NAG Library Function Document

nag_sparse_nherm_fac_sol (f11dqc)

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.

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

Restarted generalized minimum residual method.

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

Conjugate gradient squared method.

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

Bi-conjugate gradient stabilized ($\ell$) method.

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

Transpose-free quasi-minimal residual method.

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

2:    $\mathbf{n}$ – IntegerInput

On entry: $n$, the order of the matrix $A$. This must be the same value as was supplied in the preceding call to nag_sparse_nherm_fac (f11dnc).

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_nherm_fac (f11dnc).

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

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

On entry: the values returned in the array a by a previous call to nag_sparse_nherm_fac (f11dnc).

5:    $\mathbf{la}$ – IntegerInput

On entry: the dimension of the arrays a, irow and icol. This must be the same value as was supplied in the preceding call to nag_sparse_nherm_fac (f11dnc).

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 arrays irow, icol, ipivp, ipivq, istr and idiag by a previous call to nag_sparse_nherm_fac (f11dnc).

ipivp and ipivq are restored on exit.

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

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

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 $\mathbf{tol}\le 0.0$, $\tau =\max \left(\sqrt{\epsilon },10\epsilon ,\sqrt{n}\epsilon \right)$ is used, where $\epsilon$ is the machine precision. Otherwise $\tau =\max \left(\mathbf{tol},10\epsilon ,\sqrt{n}\epsilon \right)$ 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:  $\mathbf{x}\left[\mathbf{n}\right]$ – ComplexInput/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:  $\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ACCURACY

The required accuracy could not be obtained. However, a reasonable accuracy may have been achieved.

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.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.

NE_BAD_PARAM

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.

NE_CONVERGENCE

The solution has not converged after $〈\mathit{\text{value}}〉$ iterations.

NE_INT

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

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

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

NE_INT_2

On entry, $\mathbf{la}=〈\mathit{\text{value}}〉$ and $\mathbf{nnz}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{la}\ge 2\times \mathbf{nnz}$.

On entry, $\mathbf{m}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{m}\ge 1$ and $\mathbf{m}\le \min \left(\mathbf{n},〈\mathit{\text{value}}〉\right)$.

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

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.

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 2.7.6 in How to Use the NAG Library and its Documentation for further information.

NE_INVALID_CS

On entry, $i=〈\mathit{\text{value}}〉$, $\mathbf{icol}\left[i-1\right]=〈\mathit{\text{value}}〉$, and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{icol}\left[i-1\right]\ge 1$ and $\mathbf{icol}\left[i-1\right]\le \mathbf{n}$.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nherm_fac_sol (f11dqc) and nag_sparse_nherm_fac (f11dnc).

On entry, $i=〈\mathit{\text{value}}〉$, $\mathbf{irow}\left[i-1\right]=〈\mathit{\text{value}}〉$, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{irow}\left[i-1\right]\ge 1$ and $\mathbf{irow}\left[i-1\right]\le \mathbf{n}$.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nherm_fac_sol (f11dqc) and nag_sparse_nherm_fac (f11dnc).

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_fac (f11dnc) and nag_sparse_nherm_fac_sol (f11dqc).

NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

NE_NOT_STRICTLY_INCREASING

On entry, $\mathbf{a}\left[i-1\right]$ is out of order: $i=〈\mathit{\text{value}}〉$.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nherm_fac_sol (f11dqc) and nag_sparse_nherm_fac (f11dnc).

On entry, the location ($\mathbf{irow}\left[i-1\right],\mathbf{icol}\left[i-1\right]$) is a duplicate: $i=〈\mathit{\text{value}}〉$.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nherm_fac_sol (f11dqc) and nag_sparse_nherm_fac (f11dnc).

NE_REAL

On entry, $\mathbf{tol}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{tol}<1.0$.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f11dqce.c)

10.2

Program Data

Program Data (f11dqce.d)

10.3

Program Results

Program Results (f11dqce.r)