# NAG Library Routine Document

## f12aqf (complex_proc)

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 the option setting routine f12arf need not be called. If, however, you wish to reset some or all of the settings please refer to Section 11 in f12arf for a detailed description of the specification of the optional parameters.

## 1Purpose

f12aqf is a post-processing routine in a suite of routines consisting of f12anf, f12apf, f12aqf, f12arf and f12asf, that must be called following a final exit from f12aqf.

## 2Specification

Fortran Interface
 Subroutine f12aqf ( d, z, ldz, v, ldv, comm,
 Integer, Intent (In) :: ldz, ldv Integer, Intent (Inout) :: icomm(*), ifail Integer, Intent (Out) :: nconv Complex (Kind=nag_wp), Intent (In) :: sigma, resid(*) Complex (Kind=nag_wp), Intent (Inout) :: d(*), z(ldz,*), v(ldv,*), comm(*)
#include <nagmk26.h>
 void f12aqf_ (Integer *nconv, Complex d[], Complex z[], const Integer *ldz, const Complex *sigma, const Complex resid[], Complex v[], const Integer *ldv, Complex comm[], Integer icomm[], Integer *ifail)

## 3Description

The suite of routines is designed to calculate some of the eigenvalues, $\lambda$, (and optionally the corresponding eigenvectors, $x$) of a standard eigenvalue problem $Ax=\lambda x$, or of a generalized eigenvalue problem $Ax=\lambda Bx$ of order $n$, where $n$ is large and the coefficient matrices $A$ and $B$ are sparse, complex and nonsymmetric. The suite can also be used to find selected eigenvalues/eigenvectors of smaller scale dense, complex and nonsymmetric problems.
Following a call to f12apf, f12aqf returns the converged approximations to eigenvalues and (optionally) the corresponding approximate eigenvectors and/or an orthonormal basis for the associated approximate invariant subspace. The eigenvalues (and eigenvectors) are selected from those of a standard or generalized eigenvalue problem defined by complex nonsymmetric matrices. There is negligible additional cost to obtain eigenvectors; an orthonormal basis is always computed, but there is an additional storage cost if both are requested.
f12aqf is based on the routine zneupd from the ARPACK package, which uses the Implicitly Restarted Arnoldi iteration method. The method is described in Lehoucq and Sorensen (1996) and Lehoucq (2001) while its use within the ARPACK software is described in great detail in Lehoucq et al. (1998). An evaluation of software for computing eigenvalues of sparse nonsymmetric matrices is provided in Lehoucq and Scott (1996). This suite of routines offers the same functionality as the ARPACK software for complex nonsymmetric problems, but the interface design is quite different in order to make the option setting clearer and to simplify some of the interfaces.
f12aqf is a post-processing routine that must be called following a successful final exit from f12apf. f12aqf uses data returned from f12apf and options set either by default or explicitly by calling f12arf, to return the converged approximations to selected eigenvalues and (optionally):
 – the corresponding approximate eigenvectors; – an orthonormal basis for the associated approximate invariant subspace; – both.

