Parameters

NOTE: The remaining arguments BMAT, N, WHICH, NEV, TOL, RESID, NCV, V, IPARAM, IPNTR, WORKD, WORKL, LWORKL, INFO must be passed directly to DSEUPD following the last call to DSAUPD .

These arguments MUST NOT BE MODIFIED between the the last call to DSAUPD and the call to DSEUPD.

Two of these parameters (WORKL, INFO) are also output parameters.

Description

This subroutine returns the converged approximations to eigenvalues of A*z = lambda*B*z and (optionally):

There is negligible additional cost to obtain eigenvectors. An orthonormal (Lanczos) basis is always computed. There is an additional storage cost of n*nev if both are requested (in this case a separate array Z must be supplied).

These quantities are obtained from the Lanczos factorization computed by DSAUPD for the linear operator OP prescribed by the MODE selection (see IPARAM(7) in DSAUPD documentation.) DSAUPD must be called before this routine is called.

These approximate eigenvalues and vectors are commonly called Ritz values and Ritz vectors respectively. They are referred to as such in the comments that follow.

The computed orthonormal basis for the invariant subspace corresponding to these Ritz values is referred to as a Lanczos basis.

See documentation in the header of the subroutine DSAUPD for a definition of OP as well as other terms and the relation of computed Ritz values and vectors of OP with respect to the given problem A*z = lambda*B*z.

The approximate eigenvalues of the original problem are returned in ascending algebraic order.

The user may elect to call this routine once for each desired Ritz vector and store it peripherally if desired. There is also the option of computing a selected set of these vectors with a single call.

Remarks

1. The converged Ritz values are always returned in increasing (algebraic) order. c 2. Currently only HOWMNY = 'A' is implemented. It is included at this stage for the user who wants to incorporate it.

Example

 

// The following sets dimensions for this problem.

nx    = 10;

nev   = 3;
ncv   = 6;
bmat  = 'I';
which = 'LM';

maxiter = 10;

// Local Arrays

iparam  = zeros(11,1);
ipntr   = zeros(14,1);
_select = zeros(ncv,1);
d       = zeros(nev+1,1);
resid   = zeros(nx,1); 
v       = zeros(nx,ncv);
workd   = zeros(3*nx+1,1); 
workl   = zeros(ncv*ncv+8*ncv,1);

// Build the symmetric test matrix

A            = diag(10*ones(nx,1));
A(1:$-1,2:$) = A(1:$-1,2:$) + diag(6*ones(nx-1,1));
A(2:$,1:$-1) = A(2:$,1:$-1) + diag(6*ones(nx-1,1));

tol    = 0;
ido    = 0;

ishfts = 1;
maxitr = 300;
mode1  = 1;

iparam(1) = ishfts;
iparam(3) = maxitr;
iparam(7) = mode1;

sigma = 0; // the real part of the shift

// M A I N   L O O P (Reverse communication)

iter = 0;
while(iter<maxiter)
  info_dsaupd = 0;
  iter = iter + 1;
  // Repeatedly call the routine DSAUPD and take actions indicated by parameter IDO until
  // either convergence is indicated or maxitr has been exceeded.

  [ido,resid,v,iparam,ipntr,workd,workl,info_dsaupd] = dsaupd(ido,bmat,nx,which,nev,tol,resid,ncv,v,iparam,ipntr,workd,workl,info_dsaupd);
  
  if (ido==99) then
    // BE CAREFUL: DON'T CALL dseupd IF ido == 99 !!
    break;
  end
  
  if (ido==-1 | ido==1) then
    // Perform matrix vector multiplication 
    workd(ipntr(2)+1:ipntr(2)+nx) = A*workd(ipntr(1)+1:ipntr(1)+nx);
    // L O O P   B A C K to call DSAUPD again.
    continue
  end
  
  if (info_dsaupd < 0) then
    printf('\nError with dsaupd, info = %d\n',info_dsaupd);
    printf('Check the documentation of dsaupd\n\n');
  else 
    // Post-Process using DSEUPD.

    rvec    = 1;
    howmany = 'A';

    info_dseupd = 0;

    [d,d(1,2),v,resid,v,iparam,ipntr,workd,workl,info_dseupd] = dseupd(rvec,howmany,_select,d,d(1,2),v,sigma, ...
                                                                       bmat,nx,which,nev,tol,resid,ncv,v, ...
                                                                       iparam,ipntr,workd,workl,info_dseupd);

    if (info_dseupd~=0) then
      printf('\nError with dseupd, info = %d\n', info_dseupd);
      printf('Check the documentation of dseupd.\n\n');
    end

    if (info_dseupd==1) then
      printf('\nMaximum number of iterations reached.\n\n');
    elseif (info_dseupd==3) then
      printf('\nNo shifts could be applied during implicit Arnoldi update, try increasing NCV.\n\n');
    end
  end
end

// Done with program dssimp.
printf('\nDSSIMP\n');
printf('======\n');
printf('\n');
printf('Iterations is %d\n', iter);
printf('Size of the matrix is %d\n', nx);
printf('The number of Ritz values requested is %d\n', nev);
printf('The number of Arnoldi vectors generated (NCV) is %d\n', ncv);
printf('What portion of the spectrum: %s\n', which);
printf('The number of Implicit Arnoldi update iterations taken is %d\n', iparam(3));
printf('The number of OP*x is %d\n', iparam(9));
printf('The convergence criterion is %d\n', tol);
 

      

See Also

Authors

Bibliography

1. D.C. Sorensen, "Implicit Application of Polynomial Filters in k-Step Arnoldi Method", SIAM J. Matr. Anal. Apps., 13 (1992), pp 357-385.

2. R.B. Lehoucq, "Analysis and Implementation of an Implicitly Restarted Arnoldi Iteration", Rice University Technical Report TR95-13, Department of Computational and Applied Mathematics.

3. B.N. Parlett and Y. Saad, "Complex Shift and Invert Strategies for Real Matrices", Linear Algebra and its Applications, vol 88/89, pp 575-595, (1987).

Used Functions

Based on ARPACK routine dsaupd