NAG Library Function Document

nag_sparse_nsym_sol (f11dec)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

1:    $\mathbf{method}$ – Nag_SparseNsym_MethodInput

On entry: specifies the iterative method to be used.

$\mathbf{method}=\mathrm{Nag\_SparseNsym\_RGMRES}$

The restarted generalized minimum residual method is used.

$\mathbf{method}=\mathrm{Nag\_SparseNsym\_CGS}$

The conjugate gradient squared method is used.

$\mathbf{method}=\mathrm{Nag\_SparseNsym\_BiCGSTAB}$

The bi-conjugate gradient stabilised $\left(\ell \right)$ method is used.

Constraint: $\mathbf{method}=\mathrm{Nag\_SparseNsym\_RGMRES}$, $\mathrm{Nag\_SparseNsym\_CGS}$ or $\mathrm{Nag\_SparseNsym\_BiCGSTAB}$.

2:    $\mathbf{precon}$ – Nag_SparseNsym_PrecTypeInput

On entry: specifies the type of preconditioning to be used.

$\mathbf{precon}=\mathrm{Nag\_SparseNsym\_NoPrec}$

No preconditioning.

$\mathbf{precon}=\mathrm{Nag\_SparseNsym\_SSORPrec}$

Symmetric successive-over-relaxation.

$\mathbf{precon}=\mathrm{Nag\_SparseNsym\_JacPrec}$

Jacobi.

Constraint: $\mathbf{precon}=\mathrm{Nag\_SparseNsym\_NoPrec}$, $\mathrm{Nag\_SparseNsym\_SSORPrec}$ or $\mathrm{Nag\_SparseNsym\_JacPrec}$.

3:    $\mathbf{n}$ – IntegerInput

On entry: the order of the matrix $A$.

Constraint: $\mathbf{n}\ge 1$.

4:    $\mathbf{nnz}$ – IntegerInput

On entry: the number of nonzero elements in the matrix $A$.

Constraint: $1\le \mathbf{nnz}\le {\mathbf{n}}^2$.

5:    $\mathbf{a}\left[\mathbf{nnz}\right]$ – const doubleInput

On entry: the nonzero elements of the matrix $A$, ordered by increasing row index, and by increasing column index within each row. Multiple entries for the same row and column indices are not permitted. The function nag_sparse_nsym_sort (f11zac) may be used to order the elements in this way.

6:    $\mathbf{irow}\left[\mathbf{nnz}\right]$ – const IntegerInput

7:    $\mathbf{icol}\left[\mathbf{nnz}\right]$ – const IntegerInput

On entry: the row and column indices of the nonzero elements supplied in a.

Constraints:

• irow and icol must satisfy the following constraints (which may be imposed by a call to nag_sparse_nsym_sort (f11zac)):;
• $1\le \mathbf{irow}\left[\mathit{i}\right]\le \mathbf{n}$ and $1\le \mathbf{icol}\left[\mathit{i}\right]\le \mathbf{n}$, for $\mathit{i}=0,1,\dots ,\mathbf{nnz}-1$;
• $\mathbf{irow}\left[\mathit{i}-1\right]<\mathbf{irow}\left[\mathit{i}\right]$ or $\mathbf{irow}\left[\mathit{i}-1\right]=\mathbf{irow}\left[\mathit{i}\right]$ and $\mathbf{icol}\left[\mathit{i}-1\right]<\mathbf{icol}\left[\mathit{i}\right]$, for $\mathit{i}=1,2,\dots ,\mathbf{nnz}-1$.

8:    $\mathbf{omega}$ – doubleInput

On entry: if $\mathbf{precon}=\mathrm{Nag\_SparseNsym\_SSORPrec}$, omega is the relaxation argument $\omega$ to be used in the SSOR method. Otherwise omega need not be initialized and is not referenced.

Constraint: $0.0<\mathbf{omega}<2.0$.

9:    $\mathbf{b}\left[\mathbf{n}\right]$ – const doubleInput

On entry: the right-hand side vector $b$.

10:  $\mathbf{m}$ – IntegerInput

On entry: if $\mathbf{method}=\mathrm{Nag\_SparseNsym\_RGMRES}$, m is the dimension of the restart subspace.

If $\mathbf{method}=\mathrm{Nag\_SparseNsym\_BiCGSTAB}$, m is the order $\ell$ of the polynomial Bi-CGSTAB method; otherwise m is not referenced.

Constraints:

• if $\mathbf{method}=\mathrm{Nag\_SparseNsym\_RGMRES}$, $0<\mathbf{m}\le \min \left(\mathbf{n},50\right)$;
• if $\mathbf{method}=\mathrm{Nag\_SparseNsym\_BiCGSTAB}$, $0<\mathbf{m}\le \min \left(\mathbf{n},10\right)$.

