NAG Library Routine Document
F12FGF
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
F12FGF is the main solver routine in a suite of routines which includes
F12FDF and
F12FFF. F12FGF must be called following an initial call to
F12FFF and following any calls to
F12FDF.
F12FGF returns approximations to selected eigenvalues, and (optionally) the corresponding eigenvectors, of a standard or generalized eigenvalue problem defined by real banded symmetric matrices. The banded matrix must be stored using the LAPACK storage format for real banded nonsymmetric matrices.
2 Specification
SUBROUTINE F12FGF ( 
KL, KU, AB, LDAB, MB, LDMB, SIGMA, NCONV, D, 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,*), SIGMA, D(*), 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 symmetric.
Following a call to the initialization routine
F12FFF, F12FGF 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 symmetric 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 storage format for banded nonsymmetric matrices; please refer to
Section 3.3.2 in the F07 Chapter Introduction for details on this storage format.
F12FGF is based on the banded driver routines
dsbdr1 to
dsbdr6 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). This suite of routines offers the same functionality as the ARPACK banded driver software for real symmetric 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.
F12FGF, is a general purpose direct communication routine that must be called following initialization by
F12FFF. F12FGF uses 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 Arguments
 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 $B$. Since $A$ and $B$ are symmetric, the normal case is ${\mathbf{KU}}={\mathbf{KL}}$.
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
F12FFF).
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 F12FGF 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
F12FFF).
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 F12FGF is called.
Constraint:
${\mathbf{LDMB}}\ge 2\times {\mathbf{KL}}+{\mathbf{KU}}+1$.
 7: $\mathrm{SIGMA}$ – 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.
 8: $\mathrm{NCONV}$ – INTEGEROutput

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

Note: the dimension of the array
D
must be at least
${\mathbf{NCV}}$ (see
F12FFF).
On exit: the first
NCONV locations of the array
D contain the converged approximate eigenvalues.
 10: $\mathrm{Z}\left({\mathbf{LDZ}},*\right)$ – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array
Z
must be at least
${\mathbf{NCV}}+1$ 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
F12FFF).
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 eigenvalue
$\mathit{i}1$, for
$\mathit{i}=1,2,\dots ,{\mathbf{NCONV}}$, is stored in the
$\mathit{i}$th column of
Z.
 11: $\mathrm{LDZ}$ – INTEGERInput

On entry: the first dimension of the array
Z as declared in the (sub)program from which F12FGF 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, ${\mathbf{LDZ}}\ge 1$.
 12: $\mathrm{RESID}\left(*\right)$ – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array
RESID
must be at least
${\mathbf{N}}$ (see
F12FFF).
On entry: need not be set unless the option
Initial Residual has been set in a prior call to
F12FDF in which case
RESID must contain an initial residual vector.
On exit: contains the final residual vector.
 13: $\mathrm{V}\left({\mathbf{LDV}},*\right)$ – 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
F12FFF).
On exit: if the option
Vectors (see
F12FDF) has been set to Schur or Ritz and a separate array
Z has been passed then the first
${\mathbf{NCONV}}\times n$ elements of
V will contain approximate Schur vectors that span the desired invariant subspace.
The
$j$th Schur vector is stored in the
$i$th column of
V.
 14: $\mathrm{LDV}$ – INTEGERInput

On entry: the first dimension of the array
V as declared in the (sub)program from which F12FGF is called.
Constraint:
${\mathbf{LDV}}\ge n$.
 15: $\mathrm{COMM}\left(*\right)$ – 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
F12FFF).
On initial entry: must remain unchanged from the prior call to
F12FDF and
F12FFF.
On exit: contains no useful information.
 16: $\mathrm{ICOMM}\left(*\right)$ – 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
F12FFF).
On initial entry: must remain unchanged from the prior call to
F12FBF and
F12FDF.
On exit: contains no useful information.
 17: $\mathrm{IFAIL}$ – 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).
6 Error 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, ${\mathbf{KL}}<0$.
 ${\mathbf{IFAIL}}=2$

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

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

${\mathbf{Iteration\; Limit}}<0$.
 ${\mathbf{IFAIL}}=5$

The options
Generalized and
Regular are incompatible.
 ${\mathbf{IFAIL}}=6$

Eigenvalues from
Both Ends of the spectrum were requested, but only one eigenvalue (
NEV) is requested.
 ${\mathbf{IFAIL}}=7$

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

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.
 ${\mathbf{IFAIL}}=9$

On entry, the option ${\mathbf{Vectors}}=\text{Select}$ was selected, but this is not yet implemented.
 ${\mathbf{IFAIL}}=10$

The number of eigenvalues found to sufficient accuracy is zero.
 ${\mathbf{IFAIL}}=11$

Could not build a Lanczos factorization. Consider changing
NCV or
NEV in the initialization routine (see
Section 5 in F12FAF for details of these arguments).
 ${\mathbf{IFAIL}}=12$

Unexpected error in internal call to compute eigenvalues and corresponding error bounds of the current symmetric tridiagonal matrix. Please contact
NAG.
 ${\mathbf{IFAIL}}=13$

Unexpected error during calculation of a real Schur form: there was a failure to compute all the converged eigenvalues. Please contact
NAG.
 ${\mathbf{IFAIL}}=14$

Failure during internal factorization of real banded matrix. Please contact
NAG.
 ${\mathbf{IFAIL}}=15$

Failure during internal solution of real banded system. Please contact
NAG.
 ${\mathbf{IFAIL}}=16$

The maximum number of iterations has been reached. Some Ritz values may have converged;
NCONV returns the number of converged values.
 ${\mathbf{IFAIL}}=17$

No shifts could be applied during a cycle of the implicitly restarted Lanczos iteration. One possibility is to increase the size of
NCV relative to
NEV (see
Section 5 in F12FFF for details of these arguments).
 ${\mathbf{IFAIL}}=18$

Either an initial call to the setup routine has not been made or the communication arrays have become corrupted.
 ${\mathbf{IFAIL}}=19$

The routine was unable to dynamically allocate sufficient internal workspace. Please contact
NAG.
 ${\mathbf{IFAIL}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
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.
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
F12FGF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F12FGF 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 $Ax=\lambda x$ in regular mode, where $A$ is obtained from the standard central difference discretization of the twodimensional convectiondiffusion operator $\frac{{d}^{2}u}{d{x}^{2}}+\frac{{d}^{2}u}{d{y}^{2}}=\rho \frac{du}{dx}$ on the unit square with zero Dirichlet boundary conditions. $A$ is stored in LAPACK banded storage format.
10.1 Program Text
Program Text (f12fgfe.f90)
10.2 Program Data
Program Data (f12fgfe.d)
10.3 Program Results
Program Results (f12fgfe.r)