NAG Library Function Document

nag_sparse_nsym_fac_sol (f11dcc)

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.

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

Then 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:    $\mathbf{n}$ – IntegerInput

On entry: the order of the matrix $A$. This must be the same value as was supplied in the preceding call to nag_sparse_nsym_fac (f11dac).

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

3:    $\mathbf{nnz}$ – IntegerInput

On entry: the number of nonzero-elements in the matrix $A$. This must be the same value as was supplied in the preceding call to nag_sparse_nsym_fac (f11dac).

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

4:    $\mathbf{a}\left[\mathbf{la}\right]$ – const doubleInput

On entry: the values returned in the array a by a previous call to nag_sparse_nsym_fac (f11dac).

5:    $\mathbf{la}$ – IntegerInput

On entry: the second dimension of the arrays a, irow and icol.This must be the same value as returned by a previous call to nag_sparse_nsym_fac (f11dac).

Constraint: $\mathbf{la}\ge 2\times \mathbf{nnz}$.

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

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

8:    $\mathbf{ipivp}\left[\mathbf{n}\right]$ – const IntegerInput

9:    $\mathbf{ipivq}\left[\mathbf{n}\right]$ – const IntegerInput

10:  $\mathbf{istr}\left[\mathbf{n}+1\right]$ – const IntegerInput

11:  $\mathbf{idiag}\left[\mathbf{n}\right]$ – const IntegerInput

On entry: the values returned in the arrays irow, icol, ipivp, ipivq, istr and idiag by a previous call to nag_sparse_nsym_fac (f11dac).

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

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

13:  $\mathbf{m}$ – 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 $\left(\ell \right)$ 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 $\mathbf{method}=\mathrm{Nag\_SparseNsym\_BiCGSTAB}$, $0<\mathbf{m}\le \min \left(\mathbf{n},10\right)$.

14:  $\mathbf{tol}$ – 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: $\mathbf{tol}<1.0$.

15:  $\mathbf{maxitn}$ – IntegerInput

On entry: the maximum number of iterations allowed.

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

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

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

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

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

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

18:  $\mathbf{itn}$ – Integer *Output

On exit: the number of iterations carried out.

19:  $\mathbf{comm}$ – 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.

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

The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6  Error Indicators and Warnings

NE_2_INT_ARG_LT

On entry, $\mathbf{la}=〈\mathit{\text{value}}〉$ while $\mathbf{nnz}=〈\mathit{\text{value}}〉$. These arguments must satisfy $\mathbf{la}\ge 2\times \mathbf{nnz}$.

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_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 method 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_INVALID_CS

On entry, the CS representation of $A$ is invalid. Check that the call to nag_sparse_nsym_fac_sol (f11dcc) has been preceded by a valid call to nag_sparse_nsym_fac (f11dac), and that the arrays a, irow and icol have not been corrupted between the two calls.

NE_INVALID_CS_PRECOND

On entry, the CS representation of the preconditioning matrix M is invalid. Check that the call to nag_sparse_nsym_fac_sol (f11dcc) has been preceded by a valid call to nag_sparse_nsym_fac (f11dac), and that the arrays a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between the two calls.

NE_NOT_REQ_ACC

The required accuracy has not been obtained in maxitn iterations.

NE_REAL_ARG_GE

On entry, tol must not be greater than or equal to 1.0: $\mathbf{tol}=〈\mathit{\text{value}}〉$.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (f11dcce.c)

10.2  Program Data

Program Data (f11dcce.d)

10.3  Program Results

Program Results (f11dcce.r)