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NAG Toolbox: nag_sparseig_real_band_solve (f12ag)
Purpose
nag_sparseig_real_band_solve (f12ag) is the main solver function in a suite of functions consisting of
nag_sparseig_real_option (f12ad),
nag_sparseig_real_band_init (f12af) and
nag_sparseig_real_band_solve (f12ag). It must be called following an initial call to
nag_sparseig_real_band_init (f12af) and following any calls to
nag_sparseig_real_option (f12ad).
nag_sparseig_real_band_solve (f12ag) 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.
Syntax
[
nconv,
dr,
di,
z,
resid,
v,
comm,
icomm,
ifail] = f12ag(
kl,
ku,
ab,
mb,
sigmar,
sigmai,
resid,
comm,
icomm)
[
nconv,
dr,
di,
z,
resid,
v,
comm,
icomm,
ifail] = nag_sparseig_real_band_solve(
kl,
ku,
ab,
mb,
sigmar,
sigmai,
resid,
comm,
icomm)
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 nonsymmetric.
Following a call to the initialization function
nag_sparseig_real_band_init (f12af),
nag_sparseig_real_band_solve (f12ag) 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
Packed storage in the F07 Chapter Introduction for details on this storage format.
nag_sparseig_real_band_solve (f12ag) is based on the banded driver functions
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 functions 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 function.
nag_sparseig_real_band_solve (f12ag), is a general purpose function that must be called following initialization by
nag_sparseig_real_band_init (f12af).
nag_sparseig_real_band_solve (f12ag) uses options, set either by default or explicitly by calling
nag_sparseig_real_option (f12ad), 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).
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
Parameters
Compulsory Input Parameters
 1:
$\mathrm{kl}$ – int64int32nag_int scalar

The number of subdiagonals of the matrices $A$ and $B$.
Constraint:
${\mathbf{kl}}\ge 0$.
 2:
$\mathrm{ku}$ – int64int32nag_int scalar

The number of superdiagonals of the matrices $A$ and $B$.
Constraint:
${\mathbf{ku}}\ge 0$.
 3:
$\mathrm{ab}\left(\mathit{ldab},:\right)$ – double array

The first dimension of the array
ab must be at least
$2\times {\mathbf{kl}}+{\mathbf{ku}}+1$.
The second dimension of the array
ab must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
Must contain the matrix
$A$ in LAPACK banded storage format for nonsymmetric matrices (see
Band storage in the F07 Chapter Introduction).
 4:
$\mathrm{mb}\left(\mathit{ldmb},:\right)$ – double array

The first dimension of the array
mb must be at least
$2\times {\mathbf{kl}}+{\mathbf{ku}}+1$.
The second dimension of the array
mb must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
Must contain the matrix
$B$ in LAPACK banded storage format for nonsymmetric matrices (see
Band storage in the F07 Chapter Introduction).
 5:
$\mathrm{sigmar}$ – double scalar

If one of the
Shifted Inverse Real modes (see
nag_sparseig_real_option (f12ad)) have been selected then
sigmar must contain the real part of the shift used; otherwise
sigmar is not referenced.
Computational modes for nonHermitian problems in the F12 Chapter Introduction describes the use of shift and inverse transformations.
 6:
$\mathrm{sigmai}$ – double scalar

If one of the
Shifted Inverse Real modes (see
nag_sparseig_real_option (f12ad)) have been selected then
sigmai must contain the imaginary part of the shift used; otherwise
sigmai is not referenced.
Computational modes for nonHermitian problems in the F12 Chapter Introduction describes the use of shift and inverse transformations.
 7:
$\mathrm{resid}\left(:\right)$ – double array

The dimension of the array
resid
must be at least
${\mathbf{n}}$ (see
nag_sparseig_real_band_init (f12af))
Need not be set unless the option
Initial Residual has been set in a prior call to
nag_sparseig_real_option (f12ad) in which case
resid must contain an initial residual vector.
 8:
$\mathrm{comm}\left(:\right)$ – double array

 9:
$\mathrm{icomm}\left(:\right)$ – int64int32nag_int array

Optional Input Parameters
None.
Output Parameters
 1:
$\mathrm{nconv}$ – int64int32nag_int scalar

The number of converged eigenvalues.
 2:
$\mathrm{dr}\left(:\right)$ – double array

The dimension of the array
dr will be
${\mathbf{nev}}+1$ (see
nag_sparseig_real_band_init (f12af))
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.
 3:
$\mathrm{di}\left(:\right)$ – double array

