Integer, Intent (In)	::	n, la, irow(la), icol(la), istr(n+1), idiag(n)
Integer, Intent (Inout)	::	ipivp(n), ipivq(n), ifail
Real (Kind=nag_wp), Intent (In)	::	a(la), y(n)
Real (Kind=nag_wp), Intent (Out)	::	x(n)
Character (1), Intent (In)	::	trans, check

C Header Interface

#include nagmk26.h

void

f11dbf_ ( const char *trans, const Integer *n, const double a[], const Integer *la, const Integer irow[], const Integer icol[], Integer ipivp[], Integer ipivq[], const Integer istr[], const Integer idiag[], const char *check, const double y[], double x[], Integer *ifail, const Charlen length_trans, const Charlen length_check)

3

Description

f11dbf solves a system of linear equations

\begin{matrix} M x = y, & or & M^{T} x = y, \end{matrix}

according to the value of the argument trans, where the matrix

M = P L D U Q

, corresponds to an incomplete

L U

decomposition of a sparse matrix stored in coordinate storage (CS) format (see Section 2.1.1 in the F11 Chapter Introduction), as generated by f11daf.

In the above decomposition

L

is a lower triangular sparse matrix with unit diagonal elements,

D

is a diagonal matrix,

U

is an upper triangular sparse matrix with unit diagonal elements and,

P

and

Q

are permutation matrices.

L

D

and

U

are supplied to f11dbf through the matrix

C = L + D^{- 1} + U - 2 I

which is an n by n sparse matrix, stored in CS format, as returned by f11daf. The permutation matrices

P

and

Q

are returned from f11daf via the arrays ipivp and ipivq.

It is envisaged that a common use of f11dbf will be to carry out the preconditioning step required in the application of f11bef to sparse linear systems. f11dbf is used for this purpose by the Black Box routine f11dcf.

f11dbf may also be used in combination with f11daf to solve a sparse system of linear equations directly (see Section 9.5 in f11daf). This use of f11dbf is demonstrated in Section 10.

4

References

None.

5

Arguments

1: $trans$ – Character(1)Input

On entry: specifies whether or not the matrix

M

is transposed.

$trans ='N'$: $M x = y$ is solved.
$trans ='T'$: $M^{T} x = y$ is solved.

Constraint:

trans ='N'

'T'

2: $n$ – IntegerInput

On entry:

n

, the order of the matrix

M

. This must be the same value as was supplied in the preceding call to f11daf.

Constraint:

n \geq 1

3: $a (la)$ – Real (Kind=nag_wp) arrayInput

On entry: the values returned in the array a by a previous call to f11daf.

4: $la$ – IntegerInput

On entry: the dimension of the arrays a, irow and icol as declared in the (sub)program from which f11dbf is called. This must be the same value returned by the preceding call to f11daf.

5: $irow (la)$ – Integer arrayInput

6: $icol (la)$ – Integer arrayInput

7: $ipivp (n)$ – Integer arrayInput

8: $ipivq (n)$ – Integer arrayInput

9: $istr (n + 1)$ – Integer arrayInput

10: $idiag (n)$ – Integer arrayInput

On entry: the values returned in arrays irow, icol, ipivp, ipivq, istr and idiag by a previous call to f11daf.

11: $check$ – Character(1)Input

On entry: specifies whether or not the CS representation of the matrix

M

should be checked.

$check ='C'$: Checks are carried on the values of n, irow, icol, ipivp, ipivq, istr and idiag.
$check ='N'$: None of these checks are carried out.

6

Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$

On entry,	$trans \neq'N'$ or $'T'$ ,
or	$check \neq'C'$ or $'N'$ .

$ifail = 2$

On entry,

n < 1

$ifail = 3$: On entry, the CS representation of the preconditioning matrix $M$ is invalid. Further details are given in the error message. Check that the call to f11dbf has been preceded by a valid call to f11daf and that the arrays a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between the two calls.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

trans ='N'

the computed solution

x

is the exact solution of a perturbed system of equations

(M + δ M) x = y

, where

|δ M| \leq c (n) ε P |L| |D| |U| Q,

c (n)

is a modest linear function of

n

, and

ε

is the machine precision. An equivalent result holds when

trans ='T'

8

Parallelism and Performance

f11dbf is not threaded in any implementation.

9

Further Comments

9.1

Timing

The time taken for a call to f11dbf is proportional to the value of nnzc returned from f11daf.

9.2

Use of check

It is expected that a common use of f11dbf will be to carry out the preconditioning step required in the application of f11bef to sparse linear systems. In this situation f11dbf is likely to be called many times with the same matrix

M

. In the interests of both reliability and efficiency, you are recommended to set

check ='C'

for the first of such calls, and for all subsequent calls set

check ='N'

10

Example

This example reads in a sparse nonsymmetric matrix

A

and a vector

y

. It then calls f11daf, with

lfill = - 1

and

dtol = 0.0

, to compute the complete

L U

decomposition

A = P L D U Q .

Finally it calls f11dbf to solve the system

P L D U Q x = y .

NAG Library Routine Document

f11dbf (real_gen_precon_ilu_solve)

▸▿ 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

9.1 Timing

9.2 Use of check

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification