# NAG C Library Function Document

## 1Purpose

nag_sparse_sym_basic_diagnostic (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)). nag_sparse_sym_basic_diagnostic (f11gfc) returns information about the computations during an iteration and/or after this has been completed. The first function of the suite, nag_sparse_sym_basic_setup (f11gdc), is a setup function, the second function, nag_sparse_sym_basic_solver (f11gec) is the proper iterative solver.
These three functions are suitable for the solution of large sparse symmetric systems of equations.

## 2Specification

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

## 3Description

nag_sparse_sym_basic_diagnostic (f11gfc) returns information about the solution process. It can be called both during a monitoring step of the solver nag_sparse_sym_basic_solver (f11gec), or after this solver has completed its tasks. Calling nag_sparse_sym_basic_diagnostic (f11gfc) at any other time will result in an error condition being raised.
For further information you should read the documentation for nag_sparse_sym_basic_setup (f11gdc) and nag_sparse_sym_basic_solver (f11gec).

## 4References

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

## 5Arguments

1:    $\mathbf{itn}$Integer *Output
On exit: the number of iterations carried out by nag_sparse_sym_basic_solver (f11gec).
2:    $\mathbf{stplhs}$double *Output
On exit: the current value of the left-hand side of the termination criterion used by nag_sparse_sym_basic_solver (f11gec).
3:    $\mathbf{stprhs}$double *Output
On exit: the current value of the right-hand side of the termination criterion used by nag_sparse_sym_basic_solver (f11gec).
4:    $\mathbf{anorm}$double *Output
On exit: for CG and SYMMLQ methods, the norm ${‖A‖}_{1}={‖A‖}_{\infty }$ when either it has been supplied to nag_sparse_sym_basic_setup (f11gdc) or it has been estimated by nag_sparse_sym_basic_solver (f11gec) (see also Sections 3 and 5 in nag_sparse_sym_basic_setup (f11gdc)). Otherwise, ${\mathbf{anorm}}=0.0$ is returned.
For MINRES method, an estimate of the infinity norm of the preconditioned matrix operator.
5:    $\mathbf{sigmax}$double *Output
On exit: for CG and SYMMLQ methods, the current estimate of the largest singular value ${\sigma }_{1}\left(\stackrel{-}{A}\right)$ of the preconditioned iteration matrix $\stackrel{-}{A}={E}^{-1}A{E}^{-\mathrm{T}}$, when either it has been supplied to nag_sparse_sym_basic_setup (f11gdc) or it has been estimated by nag_sparse_sym_basic_solver (f11gec) (see also Sections 3 and 5 in nag_sparse_sym_basic_setup (f11gdc)). Note that if ${\mathbf{its}}<{\mathbf{itn}}$ then sigmax contains the final estimate. If, on final exit from nag_sparse_sym_basic_solver (f11gec), ${\mathbf{its}}={\mathbf{itn}}$, the estimation of ${\sigma }_{1}\left(\stackrel{-}{A}\right)$ may have not converged; in this case you should look at the value returned in sigerr. Otherwise, ${\mathbf{sigmax}}=0.0$ is returned.
For MINRES method, an estimate of the final transformed residual.
6:    $\mathbf{its}$Integer *Output
On exit: for CG and SYMMLQ methods, the number of iterations employed so far in the computation of the estimate of ${\sigma }_{1}\left(\stackrel{-}{A}\right)$, the largest singular value of the preconditioned matrix $\stackrel{-}{A}={E}^{-1}A{E}^{-\mathrm{T}}$, when ${\sigma }_{1}\left(\stackrel{-}{A}\right)$ has been estimated by nag_sparse_sym_basic_solver (f11gec) using the bisection method (see also Sections 3, 5 and 9 in nag_sparse_sym_basic_setup (f11gdc)). Otherwise, ${\mathbf{its}}=0$ is returned.
7:    $\mathbf{sigerr}$double *Output
On exit: for CG and SYMMLQ methods, if ${\sigma }_{1}\left(\stackrel{-}{A}\right)$ has been estimated by nag_sparse_sym_basic_solver (f11gec) using bisection,
 $sigerr=maxσ1k-σ1k-1σ1k,σ1k-σ1k-2σ1k ,$
where $k={\mathbf{its}}$ denotes the iteration number. The estimation has converged if ${\mathbf{sigerr}}\le {\mathbf{sigtol}}$ where sigtol is an input argument to nag_sparse_sym_basic_setup (f11gdc). Otherwise, ${\mathbf{sigerr}}=0.0$ is returned.
For MINRES method, an estimate of the condition number of the preconditioned matrix.
8:    $\mathbf{work}\left[{\mathbf{lwork}}\right]$const doubleCommunication Array
On entry: the array work as returned by nag_sparse_sym_basic_solver (f11gec) (see also Section 3 in nag_sparse_sym_basic_solver (f11gec)).
9:    $\mathbf{lwork}$IntegerInput
On entry: the dimension of the array work (see also Section 5 in nag_sparse_sym_basic_setup (f11gdc)).
Constraint: ${\mathbf{lwork}}\ge 120$.
Note:  although the minimum value of lwork ensures the correct functioning of nag_sparse_sym_basic_diagnostic (f11gfc), a larger value is required by the iterative solver nag_sparse_sym_basic_solver (f11gec) (see also Section 5 in nag_sparse_sym_basic_setup (f11gdc)).
10:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

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.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{lwork}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lwork}}\ge 120$.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
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_OUT_OF_SEQUENCE
nag_sparse_sym_basic_diagnostic (f11gfc) has been called out of sequence.

Not applicable.

## 8Parallelism and Performance

nag_sparse_sym_basic_diagnostic (f11gfc) is not threaded in any implementation.