NAG Library Routine Document

f02fkf (real_symm_sparse_arnoldi)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{n}$ – IntegerInput

On entry: $n$, the order of the matrix $A$.

Constraint: $\mathbf{n}>0$.

2:     $\mathbf{nnz}$ – IntegerInput

On entry: the dimension of the array a as declared in the (sub)program from which f02fkf is called. The number of nonzero elements in the lower triangular part of the matrix $A$.

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

3:     $\mathbf{a}\left(\mathbf{nnz}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the array of nonzero elements of the lower triangular part of the $n$ by $n$ symmetric matrix $A$.

4:     $\mathbf{irow}\left(\mathbf{nnz}\right)$ – Integer arrayInput

5:     $\mathbf{icol}\left(\mathbf{nnz}\right)$ – Integer arrayInput

On entry: the row and column indices of the elements supplied in array a.

If $\mathbf{irow}\left(k\right)=i$ and $\mathbf{icol}\left(k\right)=j$ then $A_{ij}$ is stored in $\mathbf{a}\left(k\right)$. irow does not need to be ordered: an internal sort will force the correct ordering.

Constraint:

irow and icol must satisfy these constraints:

$1\le \mathbf{irow}\left(\mathit{i}\right)\le \mathbf{n}$ and $1\le \mathbf{icol}\left(\mathit{i}\right)\le \mathbf{irow}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nnz}$.

6:     $\mathbf{nev}$ – IntegerInput

On entry: the number of eigenvalues to be computed.

Constraint: $0<\mathbf{nev}<\mathbf{n}-1$.

7:     $\mathbf{ncv}$ – IntegerInput

On entry: the dimension of the array w as declared in the (sub)program from which f02fkf is called. The number of Arnoldi basis vectors to use during the computation.

At present there is no a priori analysis to guide the selection of ncv relative to nev. However, it is recommended that $\mathbf{ncv}\ge 2\times \mathbf{nev}+1$. If many problems of the same type are to be solved, you should experiment with increasing ncv while keeping nev fixed for a given test problem. This will usually decrease the required number of matrix-vector operations but it also increases the work and storage required to maintain the orthogonal basis vectors. The optimal ‘cross-over’ with respect to computation time is problem dependent and must be determined empirically.

Constraint: $\mathbf{nev}<\mathbf{ncv}\le \mathbf{n}$.

8:     $\mathbf{sigma}$ – Real (Kind=nag_wp)Input

On entry: if the Shifted Inverse mode has been selected then sigma contains the real shift used; otherwise sigma is not referenced. This mode can be selected by setting the appropriate options in the user-supplied subroutine option.

9:     $\mathbf{monit}$ – Subroutine, supplied by the NAG Library or the user.External Procedure

monit is used to monitor the progress of f02fkf. monit may be the dummy subroutine f02fkz if no monitoring is actually required. (f02fkz is included in the NAG Library.) monit is called after the solution of each eigenvalue sub-problem and also just prior to return from f02fkf.

The specification of monit is:

Fortran Interface

Subroutine monit ( ncv, niter, nconv, w, rzest, istat, iuser, ruser) Integer, Intent (In) :: ncv, niter, nconv Integer, Intent (Inout) :: istat, iuser(*) Real (Kind=nag_wp), Intent (In) :: w(ncv), rzest(ncv) Real (Kind=nag_wp), Intent (Inout) :: ruser(*)

C Header Interface

#include <nagmk26.h> void  monit (const Integer *ncv, const Integer *niter, const Integer *nconv, const double w[], const double rzest[], Integer *istat, Integer iuser[], double ruser[])

1:     $\mathbf{ncv}$ – IntegerInput

On entry: the dimension of the arrays w and rzest. The number of Arnoldi basis vectors used during the computation.

2:     $\mathbf{niter}$ – IntegerInput

On entry: the number of the current Arnoldi iteration.

3:     $\mathbf{nconv}$ – IntegerInput

On entry: the number of converged eigenvalues so far.

