NAG Library Function Document

nag_sparse_nherm_precon_bdilu_solve (f11duc)

+− 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

nag_sparse_nherm_precon_bdilu_solve (f11duc) solves a complex sparse non-Hermitian system of linear equations, represented in coordinate storage format, using a restarted generalized minimal residual (RGMRES), conjugate gradient squared (CGS), stabilized bi-conjugate gradient (Bi-CGSTAB), or transpose-free quasi-minimal residual (TFQMR) method, with block Jacobi or additive Schwarz preconditioning.

2 Specification

#include <nag.h>

#include <nagf11.h>

void

nag_sparse_nherm_precon_bdilu_solve (Nag_SparseNsym_Method method, Integer n, Integer nnz, const Complex a[], Integer la, const Integer irow[], const Integer icol[], Integer nb, const Integer istb[], const Integer indb[], Integer lindb, const Integer ipivp[], const Integer ipivq[], const Integer istr[], const Integer idiag[], const Complex b[], Integer m, double tol, Integer maxitn, Complex x[], double *rnorm, Integer *itn, NagError *fail)

3 Description

nag_sparse_nherm_precon_bdilu_solve (f11duc) solves a complex sparse non-Hermitian linear system of equations

A x = b,

using a preconditioned RGMRES (see Saad and Schultz (1986)), CGS (see Sonneveld (1989)), Bi-CGSTAB(

ℓ

) (see Van der Vorst (1989) and Sleijpen and Fokkema (1993)), or TFQMR (see Freund and Nachtigal (1991) and Freund (1993)) method.

nag_sparse_nherm_precon_bdilu_solve (f11duc) uses the incomplete (possibly overlapping) block

L U

factorization determined by nag_sparse_nherm_precon_bdilu (f11dtc) as the preconditioning matrix. A call to nag_sparse_nherm_precon_bdilu_solve (f11duc) must always be preceded by a call to nag_sparse_nherm_precon_bdilu (f11dtc). Alternative preconditioners for the same storage scheme are available by calling nag_sparse_nherm_fac_sol (f11dqc) or nag_sparse_nherm_sol (f11dsc).

The matrix

A

, and the preconditioning matrix

M

, are represented in coordinate storage (CS) format (see Section 2.1.1 in the f11 Chapter Introduction) in the arrays a, irow and icol, as returned from nag_sparse_nherm_precon_bdilu (f11dtc). The array a holds the nonzero entries in these matrices, while irow and icol hold the corresponding row and column indices.

nag_sparse_nherm_precon_bdilu_solve (f11duc) is a Black Box function which calls nag_sparse_nherm_basic_setup (f11brc), nag_sparse_nherm_basic_solver (f11bsc) and nag_sparse_nherm_basic_diagnostic (f11btc). If you wish to use an alternative storage scheme, preconditioner, or termination criterion, or require additional diagnostic information, you should call these underlying functions directly.

4 References

Freund R W (1993) A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems SIAM J. Sci. Comput. 14 470–482

Freund R W and Nachtigal N (1991) QMR: a Quasi-Minimal Residual Method for Non-Hermitian Linear Systems Numer. Math. 60 315–339

Saad Y and Schultz M (1986) GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems SIAM J. Sci. Statist. Comput. 7 856–869

Sleijpen G L G and Fokkema D R (1993) BiCGSTAB

(ℓ)

for linear equations involving matrices with complex spectrum ETNA 1 11–32

Sonneveld P (1989) CGS, a fast Lanczos-type solver for nonsymmetric linear systems SIAM J. Sci. Statist. Comput. 10 36–52

Van der Vorst H (1989) Bi-CGSTAB, a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems SIAM J. Sci. Statist. Comput. 13 631–644

5 Arguments

1: method – Nag_SparseNsym_MethodInput

On entry: specifies the iterative method to be used.

$method = Nag_SparseNsym_RGMRES$: Restarted generalized minimum residual method.
$method = Nag_SparseNsym_CGS$: Conjugate gradient squared method.
$method = Nag_SparseNsym_BiCGSTAB$: Bi-conjugate gradient stabilized ( $ℓ$ ) method.
$method = Nag_SparseNsym_TFQMR$: Transpose-free quasi-minimal residual method.

Constraint:

method = Nag_SparseNsym_RGMRES

Nag_SparseNsym_CGS

Nag_SparseNsym_BiCGSTAB

Nag_SparseNsym_TFQMR

2: n – IntegerInput

3: nnz – IntegerInput

4: a[la] – const ComplexInput

5: la – IntegerInput

6: irow[la] – const IntegerInput

7: icol[la] – const IntegerInput

8: nb – IntegerInput

9: istb[ $nb + 1$ ] – const IntegerInput

10: indb[lindb] – const IntegerInput

11: lindb – IntegerInput

12: ipivp[lindb] – const IntegerInput

13: ipivq[lindb] – const IntegerInput

14: istr[ $lindb + 1$ ] – const IntegerInput

15: idiag[lindb] – const IntegerInput

On entry: the values returned in arrays irow, icol, ipivp, ipivq, istr and idiag by a previous call to nag_sparse_nherm_precon_bdilu (f11dtc).

The arrays istb, indb and a together with the scalars n, nnz, la, nb and lindb must be the same values that were supplied in the preceding call to nag_sparse_nherm_precon_bdilu (f11dtc).

On entry: the values returned in arrays irow, icol, ipivp, ipivq, istr and idiag by a previous call to nag_sparse_nherm_precon_bdilu (f11dtc).

