f12ad:: Large Scale Eigenproblems (NAG Toolbox)

Each optional parameter is defined by a single character string consisting of one or more items. The items associated with a given option must be separated by spaces, or equals signs

[=]

. Alphabetic characters may be upper or lower case. The string

'Vectors = None'

is an example of a string used to set an optional parameter. For each option the string contains one or more of the following items:

–	a mandatory keyword;
–	a phrase that qualifies the keyword;
–	a number that specifies an integer or double value. Such numbers may be up to $16$ contiguous characters in Fortran's I, F, E or D format.

nag_sparseig_real_option (f12ad) does not have an equivalent function from the ARPACK package which passes options by directly setting values to scalar arguments or to specific elements of array arguments. nag_sparseig_real_option (f12ad) is intended to make the passing of options more transparent and follows the same principle as the single option setting functions in Chapter E04.

The setup function nag_sparseig_real_init (f12aa) must be called prior to the first call to nag_sparseig_real_option (f12ad) and all calls to nag_sparseig_real_option (f12ad) must precede the first call to nag_sparseig_real_iter (f12ab), the reverse communication iterative solver.

A complete list of optional parameters, their abbreviations, synonyms and default values is given in Optional Parameters.

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

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Errors or warnings detected by the function:

Accuracy

Further Comments

Example

This example solves

A x = λ B x

in shifted-inverse mode, where

A

and

B

are derived from the finite element discretization of the one-dimensional convection-diffusion operator

\frac{d^{2} u}{d x^{2}} + ρ \frac{d u}{d x}

on the interval

[0, 1]

, with zero Dirichlet boundary conditions.

The shift

σ

is a real number, and the operator used in the shifted-inverse iterative process is

OP = {(A - σ B)}^{- 1} B

function f12ad_example


fprintf('f12ad example results\n\n');

n   = int64(100);
nev = int64(4);
ncv = int64(20);

h = 1/(double(n)+1);
rho = 10;
md = repmat(4*h, double(n), 1);
me = repmat(h, double(n-1), 1);

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

dd = 2/h;
dl = -1/h - rho/2;
du = -1/h + rho/2;
y = zeros(n,1);

[icomm, comm, ifail] = f12aa( ...
                              n, nev, ncv);
[icomm, comm, ifail] = f12ad( ...
                              'Regular Inverse', icomm, comm);
[icomm, comm, ifail] = f12ad( ...
                              'Generalized', icomm, comm);

% Construct m and factorise
[md, me, info] = f07jd(md, me);

while (irevcm ~= 5)
  [irevcm, resid, v, x, mx, nshift, comm, icomm, ifail] = ...
    f12ab( ...
           irevcm, resid, v, x, mx, comm, icomm);
  if (irevcm == -1 || irevcm == 1)
    y(1) = dd*x(1) + du*x(2);
    for i = 2:n-1
      y(i) = dl*x(i-1) + dd*x(i) + du*x(i+1);
    end
    y(n) = dl*x(n-1) + dd*x(n);
    [x, info] = f07je(md, me, y);
  elseif (irevcm == 2)
    y(1) = 4*x(1) + x(2);
    for i=2:n-1
      y(i) = x(i-1) + 4*x(i) + x(i+1);
    end
    y(n) = x(n-1) + 4*x(n);
    x = h*y;
  elseif (irevcm == 4)
    [niter, nconv, ritzr, ritzi, rzest] = ...
    f12ae(icomm, comm);
    if (niter == 1)
      fprintf('\n');
    end
    fprintf('Iteration %2d No. converged = %d ', niter, nconv);
    fprintf('Norm of estimates = %10.2e\n', norm(rzest));
  end
end

[nconv, dr, di, z, v, comm, icomm, ifail] = ...
f12ac(0, 0, resid, v, comm, icomm);

fprintf('\nThe %4d Ritz values of largest magnitude are:\n\n',nconv);
fprintf('%9.4f %+9.4fi\n', [dr di]');

f12ad example results


Iteration  1 No. converged = 0 Norm of estimates =   5.56e+03
Iteration  2 No. converged = 0 Norm of estimates =   5.45e+03
Iteration  3 No. converged = 0 Norm of estimates =   5.30e+03
Iteration  4 No. converged = 0 Norm of estimates =   6.24e+03
Iteration  5 No. converged = 0 Norm of estimates =   7.16e+03
Iteration  6 No. converged = 0 Norm of estimates =   5.45e+03
Iteration  7 No. converged = 0 Norm of estimates =   6.43e+03
Iteration  8 No. converged = 0 Norm of estimates =   5.11e+03
Iteration  9 No. converged = 0 Norm of estimates =   7.19e+03
Iteration 10 No. converged = 1 Norm of estimates =   5.78e+03
Iteration 11 No. converged = 2 Norm of estimates =   4.73e+03
Iteration 12 No. converged = 3 Norm of estimates =   5.00e+03

The    4 Ritz values of largest magnitude are:

20383.0384   +0.0000i
20338.7563   +0.0000i
20265.2844   +0.0000i
20163.1142   +0.0000i

Optional Parameters

Several optional parameters for the computational functions nag_sparseig_real_iter (f12ab) and nag_sparseig_real_proc (f12ac) define choices in the problem specification or the algorithm logic. In order to reduce the number of formal arguments of nag_sparseig_real_iter (f12ab) and nag_sparseig_real_proc (f12ac) these optional parameters have associated default values that are appropriate for most problems. Therefore, you need only specify those optional parameters whose values are to be different from their default values.

