NAG Library Routine Document
F12AQF
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 10 in F12ARF
for a detailed description of the specification of the optional parameters.
1 Purpose
F12AQF is a postprocessing routine in a suite of routines consisting of
F12ANF,
F12APF, F12AQF,
F12ARF and
F12ASF, that must be called following a final exit from F12AQF.
2 Specification
SUBROUTINE F12AQF ( 
NCONV, D, Z, LDZ, SIGMA, RESID, V, LDV, COMM, ICOMM, IFAIL) 
INTEGER 
NCONV, LDZ, LDV, ICOMM(*), IFAIL 
COMPLEX (KIND=nag_wp) 
D(*), Z(LDZ,*), SIGMA, RESID(*), V(LDV,*), COMM(*) 

3 Description
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 postprocessing 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. 
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 MCSP5471195 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 Largescale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods SIAM, Philidelphia
5 Parameters
 1: NCONV – INTEGEROutput
On exit: the number of converged eigenvalues as found by
F12ARF.
 2: D($*$) – 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: Z(LDZ,$*$) – 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: 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: 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: RESID($*$) – 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: V(LDV,$*$) – 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: 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: COMM($*$) – 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: ICOMM($*$) – 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: IFAIL – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. 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
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this parameter, 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
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, 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 parameter
ICOMM, is zero.
 ${\mathbf{IFAIL}}=4$
The number of converged eigenvalues as calculated by
F12APF differ from the value passed to it through the parameter
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.
 ${\mathbf{IFAIL}}=7$
Unexpected error in internal call while calculating eigenvectors. Please contact
NAG.
 ${\mathbf{IFAIL}}=8$
Either the solver routine
F12APF has not been called prior to the call of this routine or a communication array has become corrupted.
 ${\mathbf{IFAIL}}=9$
The routine was unable to dynamically allocate sufficient internal workspace. Please contact
NAG.
 ${\mathbf{IFAIL}}=10$
An unexpected error has occurred. Please contact
NAG.
7 Accuracy
The relative accuracy of a Ritz value,
$\lambda $, is considered acceptable if its Ritz estimate
$\le {\mathbf{Tolerance}}\times \left\lambda \right$. The default
Tolerance used is the
machine precision given by
X02AJF.
None.
9 Example
This example solves $Ax=\lambda Bx$ in regularinvert mode, where $A$ and $B$ are derived from the standard central difference discretization of the onedimensional convectiondiffusion operator $\frac{{d}^{2}u}{d{x}^{2}}+\rho \frac{du}{dx}$ on $\left[0,1\right]$, with zero Dirichlet boundary conditions.
9.1 Program Text
Program Text (f12aqfe.f90)
9.2 Program Data
Program Data (f12aqfe.d)
9.3 Program Results
Program Results (f12aqfe.r)