nag_superlu_lu_factorize (f11mec) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_superlu_lu_factorize (f11mec)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_superlu_lu_factorize (f11mec) computes the LU  factorization of a real sparse matrix in compressed column (Harwell–Boeing), column-permuted format.

2  Specification

#include <nag.h>
#include <nagf11.h>
void  nag_superlu_lu_factorize (Integer n, const Integer irowix[], const double a[], Integer iprm[], double thresh, Integer nzlmx, Integer *nzlumx, Integer nzumx, Integer il[], double lval[], Integer iu[], double uval[], Integer *nnzl, Integer *nnzu, double *flop, NagError *fail)

3  Description

Given a real sparse matrix A, nag_superlu_lu_factorize (f11mec) computes an LU  factorization of A with partial pivoting, Pr A Pc = LU , where Pr is a row permutation matrix (computed by nag_superlu_lu_factorize (f11mec)), Pc is a (supplied) column permutation matrix, L is unit lower triangular and U is upper triangular. The column permutation matrix, Pc, must be computed by a prior call to nag_superlu_column_permutation (f11mdc). The matrix A must be presented in the column permuted, compressed column (Harwell–Boeing) format.
The LU  factorization is output in the form of four one-dimensional arrays: integer arrays il and iu and real-valued arrays lval and uval. These describe the sparsity pattern and numerical values in the L and U matrices. The minimum required dimensions of these arrays cannot be given as a simple function of the size arguments (order and number of nonzero values) of the matrix A. This is due to unpredictable fill-in created by partial pivoting. nag_superlu_lu_factorize (f11mec) will, on return, indicate which dimensions of these arrays were not adequate for the computation or (in the case of one of them) give a firm bound. You should then allocate more storage and try again.

4  References

Demmel J W, Eisenstat S C, Gilbert J R, Li X S and Li J W H (1999) A supernodal approach to sparse partial pivoting SIAM J. Matrix Anal. Appl. 20 720–755
Demmel J W, Gilbert J R and Li X S (1999) An asynchronous parallel supernodal algorithm for sparse gaussian elimination SIAM J. Matrix Anal. Appl. 20 915–952

5  Arguments

1:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
2:     irowix[dim]const IntegerInput
Note: the dimension, dim, of the array irowix must be at least nnz, the number of nonzeros of the sparse matrix A.
On entry: the row index array of sparse matrix A.
3:     a[dim]const doubleInput
Note: the dimension, dim, of the array a must be at least nnz, the number of nonzeros of the sparse matrix A.
On entry: the array of nonzero values in the sparse matrix A.
4:     iprm[7×n]IntegerInput/Output
On entry: contains the column permutation which defines the permutation Pc and associated data structures as computed by function nag_superlu_column_permutation (f11mdc).
On exit: part of the array is modified to record the row permutation Pr determined by pivoting.
5:     threshdoubleInput
On entry: the diagonal pivoting threshold, t. At step j of the Gaussian elimination, if AjjtmaxijAij, use Ajj as a pivot, otherwise use maxijAij. A value of t=1 corresponds to partial pivoting, a value of t=0 corresponds to always choosing the pivot on the diagonal (unless it is zero).
Suggested value: thresh=1.0. Smaller values may result in a faster factorization, but the benefits are likely to be small in most cases. It might be possible to use thresh=0.0 if you are confident about the stability of the factorization, for example, if A is diagonally dominant.
Constraint: 0.0thresh1.0.
6:     nzlmxIntegerInput
On entry: indicates the available size of array il. The dimension of il should be at least 7×n+nzlmx+4. A good range for nzlmx that works for many problems is nnz to 8×nnz, where nnz is the number of nonzeros in the sparse matrix A. If, on exit, fail.code= NE_NZLMX_TOO_SMALL, the given nzlmx was too small and you should attempt to provide more storage and call the function again.
Constraint: nzlmx1.
7:     nzlumxInteger *Input/Output
On entry: indicates the available size of array lval. The dimension of lval should be at least nzlumx.
Constraint: nzlumx1.
On exit: if fail.code= NE_NZLUMX_TOO_SMALL, the given nzlumx was too small and is reset to a value that will be sufficient. You should then provide the indicated storage and call the function again.
8:     nzumxIntegerInput
On entry: indicates the available sizes of arrays iu and uval. The dimension of iu should be at least 2×n+nzumx+1 and the dimension of uval should be at least nzumx. A good range for nzumx that works for many problems is nnz to 8×nnz, where nnz is the number of nonzeros in the sparse matrix A. If, on exit, fail.code= NE_NZUMX_TOO_SMALL, the given nzumx was too small and you should attempt to provide more storage and call the function again.
Constraint: nzumx1.
9:     il[7×n+nzlmx+4]IntegerOutput
On exit: encapsulates the sparsity pattern of matrix L.
10:   lval[nzlumx]doubleOutput
On exit: records the nonzero values of matrix L and some of the nonzero values of matrix U.
11:   iu[2×n+nzumx+1]IntegerOutput
On exit: encapsulates the sparsity pattern of matrix U.
12:   uval[nzumx]doubleOutput
On exit: records some of the nonzero values of matrix U.
13:   nnzlInteger *Output
On exit: the number of nonzero values in the matrix L.
14:   nnzuInteger *Output
On exit: the number of nonzero values in the matrix U.
15:   flopdouble *Output
On exit: the number of floating point operations performed.
16:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument number value had an illegal value.
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, nzlmx=value.
Constraint: nzlmx1.
On entry, nzlumx=value.
Constraint: nzlumx1.
On entry, nzumx=value.
Constraint: nzumx1.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_NZLMX_TOO_SMALL
Insufficient nzlmx.
NE_NZLUMX_TOO_SMALL
Insufficient nzlumx.
NE_NZUMX_TOO_SMALL
Insufficient nzumx.
NE_REAL
On entry, thresh=value.
Constraint: 0.0thresh1.0.
NE_SINGULAR_MATRIX
The matrix is singular – no factorization possible.

7  Accuracy

The computed factors L and U are the exact factors of a perturbed matrix A+E, where
E c n ε L U ,
cn is a modest linear function of n, and ε is the machine precision, when partial pivoting is used. If no partial pivoting is used, the factorization accuracy can be considerably worse. A call to nag_superlu_diagnostic_lu (f11mmc) after nag_superlu_lu_factorize (f11mec) can help determine the quality of the factorization.

8  Further Comments

The total number of floating point operations depends on the sparsity pattern of the matrix A.
A call to nag_superlu_lu_factorize (f11mec) may be followed by calls to the functions:

9  Example

This example computes the LU factorization of the matrix A, where
A= 2.00 1.00 0 0 0 0 0 1.00 -1.00 0 4.00 0 1.00 0 1.00 0 0 0 1.00 2.00 0 -2.00 0 0 3.00 .

9.1  Program Text

Program Text (f11mece.c)

9.2  Program Data

Program Data (f11mece.d)

9.3  Program Results

Program Results (f11mece.r)


nag_superlu_lu_factorize (f11mec) (PDF version)
f11 Chapter Contents
f11 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012