NAG CL Interface
f12auc (complex_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 f12arc need not be called.
If, however, you wish to reset some or all of the settings please refer to Section 11 in f12arc for a detailed description of the specification of the optional parameters.
1
Purpose
f12auc is the main solver function in a suite of functions consisting of
f12arc,
f12atc and
f12auc. It must be called following an initial call to
f12atc and following any calls to
f12arc.
f12auc returns approximations to selected eigenvalues, and (optionally) the corresponding eigenvectors, of a standard or generalized eigenvalue problem defined by complex banded nonHermitian matrices. The banded matrix must be stored using the LAPACK
column ordered
storage format for complex banded nonHermitian
(see
Section 3.4.4 in the
F07 Chapter Introduction).
2
Specification
void 
f12auc (Integer kl,
Integer ku,
const Complex ab[],
const Complex mb[],
Complex sigma,
Integer *nconv,
Complex d[],
Complex z[],
Complex resid[],
Complex v[],
Complex comm[],
Integer icomm[],
NagError *fail) 

The function may be called by the names: f12auc, nag_sparseig_complex_band_solve or nag_complex_banded_eigensystem_solve.
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, complex and nonHermitian.
Following a call to the initialization function
f12atc,
f12auc returns the converged approximations to eigenvalues and (optionally) the corresponding approximate eigenvectors and/or a unitary 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 banded nonHermitian matrices. There is negligible additional computational cost to obtain eigenvectors; a unitary 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 nonHermitian matrices; please refer to
Section 3.4.4 in the
F07 Chapter Introduction for details on this storage format.
f12auc is based on the banded driver functions
znbdr1 to
znbdr4 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 nonHermitian matrices is provided in
Lehoucq and Scott (1996). This suite of functions offers the same functionality as the ARPACK banded driver software for complex nonHermitian 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.
f12auc, is a general purpose function that must be called following initialization by
f12atc.
f12auc uses options, set either by default or explicitly by calling
f12arc, to return the converged approximations to selected eigenvalues and (optionally):

–the corresponding approximate eigenvectors;

–a unitary 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
Note: in the following description
n,
nev and
ncv appears. In every case they should be interpretted as the value associated with the identically named argument in a prior call to
f12atc.

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 $B$.
Constraint:
${\mathbf{ku}}\ge 0$.

3:
$\mathbf{ab}\left[\mathit{dim}\right]$ – const Complex
Input

Note: the dimension,
dim, of the array
ab
must be at least
${\mathbf{n}}\times \left(2\times {\mathbf{kl}}+{\mathbf{ku}}+1\right)$.
On entry: must contain the matrix
$A$ in LAPACK columnordered banded storage format for nonHermitian matrices; that is, element
${a}_{ij}$ is stored in
${\mathbf{ab}}\left[\left(j1\right)\times \left(2\times {\mathbf{kl}}+{\mathbf{ku}}+1\right)+{\mathbf{kl}}+{\mathbf{ku}}+ij\right]$, which may be written as
${\mathbf{ab}}\left[\left(2\times j1\right)\times {\mathbf{kl}}+j\times {\mathbf{ku}}+i1\right]$, for
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j{\mathbf{ku}}\right)\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{\mathbf{kl}}\right)$ and
$j=1,2,\dots ,n$, (see
Section 3.4.4 in the
F07 Chapter Introduction).

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

Note: the dimension,
dim, of the array
mb
must be at least
${\mathbf{n}}\times \left(2\times {\mathbf{kl}}+{\mathbf{ku}}+1\right)$.
On entry: must contain the matrix
$B$ in LAPACK columnordered banded storage format for nonHermitian matrices; that is, element
${a}_{ij}$ is stored in
${\mathbf{mb}}\left[\left(j1\right)\times \left(2\times {\mathbf{kl}}+{\mathbf{ku}}+1\right)+{\mathbf{kl}}+{\mathbf{ku}}+ij\right]$, which may be written as
${\mathbf{mb}}\left[\left(2\times j1\right)\times {\mathbf{kl}}+j\times {\mathbf{ku}}+i1\right]$, for
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j{\mathbf{ku}}\right)\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{\mathbf{kl}}\right)$ and
$j=1,2,\dots ,n$, (see
Section 3.4.4 in the
F07 Chapter Introduction).

