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

Fortran Interface

Subroutine f11jqf (

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)	::	tol
Real (Kind=nag_wp), Intent (Out)	::	rnorm
Complex (Kind=nag_wp), Intent (In)	::	a(la), b(n)
Complex (Kind=nag_wp), Intent (Inout)	::	x(n)
Complex (Kind=nag_wp), Intent (Out)	::	work(lwork)
Character (*), Intent (In)	::	method

C Header Interface

#include <nag.h>

void

f11jqf_ (const char *method, const Integer *n, const Integer *nnz, const Complex a[], const Integer *la, const Integer irow[], const Integer icol[], Integer ipiv[], const Integer istr[], const Complex b[], const double *tol, const Integer *maxitn, Complex x[], double *rnorm, Integer *itn, Complex work[], const Integer *lwork, Integer *ifail, const Charlen length_method)

The routine may be called by the names f11jqf or nagf_sparse_complex_herm_solve_ilu.

3 Description

f11jqf solves a complex sparse Hermitian 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).

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 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

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: $n$ – Integer Input

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:

n \geq 1

3: $nnz$ – Integer Input

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

4: $a (la)$ – Complex (Kind=nag_wp) array Input

On entry: the values returned in the array a by a previous call to f11jnf.

5: $la$ – Integer Input

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:

la \geq 2 \times nnz

6: $irow (la)$ – Integer array Input

7: $icol (la)$ – Integer array Input

8: $ipiv (n)$ – Integer array Input

9: $istr (n + 1)$ – Integer array Input

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) array Input

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: $maxitn$ – Integer Input

On entry: the maximum number of iterations allowed.

Constraint:

maxitn \geq 1

13: $x (n)$ – Complex (Kind=nag_wp) array Input/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: $itn$ – Integer Output

On exit: the number of iterations carried out.

16: $work (lwork)$ – Complex (Kind=nag_wp) array Workspace

17: $lwork$ – Integer Input

On entry: the dimension of the array work as declared in the (sub)program from which f11jqf is called.

Constraints:

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

18: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. 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, $i = ⟨ value ⟩$ , $irow (i) = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
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 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.

$ifail = 3$: On entry, istr appears to be invalid.
Check that a, irow, icol, ipiv and istr have not been corrupted between calls to f11jnf.

$ifail = 4$: 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, code $⟨ value ⟩$ , has occurred in an internal call. 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 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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

Background information to multithreading can be found in the Multithreading documentation.

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 implementation-specific information.

9 Further Comments

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

\bar{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.

f11jq: FL CL CPP AD PY MB

NAG FL Interfacef11jqf (complex_​herm_​solve_​ilu)

▸▿ 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

NAG FL Interface
f11jqf (complex_herm_solve_ilu)