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# dsaupd

Interface for the Implicitly Restarted Arnoldi Iteration, to compute approximations to the converged approximations to eigenvalues of A*z = lambda*B*z

### Calling Sequence

[D,Z,RESID,V,IPARAM,IPNTR,WORKD,WORKL,INFO] = dseupd(RVEC,HOWMANY,SELECT,D,Z,SIGMA,BMAT,N,WHICH, NEV,TOL,RESID,NCV,V,IPARAM,IPNTR,WORKD,WORKL,INFO)

### Arguments

- RVEC
Integer (INPUT)

Specifies whether Ritz vectors corresponding to the Ritz value approximations to the eigenproblem A*z = lambda*B*z are computed.

RVEC = 0 Compute Ritz values only.

RVEC = 1 Compute Ritz vectors.

- HOWMNY
Character*1 (INPUT)

Specifies how many Ritz vectors are wanted and the form of Z the matrix of Ritz vectors. See remark 1 below.

'A': compute NEV Ritz vectors;

'S': compute some of the Ritz vectors, specified by the integer array SELECT.

- SELECT
Integer array of dimension NCV. (INPUT/WORKSPACE)

If HOWMANY = 'S', SELECT specifies the Ritz vectors to be computed. To select the Ritz vector corresponding to a Ritz value D(j), SELECT(j) must be set to 1.

If HOWMANY = 'A' , SELECT is used as a workspace for reordering the Ritz values.

- D
Double precision array of dimension NEV. (OUTPUT)

On exit, D contains the Ritz value approximations to the eigenvalues of A*z = lambda*B*z. The values are returned in ascending order.

If IPARAM(7) = 3,4,5 then D represents the Ritz values of OP computed by dsaupd transformed to those of the original eigensystem A*z = lambda*B*z.

If IPARAM(7) = 1,2 then the Ritz values of OP are the same as the those of A*z = lambda*B*z.

- Z
Double precision N by NEV array.

If HOWMNY = 'A'. (OUTPUT) On exit, Z contains the B-orthonormal Ritz vectors of the eigensystem A*z = lambda*B*z corresponding to the Ritz value approximations.

If RVEC = 0 then Z is not referenced.

NOTE: The array Z may be set equal to first NEV columns of the Arnoldi/Lanczos basis array V computed by DSAUPD .

- SIGMA
Double precision (INPUT)

If IPARAM(7) = 3,4,5 represents the shift. Not referenced if IPARAM(7) = 1 or 2.

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.

- WORKL
Double precision work array of length LWORKL. (OUTPUT/WORKSPACE)

WORKL(1:4*ncv) contains information obtained in dsaupd. They are not changed by dseupd.

WORKL(4*ncv+1:ncv*ncv+8*ncv) holds the untransformed Ritz values, the computed error estimates, and the associated eigenvector matrix of H.

Note: IPNTR(8:10) contains the pointer into WORKL for addresses of the above information computed by dseupd .

IPNTR(8): pointer to the NCV RITZ values of the original system.

IPNTR(9): pointer to the NCV corresponding error bounds.

IPNTR(10): pointer to the NCV by NCV matrix of eigenvectors of the tridiagonal matrix T. Only referenced by dseupd if RVEC = 1 See Remarks.

- INFO
Integer. (OUTPUT)

Error flag on output.

0: Normal exit.

-1: N must be positive.

-2: NEV must be positive.

-3: NCV must be greater than NEV and less than or equal to N.

-5: WHICH must be one of 'LM', 'SM', 'LA', 'SA' or 'BE'.

-6: BMAT must be one of 'I' or 'G'.

-7: Length of private work WORKL array is not sufficient.

-8: Error return from trid. eigenvalue calculation; Information error from LAPACK routine dsteqr .

-9: Starting vector is zero.

-10: IPARAM(7) must be 1,2,3,4,5.

-11: IPARAM(7) = 1 and BMAT = 'G' are incompatible.

-12: NEV and WHICH = 'BE' are incompatible.

-14: DSAUPD did not find any eigenvalues to sufficient accuracy.

-15: HOWMNY must be one of 'A' or 'S' if RVEC = 1

-16: HOWMNY = 'S' not yet implemented

-17: DSEUPD got a different count of the number of converged Ritz values than DSAUPD got. This indicates the user probably made an error in passing data from DSAUPD to DSEUPD or that the data was modified before entering DSEUPD.

### Description

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

the corresponding approximate eigenvectors,

an orthonormal (Lanczos) basis for the associated approximate invariant subspace,

Both.

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);

### Authors

- Danny Sorensen, Richard Lehoucq, Phuong Vu
CRPC / Rice University Applied Mathematics Rice University Houston, Texas

### 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

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