f11dkc computes the approximate solution of a real, symmetric or nonsymmetric, sparse system of linear equations applying a number of Jacobi iterations. It is expected that f11dkc will be used as a preconditioner for the iterative solution of real sparse systems of equations.

2 Specification

#include <nag.h>

void	f11dkc (Nag_SparseNsym_Store store, Nag_TransType trans, Nag_InitializeA init, Integer niter, Integer n, Integer nnz, const double a[], const Integer irow[], const Integer icol[], Nag_SparseNsym_CheckData check, const double b[], double x[], double diag[], NagError *fail)

The function may be called by the names: f11dkc, nag_sparse_real_gen_precon_jacobi or nag_sparse_nsym_jacobi.

3 Description

f11dkc computes the approximate solution of the real 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

f11dkc can be used both for nonsymmetric and symmetric systems of equations. For symmetric 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 f11dkc will be used as a preconditioner for the iterative solution of real sparse systems of equations. This may be with either the symmetric or nonsymmetric suites of functions.

For symmetric systems the suite consists of:

For nonsymmetric 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 symmetric matrix

A

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

$store = Nag_SparseNsym_StoreCS$: The complete matrix $A$ is stored in CS format.
$store = Nag_SparseNsym_StoreSCS$: The lower triangle of the symmetric 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^{T} x = b

is required.

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

Suggested value: if the matrix

A

is symmetric 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 double 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 f11zac or f11zbc 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

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.

f11dkc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f11dkc 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 f11dkc is proportional to

niter \times nnz

9.2 Use of check

It is expected that a common use of f11dkc will be as preconditioner for the iterative solution of real, symmetric or nonsymmetric, linear systems. In this situation, f11dkc 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 real sparse nonsymmetric system of equations

A x = b

iteratively using f11dkc as a preconditioner.

f11dk: FL CL CPP AD PY MB

NAG CL Interfacef11dkc (real_​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
f11dkc (real_gen_precon_jacobi)