NAG Library Routine Document

f11jcf solves a real sparse symmetric system of linear equations, represented in symmetric coordinate storage format, using a conjugate gradient or Lanczos method, with incomplete Cholesky preconditioning.

2

Specification

Fortran Interface

Subroutine f11jcf (

method, n, nnz, a, la, irow, icol, ipiv, istr, b, tol, maxitn, x, rnorm, itn, work, lwork, ifail)

Integer, Intent (In)	::	n, nnz, la, irow(la), icol(la), istr(n+1), maxitn, lwork
Integer, Intent (Inout)	::	ipiv(n), ifail
Integer, Intent (Out)	::	itn
Real (Kind=nag_wp), Intent (In)	::	a(la), b(n), tol
Real (Kind=nag_wp), Intent (Inout)	::	x(n)
Real (Kind=nag_wp), Intent (Out)	::	rnorm, work(lwork)
Character (*), Intent (In)	::	method

C Header Interface

#include <nagmk26.h>

void

f11jcf_ (const char *method, const Integer *n, const Integer *nnz, const double a[], const Integer *la, const Integer irow[], const Integer icol[], Integer ipiv[], const Integer istr[], const double b[], const double *tol, const Integer *maxitn, double x[], double *rnorm, Integer *itn, double work[], const Integer *lwork, Integer *ifail, const Charlen length_method)

3

Description

f11jcf solves a real sparse symmetric linear system of equations

A x = b,

using a preconditioned conjugate gradient method (see Meijerink and Van der Vorst (1977)), or a preconditioned Lanczos method based on the algorithm SYMMLQ (see Paige and Saunders (1975)). The conjugate gradient method is more efficient if

A

is positive definite, but may fail to converge for indefinite matrices. In this case the Lanczos method should be used instead. For further details see Barrett et al. (1994).

f11jcf uses the incomplete Cholesky factorization determined by f11jaf as the preconditioning matrix. A call to f11jcf must always be preceded by a call to f11jaf. Alternative preconditioners for the same storage scheme are available by calling f11jef.

The matrix

A

, and the preconditioning matrix

M

, are represented in symmetric coordinate storage (SCS) format (see Section 2.1.2 in the F11 Chapter Introduction) in the arrays a, irow and icol, as returned from f11jaf. The array a holds the nonzero entries in the lower triangular parts of these matrices, while irow and icol hold the corresponding row and column indices.

4

References

Barrett R, Berry M, Chan T F, Demmel J, Donato J, Dongarra J, Eijkhout V, Pozo R, Romine C and Van der Vorst H (1994) Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods SIAM, Philadelphia

Meijerink J and Van der Vorst H (1977) An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix Math. Comput. 31 148–162

Paige C C and Saunders M A (1975) Solution of sparse indefinite systems of linear equations SIAM J. Numer. Anal. 12 617–629

Salvini S A and Shaw G J (1995) An evaluation of new NAG Library solvers for large sparse symmetric linear systems NAG Technical Report TR1/95

5

Arguments

1: $method$ – Character(*)Input

On entry: specifies the iterative method to be used.

$method ='CG'$: Conjugate gradient method.
$method ='SYMMLQ'$: Lanczos method (SYMMLQ).

Constraint:

method ='CG'

'SYMMLQ'

2: $n$ – IntegerInput

On entry:

n

, the order of the matrix

A

. This must be the same value as was supplied in the preceding call to f11jaf.

Constraint:

n \geq 1

3: $nnz$ – IntegerInput

On entry: the number of nonzero elements in the lower triangular part of the matrix

A

. This must be the same value as was supplied in the preceding call to f11jaf.

Constraint:

1 \leq nnz \leq n \times (n + 1) / 2

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

On entry: the values returned in the array a by a previous call to f11jaf.

5: $la$ – IntegerInput

On entry: the dimension of the arrays a, irow and icol as declared in the (sub)program from which f11jcf is called. This must be the same value as was supplied in the preceding call to f11jaf.

Constraint:

la \geq 2 \times nnz

6: $irow (la)$ – Integer arrayInput

7: $icol (la)$ – Integer arrayInput

8: $ipiv (n)$ – Integer arrayInput

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

On entry: the values returned in arrays irow, icol, ipiv and istr by a previous call to f11jaf.

10: $b (n)$ – Real (Kind=nag_wp) arrayInput

On entry: the right-hand side vector

b

11: $tol$ – Real (Kind=nag_wp)Input

On entry: the required tolerance. Let

x_{k}

denote the approximate solution at iteration

k

, and

r_{k}

the corresponding residual. The algorithm is considered to have converged at iteration

k

{‖r_{k}‖}_{\infty} \leq τ \times ({‖b‖}_{\infty} + {‖A‖}_{\infty} {‖x_{k}‖}_{\infty}) .

tol \leq 0.0

τ = \max \sqrt{ε}, 10 ε, \sqrt{n} ε

is used, where

ε

is the machine precision. Otherwise

τ = \max (tol, 10 ε, \sqrt{n} ε)

is used.

