NAG Library Function Document

nag_sparse_nherm_fac_sol (f11dqc)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

1:     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:     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:     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:     a[la] – const ComplexInput

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

5:     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:     irow[la] – const IntegerInput

7:     icol[la] – const IntegerInput

8:     ipivp[n] – const IntegerInput

9:     ipivq[n] – const IntegerInput

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

11:   idiag[n] – 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:   b[n] – const ComplexInput

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

13:   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:   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 $\mathbf{tol}\le 0.0$, $\tau =\max \left(\sqrt{\epsilon },\sqrt{n}\epsilon \right)$ 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:   maxitn – IntegerInput

On entry: the maximum number of iterations allowed.

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

16:   x[n] – ComplexInput/Output

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

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

17:   rnorm – double *Output

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

18:   itn – Integer *Output

On exit: the number of iterations carried out.

19:   fail – NagError *Input/Output

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

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.

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.

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, $i=〈\mathit{\text{value}}〉$, $\mathbf{irow}\left[i-1\right]=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
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_sol (f11dqc) and nag_sparse_nherm_fac (f11dnc).

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  Further Comments

9  Example

9.1  Program Text

Program Text (f11dqce.c)

9.2  Program Data

Program Data (f11dqce.d)

9.3  Program Results

Program Results (f11dqce.r)