NAG Library Routine Document
F12AGF
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.
1 Purpose
F12AGF is the main solver routine in a suite of routines consisting of
F12ADF,
F12AFF and F12AGF. It must be called following an initial call to
F12AFF and following any calls to
F12ADF.
F12AGF returns approximations to selected eigenvalues, and (optionally) the corresponding eigenvectors, of a standard or generalized eigenvalue problem defined by real banded nonsymmetric matrices. The banded matrix must be stored using the LAPACK
storage format for real banded nonsymmetric matrices.
2 Specification
SUBROUTINE F12AGF ( 
KL, KU, AB, LDAB, MB, LDMB, SIGMAR, SIGMAI, NCONV, DR, DI, Z, LDZ, RESID, V, LDV, COMM, ICOMM, IFAIL) 
INTEGER 
KL, KU, LDAB, LDMB, NCONV, LDZ, LDV, ICOMM(*), IFAIL 
REAL (KIND=nag_wp) 
AB(LDAB,*), MB(LDMB,*), SIGMAR, SIGMAI, DR(*), DI(*), Z(LDZ,*), 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 banded, real and nonsymmetric.
Following a call to the initialization routine
F12AFF, F12AGF 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 banded nonsymmetric matrices. There is negligible additional computational cost to obtain eigenvectors; an orthonormal basis is always computed, but there is an additional storage cost if both are requested.
The banded matrices
$A$ and
$B$ must be stored using the LAPACK column ordered storage format for banded nonsymmetric matrices; please refer to
Section 3.3.2 in the F07 Chapter Introduction for details on this storage format.
F12AGF is based on the banded driver routines
dnbdr1 to
dnbdr6 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 banded driver software for real nonsymmetric problems, but the interface design is quite different in order to make the option setting clearer and to combine the different drivers into a general purpose routine.
F12AGF, is a general purpose routine that must be called following initialization by
F12AFF. F12AGF uses 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. 
Please note that for
Generalized problems, the
Shifted Inverse Imaginary and
Shifted Inverse Real inverse modes are only appropriate if either
$A$ or
$B$ is symmetric semidefinite. Otherwise, if
$A$ or
$B$ is nonsingular, the
Standard problem can be solved using the matrix
${B}^{1}A$ (say).
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: $\mathrm{KL}$ – INTEGERInput

On entry: the number of subdiagonals of the matrices $A$ and $B$.
Constraint:
${\mathbf{KL}}\ge 0$.
 2: $\mathrm{KU}$ – INTEGERInput

On entry: the number of superdiagonals of the matrices $A$ and $B$.
Constraint:
${\mathbf{KU}}\ge 0$.
 3: $\mathrm{AB}\left({\mathbf{LDAB}},*\right)$ – REAL (KIND=nag_wp) arrayInput

Note: the second dimension of the array
AB
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$ (see
F12AFF).
On entry: must contain the matrix
$A$ in LAPACK banded storage format for nonsymmetric matrices (see
Section 3.3.4 in the F07 Chapter Introduction).
 4: $\mathrm{LDAB}$ – INTEGERInput

On entry: the first dimension of the array
AB as declared in the (sub)program from which F12AGF is called.
Constraint:
${\mathbf{LDAB}}\ge 2\times {\mathbf{KL}}+{\mathbf{KU}}+1$.
 5: $\mathrm{MB}\left({\mathbf{LDMB}},*\right)$ – REAL (KIND=nag_wp) arrayInput

Note: the second dimension of the array
MB
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$ (see
F12AFF).
On entry: must contain the matrix
$B$ in LAPACK banded storage format for nonsymmetric matrices (see
Section 3.3.4 in the F07 Chapter Introduction).
 6: $\mathrm{LDMB}$ – INTEGERInput

On entry: the first dimension of the array
MB as declared in the (sub)program from which F12AGF is called.
Constraint:
${\mathbf{LDMB}}\ge 2\times {\mathbf{KL}}+{\mathbf{KU}}+1$.
 7: $\mathrm{SIGMAR}$ – REAL (KIND=nag_wp)Input

