NAG Library Routine Document

F11JQF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     METHOD – CHARACTER(*)Input

On entry: specifies the iterative method to be used.

$\mathbf{METHOD}=\text{'CG'}$

Conjugate gradient method.

$\mathbf{METHOD}=\text{'SYMMLQ'}$

Lanczos method (SYMMLQ).

Constraint: $\mathbf{METHOD}=\text{'CG'}$ or $\text{'SYMMLQ'}$.

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 F11JNF.

Constraint: $\mathbf{N}\ge 1$.

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 F11JNF.

Constraint: $1\le \mathbf{NNZ}\le \mathbf{N}\times \left(\mathbf{N}+1\right)/2$.

4:     A(LA) – COMPLEX (KIND=nag_wp) arrayInput

On entry: the values returned in the array A by a previous call to F11JNF.

5:     LA – INTEGERInput

On entry: the dimension of the arrays A, IROW and ICOL as declared in the (sub)program from which F11JQF is called. This must be the same value as was supplied in the preceding call to F11JNF.

Constraint: $\mathbf{LA}\ge 2\times \mathbf{NNZ}$.

6:     IROW(LA) – INTEGER arrayInput

7:     ICOL(LA) – INTEGER arrayInput

8:     IPIV(N) – INTEGER arrayInput

9:     ISTR($\mathbf{N}+1$) – INTEGER arrayInput

On entry: the values returned in arrays IROW, ICOL, IPIV and ISTR by a previous call to F11JNF.

10:   B(N) – COMPLEX (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 $k$ if

$${‖r_k‖}_{\infty }\le \tau \times \left({‖b‖}_{\infty }+{‖A‖}_{\infty }{‖x_k‖}_{\infty }\right)\text{.}$$

If $\mathbf{TOL}\le 0.0$, $\tau =\max \left(\sqrt{\epsilon },\sqrt{n}\epsilon \right)$ is used, where $\epsilon$ is the machine precision. Otherwise $\tau =\max \left(\mathbf{TOL},10\epsilon ,\sqrt{n}\epsilon \right)$ is used.

Constraint: $\mathbf{TOL}<1.0$.

12:   MAXITN – INTEGERInput

On entry: the maximum number of iterations allowed.

Constraint: $\mathbf{MAXITN}\ge 1$.

13:   X(N) – COMPLEX (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) – COMPLEX (KIND=nag_wp) arrayWorkspace

17:   LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F11JQF is called.

Constraints:

• if $\mathbf{METHOD}=\text{'CG'}$, $\mathbf{LWORK}\ge 6\times \mathbf{N}+120$;
• if $\mathbf{METHOD}=\text{'SYMMLQ'}$, $\mathbf{LWORK}\ge 7\times \mathbf{N}+120$.

18:   IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.

On exit: $\mathbf{IFAIL}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

$\mathbf{IFAIL}=1$

On entry,	$\mathbf{METHOD}\ne \text{'CG'}$ or $\text{'SYMMLQ'}$,
or	$\mathbf{N}<1$,
or	$\mathbf{NNZ}<1$,
or	$\mathbf{NNZ}>\mathbf{N}\times \left(\mathbf{N}+1\right)/2$,
or	LA too small,
or	$\mathbf{TOL}\ge 1.0$,
or	$\mathbf{MAXITN}<1$,
or	LWORK too small.

$\mathbf{IFAIL}=2$

On entry, the SCS representation of $A$ is invalid. Further details are given in the error message. Check that the call to F11JQF has been preceded by a valid call to F11JNF, and that the arrays A, IROW, and ICOL have not been corrupted between the two calls.

$\mathbf{IFAIL}=3$

On entry, the SCS representation of $M$ is invalid. Further details are given in the error message. Check that the call to F11JQF has been preceded by a valid call to F11JNF, and that the arrays A, IROW, ICOL, IPIV and ISTR have not been corrupted between the two calls.

$\mathbf{IFAIL}=4$

The required accuracy could not be obtained. However, a reasonable accuracy has been obtained and further iterations could not improve the result.

$\mathbf{IFAIL}=5$

Required accuracy not obtained in MAXITN iterations.

$\mathbf{IFAIL}=6$

The preconditioner appears not to be positive definite.

$\mathbf{IFAIL}=7$

The matrix of the coefficients appears not to be positive definite (conjugate gradient method only).

$\mathbf{IFAIL}=8$

A serious error has occurred in an internal call to an auxiliary routine. Check all subroutine calls and array sizes. Seek expert help.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (f11jqfe.f90)

9.2  Program Data

Program Data (f11jqfe.d)

9.3  Program Results

Program Results (f11jqfe.r)