f11dxc computes the approximate solution of a complex, Hermitian or non-Hermitian, sparse system of linear equations applying a number of Jacobi iterations. It is expected that f11dxc will be used as a preconditioner for the iterative solution of complex sparse systems of equations.

2 Specification

#include <nag.h>

void	f11dxc (Nag_SparseNsym_Store store, Nag_TransType trans, Nag_InitializeA init, Integer niter, Integer n, Integer nnz, const Complex a[], const Integer irow[], const Integer icol[], Nag_SparseNsym_CheckData check, const Complex b[], Complex x[], Complex diag[], NagError *fail)

The function may be called by the names: f11dxc, nag_sparse_complex_gen_precon_jacobi or nag_sparse_nherm_jacobi.

3 Description

f11dxc computes the approximate solution of the complex sparse system of linear equations

A x = b

using niter iterations of the Jacobi algorithm (see also Golub and Van Loan (1996) and Young (1971)):

x_{k + 1} = x_{k} + D^{- 1} (b - A x_{k})

(1)

where

k = 1, 2, \dots, niter

and

x_{0} = 0

f11dxc can be used both for non-Hermitian and Hermitian systems of equations. For Hermitian matrices, either all nonzero elements of the matrix

A

can be supplied using coordinate storage (CS), or only the nonzero elements of the lower triangle of

A

, using symmetric coordinate storage (SCS) (see the F11 Chapter Introduction).

It is expected that f11dxc will be used as a preconditioner for the iterative solution of complex sparse systems of equations. This may be with either the Hermitian or non-Hermitian suites of functions.

For Hermitian systems the suite consists of:

For non-Hermitian systems the suite consists of:

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

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

5 Arguments

1: $store$ – Nag_SparseNsym_Store Input

On entry: specifies whether the matrix

A

is stored using symmetric coordinate storage (SCS) (applicable only to a Hermitian matrix

A

) or coordinate storage (CS) (applicable to both Hermitian and non-Hermitian matrices).

$store = Nag_SparseNsym_StoreCS$: The complete matrix $A$ is stored in CS format.
$store = Nag_SparseNsym_StoreSCS$: The lower triangle of the Hermitian matrix $A$ is stored in SCS format.

Constraint:

store = Nag_SparseNsym_StoreCS

Nag_SparseNsym_StoreSCS

2: $trans$ – Nag_TransType Input

On entry: if

store = Nag_SparseNsym_StoreCS

, specifies whether the approximate solution of

A x = b

or of

A^{H} x = b

is required.

$trans = Nag_NoTrans$: The approximate solution of $A x = b$ is calculated.
$trans = Nag_Trans$: The approximate solution of $A^{H} x = b$ is calculated.

Suggested value: if the matrix

A

is Hermitian and stored in CS format, it is recommended that

trans = Nag_NoTrans

for reasons of efficiency.

Constraint:

trans = Nag_NoTrans

Nag_Trans

3: $init$ – Nag_InitializeA Input

On entry: on first entry, init should be set to

Nag_InitializeI

, unless the diagonal elements of

A

are already stored in the array diag. If diag already contains the diagonal of

A

, it must be set to

Nag_InputA

$init = Nag_InputA$: diag must contain the diagonal of $A$ .
$init = Nag_InitializeI$: diag will store the diagonal of $A$ on exit.

Suggested value:

init = Nag_InitializeI

on first entry;

init = Nag_InputA

, subsequently, unless diag has been overwritten.

Constraint:

init = Nag_InputA

Nag_InitializeI

4: $niter$ – Integer Input

On entry: the number of Jacobi iterations requested.

Constraint:

niter \geq 1

5: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 1

6: $nnz$ – Integer Input

On entry: if

store = Nag_SparseNsym_StoreCS

, the number of nonzero elements in the matrix

A

store = Nag_SparseNsym_StoreSCS

, the number of nonzero elements in the lower triangle of the matrix

A

Constraints:

if $store = Nag_SparseNsym_StoreCS$ , $1 \leq nnz \leq n^{2}$ ;
if $store = Nag_SparseNsym_StoreSCS$ , $1 \leq nnz \leq n \times (n + 1) / 2$ .

7: $a [nnz]$ – const Complex Input

On entry: if

store = Nag_SparseNsym_StoreCS

, the nonzero elements in the matrix

A

(CS format).

store = Nag_SparseNsym_StoreSCS

, the nonzero elements in the lower triangle of the matrix

A

(SCS format).

