ComplexSuperLU Class

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

For a list of all members of this type, see ComplexSuperLU Members.

System.Object
Imsl.Math.ComplexSuperLU

public class ComplexSuperLU

Thread Safety

Public static (Shared in Visual Basic) members of this type are safe for multithreaded operations. Instance members are not guaranteed to be thread-safe.

Remarks

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 Complex.

Gaussian elimination, applied to the system above, can be shortly described as follows:

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.
Solve 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 . Multiplying means permuting the rows of . 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 is solved the following steps are performed:

Equilibrate matrix A, i.e. compute diagonal matrices and 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.
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 is a column permutation matrix.
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.
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 denote the j-th column of matrices $\tilde{A}$ and U, respectively.
Estimate the reciprocal of the condition number of matrix $\tilde{A}$ .

Method Solve uses this information to perform the following steps:

Solve the system using the computed triangular factors.
Iteratively refine the solution, again using the computed triangular factors. This is equivalent to Newton's method.
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 and properties 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).

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Requirements

Namespace: Imsl.Math

Assembly: ImslCS (in ImslCS.dll)

ComplexSuperLU Class

Thread Safety

Remarks

Requirements

See Also