Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_sparseig_real_symm_iter (f12fb)

## Purpose

nag_sparseig_real_symm_iter (f12fb) is an iterative solver in a suite of functions consisting of nag_sparseig_real_symm_init (f12fa), nag_sparseig_real_symm_iter (f12fb), nag_sparseig_real_symm_proc (f12fc), nag_sparseig_real_symm_option (f12fd) and nag_sparseig_real_symm_monit (f12fe). It is used to find some of the eigenvalues (and optionally the corresponding eigenvectors) of a standard or generalized eigenvalue problem defined by real symmetric matrices.

## Syntax

[irevcm, resid, v, x, mx, nshift, comm, icomm, ifail] = f12fb(irevcm, resid, v, x, mx, comm, icomm)
[irevcm, resid, v, x, mx, nshift, comm, icomm, ifail] = nag_sparseig_real_symm_iter(irevcm, resid, v, x, mx, 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 sparse, real and symmetric. The suite can also be used to find selected eigenvalues/eigenvectors of smaller scale dense, real and symmetric problems.
nag_sparseig_real_symm_iter (f12fb) is a reverse communication function, based on the ARPACK routine dsaupd, using the Implicitly Restarted Arnoldi iteration method, which for symmetric problems reduces to a variant of the Lanczos 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 symmetric matrices is provided in Lehoucq and Scott (1996). This suite of functions offers the same functionality as the ARPACK software for real symmetric problems, but the interface design is quite different in order to make the option setting clearer and to simplify the interface of nag_sparseig_real_symm_iter (f12fb).
The setup function nag_sparseig_real_symm_init (f12fa) must be called before nag_sparseig_real_symm_iter (f12fb), the reverse communication iterative solver. Options may be set for nag_sparseig_real_symm_iter (f12fb) by prior calls to the option setting function nag_sparseig_real_symm_option (f12fd) and a post-processing function nag_sparseig_real_symm_proc (f12fc) must be called following a successful final exit from nag_sparseig_real_symm_iter (f12fb). nag_sparseig_real_symm_monit (f12fe), may be called following certain flagged, intermediate exits from nag_sparseig_real_symm_iter (f12fb) to provide additional monitoring information about the computation.
nag_sparseig_real_symm_iter (f12fb) uses reverse communication, i.e., it returns repeatedly to the calling program with the argument irevcm (see Arguments) set to specified values which require the calling program to carry out one of the following tasks:
 – compute the matrix-vector product $y=\mathrm{OP}x$, where $\mathrm{OP}$ is defined by the computational mode; – compute the matrix-vector product $y=Bx$; – notify the completion of the computation; – allow the calling program to monitor the solution.
The problem type to be solved (standard or generalized), the spectrum of eigenvalues of interest, the mode used (regular, regular inverse, shifted inverse, Buckling or Cayley) and other options can all be set using the option setting function nag_sparseig_real_symm_option (f12fd).

## 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

Note: this function uses reverse communication. Its use involves an initial entry, intermediate exits and re-entries, and a final exit, as indicated by the argument irevcm. Between intermediate exits and re-entries, all arguments other than x, mx and comm must remain unchanged.