The remainder of this section can be skipped if you wish to use the default values for all optional parameters.

The following is a list of the optional parameters available. A full description of each optional parameter is provided in Description of the s.

Optional parameters may be specified by calling nag_sparseig_real_option (f12ad) before a call to nag_sparseig_real_iter (f12ab), but after a call to nag_sparseig_real_init (f12aa). One call is necessary for each optional parameter.

All optional parameters you do not specify are set to their default values. Optional parameters you specify are unaltered by nag_sparseig_real_iter (f12ab) and nag_sparseig_real_proc (f12ac) (unless they define invalid values) and so remain in effect for subsequent calls unless you alter them.

Description of the Optional Parameters

For each option, we give a summary line, a description of the optional parameter and details of constraints.

The summary line contains:

Keywords and character values are case and white space insensitive.

The destination for advisory messages.

This special keyword may be used to reset all optional parameters to their default values.

During the Arnoldi iterative process, shifts are applied internally as part of the implicit restarting scheme. The shift strategy used by default and selected by the Exact Shifts is strongly recommended over the alternative Supplied Shifts (see Lehoucq et al. (1998) for details of shift strategies).

If Exact Shifts are used then these are computed internally by the algorithm in the implicit restarting scheme.

If Supplied Shifts are used then, during the Arnoldi iterative process, you must supply shifts through array arguments of nag_sparseig_real_iter (f12ab) when nag_sparseig_real_iter (f12ab) returns with

irevcm = 3

; the real and imaginary parts of the shifts are supplied in x and mx respectively (or in comm when the option

Pointers = YES

is set). This option should only be used if you are an experienced user since this requires some algorithmic knowledge and because more operations are usually required than for the implicit shift scheme. Details on the use of explicit shifts and further references on shift strategies are available in Lehoucq et al. (1998).

The limit on the number of Arnoldi iterations that can be performed before nag_sparseig_real_iter (f12ab) exits. If not all requested eigenvalues have converged to within Tolerance and the number of Arnoldi iterations has reached this limit then nag_sparseig_real_iter (f12ab) exits with an error; nag_sparseig_real_proc (f12ac) can still be called subsequently to return the number of converged eigenvalues, the converged eigenvalues and, if requested, the corresponding eigenvectors.

The Arnoldi iterative method converges on a number of eigenvalues with given properties. The default is for nag_sparseig_real_iter (f12ab) to compute the eigenvalues of largest magnitude using Largest Magnitude. Alternatively, eigenvalues may be chosen which have Largest Real part, Largest Imaginary part,Smallest Magnitude, Smallest Real part or Smallest Imaginary part.

Note that these options select the eigenvalue properties for eigenvalues of

OP

(and

B

for Generalized problems), the linear operator determined by the computational mode and problem type.

Normally each optional parameter specification is not printed to the advisory channel as it is supplied. Optional parameter List may be used to enable printing and optional parameter Nolist may be used to suppress the printing.

i > 0

, monitoring information is output to channel number

i

during the solution of each problem; this may be the same as the Advisory channel number. The type of information produced is dependent on the value of Print Level, see the description of the optional parameter Print Level for details of the information produced. Please see nag_file_open (x04ac) to associate a file with a given channel number.

During the iterative process and reverse communication calls to nag_sparseig_real_iter (f12ab), required data can be communicated to and from nag_sparseig_real_iter (f12ab) in one of two ways. When

Pointers = NO

is selected (the default) then the array arguments x and mx are used to supply you with required data and used to return computed values back to nag_sparseig_real_iter (f12ab). For example, when

irevcm = 1

, nag_sparseig_real_iter (f12ab) returns the vector

x

in x and the matrix-vector product

B x

in mx and expects the result or the linear operation

OP (x)

to be returned in x.

Pointers = YES

is selected then the data is passed through sections of the array argument comm. The section corresponding to x when

Pointers = NO

begins at a location given by the first element of icomm; similarly the section corresponding to mx begins at a location given by the second element of icomm. This option allows nag_sparseig_real_iter (f12ab) to perform fewer copy operations on each intermediate exit and entry, but can also lead to less elegant code in the calling program.

This controls the amount of printing produced by nag_sparseig_real_option (f12ad) as follows.