5:
$\mathbf{sigma}$ – Complex
Input

On entry: if the
${\mathbf{Shifted\; Inverse}}$ mode (see
f12arc) has been selected then
sigma must contain the shift used; otherwise
sigma is not referenced.
Section 4.2 in the
F12 Chapter Introduction describes the use of shift and invert transformations.

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

On exit: the number of converged eigenvalues.

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

Note: the dimension,
dim, of the array
d
must be at least
${\mathbf{nev}}$ (see
f12atc).
On exit: the first
nconv locations of the array
d contain the converged approximate eigenvalues.

8:
$\mathbf{z}\left[\mathit{dim}\right]$ – Complex
Output

Note: the dimension,
dim, of the array
z
must be at least
${\mathbf{n}}\times {\mathbf{nev}}$ if the default option
${\mathbf{Vectors}}=\mathrm{RITZ}$ (see
f12arc) has been selected (see
f12atc).
On exit: if the default option
${\mathbf{Vectors}}=\mathrm{RITZ}$ (see
f12arc) has been selected then
z contains the final set of eigenvectors corresponding to the eigenvalues held in
d, otherwise
z is not referenced and may be
NULL. The complex eigenvector associated with an eigenvalue
${\mathbf{d}}\left[j\right]$ is stored in the corresponding array section of
z, namely
${\mathbf{z}}\left[{\mathbf{n}}\times \left(\mathit{j}1\right)+\mathit{i}1\right]$, for
$\mathit{i}=1,2,\dots ,{\mathbf{n}}$ and
$\mathit{j}=1,2,\dots ,{\mathbf{nconv}}$.

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

Note: the dimension,
dim, of the array
resid
must be at least
${\mathbf{n}}$ (see
f12atc).
On entry: need not be set unless the option
${\mathbf{Initial\; Residual}}$ has been set in a prior call to
f12arc in which case
resid must contain an initial residual vector.
On exit: contains the final residual vector. This can be used as the starting residual to improve convergence on the solution of a closely related eigenproblem. This has no relation to the error residual $Ax\lambda x$ or $Ax\lambda Bx$.

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

On exit: if the option
${\mathbf{Vectors}}=\mathrm{SCHUR}$ or
$\mathrm{RITZ}$ (see
f12arc) has been set and a separate array
z has been passed (i.e.,
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:
$\mathbf{comm}\left[60\right]$ – Complex
Communication Array

On entry: must remain unchanged from the prior call to
f12arc and
f12atc.

12:
$\mathbf{icomm}\left[140\right]$ – Integer
Communication Array

On entry: must remain unchanged from the prior call to
f12arc and
f12atc.

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_COMP_BAND_FAC

Failure during internal factorization of complex banded matrix. Please contact
NAG.
 NE_COMP_BAND_SOL

Failure during internal solution of complex banded matrix. Please contact
NAG.
 NE_EIGENVALUES

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

Either the initialization function has not been called prior to the first call of this function or a communication array has 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: ${\mathbf{ku}}\ge 0$.
 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_EIGVEC_FAIL

Error in internal call to compute eigenvectors. 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.
The maximum number of iterations $\text{}\le 0$, the option ${\mathbf{Iteration\; Limit}}$ has been set to $\u2329\mathit{\text{value}}\u232a$.
 NE_NO_ARNOLDI_FAC

Could not build an Arnoldi factorization. The size of the current Arnoldi 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 Arnoldi iteration.
 NE_OPT_INCOMPAT

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

Overflow occurred during transformation of Ritz values to those of the original problem.
 NE_REAL_BAND_FAC

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

Failure during internal solution of real banded matrix. Please contact
NAG.
 NE_SCHUR_EIG_FAIL

During calculation of a Schur form, there was a failure to compute a number of eigenvalues Please contact
NAG.
 NE_SCHUR_REORDER

The computed Schur form could not be reordered by an internal call. Please contact
NAG.
 NE_TOO_MANY_ITER

The maximum number of iterations has been reached. The maximum number of $\text{iterations}=\u2329\mathit{\text{value}}\u232a$. The number of converged eigenvalues $\text{}=\u2329\mathit{\text{value}}\u232a$.
 NE_ZERO_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
f12auc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f12auc 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 constructs the matrices $A$ and $B$ using LAPACK band storage format and solves $Ax=\lambda Bx$ in shifted inverse mode using the complex shift $\sigma $.
10.1
Program Text
10.2
Program Data
10.3
Program Results