psb_krylov -- Krylov Methods Driver Routine

This subroutine is a driver that provides a general interface for all the Krylov-Subspace family methods implemented in PSBLAS version 2.

The stopping criterion can take the following values:

1

normwise backward error in the infinity norm; the iteration is stopped when

$$err = \frac{\Vert r_i\Vert}{(\Vert A\Vert\Vert x_i\Vert+\Vert b\Vert)} < eps$$

2

Relative residual in the 2-norm; the iteration is stopped when

$$err = \frac{\Vert r_i\Vert}{\Vert b\Vert _2} < eps$$

3

Relative residual reduction in the 2-norm; the iteration is stopped when

$$err = \frac{\Vert r_i\Vert}{\Vert r_0\Vert _2} < eps$$

The behaviour is controlled by the istop argument (see later). In the above formulae, $x_i$ is the tentative solution and $r_i=b-Ax_i$ the corresponding residual at the $i$-th iteration.

Type:

Synchronous.

On Entry

method

a string that defines the iterative method to be used. Supported values are:

CG:

the Conjugate Gradient method;

CGS:

the Conjugate Gradient Stabilized method;

GCR:

the Generalized Conjugate Residual method;

FCG:

the Flexible Conjugate Gradient method⁵;

BICG:

the Bi-Conjugate Gradient method;

BICGSTAB:

the Bi-Conjugate Gradient Stabilized method;

BICGSTABL:

the Bi-Conjugate Gradient Stabilized method with restarting;

RGMRES:

the Generalized Minimal Residual method with restarting.

a

the local portion of global sparse matrix $A$.
Scope: local
Type: required
Intent: in.
Specified as: a structured data of type spdatapsb_Tspmat_type.

prec

The data structure containing the preconditioner.
Scope: local
Type: required
Intent: in.
Specified as: a structured data of type precdatapsb_prec_type.

b

The RHS vector.
Scope: local
Type: required
Intent: in.
Specified as: a rank one array or an object of type vdatapsb_T_vect_type.

x

The initial guess.
Scope: local
Type: required
Intent: inout.
Specified as: a rank one array or an object of type vdatapsb_T_vect_type.

eps

The stopping tolerance.
Scope: global
Type: required
Intent: in.
Specified as: a real number.

desc_a

contains data structures for communications.
Scope: local
Type: required
Intent: in.
Specified as: a structured data of type descdatapsb_desc_type.

itmax

The maximum number of iterations to perform.
Scope: global
Type: optional
Intent: in.
Default: $itmax = 1000$.
Specified as: an integer variable $itmax \ge 1$.

itrace

If $>0$ print out an informational message about convergence every $itrace$ iterations.
Scope: global
Type: optional
Intent: in.

irst

An integer specifying the restart parameter.
Scope: global
Type: optional.
Intent: in.
Values: $irst>0$. This is employed for the BiCGSTABL or RGMRES methods, otherwise it is ignored.

istop

An integer specifying the stopping criterion.
Scope: global
Type: optional.
Intent: in.
Values: 1: use the normwise backward error, 2: use the scaled 2-norm of the residual, 3: use the residual reduction in the 2-norm. Default: 2.

On Return

x

The computed solution.
Scope: local
Type: required
Intent: inout.
Specified as: a rank one array or an object of type vdatapsb_T_vect_type.

iter

The number of iterations performed.
Scope: global
Type: optional
Intent: out.
Returned as: an integer variable.

err

The convergence estimate on exit.
Scope: global
Type: optional
Intent: out.
Returned as: a real number.

cond

An estimate of the condition number of matrix $A$; only available with the $CG$ method on real data.
Scope: global
Type: optional
Intent: out.
Returned as: a real number. A correct result will be greater than or equal to one; if specified for non-real data, or an error occurred, zero is returned.

info

Error code.
Scope: local
Type: required
Intent: out.
An integer value; 0 means no error has been detected.