f11jsc solves a complex sparse Hermitian system of linear equations, represented in symmetric coordinate storage format, using a conjugate gradient or Lanczos method, without preconditioning, with Jacobi or with SSOR preconditioning.

2 Specification

#include <nag.h>

void	f11jsc (Nag_SparseSym_Method method, Nag_SparseSym_PrecType precon, Integer n, Integer nnz, const Complex a[], const Integer irow[], const Integer icol[], double omega, const Complex b[], double tol, Integer maxitn, Complex x[], double rnorm, Integer itn, double rdiag[], NagError *fail)

The function may be called by the names: f11jsc, nag_sparse_complex_herm_solve_jacssor or nag_sparse_herm_sol.

3 Description

f11jsc solves a complex sparse Hermitian linear system of equations

A x = b,

using a preconditioned conjugate gradient method (see Barrett et al. (1994)), 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).

f11jsc allows the following choices for the preconditioner:

–no preconditioning;
–Jacobi preconditioning (see Young (1971));
–symmetric successive-over-relaxation (SSOR) preconditioning (see Young (1971)).

For incomplete Cholesky (IC) preconditioning see f11jqc.

The matrix

A

is represented in symmetric coordinate storage (SCS) format (see Section 2.1.2 in the F11 Chapter Introduction) in the arrays a, irow and icol. The array a holds the nonzero entries in the lower triangular part of the matrix, 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

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

Young D (1971) Iterative Solution of Large Linear Systems Academic Press, New York

5 Arguments

1: $method$ – Nag_SparseSym_Method Input

On entry: specifies the iterative method to be used.

$method = Nag_SparseSym_CG$: Conjugate gradient method.
$method = Nag_SparseSym_SYMMLQ$: Lanczos method (SYMMLQ).

Constraint:

method = Nag_SparseSym_CG

Nag_SparseSym_SYMMLQ

2: $precon$ – Nag_SparseSym_PrecType Input

On entry: specifies the type of preconditioning to be used.

$precon = Nag_SparseSym_NoPrec$: No preconditioning.
$precon = Nag_SparseSym_JacPrec$: Jacobi.
$precon = Nag_SparseSym_SSORPrec$: Symmetric successive-over-relaxation (SSOR).

Constraint:

precon = Nag_SparseSym_NoPrec

Nag_SparseSym_JacPrec

Nag_SparseSym_SSORPrec

3: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 1

4: $nnz$ – Integer Input

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

A

Constraint:

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

5: $a [nnz]$ – const Complex Input

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

A

, ordered by increasing row index, and by increasing column index within each row. Multiple entries for the same row and column indices are not permitted. The function f11zpc may be used to order the elements in this way.

6: $irow [nnz]$ – const Integer Input

7: $icol [nnz]$ – const Integer Input

On entry: the row and column indices of the nonzero elements supplied in array a.

Constraints:

irow and icol must satisfy these constraints (which may be imposed by a call to f11zpc):

$1 \leq irow [i] \leq n$ and $1 \leq icol [i] \leq irow [i]$ , for $i = 0, 1, \dots, nnz - 1$ ;
$irow [i - 1] < irow [i]$ or $irow [i - 1] = irow [i]$ and $icol [i - 1] < icol [i]$ , for $i = 1, 2, \dots, nnz - 1$ .

8: $omega$ – double Input

On entry: if

precon = Nag_SparseSym_SSORPrec

, omega is the relaxation parameter

ω

to be used in the SSOR method. Otherwise omega need not be initialized.

Constraint:

0.0 < omega < 2.0

9: $b [n]$ – const Complex Input

On entry: the right-hand side vector

b

10: $tol$ – double 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

11: $maxitn$ – Integer Input

On entry: the maximum number of iterations allowed.

Constraint:

maxitn \geq 1

12: $x [n]$ – Complex Input/Output

On entry: an initial approximation to the solution vector

x

On exit: an improved approximation to the solution vector

x

13: $rnorm$ – double * Output

On exit: the final value of the residual norm

‖ r_{k} ‖

, where

k

is the output value of itn.

14: $itn$ – Integer * Output

On exit: the number of iterations carried out.

15: $rdiag [n]$ – double Output

On exit: the elements of the diagonal matrix

D^{- 1}

, where

D

is the diagonal part of

A

. Note that since

A

is Hermitian the elements of

D^{- 1}

are necessarily real.

16: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

A nonzero element has been supplied which does not lie in the lower triangular part of $A$ , is out of order, or has duplicate row and column indices. Consider calling f11zpc to reorder and sum or remove duplicates.
NE_ACCURACY: The required accuracy could not be obtained. However, a reasonable accuracy has been achieved and further iterations could not improve the result.
NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_COEFF_NOT_POS_DEF: The matrix of the coefficients a appears not to be positive definite. The computation cannot continue.
NE_CONVERGENCE: The solution has not converged after $⟨ value ⟩$ iterations.
NE_INT: On entry, $maxitn = ⟨ value ⟩$ .
Constraint: $maxitn \geq 1$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .

On entry, $nnz = ⟨ value ⟩$ .
Constraint: $nnz \geq 1$ .
NE_INT_2: On entry, $nnz = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $nnz \leq n \times (n + 1) / 2$
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.

A serious error, code $⟨ value ⟩$ , has occurred in an internal call. Check all function calls and array sizes. Seek expert help.
NE_INVALID_SCS: On entry, $I = ⟨ value ⟩$ , $icol [I - 1] = ⟨ value ⟩$ and $irow [I - 1] = ⟨ value ⟩$ .
Constraint: $icol [I - 1] \geq 1$ and $icol [I - 1] \leq irow [I - 1]$ .

On entry, $i = ⟨ value ⟩$ , $irow [i - 1] = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $irow [i - 1] \geq 1$ and $irow [i - 1] \leq n$ .
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_NOT_STRICTLY_INCREASING: On entry, $a [i - 1]$ is out of order: $i = ⟨ value ⟩$ .

On entry, the location ( $irow [I - 1], icol [I - 1]$ ) is a duplicate: $I = ⟨ value ⟩$ .
NE_PRECOND_NOT_POS_DEF: The preconditioner appears not to be positive definite. The computation cannot continue.
NE_REAL: On entry, $omega = ⟨ value ⟩$ .
Constraint: $0.0 < omega < 2.0$ .

On entry, $tol = ⟨ value ⟩$ .
Constraint: $tol < 1.0$ .
NE_ZERO_DIAG_ELEM: The matrix $A$ has a non-real diagonal entry in row $⟨ value ⟩$ .

The matrix $A$ has a zero diagonal entry in row $⟨ value ⟩$ .

The matrix $A$ has no diagonal entry in row $⟨ value ⟩$ .

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.

f11jsc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f11jsc 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The time taken by f11jsc for each iteration is roughly proportional to nnz. 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 SSOR preconditioning.

f11js: FL CL CPP AD PY MB

NAG CL Interfacef11jsc (complex_​herm_​solve_​jacssor)

▸▿ 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 CL Interface
f11jsc (complex_herm_solve_jacssor)