NAG CL Interface
f11mec (direct_​real_​gen_​lu)

1 Purpose

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

2 Specification

#include <nag.h>
void  f11mec (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)
The function may be called by the names: f11mec, nag_sparse_direct_real_gen_lu or nag_superlu_lu_factorize.

3 Description

Given a real sparse matrix A, f11mec computes an LU factorization of A with partial pivoting, Pr A Pc = LU , where Pr is a row permutation matrix (computed by 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 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. 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: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
2: irowix[dim] const Integer Input
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 double Input
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] Integer Input/Output
On entry: contains the column permutation which defines the permutation Pc and associated data structures as computed by function f11mdc.
On exit: part of the array is modified to record the row permutation Pr determined by pivoting.
5: thresh double Input
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: nzlmx Integer Input
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: nzlumx Integer * 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: nzumx Integer Input
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] Integer Output
On exit: encapsulates the sparsity pattern of matrix L.
10: lval[nzlumx] double Output
On exit: records the nonzero values of matrix L and some of the nonzero values of matrix U.
11: iu[2×n+nzumx+1] Integer Output
On exit: encapsulates the sparsity pattern of matrix U.
12: uval[nzumx] double Output
On exit: records some of the nonzero values of matrix U.
13: nnzl Integer * Output
On exit: the number of nonzero values in the matrix L.
14: nnzu Integer * Output
On exit: the number of nonzero values in the matrix U.
15: flop double * Output
On exit: the number of floating-point operations performed.
16: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
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 f11mmc after f11mec can help determine the quality of the factorization.

8 Parallelism and Performance

f11mec is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f11mec makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

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

10 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 .  

10.1 Program Text

Program Text (f11mece.c)

10.2 Program Data

Program Data (f11mece.d)

10.3 Program Results

Program Results (f11mece.r)