### Compulsory Input Parameters

1:     $\mathrm{irevcm}$int64int32nag_int scalar
On initial entry: ${\mathbf{irevcm}}=0$, otherwise an error condition will be raised.
On intermediate re-entry: must be unchanged from its previous exit value. Changing irevcm to any other value between calls will result in an error.
Constraint: on initial entry, ${\mathbf{irevcm}}=0$; on re-entry irevcm must remain unchanged.
2:     $\mathrm{resid}\left(:\right)$ – double array
The dimension of the array resid must be at least ${\mathbf{n}}$ (see nag_sparseig_real_symm_init (f12fa))
On initial entry: need not be set unless the option Initial Residual has been set in a prior call to nag_sparseig_real_symm_option (f12fd) in which case resid should contain an initial residual vector, possibly from a previous run.
On intermediate re-entry: must be unchanged from its previous exit. Changing resid to any other value between calls may result in an error exit.
3:     $\mathrm{v}\left(\mathit{ldv},:\right)$ – double array
The first dimension of the array v must be at least ${\mathbf{n}}$.
The second dimension of the array v must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{ncv}}\right)$.
On initial entry: need not be set.
On intermediate re-entry: must be unchanged from its previous exit.
4:     $\mathrm{x}\left(:\right)$ – double array
The dimension of the array x must be at least ${\mathbf{n}}$ (see nag_sparseig_real_symm_init (f12fa))
On initial entry: need not be set, it is used as a convenient mechanism for accessing elements of comm.
On intermediate re-entry: if ${\mathbf{Pointers}}=\mathrm{YES}$, x need not be set.
If ${\mathbf{Pointers}}=\mathrm{NO}$, x must contain the result of $y=\mathrm{OP}x$ when irevcm returns the value $-1$ or $+1$. It must return the real parts of the computed shifts when irevcm returns the value $3$.
5:     $\mathrm{mx}\left(:\right)$ – double array
The dimension of the array mx must be at least ${\mathbf{n}}$ (see nag_sparseig_real_symm_init (f12fa))
On initial entry: need not be set, it is used as a convenient mechanism for accessing elements of comm.
On intermediate re-entry: if ${\mathbf{Pointers}}=\mathrm{YES}$, mx need not be set.
If ${\mathbf{Pointers}}=\mathrm{NO}$, mx must contain the result of $y=Bx$ when irevcm returns the value $2$. It must return the imaginary parts of the computed shifts when irevcm returns the value $3$.
6:     $\mathrm{comm}\left(:\right)$ – double array
The dimension of the array comm must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lcomm}}\right)$ (see nag_sparseig_real_symm_init (f12fa))
On initial entry: must remain unchanged following a call to the setup function nag_sparseig_real_symm_init (f12fa).
7:     $\mathrm{icomm}\left(:\right)$int64int32nag_int array
The dimension of the array icomm must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{licomm}}\right)$ (see nag_sparseig_real_symm_init (f12fa))
On initial entry: must remain unchanged following a call to the setup function nag_sparseig_real_symm_init (f12fa).

None.

### Output Parameters

1:     $\mathrm{irevcm}$int64int32nag_int scalar
On intermediate exit: has the following meanings.
${\mathbf{irevcm}}=-1$
The calling program must compute the matrix-vector product $y=\mathrm{OP}x$, where $x$ is stored in x (by default) or in the array comm (starting from the location given by the first element of icomm) when the option ${\mathbf{Pointers}}=\mathrm{YES}$ is set in a prior call to nag_sparseig_real_symm_option (f12fd). The result $y$ is returned in x (by default) or in the array comm (starting from the location given by the second element of icomm) when the option ${\mathbf{Pointers}}=\mathrm{YES}$ is set.
${\mathbf{irevcm}}=1$
The calling program must compute the matrix-vector product $y=\mathrm{OP}x$. This is similar to the case ${\mathbf{irevcm}}=-1$ except that the result of the matrix-vector product $Bx$ (as required in some computational modes) has already been computed and is available in mx (by default) or in the array comm (starting from the location given by the third element of icomm) when the option ${\mathbf{Pointers}}=\mathrm{YES}$ is set.
${\mathbf{irevcm}}=2$
The calling program must compute the matrix-vector product $y=Bx$, where $x$ is stored in x and $y$ is returned in mx (by default) or in the array comm (starting from the location given by the second element of icomm) when the option ${\mathbf{Pointers}}=\mathrm{YES}$ is set.
${\mathbf{irevcm}}=3$
Compute the nshift real and imaginary parts of the shifts where the real parts are to be returned in the first nshift locations of the array x and the imaginary parts are to be returned in the first nshift locations of the array mx. Only complex conjugate pairs of shifts may be applied and the pairs must be placed in consecutive locations. This value of irevcm will only arise if the optional parameter Supplied Shifts is set in a prior call to nag_sparseig_real_symm_option (f12fd) which is intended for experienced users only; the default and recommended option is to use exact shifts (see Lehoucq et al. (1998) for details and guidance on the choice of shift strategies).
${\mathbf{irevcm}}=4$
Monitoring step: a call to nag_sparseig_real_symm_monit (f12fe) can now be made to return the number of Arnoldi iterations, the number of converged Ritz values, their real and imaginary parts, and the corresponding Ritz estimates.
On final exit: ${\mathbf{irevcm}}=5$: nag_sparseig_real_symm_iter (f12fb) has completed its tasks. The value of ifail determines whether the iteration has been successfully completed, or whether errors have been detected. On successful completion nag_sparseig_real_symm_proc (f12fc) must be called to return the requested eigenvalues and eigenvectors (and/or Schur vectors).
2:     $\mathrm{resid}\left(:\right)$ – double array
The dimension of the array resid will be ${\mathbf{n}}$ (see nag_sparseig_real_symm_init (f12fa))
On intermediate exit: contains the current residual vector.
On final exit: contains the final residual vector.
3:     $\mathrm{v}\left(\mathit{ldv},:\right)$ – double array
The first dimension of the array v will be ${\mathbf{n}}$.
The second dimension of the array v will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{ncv}}\right)$.
On intermediate exit: contains the current set of Arnoldi basis vectors.
On final exit: contains the final set of Arnoldi basis vectors.
4:     $\mathrm{x}\left(:\right)$ – double array
The dimension of the array x will be ${\mathbf{n}}$ (see nag_sparseig_real_symm_init (f12fa))
On intermediate exit: if ${\mathbf{Pointers}}=\mathrm{YES}$, x is not referenced.
If ${\mathbf{Pointers}}=\mathrm{NO}$, x contains the vector $x$ when irevcm returns the value $-1$ or $+1$.
On final exit: does not contain useful data.
5:     $\mathrm{mx}\left(:\right)$ – double array
The dimension of the array mx will be ${\mathbf{n}}$ (see nag_sparseig_real_symm_init (f12fa))
On intermediate exit: if ${\mathbf{Pointers}}=\mathrm{YES}$, mx is not referenced.
If ${\mathbf{Pointers}}=\mathrm{NO}$, mx contains the vector $Bx$ when irevcm returns the value $+1$.
On final exit: does not contain any useful data.
6:     $\mathrm{nshift}$int64int32nag_int scalar
On intermediate exit: if the option Supplied Shifts is set and irevcm returns a value of $3$, nshift returns the number of complex shifts required.
7:     $\mathrm{comm}\left(:\right)$ – double array
The dimension of the array comm will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lcomm}}\right)$ (see nag_sparseig_real_symm_init (f12fa))
Contains data defining the current state of the iterative process.
8:     $\mathrm{icomm}\left(:\right)$int64int32nag_int array
The dimension of the array icomm will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{licomm}}\right)$ (see nag_sparseig_real_symm_init (f12fa))
Contains data defining the current state of the iterative process.
9:     $\mathrm{ifail}$int64int32nag_int scalar
On final exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

