

ROW_SCALING

COLUMN_SCALING

ROW_AND_COLUMN_SCALING

Constructor Detail

ComplexSuperLU

Method Detail

solve

solveTranspose

solveConjugateTranspose

getEquilibrationMethod

setEquilibrate

setColumnPermutationMethod

getColumnPermutationMethod

setSymmetricMode

getSymmetricMode

setIterativeRefinement

getIterativeRefinement

setDiagonalPivotThreshold

getDiagonalPivotThreshold

setPivotGrowth

getPivotGrowth

getReciprocalPivotGrowthFactor

getConditionNumber

getForwardErrorBound

getRelativeBackwardError

getEquilibrate

setPerformanceTuningParameters

getPerformanceTuningParameters