Constraint:

tol < 1.0

12: $maxitn$ – IntegerInput

On entry: the maximum number of iterations allowed.

Constraint:

maxitn \geq 1

13: $x (n)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: an initial approximation to the solution vector

x

On exit: an improved approximation to the solution vector

x

14: $rnorm$ – Real (Kind=nag_wp)Output

On exit: the final value of the residual norm

{‖r_{k}‖}_{\infty}

, where

k

is the output value of itn.

15: $itn$ – IntegerOutput

On exit: the number of iterations carried out.

16: $work (lwork)$ – Real (Kind=nag_wp) arrayWorkspace

17: $lwork$ – IntegerInput

On entry: the dimension of the array work as declared in the (sub)program from which f11jcf is called.

Constraints:

if $method ='CG'$ , $lwork \geq 6 \times n + 120$ ;
if $method ='SYMMLQ'$ , $lwork \geq 7 \times n + 120$ .

18: $ifail$ – IntegerInput/Output

On entry: ifail must be set to

0

- 1 or 1

. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 or 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this argument, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

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, $la = 〈value〉$ and $nnz = 〈value〉$ .
Constraint: $la \geq 2 \times nnz$ .

On entry, lwork is too small: $lwork = 〈value〉$ . Minimum required value of $lwork = 〈value〉$ .

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

On entry, $method = 〈value〉$ .
Constraint: $method ='CG'$ or $'SYMMLQ'$ .

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

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

On entry, $nnz = 〈value〉$ and $n = 〈value〉$ .
Constraint: $nnz \leq n \times (n + 1) / 2$ .

On entry, $tol = 〈value〉$ .
Constraint: $tol < 1.0$ .

$ifail = 2$: On entry, $a (i)$ is out of order: $i = 〈value〉$ .

On entry, $i = 〈value〉$ , $icol (i) = 〈value〉$ , $irow (i) = 〈value〉$ .
Constraint: $icol (i) \geq 1$ and $icol (i) \leq irow (i)$ .

On entry, $i = 〈value〉$ , $irow (i) = 〈value〉$ , $n = 〈value〉$ .
Constraint: $irow (i) \geq 1$ and $irow (i) \leq n$ .

On entry, the location ( $irow (i), icol (i)$ ) is a duplicate: $i = 〈value〉$ .
Check that a, irow, icol, ipiv and istr have not been corrupted between calls to f11jaf and f11jcf.

$ifail = 3$: The SCS representation of the preconditioner is invalid.
Check that a, irow, icol, ipiv and istr have not been corrupted between calls to f11jaf and f11jcf.

$ifail = 4$: The required accuracy could not be obtained. However a reasonable accuracy has been achieved.

$ifail = 5$: The solution has not converged after $〈value〉$ iterations.

$ifail = 6$: The preconditioner appears not to be positive definite. The computation cannot continue.

$ifail = 7$: The matrix of the coefficients a appears not to be positive definite. The computation cannot continue.

$ifail = 8$: A serious error has occurred in an internal call: $ifail = 〈value〉$ . Check all subroutine calls and array sizes. Seek expert help.

A serious error has occurred in an internal call: $IREVCM = 〈value〉$ . Check all subroutine calls and array sizes. Seek expert help.

$ifail = - 99$: 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

On successful termination, the final residual

r_{k} = b - A x_{k}

, where

k = itn

, satisfies the termination criterion

{‖r_{k}‖}_{\infty} \leq τ \times ({‖b‖}_{\infty} + {‖A‖}_{\infty} {‖x_{k}‖}_{\infty}) .

The value of the final residual norm is returned in rnorm.

8

Parallelism and Performance

f11jcf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f11jcf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9

Further Comments

The time taken by f11jcf for each iteration is roughly proportional to the value of nnzc returned from the preceding call to f11jaf. One iteration with the Lanczos method (SYMMLQ) requires a slightly larger number of operations than one iteration with the conjugate gradient method.

The number of iterations required to achieve a prescribed accuracy cannot be easily determined a priori, as it can depend dramatically on the conditioning and spectrum of the preconditioned matrix of the coefficients

\bar{A} = M^{- 1} A

Some illustrations of the application of f11jcf to linear systems arising from the discretization of two-dimensional elliptic partial differential equations, and to random-valued randomly structured symmetric positive definite linear systems, can be found in Salvini and Shaw (1995).

10

Example

This example solves a symmetric positive definite system of equations using the conjugate gradient method, with incomplete Cholesky preconditioning.

NAG Library Routine Document

f11jcf (real_symm_solve_ichol)

▸▿ 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