4:     $\mathbf{w}\left(\mathbf{ncv}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the first nconv elements of w contain the converged approximate eigenvalues.

5:     $\mathbf{rzest}\left(\mathbf{ncv}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the first nconv elements of rzest contain the Ritz estimates (error bounds) on the converged approximate eigenvalues.

6:     $\mathbf{istat}$ – IntegerInput/Output

On entry: set to zero.

On exit: if set to a nonzero value f02fkf returns immediately with $\mathbf{ifail}=\mathbf{9}$.

7:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace

8:     $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace

monit is called with the arguments iuser and ruser as supplied to f02fkf. You should use the arrays iuser and ruser to supply information to monit.

monit must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which f02fkf is called. Arguments denoted as Input must not be changed by this procedure.

10:   $\mathbf{option}$ – Subroutine, supplied by the NAG Library or the user.External Procedure

You can supply non-default options to the Arnoldi eigensolver by repeated calls to f12fdf from within option. (Please note that it is only necessary to call f12fdf; no call to f12faf is required from within option.) For example, you can set the mode to Shifted Inverse, you can increase the Iteration Limit beyond its default and you can print varying levels of detail on the iterative process using Print Level.

If only the default options (including that the eigenvalues of largest magnitude are sought) are to be used then option may be the dummy subroutine f02eky (f02eky is included in the NAG Library). See Section 10 for an example of using option to set some non-default options.

The specification of option is:

Fortran Interface

Subroutine option ( icomm, comm, istat, iuser, ruser) Integer, Intent (Inout) :: icomm(*), istat, iuser(*) Real (Kind=nag_wp), Intent (Inout) :: comm(*), ruser(*)

C Header Interface

#include <nagmk26.h> void  option (Integer icomm[], double comm[], Integer *istat, Integer iuser[], double ruser[])

1:     $\mathbf{icomm}\left(*\right)$ – Integer arrayCommunication Array

On entry: contains details of the default option set. This array must be passed as argument icomm in any call to f12fdf.

On exit: contains data on the current options set which may be altered from the default set via calls to f12fdf.

2:     $\mathbf{comm}\left(*\right)$ – Real (Kind=nag_wp) arrayCommunication Array

On entry: contains details of the default option set. This array must be passed as argument comm in any call to f12fdf.

On exit: contains data on the current options set which may be altered from the default set via calls to f12fdf.

3:     $\mathbf{istat}$ – IntegerInput/Output

On entry: set to zero.

On exit: if set to a nonzero value f02fkf returns immediately with $\mathbf{ifail}=\mathbf{10}$.

4:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace

5:     $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace

option is called with the arguments iuser and ruser as supplied to f02fkf. You should use the arrays iuser and ruser to supply information to option.

option must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which f02fkf is called.

11:   $\mathbf{nconv}$ – IntegerOutput

On exit: the number of converged approximations to the selected eigenvalues. On successful exit, this will normally be nev.

12:   $\mathbf{w}\left(\mathbf{ncv}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: the first nconv elements contain the converged approximations to the selected eigenvalues.

13:   $\mathbf{v}\left(\mathbf{ldv},*\right)$ – Real (Kind=nag_wp) arrayOutput

Note: the second dimension of the array v must be at least $\mathbf{ncv}$.

On exit: contains the eigenvectors associated with the eigenvalue ${\lambda }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathbf{nconv}$ (stored in w). For eigenvalue, ${\lambda }_j$, the corresponding eigenvector is stored in $\mathbf{v}\left(\mathit{i},j\right)$, for $\mathit{i}=1,2,\dots ,n$.

14:   $\mathbf{ldv}$ – IntegerInput

On entry: the first dimension of the array v as declared in the (sub)program from which f02fkf is called.