The dimension of the array
di will be
${\mathbf{nev}}+1$ (see
nag_sparseig_real_band_init (f12af))
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.
 4:
$\mathrm{z}\left({\mathbf{n}}\times \left({\mathbf{nev}}+1\right)\right)$ – double array

The second dimension of the array
z will be
${\mathbf{n}}\times \left({\mathbf{nev}}+1\right)$.
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.
 5:
$\mathrm{resid}\left(:\right)$ – double array

The dimension of the array
resid will be
${\mathbf{n}}$ (see
nag_sparseig_real_band_init (f12af))
Contains the final residual vector.
 6:
$\mathrm{v}\left({\mathbf{n}}\times {\mathbf{ncv}}\right)$ – double array

If the option
Vectors (see
nag_sparseig_real_option (f12ad)) 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.
 7:
$\mathrm{comm}\left(:\right)$ – double array

Contains no useful information.
 8:
$\mathrm{icomm}\left(:\right)$ – int64int32nag_int array

Contains no useful information.
 9:
$\mathrm{ifail}$ – int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see
Error Indicators and Warnings).
Error Indicators and Warnings
Errors or warnings detected by the function:
 ${\mathbf{ifail}}=1$

On entry, ${\mathbf{kl}}<0$.
 ${\mathbf{ifail}}=2$

On entry, ${\mathbf{ku}}<0$.
 ${\mathbf{ifail}}=3$

On entry, $\mathit{ldab}<2\times {\mathbf{kl}}+{\mathbf{ku}}+1$.
 ${\mathbf{ifail}}=4$

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 function
nag_sparseig_real_band_init (f12af) has not been called prior to the first call of this function or a communication array has become corrupted.
 ${\mathbf{ifail}}=9$

On entry, $\mathit{ldz}<{\mathbf{n}}$ or $\mathit{ldz}<1$ 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 function (see
Arguments in
nag_sparseig_real_band_init (f12af) for details of these arguments).
 ${\mathbf{ifail}}=13$

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
Arguments in
nag_sparseig_real_band_init (f12af) for details of these arguments).
 ${\mathbf{ifail}}=23$

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

The function 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.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
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
nag_machine_precision (x02aj).
Further Comments
None.
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 $.
Open in the MATLAB editor:
f12ag_example
function f12ag_example
fprintf('f12ag example results\n\n');
nx = int64(10);
n = nx*nx;
nev = int64(4);
ncv = nx;
kl = nx;
ku = nx;
ab = zeros(2*kl+ku+1,n);
mb = zeros(2*kl+ku+1,n);
idiag = kl + ku + 1;
for j=1:n
ab(idiag,j) = 4;
mb(idiag,j) = 4;
end
isup = kl + ku;
isub = kl + ku + 2;
rho = 100;
h = 1/double(nx+1);
for i=1:nx
lo = (i1)*nx;
for j=lo+1:lo+nx1
ab(isub,j+1) = 1 + 0.5*h*rho;
ab(isup,j) = 1  0.5*h*rho;
end
end
for j = 1:n  1
mb(isub,j+1) = 1;
mb(isup,j) = 1;
end
isup = kl + 1;
isub = 2*kl + ku + 1;
for i = 1:nx  1
lo = (i1)*nx;
for j = lo + 1:lo + nx
ab(isup,nx+j) = 1;
ab(isub,j) = 1;
end
end
sigmar = 0.4;
sigmai = 0.6;
resid = zeros(100,1);
[icomm, comm, ifail] = f12af( ...
n, nev, ncv);
[icomm, comm, ifail] = f12ad( ...
'Shifted imaginary', icomm, comm);
[icomm, comm, ifail] = f12ad( ...
'Generalized', icomm, comm);
[nconv, dr, di, z, resid, v, comm, icomm, ifail] = ...
f12ag( ...
kl, ku, ab, mb, sigmar, sigmai, resid, comm, icomm);
fprintf('The %4d Ritz values closest to %8.2f %+8.2fi are:\n\n', ...
nconv, sigmar, sigmai);
fprintf('%9.4f %+9.4fi\n', [dr(1:nconv) di(1:nconv)]');
f12ag example results
The 4 Ritz values closest to 0.40 +0.60i are:
0.3610 +0.7223i
0.3610 0.7223i
0.4598 0.7199i
0.4598 +0.7199i
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