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>

void	f11jqc (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)

The function may be called by the names: f11jqc, nag_sparse_complex_herm_solve_ilu or nag_sparse_herm_chol_sol.

3 Description

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).

f11jqc uses the incomplete Cholesky factorization determined by f11jnc as the preconditioning matrix. A call to f11jqc must always be preceded by a call to f11jnc. Alternative preconditioners for the same storage scheme are available by calling 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 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_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: $n$ – 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 f11jnc.

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 f11jnc.

Constraint:

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

4: $a [la]$ – const Complex Input

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

5: $la$ – Integer Input

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 f11jnc.

Constraint:

la \geq 2 \times nnz

6: $irow [la]$ – const Integer Input

7: $icol [la]$ – const Integer Input

8: $ipiv [n]$ – const Integer Input

9: $istr [n + 1]$ – const Integer Input

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

10: $b [n]$ – const Complex Input

On entry: the right-hand side vector

b

11: $tol$ – 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

12: $maxitn$ – Integer Input

On entry: the maximum number of iterations allowed.

Constraint:

maxitn \geq 1

13: $x [n]$ – Complex Input/Output

On entry: an initial approximation to the solution vector

x

On exit: an improved approximation to the solution vector

x

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 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Check that the call to f11jqc has been preceded by a valid call to 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 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 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, $la = ⟨ value ⟩$ 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 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, $i = ⟨ value ⟩$ , $icol [i - 1] = ⟨ value ⟩$ , $irow [i - 1] = ⟨ value ⟩$ .
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 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, $tol = ⟨ value ⟩$ .
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

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

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 f11jqc for each iteration is roughly proportional to the value of nnzc returned from the preceding call to 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.

f11jq: FL CL CPP AD

NAG CL Interfacef11jqc (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 CL Interface
f11jqc (complex_herm_solve_ilu)