NAG Library Function Document

nag_sparse_herm_chol_sol (f11jqc)

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:    $\mathbf{n}$ – 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_herm_chol_fac (f11jnc).

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

3:    $\mathbf{nnz}$ – IntegerInput

On entry: the number of nonzero elements in the lower triangular part of the matrix $A$. This must be the same value as was supplied in the preceding call to nag_sparse_herm_chol_fac (f11jnc).

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

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_herm_chol_fac (f11jnc).

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_herm_chol_fac (f11jnc).

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

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

On entry: the values returned in arrays irow, icol, ipiv and istr by a previous call to nag_sparse_herm_chol_fac (f11jnc).

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

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

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

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

12:  $\mathbf{maxitn}$ – IntegerInput

On entry: the maximum number of iterations allowed.

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

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

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

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

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:  $\mathbf{fail}$ – 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.

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

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, $i=〈\mathit{\text{value}}〉$, $\mathbf{icol}\left[i-1\right]=〈\mathit{\text{value}}〉$, $\mathbf{irow}\left[i-1\right]=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{icol}\left[i-1\right]\ge 1$ and $\mathbf{icol}\left[i-1\right]\le \mathbf{irow}\left[i-1\right]$.
Check that a, irow, icol, ipiv and istr have not been corrupted between calls to nag_sparse_herm_chol_fac (f11jnc) and nag_sparse_herm_chol_sol (f11jqc).

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, ipiv and istr have not been corrupted between calls to nag_sparse_herm_chol_fac (f11jnc) and nag_sparse_herm_chol_sol (f11jqc).

NE_INVALID_SCS_PRECOND

The SCS representation of the preconditioner is invalid. Check that a, irow, icol, ipiv and istr have not been corrupted between calls to nag_sparse_herm_chol_fac (f11jnc) and nag_sparse_herm_chol_sol (f11jqc).

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, ipiv and istr have not been corrupted between calls to nag_sparse_herm_chol_fac (f11jnc) and nag_sparse_herm_chol_sol (f11jqc).

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, ipiv and istr have not been corrupted between calls to nag_sparse_herm_chol_fac (f11jnc) and nag_sparse_herm_chol_sol (f11jqc).

NE_PRECOND_NOT_POS_DEF

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

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 (f11jqce.c)

10.2

Program Data

Program Data (f11jqce.d)

10.3

Program Results

Program Results (f11jqce.r)