NAG Library Routine Document

F12ARF

Note: this routine uses optional parameters to define choices in the problem specification. If you wish to use default settings for all of the optional parameters, then this routine need not be called. If, however, you wish to reset some or all of the settings please refer to Section 10 for a detailed description of the specification of the optional parameters.

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

+− 10 Optional Parameters

10.1 Description of the Optional Parameters

1 Purpose

F12ARF is an option setting routine in a suite of routines consisting of F12ANF, F12APF, F12AQF, F12ARF and F12ASF, and may be used to supply individual optional parameters to F12APF and F12AQF. The initialization routine F12ANF must have been called prior to calling F12ARF.

2 Specification

SUBROUTINE F12ARF (

STR, ICOMM, COMM, IFAIL)

INTEGER	ICOMM(*), IFAIL
COMPLEX (KIND=nag_wp)	COMM(*)
CHARACTER(*)	STR

3 Description

F12ARF may be used to supply values for optional parameters to F12APF and F12AQF. It is only necessary to call F12ARF for those parameters whose values are to be different from their default values. One call to F12ARF sets one parameter value.

Each optional parameter is defined by a single character string consisting of one or more items. The items associated with a given option must be separated by spaces, or equals signs

[=]

. Alphabetic characters may be upper or lower case. The string

'Pointers = Yes'

is an example of a string used to set an optional parameter. For each option the string contains one or more of the following items:

–	a mandatory keyword;
–	a phrase that qualifies the keyword;
–	a number that specifies an integer or real value. Such numbers may be up to $16$ contiguous characters in Fortran's I, F, E or D format.

F12ARF does not have an equivalent routine from the ARPACK package which passes options by directly setting values to scalar parameters or to specific elements of array arguments. F12ARF is intended to make the passing of options more transparent and follows the same principle as the single option setting routines in Chapter E04 (see E04NSF for an example).

The setup routine F12ANF must be called prior to the first call to F12ARF and all calls to F12ARF must precede the first call to F12APF, the reverse communication iterative solver.

A complete list of optional parameters, their abbreviations, synonyms and default values is given in Section 10.

4 References

Lehoucq R B (2001) Implicitly restarted Arnoldi methods and subspace iteration SIAM Journal on Matrix Analysis and Applications 23 551–562

Lehoucq R B and Scott J A (1996) An evaluation of software for computing eigenvalues of sparse nonsymmetric matrices Preprint MCS-P547-1195 Argonne National Laboratory

Lehoucq R B and Sorensen D C (1996) Deflation techniques for an implicitly restarted Arnoldi iteration SIAM Journal on Matrix Analysis and Applications 17 789–821

Lehoucq R B, Sorensen D C and Yang C (1998) ARPACK Users' Guide: Solution of Large-scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods SIAM, Philidelphia

5 Parameters

1: STR – CHARACTER(*)Input: On entry: a single valid option string (as described in Section 3 and Section 10).
2: ICOMM( $*$ ) – INTEGER arrayCommunication Array: Note: the dimension of the array ICOMM must be at least $\max (1, LICOMM)$ (see F12ANF).
On initial entry: must remain unchanged following a call to the setup routine F12ANF.

On exit: contains data on the current options set.
3: COMM( $*$ ) – COMPLEX (KIND=nag_wp) arrayCommunication Array: Note: the dimension of the array COMM must be at least $\max (1, LCOMM)$ (see F12ANF).
On initial entry: must remain unchanged following a call to the setup routine F12ANF.

On exit: contains data on the current options set.
4: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$ . When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit: $IFAIL = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$: The string passed in STR contains an ambiguous keyword.

$IFAIL = 2$: The string passed in STR contains a keyword that could not be recognized.

$IFAIL = 3$: The string passed in STR contains a second keyword that could not be recognized.

$IFAIL = 4$: The initialization routine F12ANF has not been called or a communication array has become corrupted.

7 Accuracy

Not applicable.

8 Further Comments

None.

9 Example

This example solves

A x = λ B x

in shifted-inverse mode, where

A

and

B

are derived from the finite element discretization of the one-dimensional convection-diffusion operator

\frac{d^{2} u}{d x^{2}} + ρ \frac{d u}{d x}

on the interval

[0, 1]

, with zero Dirichlet boundary conditions.

10 Optional Parameters

Several optional parameters for the computational routines F12APF and F12AQF define choices in the problem specification or the algorithm logic. In order to reduce the number of formal parameters of F12APF and F12AQF these optional parameters have associated default values that are appropriate for most problems. Therefore, you need only specify those optional parameters whose values are to be different from their default values.