$= 0$	No output except error messages.
$> 0$	The set of selected options.
$= 2$	Problem and timing statistics on final exit from nag_sparseig_real_iter (f12ab).
$\geq 5$	A single line of summary output at each Arnoldi iteration.
$\geq 10$	If $Monitoring > 0$ , Monitoring is set, then at each iteration, the length and additional steps of the current Arnoldi factorization and the number of converged Ritz values; during re-orthogonalization, the norm of initial/restarted starting vector.
$\geq 20$	Problem and timing statistics on final exit from nag_sparseig_real_iter (f12ab). If $Monitoring > 0$ , Monitoring is set, then at each iteration, the number of shifts being applied, the eigenvalues and estimates of the Hessenberg matrix $H$ , the size of the Arnoldi basis, the wanted Ritz values and associated Ritz estimates and the shifts applied; vector norms prior to and following re-orthogonalization.
$\geq 30$	If $Monitoring > 0$ , Monitoring is set, then on final iteration, the norm of the residual; when computing the Schur form, the eigenvalues and Ritz estimates both before and after sorting; for each iteration, the norm of residual for compressed factorization and the compressed upper Hessenberg matrix $H$ ; during re-orthogonalization, the initial/restarted starting vector; during the Arnoldi iteration loop, a restart is flagged and the number of the residual requiring iterative refinement; while applying shifts, the indices of the shifts being applied.
$\geq 40$	If $Monitoring > 0$ , Monitoring is set, then during the Arnoldi iteration loop, the Arnoldi vector number and norm of the current residual; while applying shifts, key measures of progress and the order of $H$ ; while computing eigenvalues of $H$ , the last rows of the Schur and eigenvector matrices; when computing implicit shifts, the eigenvalues and Ritz estimates of $H$ .
$\geq 50$	If Monitoring is set, then during Arnoldi iteration loop: norms of key components and the active column of $H$ , norms of residuals during iterative refinement, the final upper Hessenberg matrix $H$ ; while applying shifts: number of shifts, shift values, block indices, updated matrix $H$ ; while computing eigenvalues of $H$ : the matrix $H$ , the computed eigenvalues and Ritz estimates.

To begin the Arnoldi iterative process, nag_sparseig_real_iter (f12ab) requires an initial residual vector. By default nag_sparseig_real_iter (f12ab) provides its own random initial residual vector; this option can also be set using optional parameter Random Residual. Alternatively, you can supply an initial residual vector (perhaps from a previous computation) to nag_sparseig_real_iter (f12ab) through the array argument resid; this option can be set using optional parameter Initial Residual.

These options define the computational mode which in turn defines the form of operation

OP (x)

to be performed when nag_sparseig_real_iter (f12ab) returns with

irevcm = - 1

1

and the matrix-vector product

B x

when nag_sparseig_real_iter (f12ab) returns with

irevcm = 2

Given a Standard eigenvalue problem in the form

A x = λ x

then the following modes are available with the appropriate operator

OP (x)

Regular	$OP = A$
Shifted Inverse Real	$OP = {(A - σ I)}^{- 1}$ where $σ$ is real

Given a Generalized eigenvalue problem in the form

A x = λ B x

then the following modes are available with the appropriate operator

OP (x)

Regular Inverse	$OP = B^{- 1} A$
Shifted Inverse Real with real shift	$OP = {(A - σ B)}^{- 1} B$ , where $σ$ is real
Shifted Inverse Real with complex shift	$OP = Real ({(A - σ B)}^{- 1} B)$ , where $σ$ is complex
Shifted Inverse Imaginary	$OP = Imag ({(A - σ B)}^{- 1} B)$ , where $σ$ is complex

The problem to be solved is either a standard eigenvalue problem,

A x = λ x

, or a generalized eigenvalue problem,

A x = λ B x

. The optional parameter Standard should be used when a standard eigenvalue problem is being solved and the optional parameter Generalized should be used when a generalized eigenvalue problem is being solved.

An approximate eigenvalue has deemed to have converged when the corresponding Ritz estimate is within Tolerance relative to the magnitude of the eigenvalue.

The function nag_sparseig_real_proc (f12ac) can optionally compute the Schur vectors and/or the eigenvectors corresponding to the converged eigenvalues. To turn off computation of any vectors the option

Vectors = NONE

should be set. To compute only the Schur vectors (at very little extra cost), the option

Vectors = SCHUR

should be set and these will be returned in the array argument v of nag_sparseig_real_proc (f12ac). To compute the eigenvectors (Ritz vectors) corresponding to the eigenvalue estimates, the option

Vectors = RITZ

should be set and these will be returned in the array argument z of nag_sparseig_real_proc (f12ac), if z is set equal to v (as in Example) then the Schur vectors in v are overwritten by the eigenvectors computed by nag_sparseig_real_proc (f12ac).

NAG Toolbox: nag_sparseig_real_option (f12ad)

▸▿ Contents

Purpose

Syntax

Description