The arrays istb, indb and the scalars nb and lindb must be the same values that were supplied in the preceding call to nag_sparse_nherm_precon_bdilu (f11dtc).

16: b[n] – const ComplexInput

On entry: the right-hand side vector

b

17: m – IntegerInput

On entry: if

method = Nag_SparseNsym_RGMRES

, m is the dimension of the restart subspace.

method = Nag_SparseNsym_BiCGSTAB

, m is the order

ℓ

of the polynomial Bi-CGSTAB method.

Otherwise, m is not referenced.

Constraints:

if $method = Nag_SparseNsym_RGMRES$ , $0 < m \leq \min (n, 50)$ ;
if $method = Nag_SparseNsym_BiCGSTAB$ , $0 < m \leq \min (n, 10)$ .

18: 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}, \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

19: maxitn – IntegerInput

On entry: the maximum number of iterations allowed.

Constraint:

maxitn \geq 1

20: x[n] – ComplexInput/Output

On entry: an initial approximation to the solution vector

x

On exit: an improved approximation to the solution vector

x

21: rnorm – double *Output

On exit: the final value of the residual norm

{‖r_{k}‖}_{\infty}

, where

k

is the output value of itn.

22: itn – Integer *Output

On exit: the number of iterations carried out.

23: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ACCURACY: The required accuracy could not be obtained. However a reasonable accuracy may have been achieved. You should check the output value of rnorm for acceptability. This error code usually implies that your problem has been fully and satisfactorily solved to within or close to the accuracy available on your system. Further iterations are unlikely to improve on this situation.
NE_ALG_FAIL: Algorithmic breakdown. A solution is returned, although it is possible that it is completely inaccurate.
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_CONVERGENCE: The solution has not converged after $⟨value⟩$ iterations.
NE_INT: On entry, $istb [0] = ⟨value⟩$ .
Constraint: $istb [0] \geq 1$ .
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, $nb = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $1 \leq nb \leq n$ .
On entry, $nnz = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $nnz \leq n^{2}$ .
NE_INT_3: On entry, $lindb = ⟨value⟩$ , $istb [nb] - 1 = ⟨value⟩$ and $nb = ⟨value⟩$ .
Constraint: $lindb \geq istb [nb] - 1$ .
On entry, $m = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: if $method = Nag_SparseNsym_RGMRES$ , $1 \leq m \leq \min (n, ⟨value⟩)$ .
If $method = Nag_SparseNsym_BiCGSTAB$ , $1 \leq m \leq \min (n, ⟨value⟩)$ .
NE_INT_ARRAY: On entry, $indb [⟨value⟩] = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $1 \leq indb [m - 1] \leq n$ , for $m = istb [0], istb [0] + 1, \dots, istb [nb] - 1$ .
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.
NE_INVALID_CS: On entry, $icol [⟨value⟩] = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $1 \leq icol [i - 1] \leq n$ , for $i = 1, 2, \dots, nnz$ .
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nherm_precon_bdilu (f11dtc) and nag_sparse_nherm_precon_bdilu_solve (f11duc).
On entry, $irow [⟨value⟩] = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $1 \leq irow [i - 1] \leq n$ , for $i = 1, 2, \dots, nnz$ .
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nherm_precon_bdilu (f11dtc) and nag_sparse_nherm_precon_bdilu_solve (f11duc).
NE_INVALID_CS_PRECOND: The CS representation of the preconditioner is invalid.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nherm_precon_bdilu (f11dtc) and nag_sparse_nherm_precon_bdilu_solve (f11duc).
NE_NOT_STRICTLY_INCREASING: On entry, element $⟨value⟩$ of a was out of order.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nherm_precon_bdilu (f11dtc) and nag_sparse_nherm_precon_bdilu_solve (f11duc).
On entry, for $b = ⟨value⟩$ , $istb [b] = ⟨value⟩$ and $istb [b - 1] = ⟨value⟩$ .
Constraint: $istb [b] > istb [b - 1]$ , for $b = 1, 2, \dots, nb$ .
On entry, location $⟨value⟩$ of $(irow, icol)$ was a duplicate.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nherm_precon_bdilu (f11dtc) and nag_sparse_nherm_precon_bdilu_solve (f11duc).
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_nherm_precon_bdilu_solve (f11duc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_sparse_nherm_precon_bdilu_solve (f11duc) 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 Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The time taken by nag_sparse_nherm_precon_bdilu_solve (f11duc) for each iteration is roughly proportional to the value of nnzc returned from the preceding call to nag_sparse_nherm_precon_bdilu (f11dtc).

The number of iterations required to achieve a prescribed accuracy cannot be easily determined a priori, as it can depend dramatically on the conditioning and spectrum of the preconditioned coefficient matrix

\bar{A} = M^{- 1} A

10 Example

This example program reads in a sparse matrix

A

and a vector

b

. It calls nag_sparse_nherm_precon_bdilu (f11dtc), with the array

lfill = 0

and the array

dtol = 0.0

, to compute an overlapping incomplete

L U

factorization. This is then used as an additive Schwarz preconditioner on a call to nag_sparse_nherm_precon_bdilu_solve (f11duc) which uses the RGMRES method to solve

A x = b

NAG Library Function Documentnag_sparse_nherm_precon_bdilu_solve (f11duc)

+− 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 Library Function Document

nag_sparse_nherm_precon_bdilu_solve (f11duc)