NAG Library Routine Document
F12FCF
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 F12FDF need not be called.
If, however, you wish to reset some or all of the settings please refer to Section 11 in F12FDF for a detailed description of the specification of the optional parameters.
1 Purpose
F12FCF is a postprocessing routine in a suite of routines which includes
F12FAF,
F12FBF,
F12FDF and
F12FEF. F12FCF must be called following a final exit from
F12FBF.
2 Specification
SUBROUTINE F12FCF ( 
NCONV, D, Z, LDZ, SIGMA, RESID, V, LDV, COMM, ICOMM, IFAIL) 
INTEGER 
NCONV, LDZ, LDV, ICOMM(*), IFAIL 
REAL (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, real and symmetric. The suite can also be used to find selected eigenvalues/eigenvectors of smaller scale dense, real and symmetric problems.
Following a call to
F12FBF, F12FCF 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 symmetric 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.
F12FCF is based on the routine
dseupd from the ARPACK package, which uses the Implicitly Restarted Lanczos 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 symmetric matrices is provided in
Lehoucq and Scott (1996). This suite of routines offers the same functionality as the ARPACK software for real symmetric problems, but the interface design is quite different in order to make the option setting clearer and to simplify some of the interfaces.
F12FCF, is a postprocessing routine that must be called following a successful final exit from
F12FBF. F12FCF uses data returned from
F12FBF and options, set either by default or explicitly by calling
F12FDF, 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: $\mathrm{NCONV}$ – INTEGEROutput

On exit: the number of converged eigenvalues as found by
F12FBF.
 2: $\mathrm{D}\left(*\right)$ – REAL (KIND=nag_wp) arrayOutput

Note: the dimension of the array
D
must be at least
${\mathbf{NCV}}$ (see
F12FAF).
On exit: the first
NCONV locations of the array
D contain the converged approximate eigenvalues.
 3: $\mathrm{Z}\left({\mathbf{LDZ}},*\right)$ – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array
Z
must be at least
${\mathbf{NCV}}$ 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
F12FAF).
On exit: if the default option
${\mathbf{Vectors}}=\mathrm{RITZ}$ (see
F12FDF) has been selected then
Z contains the final set of eigenvectors corresponding to the eigenvalues held in
D. The real eigenvector associated with an eigenvalue is stored in the corresponding column of
Z.
 4: $\mathrm{LDZ}$ – INTEGERInput

On entry: the first dimension of the array
Z as declared in the (sub)program from which F12FCF 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: $\mathrm{SIGMA}$ – REAL (KIND=nag_wp)Input

On entry: if one of the
Shifted Inverse (see
F12FDF) modes has been selected then
SIGMA contains the real shift used; otherwise
SIGMA is not referenced.
 6: $\mathrm{RESID}\left(*\right)$ – REAL (KIND=nag_wp) arrayInput

Note: the dimension of the array
RESID
must be at least
${\mathbf{N}}$ (see
F12FAF).
On entry: must not be modified following a call to
F12FBF since it contains data required by F12FCF.
 7: $\mathrm{V}\left({\mathbf{LDV}},*\right)$ – 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
F12FAF).
On entry: the
NCV columns of
V contain the Lanczos basis vectors for
$\mathrm{OP}$ as constructed by
F12FBF.
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.
 8: $\mathrm{LDV}$ – INTEGERInput

On entry: the first dimension of the array
V as declared in the (sub)program from which F12FCF is called.
Constraint:
${\mathbf{LDV}}\ge n$.
 9: $\mathrm{COMM}\left(*\right)$ – 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
F12FAF).
On initial entry: must remain unchanged from the prior call to
F12FAF.
On exit: contains data on the current state of the solution.
 10: $\mathrm{ICOMM}\left(*\right)$ – 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
F12FAF).
On initial entry: must remain unchanged from the prior call to
F12FAF.
On exit: contains data on the current state of the solution.
 11: $\mathrm{IFAIL}$ – 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 F12FCF, as communicated through the parameter
ICOMM, is zero.
 ${\mathbf{IFAIL}}=4$

The number of converged eigenvalues as calculated by
F12FBF differ from the value passed to it through the parameter
ICOMM.
 ${\mathbf{IFAIL}}=5$

Unexpected error during calculation of a tridiagonal form: there was a failure to compute all the converged eigenvalues. Please contact
NAG.
 ${\mathbf{IFAIL}}=6$

The routine was unable to dynamically allocate sufficient internal workspace. Please contact
NAG.
 ${\mathbf{IFAIL}}=7$

An unexpected error has occurred. Please contact
NAG.
 ${\mathbf{IFAIL}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.8 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.7 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=999$
Dynamic memory allocation failed.
See
Section 3.6 in the Essential Introduction for further information.
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.
8 Parallelism and Performance
F12FCF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F12FCF 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 implementationspecific information.
None.
10 Example
This example solves $Ax=\lambda Bx$ in regular mode, where $A$ and $B$ are obtained from the standard central difference discretization of the onedimensional Laplacian operator $\frac{{d}^{2}u}{d{x}^{2}}$ on $\left[0,1\right]$, with zero Dirichlet boundary conditions.
10.1 Program Text
Program Text (f12fcfe.f90)
10.2 Program Data
Program Data (f12fcfe.d)
10.3 Program Results
Program Results (f12fcfe.r)