NAG C Library Function Document

nag_sparse_herm_chol_sol (f11jqc) 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

#include <nag.h>

#include <nagf11.h>

void

nag_sparse_herm_chol_sol (Nag_SparseSym_Method method, Integer n, Integer nnz, const Complex a[], Integer la, const Integer irow[], const Integer icol[], const Integer ipiv[], const Integer istr[], const Complex b[], double tol, Integer maxitn, Complex x[], double *rnorm, Integer *itn, NagError *fail)

3

Description

nag_sparse_herm_chol_sol (f11jqc) 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).

nag_sparse_herm_chol_sol (f11jqc) uses the incomplete Cholesky factorization determined by nag_sparse_herm_chol_fac (f11jnc) as the preconditioning matrix. A call to nag_sparse_herm_chol_sol (f11jqc) must always be preceded by a call to nag_sparse_herm_chol_fac (f11jnc). Alternative preconditioners for the same storage scheme are available by calling nag_sparse_herm_sol (f11jsc).

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 nag_sparse_herm_chol_fac (f11jnc). 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: $method$ – Nag_SparseSym_MethodInput

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: $n$ – IntegerInput

On entry:

n

, the order of the matrix

A

. This must be the same value as was supplied in the preceding call to nag_sparse_herm_chol_fac (f11jnc).

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 nag_sparse_herm_chol_fac (f11jnc).

Constraint:

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

4: $a [la]$ – const ComplexInput

On entry: the values returned in the array a by a previous call to nag_sparse_herm_chol_fac (f11jnc).

5: $la$ – IntegerInput

On entry: the dimension of the arrays a, irow and icol. This must be the same value as was supplied in the preceding call to nag_sparse_herm_chol_fac (f11jnc).

Constraint:

la \geq 2 \times nnz

6: $irow [la]$ – const IntegerInput

7: $icol [la]$ – const IntegerInput

8: $ipiv [n]$ – const IntegerInput

9: $istr [n + 1]$ – const IntegerInput

On entry: the values returned in arrays irow, icol, ipiv and istr by a previous call to nag_sparse_herm_chol_fac (f11jnc).

10: $b [n]$ – const ComplexInput

On entry: the right-hand side vector

b

11: $tol$ – doubleInput

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: $x [n]$ – ComplexInput/Output

On entry: an initial approximation to the solution vector

x

On exit: an improved approximation to the solution vector

x

14: $rnorm$ – double *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: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

Check that the call to nag_sparse_herm_chol_sol (f11jqc) has been preceded by a valid call to nag_sparse_herm_chol_fac (f11jnc), and that the arrays a, irow, and icol have not been corrupted between the two calls.; Check that a, irow, icol, ipiv and istr have not been corrupted between calls to nag_sparse_herm_chol_fac (f11jnc).
NE_ACCURACY: The required accuracy could not be obtained. However a reasonable accuracy has been achieved.
NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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, $la = 〈value〉$ and $nnz = 〈value〉$ .
Constraint: $la \geq 2 \times nnz$ .

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 2.7.6 in How to Use the NAG Library and its Documentation 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, $i = 〈value〉$ , $icol [i - 1] = 〈value〉$ , $irow [i - 1] = 〈value〉$ .
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_INVALID_SCS_PRECOND: On entry, istr appears to be invalid.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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, $tol = 〈value〉$ .
Constraint: $tol < 1.0$ .

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

nag_sparse_herm_chol_sol (f11jqc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_sparse_herm_chol_sol (f11jqc) 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 nag_sparse_herm_chol_sol (f11jqc) for each iteration is roughly proportional to the value of nnzc returned from the preceding call to nag_sparse_herm_chol_fac (f11jnc). 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.

NAG C Library Function Document

nag_sparse_herm_chol_sol (f11jqc)

▸▿ 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