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

15:   $\mathbf{resid}\left(\mathbf{nev}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: the residual ${‖A{w}_{\mathit{i}}-{\lambda }_{\mathit{i}}{w}_{\mathit{i}}‖}_2$ for the estimates to the eigenpair ${\lambda }_{\mathit{i}}$ and $w_{\mathit{i}}$ is returned in $\mathbf{resid}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nconv}$.

16:   $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace

iuser is not used by f02fkf, but is passed directly to monit and option and may be used to pass information to these routines.

17:   $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace

ruser is not used by f02fkf, but is passed directly to monit and option and may be used to pass information to these routines.

18:   $\mathbf{ifail}$ – IntegerInput/Output

On entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of ifail on exit.

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

6

Error Indicators and Warnings

$\mathbf{ifail}=1$

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}>0$.

$\mathbf{ifail}=2$

On entry, $\mathbf{nnz}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nnz}>0$.

On entry, $\mathbf{nnz}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nnz}\le \mathbf{n}\times \left(\mathbf{n}+1\right)/2$.

$\mathbf{ifail}=4$

On entry, for $i=〈\mathit{\text{value}}〉$, $\mathbf{irow}\left(i\right)=〈\mathit{\text{value}}〉$.
Constraint: $1\le \mathbf{irow}\left(i\right)\le \mathbf{n}$.

$\mathbf{ifail}=5$

On entry, for $i=〈\mathit{\text{value}}〉$, $\mathbf{icol}\left(i\right)=〈\mathit{\text{value}}〉$, $\mathbf{irow}\left(i\right)=〈\mathit{\text{value}}〉$.
Constraint: $1\le \mathbf{icol}\left(i\right)\le \mathbf{irow}\left(i\right)$.

$\mathbf{ifail}=6$

On entry, $\mathbf{nev}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nev}>0$.

On entry, $\mathbf{nev}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nev}<\left(\mathbf{n}-1\right)$.

$\mathbf{ifail}=7$

On entry, $\mathbf{ncv}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ncv}\le \mathbf{n}$.

On entry, $\mathbf{ncv}=〈\mathit{\text{value}}〉$ and $\mathbf{nev}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ncv}>\mathbf{nev}$.

$\mathbf{ifail}=8$

On entry, the matrix $\left(A-\sigma I\right)$ is numerically singular and could not be inverted. Try perturbing the value of $\sigma$.

$\mathbf{ifail}=9$

User requested termination in monit, $\mathbf{istat}=〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=10$

User requested termination in option, $\mathbf{istat}=〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=14$

On entry, $\mathbf{ldv}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ldv}\ge \mathbf{n}$.

$\mathbf{ifail}=20$

The maximum number of iterations, through the optional parameter Iteration Limit, has been set to a non-positive value.

$\mathbf{ifail}=21$

The option Both Ends has been set but only $1$ eigenvalue is requested.

$\mathbf{ifail}=22$

The maximum number of iterations has been reached.
The maximum number of iterations $=〈\mathit{\text{value}}〉$.
The number of converged eigenvalues $=〈\mathit{\text{value}}〉$.
See the routine document for further details.

$\mathbf{ifail}=30$

A serious error, code $\left(〈\mathit{\text{value}}〉,〈\mathit{\text{value}}〉\right)$, has occurred in an internal call. Check all subroutine calls and array sizes. If the call is correct then please contact NAG for assistance.

$\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

9.1

Additional Licensor

10

Example

10.1

Program Text

Program Text (f02fkfe.f90)

10.2

Program Data

Program Data (f02fkfe.d)

10.3

Program Results

Program Results (f02fkfe.r)

11

Optional Parameters

• Advisory
• Both Ends
• Defaults
• Iteration Limit
• Largest Algebraic
• Largest Magnitude
• List
• Monitoring
• Nolist
• Print Level
• Regular
• Regular Inverse
• Shifted Inverse
• Smallest Algebraic
• Smallest Magnitude
• Tolerance

11.1

Description of the Optional Parameters

• 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 $\epsilon$ is a generic notation for machine precision (see x02ajf).