On entry: if one of the
Shifted Inverse Real modes (see
F12ADF) have been selected then
SIGMAR must contain the real part of the shift used; otherwise
SIGMAR is not referenced.
Section 4.3.4 in the F12 Chapter Introduction describes the use of shift and inverse transformations.
 8: $\mathrm{SIGMAI}$ – REAL (KIND=nag_wp)Input

On entry: if one of the
Shifted Inverse Real modes (see
F12ADF) have been selected then
SIGMAI must contain the imaginary part of the shift used; otherwise
SIGMAI is not referenced.
Section 4.3.4 in the F12 Chapter Introduction describes the use of shift and inverse transformations.
 9: $\mathrm{NCONV}$ – INTEGEROutput

On exit: the number of converged eigenvalues.
 10: $\mathrm{DR}\left(*\right)$ – REAL (KIND=nag_wp) arrayOutput

Note: the dimension of the array
DR
must be at least
${\mathbf{NEV}}+1$ (see
F12AFF).
On exit: the first
NCONV locations of the array
DR contain the real parts of the converged approximate eigenvalues. The number of eigenvalues returned may be one more than the number requested by
NEV since complex values occur as conjugate pairs and the second in the pair can be returned in position
${\mathbf{NEV}}+1$ of the array.
 11: $\mathrm{DI}\left(*\right)$ – REAL (KIND=nag_wp) arrayOutput

Note: the dimension of the array
DI
must be at least
${\mathbf{NEV}}+1$ (see
F12AFF).
On exit: the first
NCONV locations of the array
DI contain the imaginary parts of the converged approximate eigenvalues. The number of eigenvalues returned may be one more than the number requested by
NEV since complex values occur as conjugate pairs and the second in the pair can be returned in position
${\mathbf{NEV}}+1$ of the array.
 12: $\mathrm{Z}\left({\mathbf{LDZ}},*\right)$ – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array
Z
must be at least
${\mathbf{NEV}}+1$ if the default option
${\mathbf{Vectors}}=\text{Ritz}$ has been selected and at least
$1$ if the option
${\mathbf{Vectors}}=\text{None or Schur}$ has been selected (see
F12AAF).
On exit: if the default option
${\mathbf{Vectors}}=\text{Ritz}$ 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.
 13: $\mathrm{LDZ}$ – INTEGERInput

On entry: the first dimension of the array
Z as declared in the (sub)program from which F12AGF 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, ${\mathbf{LDZ}}\ge 1$.
 14: $\mathrm{RESID}\left(*\right)$ – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array
RESID
must be at least
${\mathbf{N}}$ (see
F12AFF).
On entry: need not be set unless the option
Initial Residual has been set in a prior call to
F12ADF in which case
RESID must contain an initial residual vector.
On exit: contains the final residual vector.
 15: $\mathrm{V}\left({\mathbf{LDV}},*\right)$ – REAL (KIND=nag_wp) arrayOutput

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
F12AFF).
On exit: if the option
Vectors (see
F12ADF) has been set to Schur or Ritz then the first
${\mathbf{NCONV}}\times n$ elements of
V will contain approximate Schur vectors that span the desired invariant subspace.
The
$i$th Schur vector is stored in the
$i$th column of
V.
 16: $\mathrm{LDV}$ – INTEGERInput

On entry: the first dimension of the array
V as declared in the (sub)program from which F12AGF is called.
Constraint:
${\mathbf{LDV}}\ge {\mathbf{N}}$.
 17: $\mathrm{COMM}\left(*\right)$ – REAL (KIND=nag_wp) arrayCommunication Array

On entry: must remain unchanged from the prior call to
F12ADF and
F12AFF.
On exit: contains no useful information.
 18: $\mathrm{ICOMM}\left(*\right)$ – INTEGER arrayCommunication Array

