NAG Library Routine Document
F11JQF
1 Purpose
F11JQF solves a complex sparse Hermitian system of linear equations, represented in symmetric coordinate storage format, using a conjugate gradient or Lanczos method, with incomplete Cholesky preconditioning.
2 Specification
SUBROUTINE F11JQF ( 
METHOD, N, NNZ, A, LA, IROW, ICOL, IPIV, ISTR, B, TOL, MAXITN, X, RNORM, ITN, WORK, LWORK, IFAIL) 
INTEGER 
N, NNZ, LA, IROW(LA), ICOL(LA), IPIV(N), ISTR(N+1), MAXITN, ITN, LWORK, IFAIL 
REAL (KIND=nag_wp) 
TOL, RNORM 
COMPLEX (KIND=nag_wp) 
A(LA), B(N), X(N), WORK(LWORK) 
CHARACTER(*) 
METHOD 

3 Description
F11JQF solves a complex sparse Hermitian linear system of equations
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).
F11JQF uses the incomplete Cholesky factorization determined by
F11JNF as the preconditioning matrix. A call to F11JQF must always be preceded by a call to
F11JNF. Alternative preconditioners for the same storage scheme are available by calling
F11JSF.
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
F11JNF. 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 Mmatrix 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
5 Parameters
 1: $\mathrm{METHOD}$ – 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: $\mathrm{N}$ – 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: $\mathrm{NNZ}$ – 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: $\mathrm{A}\left({\mathbf{LA}}\right)$ – COMPLEX (KIND=nag_wp) arrayInput

On entry: the values returned in the array
A by a previous call to
F11JNF.
 5: $\mathrm{LA}$ – 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: $\mathrm{IROW}\left({\mathbf{LA}}\right)$ – INTEGER arrayInput
 7: $\mathrm{ICOL}\left({\mathbf{LA}}\right)$ – INTEGER arrayInput
 8: $\mathrm{IPIV}\left({\mathbf{N}}\right)$ – INTEGER arrayInput
 9: $\mathrm{ISTR}\left({\mathbf{N}}+1\right)$ – INTEGER arrayInput

On entry: the values returned in arrays
IROW,
ICOL,
IPIV and
ISTR by a previous call to
F11JNF.
 10: $\mathrm{B}\left({\mathbf{N}}\right)$ – COMPLEX (KIND=nag_wp) arrayInput

On entry: the righthand side vector $b$.
 11: $\mathrm{TOL}$ – 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
If
${\mathbf{TOL}}\le 0.0$,
$\tau =\mathrm{max}\phantom{\rule{0.25em}{0ex}}\sqrt{\epsilon},10\epsilon ,\sqrt{n}\epsilon $ is used, where
$\epsilon $ is the
machine precision. Otherwise
$\tau =\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{TOL}},10\epsilon ,\sqrt{n}\epsilon \right)$ is used.
Constraint:
${\mathbf{TOL}}<1.0$.
 12: $\mathrm{MAXITN}$ – INTEGERInput

On entry: the maximum number of iterations allowed.
Constraint:
${\mathbf{MAXITN}}\ge 1$.
 13: $\mathrm{X}\left({\mathbf{N}}\right)$ – 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: $\mathrm{RNORM}$ – REAL (KIND=nag_wp)Output

On exit: the final value of the residual norm
${\Vert {r}_{k}\Vert}_{\infty}$, where
$k$ is the output value of
ITN.
 15: $\mathrm{ITN}$ – INTEGEROutput

On exit: the number of iterations carried out.
 16: $\mathrm{WORK}\left({\mathbf{LWORK}}\right)$ – COMPLEX (KIND=nag_wp) arrayWorkspace
 17: $\mathrm{LWORK}$ – 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: $\mathrm{IFAIL}$ – 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
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$

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.
 ${\mathbf{IFAIL}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.8 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.7 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=999$
Dynamic memory allocation failed.
See
Section 3.6 in the Essential Introduction for further information.
7 Accuracy
On successful termination, the final residual
${r}_{k}=bA{x}_{k}$, where
$k={\mathbf{ITN}}$, satisfies the termination criterion
The value of the final residual norm is returned in
RNORM.
8 Parallelism and Performance
F11JQF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F11JQF 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 implementationspecific information.
The time taken by F11JQF for each iteration is roughly proportional to the value of
NNZC returned from the preceding call to
F11JNF. 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 easily be determined a priori, as it can depend dramatically on the conditioning and spectrum of the preconditioned matrix of the coefficients $\stackrel{}{A}={M}^{1}A$.
10 Example
This example solves a complex sparse Hermitian positive definite system of equations using the conjugate gradient method, with incomplete Cholesky preconditioning.
10.1 Program Text
Program Text (f11jqfe.f90)
10.2 Program Data
Program Data (f11jqfe.d)
10.3 Program Results
Program Results (f11jqfe.r)