11:  $\mathbf{tol}$ – 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 $k$ if:

$${‖r_k‖}_{\infty }\le \tau \times \left({‖b‖}_{\infty }+{‖A‖}_{\infty }{‖x_k‖}_{\infty }\right)\text{.}$$

If $\mathbf{tol}\le 0.0$, $\tau =\max \left(\sqrt{\epsilon },\sqrt{\mathbf{n}},\epsilon \right)$ is used, where $\epsilon$ is the machine precision. Otherwise $\tau =\max \left(\mathbf{tol},10\epsilon ,,,\sqrt{\mathbf{n}},\epsilon \right)$ is used.

Constraint: $\mathbf{tol}<1.0$.

12:  $\mathbf{maxitn}$ – IntegerInput

On entry: the maximum number of iterations allowed.

Constraint: $\mathbf{maxitn}\ge 1$.

13:  $\mathbf{x}\left[\mathbf{n}\right]$ – doubleInput/Output

On entry: an initial approximation to the solution vector $x$.

On exit: an improved approximation to the solution vector $x$.

14:  $\mathbf{rnorm}$ – double *Output

On exit: the final value of the residual norm ${‖r_k‖}_{\infty }$, where $k$ is the output value of itn.

15:  $\mathbf{itn}$ – Integer *Output

On exit: the number of iterations carried out.

16:  $\mathbf{comm}$ – Nag_Sparse_Comm *Input/Output

On entry/exit: a pointer to a structure of type Nag_Sparse_Comm whose members are used by the iterative solver.

17:  $\mathbf{fail}$ – NagError *Input/Output

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

6  Error Indicators and Warnings

NE_ACC_LIMIT

The required accuracy could not be obtained. However, a reasonable accuracy has been obtained and further iterations cannot improve the result.

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_ALLOC_FAIL

Dynamic memory allocation failed.

NE_BAD_PARAM

On entry, argument method had an illegal value.

On entry, argument precon had an illegal value.

NE_INT_2

On entry, $\mathbf{m}=〈\mathit{\text{value}}〉$, $\min \left(\mathbf{n},10\right)=〈\mathit{\text{value}}〉$.
Constraint: $0<\mathbf{m}\le \min \left(\mathbf{n},10\right)$ when $\mathbf{method}=\mathrm{Nag\_SparseNsym\_BiCGSTAB}$.

On entry, $\mathbf{m}=〈\mathit{\text{value}}〉$, $\min \left(\mathbf{n},50\right)=〈\mathit{\text{value}}〉$.
Constraint: $0<\mathbf{m}\le \min \left(\mathbf{n},50\right)$ when $\mathbf{method}=\mathrm{Nag\_SparseNsym\_RGMRES}$.

On entry, $\mathbf{nnz}=〈\mathit{\text{value}}〉$, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $1\le \mathbf{nnz}\le {\mathbf{n}}^2$.

NE_INT_ARG_LT

On entry, $\mathbf{maxitn}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{maxitn}\ge 1$.

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}\ge 1$.

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_NONSYMM_MATRIX_DUP

A nonzero matrix element has been supplied which does not lie within the matrix $A$, is out of order or has duplicate row and column indices, i.e., one or more of the following constraints has been violated:

• $1\le \mathbf{irow}\left[\mathit{i}\right]\le \mathbf{n}$ and $1\le \mathbf{icol}\left[\mathit{i}\right]\le \mathbf{n}$, for $\mathit{i}=0,1,\dots ,\mathbf{nnz}-1$.
• $\mathbf{irow}\left[i-1\right]<\mathbf{irow}\left[i\right]$, or
• $\mathbf{irow}\left[\mathit{i}-1\right]=\mathbf{irow}\left[\mathit{i}\right]$ and $\mathbf{icol}\left[\mathit{i}-1\right]<\mathbf{icol}\left[\mathit{i}\right]$, for $\mathit{i}=1,2,\dots ,\mathbf{nnz}-1$.

Call nag_sparse_nsym_sort (f11zac) to reorder and sum or remove duplicates.

NE_NOT_REQ_ACC

The required accuracy has not been obtained in maxitn iterations.

NE_REAL

On entry, $\mathbf{omega}=〈\mathit{\text{value}}〉$.
Constraint: $0.0<\mathbf{omega}<2.0$ when $\mathbf{precon}=\mathrm{Nag\_SparseNsym\_SSORPrec}$.

NE_REAL_ARG_GE

On entry, tol must not be greater than or equal to 1: $\mathbf{tol}=〈\mathit{\text{value}}〉$.

NE_ZERO_DIAGONAL_ELEM

On entry, the matrix a has a zero diagonal element. Jacobi and SSOR preconditioners are not appropriate for this problem.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (f11dece.c)

10.2  Program Data

Program Data (f11dece.d)

10.3  Program Results

Program Results (f11dece.r)