# NAG Library Routine Document

## f12acf (real_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 f12adf need not be called. If, however, you wish to reset some or all of the settings please refer to Section 11 in f12adf for a detailed description of the specification of the optional parameters.

## 1Purpose

f12acf is a post-processing routine that must be called following a final exit from f12abf. These are part of a suite of routines for the solution of real sparse eigensystems. The suite also includes f12aaf, f12adf and f12aef.

## 2Specification

Fortran Interface
 Subroutine f12acf ( dr, di, z, ldz, v, ldv, comm,
 Integer, Intent (In) :: ldz, ldv Integer, Intent (Inout) :: icomm(*), ifail Integer, Intent (Out) :: nconv Real (Kind=nag_wp), Intent (In) :: sigmar, sigmai, resid(*) Real (Kind=nag_wp), Intent (Inout) :: dr(*), di(*), z(ldz,*), v(ldv,*), comm(*)
#include nagmk26.h
 void f12acf_ (Integer *nconv, double dr[], double di[], double z[], const Integer *ldz, const double *sigmar, const double *sigmai, const double resid[], double v[], const Integer *ldv, double 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, real and nonsymmetric. The suite can also be used to find selected eigenvalues/eigenvectors of smaller scale dense, real and nonsymmetric problems.
Following a call to f12abf, f12acf 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 real 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.
f12acf is based on the routine dneupd 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 real nonsymmetric problems, but the interface design is quite different in order to make the option setting clearer and to simplify some of the interfaces.
f12acf, is a post-processing routine that must be called following a successful final exit from f12abf. f12acf uses data returned from f12abf and options, set either by default or explicitly by calling f12adf, 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 f12abf.
2:     $\mathbf{dr}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array dr must be at least ${\mathbf{nev}}+1$ (see f12aaf).
On exit: the first nconv locations of the array dr contain the real parts of the converged approximate eigenvalues.
3:     $\mathbf{di}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array di must be at least ${\mathbf{nev}}+1$ (see f12aaf).
On exit: the first nconv locations of the array di contain the imaginary parts of the converged approximate eigenvalues.
4:     $\mathbf{z}\left({\mathbf{ldz}},*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array z must be at least ${\mathbf{nev}}+1$ 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 f12aaf).
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 dr and di. The complex eigenvector associated with the eigenvalue with positive imaginary part is stored in two consecutive columns. The first column holds the real part of the eigenvector and the second column holds the imaginary part. The eigenvector associated with the eigenvalue with negative imaginary part is simply the complex conjugate of the eigenvector associated with the positive imaginary part.
5:     $\mathbf{ldz}$ – IntegerInput
On entry: the first dimension of the array z as declared in the (sub)program from which f12acf 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, ${\mathbf{ldz}}\ge 1$.
6:     $\mathbf{sigmar}$ – Real (Kind=nag_wp)Input
On entry: if one of the Shifted Inverse Real modes have been selected then sigmar contains the real part of the shift used; otherwise sigmar is not referenced.
7:     $\mathbf{sigmai}$ – Real (Kind=nag_wp)Input
On entry: if one of the Shifted Inverse Real modes have been selected then sigmai contains the imaginary part of the shift used; otherwise sigmai is not referenced.
8:     $\mathbf{resid}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array resid must be at least ${\mathbf{n}}$ (see f12aaf).
On entry: must not be modified following a call to f12abf since it contains data required by f12acf.
9:     $\mathbf{v}\left({\mathbf{ldv}},*\right)$ – Real (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 f12aaf).
On entry: the ncv columns of v contain the Arnoldi basis vectors for $\mathrm{OP}$ as constructed by f12abf.
On exit: if the option ${\mathbf{Vectors}}=\mathrm{SCHUR}$ has been set, or the option ${\mathbf{Vectors}}=\mathrm{RITZ}$ 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.
10:   $\mathbf{ldv}$ – IntegerInput
On entry: the first dimension of the array v as declared in the (sub)program from which f12acf is called.
Constraint: ${\mathbf{ldv}}\ge {\mathbf{n}}$.
11:   $\mathbf{comm}\left(*\right)$ – Real (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 f12aaf).
On initial entry: must remain unchanged from the prior call to f12abf.
On exit: contains data on the current state of the solution.
12:   $\mathbf{icomm}\left(*\right)$ – 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 f12aaf).
On initial entry: must remain unchanged from the prior call to f12abf.
On exit: contains data on the current state of the solution.
13:   $\mathbf{ifail}$ – 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).

## 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 f12acf, as communicated through the argument icomm, is zero.
${\mathbf{ifail}}=4$
The number of converged eigenvalues as calculated by f12abf differ from the value passed to it through the argument icomm.
${\mathbf{ifail}}=5$
Unexpected error during calculation of a real 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 f12abf 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

f12acf 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 obtained 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 (f12acfe.f90)

### 10.2Program Data

Program Data (f12acfe.d)

### 10.3Program Results

Program Results (f12acfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017