NAG CL Interface
f12fgc (real_symm_band_solve)
Note: this function 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 function f12fdc need not be called.
If, however, you wish to reset some or all of the settings please refer to Section 11 in f12fdc for a detailed description of the specification of the optional parameters.
1
Purpose
f12fgc is the main solver function in a suite of functions which includes
f12fdc and
f12ffc.
f12fgc must be called following an initial call to
f12ffc and following any calls to
f12fdc.
f12fgc 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
void 
f12fgc (Integer kl,
Integer ku,
const double ab[],
const double mb[],
double sigma,
Integer *nconv,
double d[],
double z[],
double resid[],
double v[],
double comm[],
Integer icomm[],
NagError *fail) 

The function may be called by the names: f12fgc, nag_sparseig_real_symm_band_solve or nag_real_symm_banded_sparse_eigensystem_sol.
3
Description
The suite of functions 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 function
f12ffc,
f12fgc 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 3.4.2 in the
F07 Chapter Introduction for details on this storage format.
f12fgc is based on the banded driver functions
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 functions 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 function.
f12fgc, is a general purpose direct communication function that must be called following initialization by
f12ffc.
f12fgc uses options, set either by default or explicitly by calling
f12fdc, 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:
$\mathbf{kl}$ – Integer
Input

On entry: the number of subdiagonals of the matrices $A$ and $B$.
Constraint:
${\mathbf{kl}}\ge 0$.

2:
$\mathbf{ku}$ – Integer
Input

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:
$\mathbf{ab}\left[\mathit{dim}\right]$ – const double
Input

Note: the dimension,
dim, of the array
ab
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}},\times ,\left(2\times {\mathbf{kl}}+{\mathbf{ku}}+1\right)\right)$ (see
f12ffc).
On entry: must contain the matrix
$A$ in LAPACK columnordered banded storage format for nonsymmetric matrices (see
Section 3.4.4 in the
F07 Chapter Introduction).

4:
$\mathbf{mb}\left[\mathit{dim}\right]$ – const double
Input

Note: the dimension,
dim, of the array
mb
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}},\times ,\left(2\times {\mathbf{kl}}+{\mathbf{ku}}+1\right)\right)$ (see
f12ffc).
On entry: must contain the matrix
$B$ in LAPACK columnordered banded storage format for nonsymmetric matrices (see
Section 3.4.4 in the
F07 Chapter Introduction).

5:
$\mathbf{sigma}$ – double
Input

On entry: if one of the
${\mathbf{Shifted\; Inverse}}$ (see
f12fdc) modes has been selected then
sigma contains the real shift used; otherwise
sigma is not referenced.

6:
$\mathbf{nconv}$ – Integer *
Output

On exit: the number of converged eigenvalues.

7:
$\mathbf{d}\left[\mathit{dim}\right]$ – double
Output

Note: the dimension,
dim, of the array
d
must be at least
${\mathbf{ncv}}$ (see
f12ffc).
On exit: the first
nconv locations of the array
d contain the converged approximate eigenvalues.

8:
$\mathbf{z}\left[{\mathbf{n}}\times \left({\mathbf{nev}}+1\right)\right]$ – double
Output

On exit: if the default option
${\mathbf{Vectors}}=\mathrm{RITZ}$ (see
f12fdc) 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 at locations
${\mathbf{z}}\left[\mathit{i}1\times n+\mathit{j}1\right]$, for
$\mathit{j}=1,2,\dots ,n$.

9:
$\mathbf{resid}\left[\mathit{dim}\right]$ – double
Input/Output

Note: the dimension,
dim, of the array
resid
must be at least
${\mathbf{n}}$ (see
f12ffc).
On entry: need not be set unless the option
${\mathbf{Initial\; Residual}}$ has been set in a prior call to
f12fdc in which case
resid must contain an initial residual vector.
On exit: contains the final residual vector.

10:
$\mathbf{v}\left[{\mathbf{n}}\times {\mathbf{ncv}}\right]$ – double
Output

On exit: if the option
${\mathbf{Vectors}}$ (see
f12fdc) has been set to Schur or Ritz and
z does not equal
v then the first
nconv sections of
v, of length
$n$, will contain approximate Schur vectors that span the desired invariant subspace.
The $j$th Schur vector is stored in locations
${\mathbf{v}}\left[{\mathbf{n}}\times \left(\mathit{j}1\right)+\mathit{i}1\right]$, for $\mathit{j}=1,2,\dots ,{\mathbf{nconv}}$ and $\mathit{i}=1,2,\dots ,n$.

11:
$\mathbf{comm}\left[\mathit{dim}\right]$ – double
Communication Array

Note: the actual argument supplied
must be the array
comm supplied to the initialization routine
f12fdc.
On initial entry: must remain unchanged from the prior call to
f12fdc and
f12ffc.
On exit: contains no useful information.

12:
$\mathbf{icomm}\left[\mathit{dim}\right]$ – Integer
Communication Array

Note: the actual argument supplied
must be the array
icomm supplied to the initialization routine
f12ffc.
On initial entry: must remain unchanged from the prior call to
f12fbc and
f12fdc.
On exit: contains no useful information.

13:
$\mathbf{fail}$ – NagError *
Input/Output

The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_BOTH_ENDS_1

Eigenvalues from both ends of the spectrum were requested, but the number of eigenvalues (
nev in
f12ffc) requested is one.
 NE_INITIALIZATION

Either an initial call to the setup function has not been made or the communication arrays have become corrupted.
 NE_INT

On entry, ${\mathbf{kl}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{kl}}\ge 0$.
On entry, ${\mathbf{ku}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{ku}}\ge 0$.
The maximum number of iterations $\le 0$, the option ${\mathbf{Iteration\; Limit}}$ has been set to $\u2329\mathit{\text{value}}\u232a$.
 NE_INT_2

The maximum number of iterations has been reached: there have been $\u2329\mathit{\text{value}}\u232a$ iterations. There are $\u2329\mathit{\text{value}}\u232a$ converged eigenvalues.
 NE_INTERNAL_EIGVAL_FAIL

Error in internal call to compute eigenvalues and corresponding error bounds of the current upper Hessenberg matrix. Please contact
NAG.
 NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
 NE_INVALID_OPTION

On entry, ${\mathbf{Vectors}}=\text{Select}$, but this is not yet implemented.
 NE_MAX_ITER

During calculation of a tridiagonal form, there was a failure to compute $\u2329\mathit{\text{value}}\u232a$ eigenvalues in a total of $\u2329\mathit{\text{value}}\u232a$ iterations.
 NE_NO_LANCZOS_FAC

Could not build a Lanczos factorization. The size of the current Lanczos factorization $=\u2329\mathit{\text{value}}\u232a$.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
 NE_NO_SHIFTS_APPLIED

No shifts could be applied during a cycle of the implicitly restarted Lanczos iteration.
 NE_OPT_INCOMPAT

The options ${\mathbf{Generalized}}$ and ${\mathbf{Regular}}$ are incompatible.
 NE_REAL_BAND_FAC

Failure during internal factorization of banded matrix. Please contact
NAG.
 NE_REAL_BAND_SOL

Failure during internal solution of banded system. Please contact
NAG.
 NE_ZERO_EIGS_FOUND

The number of eigenvalues found to sufficient accuracy is zero.
 NE_ZERO_INIT_RESID

The option
${\mathbf{Initial\; Residual}}$ was selected but the starting vector held in
resid is zero.
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
${\mathbf{Tolerance}}$ used is the
machine precision given by
X02AJC.
8
Parallelism and Performance
f12fgc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f12fgc 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 function. 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
10.2
Program Data
10.3
Program Results