The remainder of this section can be skipped if you wish to use the default values for all optional parameters.

The following is a list of the optional parameters available. A full description of each optional parameter is provided in Section 10.1.

Optional parameters may be specified by calling F12ARF before a call to F12APF, but after a call to F12ANF. One call is necessary for each optional parameter.

Any optional parameters yoyu do not specify are set to their default values. Optional parameters you do specify are unaltered by F12APF and F12AQF (unless they define invalid values) and so remain in effect for subsequent calls unless you alter them.

10.1 Description of the Optional Parameters

For each option, we give a summary line, a description of the optional parameter and details of constraints.

The summary line contains:

the keywords, where the minimum abbreviation of each keyword is underlined;
a parameter value, where the letters $a$ , $i and r$ denote options that take character, integer and real values respectively;
the default value, where the symbol $ε$ is a generic notation for machine precision (see X02AJF).

Keywords and character values are case and white space insensitive.

Advisory

i

Default

=

the value returned by X04ABF

The destination for advisory messages.

Defaults

This special keyword may be used to reset all optional parameters to their default values.

Exact Shifts

Default

Supplied Shifts

During the Arnoldi iterative process, shifts are applied as part of the implicit restarting scheme. The shift strategy used by default and selected by the optional parameter Exact Shifts is strongly recommended over the alternative Supplied Shifts.

If Exact Shifts are used then these are computed internally by the algorithm in the implicit restarting scheme. This strategy is generally effective and cheaper to apply in terms of number of operations than using explicit shifts.

If Supplied Shifts are used then, during the Arnoldi iterative process, you must supply shifts through array arguments of F12APF when F12APF returns with

IREVCM = 3

; the complex shifts are returned in X (or in COMM when the option

Pointers = YES

is set). This option should only be used if you are an experienced user since this requires some algorithmic knowledge and because more operations are usually required than for the implicit shift scheme. Details on the use of explicit shifts and further references on shift strategies are available in Lehoucq et al. (1998).

Iteration Limit

i

Default

= 300

The limit on the number of Arnoldi iterations that can be performed before F12APF exits. If not all requested eigenvalues have converged to within Tolerance and the number of Arnoldi iterations has reached this limit then F12APF exits with an error; F12AQF can still be called subsequently to return the number of converged eigenvalues, the converged eigenvalues and, if requested, the corresponding eigenvectors.

Largest Magnitude

Default

Largest Imaginary

Largest Real

Smallest Imaginary

Smallest Magnitude

Smallest Real

The Arnoldi iterative method converges on a number of eigenvalues with given properties. The default is for F12APF to compute the eigenvalues of largest magnitude using Largest Magnitude. Alternatively, eigenvalues may be chosen which have Largest Real part, Largest Imaginary part,Smallest Magnitude, Smallest Real part or Smallest Imaginary part.

Note that these options select the eigenvalue properties for eigenvalues of

OP

(and

B

for Generalized problems), the linear operator determined by the computational mode and problem type.

Nolist

Default

List

Normally each optional parameter specification is not printed to the advisory channel as it is supplied. Optional parameter List may be used to enable printing and optional parameter Nolist may be used to suppress the printing.

Monitoring

i

Default

= - 1

i > 0

, monitoring information is output to channel number

i

during the solution of each problem; this may be the same as the Advisory channel number. The type of information produced is dependent on the value of Print Level, see the description of the optional parameter Print Level for details of the information produced. Please see X04ACF to associate a file with a given channel number.

Pointers

Default

= NO

During the iterative process and reverse communication calls to F12APF, required data can be communicated to and from F12APF in one of two ways. When

Pointers = NO

is selected (the default) then the array arguments X and MX are used to supply you with required data and used to return computed values back to F12APF. For example, when

IREVCM = 1

, F12APF returns the vector

x

in X and the matrix-vector product

B x

in MX and expects the result or the linear operation

OP (x)

to be returned in X.

Pointers = YES

is selected then the data is passed through sections of the array argument COMM. The section corresponding to X when

Pointers = NO

begins at a location given by the first element of ICOMM; similarly the section corresponding to MX begins at a location given by the second element of ICOMM. This option allows F12APF to perform fewer copy operations on each intermediate exit and entry, but can also lead to less elegant code in the calling program.

Print Level

i

Default

= 0

This controls the amount of printing produced by F12ARF as follows.