${\mathbf{ifail}}=1$
On initial entry, the maximum number of iterations $\le 0$, the option Iteration Limit has been set to a non-positive value.
${\mathbf{ifail}}=2$
The options Generalized and Regular are incompatible.
${\mathbf{ifail}}=3$
Eigenvalues from both ends of the spectrum were requested, but the number of eigenvalues requested is one.
${\mathbf{ifail}}=4$
The option Initial Residual was selected but the starting vector held in resid is zero.
W  ${\mathbf{ifail}}=5$
The maximum number of iterations has been reached. Some Ritz values may have converged; a subsequent call to nag_sparseig_real_symm_proc (f12fc) will return the number of converged values and the converged values.
${\mathbf{ifail}}=6$
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_symm_init (f12fa) for details of these arguments).
${\mathbf{ifail}}=7$
Could not build a Lanczos factorization. Consider changing ncv or nev in the initialization function (see Arguments in nag_sparseig_real_symm_init (f12fa) for details of these arguments).
${\mathbf{ifail}}=8$
Unexpected error in internal call to compute eigenvalues and corresponding error bounds of the current upper Hessenberg matrix. Please contact NAG.
${\mathbf{ifail}}=9$
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
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 $\text{}\le {\mathbf{Tolerance}}×\left|\lambda \right|$. The default Tolerance used is the machine precision given by nag_machine_precision (x02aj).

None.

