Package com.imsl.math

Class ComplexSuperLU

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.

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.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

See Also:

  • Field Details

    • NATURAL_ORDERING

      public static final int NATURAL_ORDERING
      For column ordering, use the natural ordering.
      See Also:

    • 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:

    • 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:

    • COLUMN_APPROXIMATE_MINIMUM_DEGREE

      public static final int COLUMN_APPROXIMATE_MINIMUM_DEGREE
      For column ordering, use column approximate minimum degree ordering.
      See Also:

    • PANEL_SIZE

      public static final int PANEL_SIZE
      A performance tuning parameter which can be adjusted via method setPerformanceTuningParameters.
      See Also:

    • RELAXATION_PARAMETER

      public static final int RELAXATION_PARAMETER
      A performance tuning parameter which can be adjusted via method setPerformanceTuningParameters.
      See Also:

    • MAXIMUM_SUPERNODE_SIZE

      public static final int MAXIMUM_SUPERNODE_SIZE
      A performance tuning parameter which can be adjusted via method setPerformanceTuningParameters.
      See Also:

    • MINIMUM_ROW_DIMENSION

      public static final int MINIMUM_ROW_DIMENSION
      A performance tuning parameter which can be adjusted via method setPerformanceTuningParameters.
      See Also:

    • MINIMUM_COLUMN_DIMENSION

      public static final int MINIMUM_COLUMN_DIMENSION
      A performance tuning parameter which can be adjusted via method setPerformanceTuningParameters.
      See Also:

    • FILL_FACTOR

      public static final int FILL_FACTOR
      A performance tuning parameter which can be adjusted via method setPerformanceTuningParameters.
      See Also:

    • 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:

    • 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:

    • 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:

    • 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:

  • Constructor Details

    • ComplexSuperLU

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

  • Method Details

    • 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. <th center" width="45%">method
      return value
      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.

    • 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.

    • 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.