f12ag:: Large Scale Eigenproblems (NAG Toolbox)

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

Description

The suite of functions is designed to calculate some of the eigenvalues,

λ

, (and optionally the corresponding eigenvectors,

x

) of a standard eigenvalue problem

A x = λ x

, or of a generalized eigenvalue problem

A x = λ B x

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.

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

B

is symmetric semidefinite. Otherwise, if

A

B

is non-singular, the Standard problem can be solved using the matrix

B^{- 1} A

(say).

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

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;

% Construct ab and mb
ab = zeros(2*kl+ku+1,n);
mb = zeros(2*kl+ku+1,n);

% Main diagonal of A.
idiag = kl + ku + 1;
for j=1:n
  ab(idiag,j) = 4;
  mb(idiag,j) = 4;
end

% First subdiagonal and superdiagonal of A.
isup = kl + ku;
isub = kl + ku + 2;
rho = 100;
h = 1/double(nx+1);
for i=1:nx
  lo = (i-1)*nx;
  for j=lo+1:lo+nx-1
    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

% kl-th subdiagonal and ku-th super-diagonal.
isup = kl + 1;
isub = 2*kl + ku + 1;
for i = 1:nx - 1
  lo = (i-1)*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)]');

NAG Toolbox: nag_sparseig_real_band_solve (f12ag)

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Purpose