$= 0$	No output except error messages.
$> 0$	The set of selected options.
$= 2$	Problem and timing statistics on final exit from F12APF.
$\geq 5$	A single line of summary output at each Arnoldi iteration.
$\geq 10$	If $Monitoring > 0$ , Monitoring is set, then at each iteration, the length and additional steps of the current Arnoldi factorization and the number of converged Ritz values; during re-orthogonalization, the norm of initial/restarted starting vector.
$\geq 20$	Problem and timing statistics on final exit from F12APF. If $Monitoring > 0$ , Monitoring is set, then at each iteration, the number of shifts being applied, the eigenvalues and estimates of the Hessenberg matrix $H$ , the size of the Arnoldi basis, the wanted Ritz values and associated Ritz estimates and the shifts applied; vector norms prior to and following re-orthogonalization.
$\geq 30$	If $Monitoring > 0$ , Monitoring is set, then on final iteration, the norm of the residual; when computing the Schur form, the eigenvalues and Ritz estimates both before and after sorting; for each iteration, the norm of residual for compressed factorization and the compressed upper Hessenberg matrix $H$ ; during re-orthogonalization, the initial/restarted starting vector; during the Arnoldi iteration loop, a restart is flagged and the number of the residual requiring iterative refinement; while applying shifts, the indices of the shifts being applied.
$\geq 40$	If $Monitoring > 0$ , Monitoring is set, then during the Arnoldi iteration loop, the Arnoldi vector number and norm of the current residual; while applying shifts, key measures of progress and the order of $H$ ; while computing eigenvalues of $H$ , the last rows of the Schur and eigenvector matrices; when computing implicit shifts, the eigenvalues and Ritz estimates of $H$ .
$\geq 50$	If Monitoring is set, then during Arnoldi iteration loop: norms of key components and the active column of $H$ , norms of residuals during iterative refinement, the final upper Hessenberg matrix $H$ ; while applying shifts: number of shifts, shift values, block indices, updated matrix $H$ ; while computing eigenvalues of $H$ : the matrix $H$ , the computed eigenvalues and Ritz estimates.

Random Residual

Default

Initial Residual

To begin the Arnoldi iterative process, F12APF requires an initial residual vector. By default F12APF provides its own random initial residual vector; this option can also be set using optional parameter Random Residual. Alternatively, you can supply an initial residual vector (perhaps from a previous computation) to F12APF through the array argument RESID; this option can be set using optional parameter Initial Residual.

Regular

Default

Regular Inverse

Shifted Inverse

These options define the computational mode which in turn defines the form of operation

OP (x)

to be performed when F12APF returns with

IREVCM = - 1

1

and the matrix-vector product

B x

when F12APF returns with

IREVCM = - 2

Given a Standard eigenvalue problem in the form

A x = λ x

then the following modes are available with the appropriate operator

OP (x)

Regular	$OP = A$
Shifted Inverse	$OP = {(A - σ I)}^{- 1}$

Given a Generalized eigenvalue problem in the form

A x = λ B x

then the following modes are available with the appropriate operator

OP (x)

Regular Inverse	$OP = B^{- 1} A$
Shifted Inverse	$OP = {(A - σ B)}^{- 1} B$

Standard

Default

Generalized

The problem to be solved is either a standard eigenvalue problem,

A x = λ x

, or a generalized eigenvalue problem,

A x = λ B x

. The optional parameter Standard should be used when a standard eigenvalue problem is being solved and the optional parameter Generalized should be used when a generalized eigenvalue problem is being solved.

Tolerance

r

Default

= ε

An approximate eigenvalue has deemed to have converged when the corresponding Ritz estimate is within Tolerance relative to the magnitude of the eigenvalue.

Vectors

Default

= RITZ

The routine F12AQF can optionally compute the Schur vectors and/or the eigenvectors corresponding to the converged eigenvalues. To turn off computation of any vectors the option

Vectors = NONE

should be set. To compute only the Schur vectors (at very little extra cost), the option

Vectors = SCHUR

should be set and these will be returned in the array argument V of F12AQF. To compute the eigenvectors (Ritz vectors) corresponding to the eigenvalue estimates, the option

Vectors = RITZ

should be set and these will be returned in the array argument Z of F12AQF, if Z is set equal to V (as in Section 9) then the Schur vectors in V are overwritten by the eigenvectors computed by F12AQF.

F12ARF (PDF version)

F12 Chapter Contents

F12 Chapter Introduction

NAG Library Manual