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NAG Toolbox

# NAG Toolbox: nag_sparseig_complex_monit (f12as)

## Purpose

nag_sparseig_complex_monit (f12as) can be used to return additional monitoring information during computation. It is in a suite of functions consisting of nag_sparseig_complex_init (f12an), nag_sparseig_complex_iter (f12ap), nag_sparseig_complex_proc (f12aq), nag_sparseig_complex_option (f12ar) and nag_sparseig_complex_monit (f12as).

## Syntax

[niter, nconv, ritz, rzest] = f12as(icomm, comm)
[niter, nconv, ritz, rzest] = nag_sparseig_complex_monit(icomm, comm)

## Description

The suite of functions is designed to calculate some of the eigenvalues, $\lambda$, (and optionally the corresponding eigenvectors, $x$) of a standard complex eigenvalue problem $Ax=\lambda x$, or of a generalized complex eigenvalue problem $Ax=\lambda Bx$ of order $n$, where $n$ is large and the coefficient matrices $A$ and $B$ are sparse and complex. The suite can also be used to find selected eigenvalues/eigenvectors of smaller scale dense complex problems.
On an intermediate exit from nag_sparseig_complex_iter (f12ap) with ${\mathbf{irevcm}}=4$, nag_sparseig_complex_monit (f12as) may be called to return monitoring information on the progress of the Arnoldi iterative process. The information returned by nag_sparseig_complex_monit (f12as) is:
 – the number of the current Arnoldi iteration; – the number of converged eigenvalues at this point; – the converged eigenvalues; – the error bounds on the converged eigenvalues.
nag_sparseig_complex_monit (f12as) does not have an equivalent function from the ARPACK package which prints various levels of detail of monitoring information through an output channel controlled via a argument value (see Lehoucq et al. (1998) for details of ARPACK routines). nag_sparseig_complex_monit (f12as) should not be called at any time other than immediately following an ${\mathbf{irevcm}}=4$ return from nag_sparseig_complex_iter (f12ap).

## 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 MCS-P547-1195 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 Large-scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods SIAM, Philidelphia

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{icomm}\left(:\right)$int64int32nag_int array
The dimension of the array icomm must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{licomm}}\right)$, where licomm is passed to the setup function  (see nag_sparseig_complex_init (f12an))
The array icomm output by the preceding call to nag_sparseig_complex_iter (f12ap).
2:     $\mathrm{comm}\left(:\right)$ – complex array
The dimension of the array comm must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lcomm}}\right)$, where lcomm is passed to the setup function  (see nag_sparseig_complex_init (f12an))
The array comm output by the preceding call to nag_sparseig_complex_iter (f12ap).

None.

### Output Parameters

1:     $\mathrm{niter}$int64int32nag_int scalar
The number of the current Arnoldi iteration.
2:     $\mathrm{nconv}$int64int32nag_int scalar
The number of converged eigenvalues so far.
3:     $\mathrm{ritz}\left(:\right)$ – complex array
The dimension of the array ritz will be ${\mathbf{ncv}}$ (see nag_sparseig_complex_init (f12an))
The first nconv locations of the array ritz contain the converged approximate eigenvalues.
4:     $\mathrm{rzest}\left(:\right)$ – complex array
The dimension of the array rzest will be ${\mathbf{ncv}}$ (see nag_sparseig_complex_init (f12an))
The first nconv locations of the array rzest contain the complex Ritz estimates on the converged approximate eigenvalues.

None.

## Accuracy

A Ritz value, $\lambda$, is deemed to have converged if the magnitude of its Ritz estimate $\le {\mathbf{Tolerance}}×\left|\lambda \right|$. The default Tolerance used is the machine precision given by nag_machine_precision (x02aj).

None.

## Example

This example solves $Ax=\lambda Bx$ in shifted-inverse mode, where $A$ and $B$ are obtained from the standard central difference discretization of the one-dimensional convection-diffusion operator $\frac{{d}^{2}u}{d{x}^{2}}+\rho \frac{du}{dx}$ on $\left[0,1\right]$, with zero Dirichlet boundary conditions. The shift, $\sigma$, is a complex number, and the operator used in the shifted-inverse iterative process is $\mathrm{OP}=\text{inv}\left(A-\sigma B\right)×B$.
```function f12as_example

fprintf('f12as example results\n\n');

nx  = int64(16);
n   = nx^2;
nev = int64(4);
ncv = int64(10);

irevcm = int64(0);
resid  = complex(zeros(n,1));
v  = complex(zeros(n,ncv));
x  = complex(zeros(n,1));
mx = complex(zeros(n,1));

sigma = complex(5000);
rho = complex(10);
h = 1/double(n+1);
s = rho/2;
s1 = -1/h - s - sigma*h/6;
s2 =  2/h - 4*sigma*h/6;
s3 = -1/h + s - sigma*h/6;

dl = complex(repmat(s1, n-1, 1));
dd = complex(repmat(s2, n, 1));
du = complex(repmat(s3, n-1, 1));

% Initialisation Step
[icomm, comm, ifail] = f12an( ...
n, nev, ncv);

% Set the mode
[icomm, comm, ifail] = f12ar( ...
'SHIFTED INVERSE', icomm, comm);
% Set the problem type
[icomm, comm, ifail] = f12ar( ...
'GENERALIZED', icomm, comm);

[dl, dd, du, du2, ipiv, info] = f07cr( ...
dl, dd, du);

% Solve
while (irevcm ~= 5)
[irevcm, resid, v, x, mx, nshift, comm, icomm, ifail] = ...
f12ap(...
irevcm, resid, v, x, mx, comm, icomm);
if (irevcm == -1)
x = mv(x);
[x, info] = f07cs(...
'N', dl, dd, du, du2, ipiv, x);
elseif (irevcm == 1)
[x, info] = f07cs(...
'N', dl, dd, du, du2, ipiv, mx);
elseif (irevcm == 2)
x = mv(x);
elseif (irevcm == 4)
[niter, nconv, ritz, rzest] = f12as(icomm, comm);
fprintf('Iteration %d, No. converged = %d, ', niter, nconv);
fprintf('norm of estimates = %12.2g\n', norm(rzest(1:nev),1));
end
end

% Post-process to compute eigenvalues/vectors
[nconv, d, z, v, comm, icomm, ifail] = ...
f12aq( ...
sigma, resid, v, comm, icomm);

fprintf('\n\nThe %d generalised Ritz values closest to %7.1f are:\n', ...
nconv, sigma);
for i=1:nconv
fprintf('  %8.2f  %+8.2fi\n',real(d(i)),imag(d(i)));
end

function [w] = mv(v)
n = numel(v);
w = complex(zeros(n,1));
h = 1/(n+1);

w(1) = h*(4*v(1)+v(2))/6;
for j=2:n-1
w(j)=h*(v(j-1)+4*v(j)+v(j+1))/6;
end
w(n) = h*(v(n-1)+4*v(n))/6;
```
```f12as example results

Iteration 1, No. converged = 0, norm of estimates =      7.9e-07
Iteration 2, No. converged = 1, norm of estimates =      1.7e-09
Iteration 3, No. converged = 2, norm of estimates =      3.4e-11
Iteration 4, No. converged = 2, norm of estimates =      6.2e-14
Iteration 5, No. converged = 2, norm of estimates =      8.5e-16
Iteration 6, No. converged = 3, norm of estimates =      8.2e-18

The 4 generalised Ritz values closest to  5000.0 are:
4829.85     -0.00i
5279.52     +0.00i
4400.63     +0.00i
5749.72     +0.00i
```

Chapter Contents
Chapter Introduction
NAG Toolbox

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