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

L U

factorization of

A

with partial pivoting,

P_{r} A P_{c} = L U

, where

P_{r}

is a row permutation matrix (computed by nag_superlu_lu_factorize (f11mec)),

P_{c}

is a (supplied) column permutation matrix,

L

is unit lower triangular and

U

is upper triangular. The column permutation matrix,

P_{c}

, 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

L U

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: $n$ – IntegerInput: On entry: $n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .
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: $iprm [7 \times n]$ – IntegerInput/Output: On entry: contains the column permutation which defines the permutation $P_{c}$ 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 $P_{r}$ determined by pivoting.
5: $thresh$ – doubleInput: On entry: the diagonal pivoting threshold, $t$ . At step $j$ of the Gaussian elimination, if $|A_{j j}| \geq t (\max_{i \geq j} |A_{i j}|)$ , use $A_{j j}$ as a pivot, otherwise use $\max_{i \geq j} |A_{i j}|$ . 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: $0.0 \leq thresh \leq 1.0$ .
6: $nzlmx$ – IntegerInput: On entry: indicates the available size of array il. The dimension of il should be at least $7 \times n + nzlmx + 4$ . A good range for nzlmx that works for many problems is $nnz$ to $8 \times nnz$ , 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: $nzlmx \geq 1$ .
7: $nzlumx$ – Integer *Input/Output: On entry: indicates the available size of array lval. The dimension of lval should be at least nzlumx.

Constraint: $nzlumx \geq 1$ .

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: $nzumx$ – IntegerInput: On entry: indicates the available sizes of arrays iu and uval. The dimension of iu should be at least $2 \times n + nzumx + 1$ and the dimension of uval should be at least nzumx. A good range for nzumx that works for many problems is $nnz$ to $8 \times nnz$ , 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: $nzumx \geq 1$ .
9: $il [7 \times n + nzlmx + 4]$ – 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: $iu [2 \times n + nzumx + 1]$ – 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: $nnzl$ – 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 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .

On entry, $nzlmx = 〈value〉$ .
Constraint: $nzlmx \geq 1$ .

On entry, $nzlumx = 〈value〉$ .
Constraint: $nzlumx \geq 1$ .

On entry, $nzumx = 〈value〉$ .
Constraint: $nzumx \geq 1$ .
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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: $0.0 \leq thresh \leq 1.0$ .
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| \leq c (n) ε |L| |U|,

c (n)

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

Parallelism and Performance

nag_superlu_lu_factorize (f11mec) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_superlu_lu_factorize (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 nag_superlu_lu_factorize (f11mec) may be followed by calls to the functions:

nag_superlu_solve_lu (f11mfc) to solve $A X = B$ or $A^{T} X = B$ ;
nag_superlu_condition_number_lu (f11mgc) to estimate the condition number of $A$ ;
nag_superlu_diagnostic_lu (f11mmc) to estimate the reciprocal pivot growth of the $L U$ factorization.

10

Example

This example computes the

L U

factorization of the matrix

A

, where

A = (\begin{matrix} 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 \end{matrix}) .

NAG Library Function Document

nag_superlu_lu_factorize (f11mec)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results