NAG Toolbox: nag_sparseig_real_symm_iter (f12fb)

Purpose

Syntax

Description

References

Parameters

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 $\max \left(1,\mathbf{ncv}\right)$.

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 $\max \left(1,\mathbf{lcomm}\right)$ (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 $\max \left(1,\mathbf{licomm}\right)$ (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).

Optional Input Parameters

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 $\max \left(1,\mathbf{ncv}\right)$.

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 $\max \left(1,\mathbf{lcomm}\right)$ (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 $\max \left(1,\mathbf{licomm}\right)$ (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

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.

   $\mathbf{ifail}=9$

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

Further Comments

Example

Open in the MATLAB editor: f12fb_example

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;
  ad(1:n)  = 2/h2 - sig;
  adl(1:n) = -1/h2;
  adu(1:n) = adl(1:n);

  [adl, ad, adu, adu2, ipiv, info] = f07cd( ...
                                            adl, ad, adu);

  % 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( ...
                         'N', adl, ad, adu, adu2, ipiv, x);
    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