f11gfc is the third in a suite of three functions for the iterative solution of a symmetric system of simultaneous linear equations (see Golub and Van Loan (1996)). f11gfc returns information about the computations during an iteration and/or after this has been completed. The first function of the suite, f11gdc, is a setup function, the second function, f11gec is the proper iterative solver.

These three functions are suitable for the solution of large sparse symmetric systems of equations.

2 Specification

#include <nag.h>

void	f11gfc (Integer itn, double stplhs, double stprhs, double anorm, double sigmax, Integer its, double sigerr, const double work[], Integer lwork, NagError fail)

The function may be called by the names: f11gfc, nag_sparse_real_symm_basic_diag or nag_sparse_sym_basic_diagnostic.

3 Description

f11gfc returns information about the solution process. It can be called both during a monitoring step of the solver f11gec, or after this solver has completed its tasks. Calling f11gfc at any other time will result in an error condition being raised.

For further information you should read the documentation for f11gdc and f11gec.

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: $itn$ – Integer * Output

On exit: the number of iterations carried out by f11gec.

2: $stplhs$ – double * Output

On exit: the current value of the left-hand side of the termination criterion used by f11gec.

3: $stprhs$ – double * Output

On exit: the current value of the right-hand side of the termination criterion used by f11gec.

4: $anorm$ – double * Output

On exit: for CG and SYMMLQ methods, the norm

{‖ A ‖}_{1} = {‖ A ‖}_{\infty}

when either it has been supplied to f11gdc or it has been estimated by f11gec (see also Sections 3 and 5 in f11gdc). Otherwise,

anorm = 0.0

is returned.

For MINRES method, an estimate of the infinity norm of the preconditioned matrix operator.

5: $sigmax$ – double * Output

On exit: for CG and SYMMLQ methods, the current estimate of the largest singular value

σ_{1} (\bar{A})

of the preconditioned iteration matrix

\bar{A} = E^{- 1} A E^{- T}

, when either it has been supplied to f11gdc or it has been estimated by f11gec (see also Sections 3 and 5 in f11gdc). Note that if

its < itn

then sigmax contains the final estimate. If, on final exit from f11gec,

its = itn

, the estimation of

σ_{1} (\bar{A})

may have not converged; in this case you should look at the value returned in sigerr. Otherwise,

sigmax = 0.0

is returned.

For MINRES method, an estimate of the final transformed residual.

6: $its$ – Integer * Output

On exit: for CG and SYMMLQ methods, the number of iterations employed so far in the computation of the estimate of

σ_{1} (\bar{A})

, the largest singular value of the preconditioned matrix

\bar{A} = E^{- 1} A E^{- T}

, when

σ_{1} (\bar{A})

has been estimated by f11gec using the bisection method (see also Sections 3, 5 and 9 in f11gdc). Otherwise,

its = 0

is returned.

7: $sigerr$ – double * Output

On exit: for CG and SYMMLQ methods, if

σ_{1} (\bar{A})

has been estimated by f11gec using bisection,

sigerr = \max (\frac{| σ_{1}^{(k)} - σ_{1}^{(k - 1)} |}{σ_{1}^{(k)}}, \frac{| σ_{1}^{(k)} - σ_{1}^{(k - 2)} |}{σ_{1}^{(k)}}),

where

k = its

denotes the iteration number. The estimation has converged if

sigerr \leq sigtol

where sigtol is an input argument to f11gdc. Otherwise,

sigerr = 0.0

is returned.

For MINRES method, an estimate of the condition number of the preconditioned matrix.

8: $work [lwork]$ – const double Communication Array

On entry: the array work as returned by f11gec (see also Section 3 in f11gec).

9: $lwork$ – Integer Input

On entry: the dimension of the array work (see also Section 5 in f11gdc).

Constraint:

lwork \geq 120

Note: although the minimum value of lwork ensures the correct functioning of f11gfc, a larger value is required by the iterative solver f11gec (see also Section 5 in f11gdc).

10: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_INT: On entry, $lwork = ⟨ value ⟩$ .
Constraint: $lwork \geq 120$ .
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_OUT_OF_SEQUENCE: f11gfc has been called out of sequence.