In both cases, the elements of either

A

or of its lower triangle must be ordered by increasing row index and by increasing column index within each row. Multiple entries for the same row and columns indices are not permitted. The function f11znc or f11zpc may be used to reorder the elements in this way for CS and SCS storage, respectively.

8: $irow [nnz]$ – const Integer Input

9: $icol [nnz]$ – const Integer Input

On entry: if

store = Nag_SparseNsym_StoreCS

, the row and column indices of the nonzero elements supplied in a.

store = Nag_SparseNsym_StoreSCS

, the row and column indices of the nonzero elements of the lower triangle of the matrix

A

supplied in a.

Constraints:

$1 \leq irow [i] \leq n$ , for $i = 0, 1, \dots, nnz - 1$ ;
if $store = Nag_SparseNsym_StoreCS$ , $1 \leq icol [i] \leq n$ , for $i = 0, 1, \dots, nnz - 1$ ;
if $store = Nag_SparseNsym_StoreSCS$ , $1 \leq icol [i] \leq irow [i]$ , for $i = 0, 1, \dots, nnz - 1$ ;
either $irow [i - 1] < irow [i]$ or both $irow [i - 1] = irow [i]$ and $icol [i - 1] < icol [i]$ , for $i = 1, 2, \dots, nnz - 1$ .

10: $check$ – Nag_SparseNsym_CheckData Input

On entry: specifies whether or not the CS or SCS representation of the matrix

A

should be checked.

$check = Nag_SparseNsym_Check$: Checks are carried out on the values of n, nnz, irow, icol; if $init = Nag_InputA$ , diag is also checked.
$check = Nag_SparseNsym_NoCheck$: None of these checks are carried out.

6 Error Indicators and Warnings

A nonzero element has been supplied which does not lie within the matrix $A$ , is out of order, or has duplicate row and column indices. Consider calling either f11zac or f11zbc to reorder and sum or remove duplicates when $store = Nag_SparseNsym_StoreCS$ or $store = Nag_SparseNsym_StoreSCS$ , respectively.
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_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .

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

On entry, $nnz = ⟨ value ⟩$ .
Constraint: $nnz \geq 1$ .
NE_INT_2: On entry, $nnz = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $nnz \leq n \times (n + 1) / 2$

On entry, $nnz = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $nnz \leq n^{2}$
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.
NE_INVALID_CS: 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, $i = ⟨ value ⟩$ , $icol [i - 1] = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $icol [i - 1] \geq 1$ and $icol [i - 1] \leq n$ .

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_ZERO_DIAG_ELEM: On entry, the diagonal element of the $I$ th row is zero or missing: $I = ⟨ value ⟩$ .

On entry, the element $diag [I - 1]$ is zero: $I = ⟨ value ⟩$ .

7 Accuracy

In general, the Jacobi method cannot be used on its own to solve systems of linear equations. The rate of convergence is bound by its spectral properties (see, for example, Golub and Van Loan (1996)) and as a solver, the Jacobi method can only be applied to a limited set of matrices. One condition that guarantees convergence is strict diagonal dominance.

However, the Jacobi method can be used successfully as a preconditioner to a wider class of systems of equations. The Jacobi method has good vector/parallel properties, hence it can be applied very efficiently. Unfortunately, it is not possible to provide criteria which define the applicability of the Jacobi method as a preconditioner, and its usefulness must be judged for each case.

8 Parallelism and Performance

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

f11dxc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f11dxc 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

9.1 Timing

The time taken for a call to f11dxc is proportional to

niter \times nnz

9.2 Use of check

It is expected that a common use of f11dxc will be as preconditioner for the iterative solution of complex, Hermitian or non-Hermitian, linear systems. In this situation, f11dxc is likely to be called many times. In the interests of both reliability and efficiency, you are recommended to set

check = Nag_SparseNsym_Check

for the first of such calls, and to set

check = Nag_SparseNsym_NoCheck

for all subsequent calls.

10 Example

This example solves the complex sparse non-Hermitian system of equations

A x = b

iteratively using f11dxc as a preconditioner.

f11dx: FL CL CPP AD PY MB

NAG CL Interfacef11dxc (complex_​gen_​precon_​jacobi)

▸▿ 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

9.1 Timing

9.2 Use of check

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
f11dxc (complex_gen_precon_jacobi)