On entry: must remain unchanged from the prior call to
F12ADF and
F12AFF.
On exit: contains no useful information.
 19: $\mathrm{IFAIL}$ – 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, ${\mathbf{KL}}<0$.
 ${\mathbf{IFAIL}}=2$

On entry, ${\mathbf{KU}}<0$.
 ${\mathbf{IFAIL}}=3$

On entry, ${\mathbf{LDAB}}<2\times {\mathbf{KL}}+{\mathbf{KU}}+1$.
 ${\mathbf{IFAIL}}=4$

On entry, the option
${\mathbf{Shifted\; Inverse\; Imaginary}}$ was selected, and
${\mathbf{SIGMAI}}=\text{zero}$, but
SIGMAI must be nonzero for this computational mode.
 ${\mathbf{IFAIL}}=5$

${\mathbf{Iteration\; Limit}}<0$.
 ${\mathbf{IFAIL}}=6$

The options
Generalized and
Regular are incompatible.
 ${\mathbf{IFAIL}}=7$

The
Initial Residual was selected but the starting vector held in
RESID is zero.
 ${\mathbf{IFAIL}}=8$

Either the initialization routine
F12AFF has not been called prior to the first call of this routine or a communication array has become corrupted.
 ${\mathbf{IFAIL}}=9$

On entry, ${\mathbf{LDZ}}<{\mathbf{N}}$ or ${\mathbf{LDZ}}<1$ when no vectors are required.
 ${\mathbf{IFAIL}}=10$

On entry, the option ${\mathbf{Vectors}}=\text{Select}$ was selected, but this is not yet implemented.
 ${\mathbf{IFAIL}}=11$

The number of eigenvalues found to sufficient accuracy is zero.
 ${\mathbf{IFAIL}}=12$

Could not build an Arnoldi factorization. Consider changing
NCV or
NEV in the initialization routine (see
Section 5 in F12AFF for details of these parameters).
 ${\mathbf{IFAIL}}=13$

Unexpected error in internal call to compute eigenvalues and corresponding error bounds of the current upper Hessenberg matrix. Please contact
NAG.
 ${\mathbf{IFAIL}}=14$

Unexpected error during calculation of a real Schur form: there was a failure to compute all the converged eigenvalues. Please contact
NAG.
 ${\mathbf{IFAIL}}=15$

Unexpected error: the computed Schur form could not be reordered by an internal call. Please contact
NAG.
 ${\mathbf{IFAIL}}=16$

Unexpected error in internal call while calculating eigenvectors. Please contact
NAG.
 ${\mathbf{IFAIL}}=17$

Failure during internal factorization of real banded matrix. Please contact
NAG.
 ${\mathbf{IFAIL}}=18$

Failure during internal solution of real banded system. Please contact
NAG.
 ${\mathbf{IFAIL}}=19$

Failure during internal factorization of complex banded matrix. Please contact
NAG.
 ${\mathbf{IFAIL}}=20$

Failure during internal solution of complex banded system. Please contact
NAG.
 ${\mathbf{IFAIL}}=21$

The maximum number of iterations has been reached. Some Ritz values may have converged;
NCONV returns the number of converged values.
 ${\mathbf{IFAIL}}=22$

No shifts could be applied during a cycle of the implicitly restarted Arnoldi iteration. One possibility is to increase the size of
NCV relative to
NEV (see
Section 5 in F12AFF for details of these parameters).
 ${\mathbf{IFAIL}}=23$

Overflow occurred during transformation of Ritz values to those of the original problem.
 ${\mathbf{IFAIL}}=24$

The routine was unable to dynamically allocate sufficient internal workspace. Please contact
NAG.
 ${\mathbf{IFAIL}}=25$

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
F12AGF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F12AGF 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 constructs the matrices $A$ and $B$ using LAPACK band storage format and solves $Ax=\lambda Bx$ in shifted imaginary mode using the complex shift $\sigma $.
10.1 Program Text
Program Text (f12agfe.f90)
10.2 Program Data
Program Data (f12agfe.d)
10.3 Program Results
Program Results (f12agfe.r)