NAG Library Function Document

nag_sparse_herm_sol (f11jsc)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:    $\mathbf{method}$ – Nag_SparseSym_MethodInput

On entry: specifies the iterative method to be used.

$\mathbf{method}=\mathrm{Nag\_SparseSym\_CG}$

Conjugate gradient method.

$\mathbf{method}=\mathrm{Nag\_SparseSym\_SYMMLQ}$

Lanczos method (SYMMLQ).

Constraint: $\mathbf{method}=\mathrm{Nag\_SparseSym\_CG}$ or $\mathrm{Nag\_SparseSym\_SYMMLQ}$.

2:    $\mathbf{precon}$ – Nag_SparseSym_PrecTypeInput

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

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

No preconditioning.

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

Jacobi.

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

Symmetric successive-over-relaxation (SSOR).

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

3:    $\mathbf{n}$ – IntegerInput

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

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

4:    $\mathbf{nnz}$ – IntegerInput

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

Constraint: $1\le \mathbf{nnz}\le \mathbf{n}\times \left(\mathbf{n}+1\right)/2$.

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

On entry: the nonzero elements of the lower triangular part 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_herm_sort (f11zpc) 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 array a.

Constraints:

irow and icol must satisfy these constraints (which may be imposed by a call to nag_sparse_herm_sort (f11zpc)):

• $1\le \mathbf{irow}\left[\mathit{i}\right]\le \mathbf{n}$ and $1\le \mathbf{icol}\left[\mathit{i}\right]\le \mathbf{irow}\left[\mathit{i}\right]$, 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 $\mathbf{precon}=\mathrm{Nag\_SparseSym\_SSORPrec}$, omega is the relaxation parameter $\omega$ to be used in the SSOR method. Otherwise omega need not be initialized.

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

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

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

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

11:  $\mathbf{maxitn}$ – IntegerInput

On entry: the maximum number of iterations allowed.

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

12:  $\mathbf{x}\left[\mathbf{n}\right]$ – ComplexInput/Output

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

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

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

On exit: the final value of the residual norm $‖r_k‖$, where $k$ is the output value of itn.

14:  $\mathbf{itn}$ – Integer *Output

On exit: the number of iterations carried out.

15:  $\mathbf{rdiag}\left[\mathbf{n}\right]$ – doubleOutput

On exit: the elements of the diagonal matrix $D^{-1}$, where $D$ is the diagonal part of $A$. Note that since $A$ is Hermitian the elements of $D^{-1}$ are necessarily real.

16:  $\mathbf{fail}$ – 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 has been achieved and further iterations could not improve the result.

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_COEFF_NOT_POS_DEF

The matrix of the coefficients a appears not to be positive definite. The computation cannot continue.

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, $\mathbf{nnz}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nnz}\le \mathbf{n}\times \left(\mathbf{n}+1\right)/2$ 

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

A serious error, code $〈\mathit{\text{value}}〉$, has occurred in an internal call. Check all function calls and array sizes. Seek expert help.

NE_INVALID_SCS

On entry, $\mathit{I}=〈\mathit{\text{value}}〉$, $\mathbf{icol}\left[\mathit{I}-1\right]=〈\mathit{\text{value}}〉$ and $\mathbf{irow}\left[\mathit{I}-1\right]=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{icol}\left[\mathit{I}-1\right]\ge 1$ and $\mathbf{icol}\left[\mathit{I}-1\right]\le \mathbf{irow}\left[\mathit{I}-1\right]$.

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 $\mathbf{irow}\left[i-1\right]\le \mathbf{n}$.

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: $i=〈\mathit{\text{value}}〉$.

On entry, the location ($\mathbf{irow}\left[\mathit{I}-1\right],\mathbf{icol}\left[\mathit{I}-1\right]$) is a duplicate: $\mathit{I}=〈\mathit{\text{value}}〉$. Consider calling nag_sparse_herm_sort (f11zpc) to reorder and sum or remove duplicates.

NE_PRECOND_NOT_POS_DEF

The preconditioner appears not to be positive definite. The computation cannot continue.

NE_REAL

On entry, $\mathbf{omega}=〈\mathit{\text{value}}〉$.
Constraint: $0.0<\mathbf{omega}<2.0$.

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

NE_ZERO_DIAG_ELEM

The matrix $A$ has a non-real diagonal entry in row $〈\mathit{\text{value}}〉$.

The matrix $A$ has a zero diagonal entry in row $〈\mathit{\text{value}}〉$.

The matrix $A$ has no diagonal entry in row $〈\mathit{\text{value}}〉$.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f11jsce.c)

10.2

Program Data

Program Data (f11jsce.d)

10.3

Program Results

Program Results (f11jsce.r)