## 4References

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

## 5Arguments

1:     $\mathbf{nconv}$ – IntegerOutput
On exit: the number of converged eigenvalues as found by f12arf.
2:     $\mathbf{d}\left(*\right)$ – Complex (Kind=nag_wp) arrayOutput
Note: the dimension of the array d must be at least ${\mathbf{ncv}}$ (see f12anf).
On exit: the first nconv locations of the array d contain the converged approximate eigenvalues.
3:     $\mathbf{z}\left({\mathbf{ldz}},*\right)$ – Complex (Kind=nag_wp) arrayOutput
Note: the second dimension of the array z must be at least ${\mathbf{nev}}$ if the default option ${\mathbf{Vectors}}=\mathrm{RITZ}$ has been selected and at least $1$ if the option ${\mathbf{Vectors}}=\mathrm{NONE}$ or $\mathrm{SCHUR}$ has been selected (see f12anf).
On exit: if the default option ${\mathbf{Vectors}}=\mathrm{RITZ}$ (see f12adf) has been selected then z contains the final set of eigenvectors corresponding to the eigenvalues held in d. The complex eigenvector associated with an eigenvalue is stored in the corresponding column of z.
4:     $\mathbf{ldz}$ – IntegerInput
On entry: the first dimension of the array z as declared in the (sub)program from which f12aqf is called.
Constraints:
• if the default option ${\mathbf{Vectors}}=\text{Ritz}$ has been selected, ${\mathbf{ldz}}\ge {\mathbf{n}}$;
• if the option ${\mathbf{Vectors}}=\text{None or Schur}$ has been selected, ${\mathbf{ldz}}\ge 1$.
5:     $\mathbf{sigma}$ – Complex (Kind=nag_wp)Input
On entry: if one of the Shifted Inverse (see f12arf) modes has been selected then sigma contains the shift used; otherwise sigma is not referenced.
6:     $\mathbf{resid}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput
Note: the dimension of the array resid must be at least ${\mathbf{n}}$ (see f12anf).
On entry: must not be modified following a call to f12apf since it contains data required by f12aqf.
7:     $\mathbf{v}\left({\mathbf{ldv}},*\right)$ – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array v must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{ncv}}\right)$ (see f12anf).
On entry: the ncv columns of v contain the Arnoldi basis vectors for $\mathrm{OP}$ as constructed by f12apf.
On exit: if the option ${\mathbf{Vectors}}=\mathrm{SCHUR}$ or $\mathrm{RITZ}$ has been set and a separate array z has been passed (i.e., z does not equal v), then the first nconv columns of v will contain approximate Schur vectors that span the desired invariant subspace.
8:     $\mathbf{ldv}$ – IntegerInput
On entry: the first dimension of the array v as declared in the (sub)program from which f12aqf is called.
Constraint: ${\mathbf{ldv}}\ge {\mathbf{n}}$.
9:     $\mathbf{comm}\left(*\right)$ – Complex (Kind=nag_wp) arrayCommunication Array
Note: the dimension of the array comm must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lcomm}}\right)$ (see f12anf).
On initial entry: must remain unchanged from the prior call to f12anf.
On exit: contains data on the current state of the solution.
10:   $\mathbf{icomm}\left(*\right)$ – Integer arrayCommunication Array
Note: the dimension of the array icomm must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{licomm}}\right)$ (see f12anf).
On initial entry: must remain unchanged from the prior call to f12anf.
On exit: contains data on the current state of the solution.
11:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . 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  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  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).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{ldz}}<\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ or ${\mathbf{ldz}}<1$ when no vectors are required.
${\mathbf{ifail}}=2$
On entry, the option ${\mathbf{Vectors}}=\text{Select}$ was selected, but this is not yet implemented.
${\mathbf{ifail}}=3$
The number of eigenvalues found to sufficient accuracy prior to calling f12aqf, as communicated through the argument icomm, is zero.
${\mathbf{ifail}}=4$
The number of converged eigenvalues as calculated by f12apf differ from the value passed to it through the argument icomm.
${\mathbf{ifail}}=5$
Unexpected error during calculation of a Schur form: there was a failure to compute all the converged eigenvalues. Please contact NAG.
${\mathbf{ifail}}=6$
Unexpected error: the computed Schur form could not be reordered by an internal call. Please contact NAG.
${\mathbf{ifail}}=7$
${\mathbf{ifail}}=8$
Either the solver routine f12apf has not been called prior to the call of this routine or a communication array has become corrupted.
${\mathbf{ifail}}=9$
${\mathbf{ifail}}=10$
${\mathbf{ifail}}=-99$
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.

## 7Accuracy

The relative accuracy of a Ritz value, $\lambda$, is considered acceptable if its Ritz estimate $\le {\mathbf{Tolerance}}×\left|\lambda \right|$. The default Tolerance used is the machine precision given by x02ajf.

## 8Parallelism and Performance

f12aqf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

None.

## 10Example

This example solves $Ax=\lambda Bx$ in regular-invert mode, where $A$ and $B$ are derived from the standard central difference discretization of the one-dimensional convection-diffusion operator $\frac{{d}^{2}u}{d{x}^{2}}+\rho \frac{du}{dx}$ on $\left[0,1\right]$, with zero Dirichlet boundary conditions.

### 10.1Program Text

Program Text (f12aqfe.f90)

### 10.2Program Data

Program Data (f12aqfe.d)

### 10.3Program Results

Program Results (f12aqfe.r)