## Example

For this function two examples are presented, with a main program and two example problems given in Example 1 (EX1) and Example 2 (EX2).
Example 1 (EX1)
The example solves $Ax=\lambda x$ in shift-invert mode, where $A$ is obtained from the standard central difference discretization of the one-dimensional Laplacian operator $\frac{{\partial }^{2}u}{\partial {x}^{2}}$ with zero Dirichlet boundary conditions. Eigenvalues closest to the shift $\sigma =0$ are sought.
Example 2 (EX2)
This example illustrates the use of nag_sparseig_real_symm_iter (f12fb) to compute the leading terms in the singular value decomposition of a real general matrix $A$. The example finds a few of the largest singular values ($\sigma$) and corresponding right singular values ($\nu$) for the matrix $A$ by solving the symmetric problem:
 $ATA ν=σν .$
Here $A$ is the $m$ by $n$ real matrix derived from the simplest finite difference discretization of the two-dimensional kernal $k\left(s,t\right)dt$ where
 $ks,t = st-1 if ​0≤s≤t≤1 ts-1 if ​0≤t
```function f12fb_example

fprintf('f12fb example results\n\n');

fprintf('Example 1:\n');
ex1;
fprintf('Example 2:\n\n');
ex2;

function ex1
n   = int64(100);
nev = int64(4);
ncv = int64(10);
imon = 0;

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

sig = 0;

h   = 1/double(n+1);
h2  = h*h;

% Initialisation Step
[icomm, comm, ifail] = f12fa( ...
n, nev, ncv);
[icomm, comm, ifail] = f12fd( ...
'Shifted Inverse', icomm, comm);

% Solve
while (irevcm ~= 5)
[irevcm, resid, v, x, mx, nshift, comm, icomm, ifail] = ...
f12fb( ...
irevcm, resid, v, x, mx, comm, icomm);
if (irevcm == 1 || irevcm == -1)
[x, info] = f07ce( ...
elseif (irevcm == 4 && imon==1)
[niter, nconv, ritz, rzest] = f12fe( ...
icomm, comm);
fprintf(['Iteration %2d, No. converged = %d, ', ...
'norm of estimates = %10.2e\n'], ...
niter, nconv, norm(rzest(1:nev),2));
end
end

% Post-process to compute eigenvalues/vectors
[nconv, d, z, v, comm, icomm, ifail] = ...
f12fc( ...
sig, resid, v, comm, icomm);

fprintf('\nThe %d Eigenvalues of smallest magnitude are:\n',nconv);
disp(d(1:nconv));

function ex2

m = int64(500);
n = int64(100);
nev = int64(4);
ncv = int64(10);

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

sigma = 0;

% Initialisation Step
[icomm, comm, ifail] = f12fa( ...
n, nev, ncv);

% Solve
while (irevcm ~= 5)
[irevcm, resid, v, x, mx, nshift, comm, icomm, ifail] = ...
f12fb( ...
irevcm, resid, v, x, mx, comm, icomm);
if (irevcm == 1 || irevcm == -1)
% x = A^TAx
y = f12fb_Ax(m,n,x);
x = f12fb_Atx(m,n,y);
end
end

% Post-process
[nconv, d, z, v, comm, icomm, ifail] = ...
f12fc( ...
sigma, resid, v, comm, icomm);

% Singular values (squared) are returned in the first column of D and the
% corresponding right singular vectors are in the V(:,1:nconv).

for j = 1:nconv
d(j,1) = sqrt(d(j,1));

% Compute the left singular vectors from u = Av/sigval/norm(Av).
ax = f12fb_Ax(m,n,v(:,j));
u(:,j) = ax/norm(ax);

% Compute the residual norm ||A*v - sigma*u|| for the leading terms
resid(j) = norm(ax - d(j,1)*u(:,j));
end

fprintf('Leading %d singular values and direct residuals:\n',nconv);
fprintf('%9.5f%12.2e\n',[d(1:nconv) resid(1:nconv)]');

function [y] = f12fb_Ax(m,n,x)

y = zeros(m,1);

h = 1/double(m+1);
k = 1/double(n+1);
t = 0;

for j=1:n
t = t + k;
s = 0;
for l = 1:j
s = s + h;
y(l) = y(l) + k*s*(t-1)*x(j);
end
for l = j+1:m
s = s + h;
y(l) = y(l) + k*t*(s-1)*x(j);
end
end

function [y] = f12fb_Atx(m,n,x)

y = zeros(n,1);

h = 1/double(m+1);
k = 1/double(n+1);
t = 0;

for j=1:n
t = t + k;
s = 0;
for l = 1:j
s = s + h;
y(j) = y(j) + k*s*(t-1)*x(l);
end
for l = j+1:m
s = s + h;
y(j) = y(j) + k*t*(s-1)*x(l);
end
end
```
```f12fb example results

Example 1:

The 4 Eigenvalues of smallest magnitude are:
9.8688
39.4657
88.7620
157.7101

Example 2:

Leading 4 singular values and direct residuals:
0.04101    2.74e-17
0.06049    2.83e-17
0.11784    5.62e-17
0.55723    2.28e-16
```