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# zneupd

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

### Arguments

- RVEC
Integer (INPUT)

Specifies whether a basis for the invariant subspace corresponding to the converged Ritz value approximations for the eigenproblem A*z = lambda*B*z is computed.

RVEC = 0 Compute Ritz values only.

RVEC = 1 Compute Ritz vectors or Schur vectors. See Remarks below.

- HOWMANY
Character (INPUT)

Specifies the form of the basis for the invariant subspace corresponding to the converged Ritz values that is to be computed.

'A': Compute NEV Ritz vectors;

'P': Compute NEV Schur vectors;

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

- SELECT
Integer array of dimension NCV. (INPUT)

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' or 'P', SELECT need not be initialized but it is used as internal workspace.

- D
Complex*16 array of dimension NEV+1. (OUTPUT)

On exit, D contains the Ritz approximations to the eigenvalues lambda for A*z = lambda*B*z.

- Z
Complex*16 N by NEV array (OUTPUT)

On exit,

If RVEC = 1 and HOWMANY = 'A', then the columns of Z represents approximate eigenvectors (Ritz vectors) corresponding to the NCONV=IPARAM(5) Ritz values for eigensystem A*z = lambda*B*z.

If RVEC = 0 or HOWMANY = 'P', then Z is NOT REFERENCED.

NOTE: If if RVEC = 1 and a Schur basis is not required, the array Z may be set equal to first NEV+1 columns of the Arnoldi basis array V computed by ZNAUPD. In this case the Arnoldi basis will be destroyed and overwritten with the eigenvector basis.

- SIGMA
Complex*16 (INPUT)

If IPARAM(7) = 3 then SIGMA represents the shift.

Not referenced if IPARAM(7) = 1 or 2.

- WORKev
Complex*16 work array of dimension 2*NCV. (WORKSPACE)

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

These arguments MUST NOT BE MODIFIED between the the last call to ZNAUPD and the call to ZNEUPD.

Three of these parameters (V, WORKL and INFO) are also output parameters.

- V
Complex*16 N by NCV array. (INPUT/OUTPUT)

Upon INPUT: the NCV columns of V contain the Arnoldi basis vectors for OP as constructed by ZNAUPD.

Upon OUTPUT: If RVEC = 1 the first NCONV=IPARAM(5) columns contain approximate Schur vectors that span the desired invariant subspace.

NOTE: If the array Z has been set equal to first NEV+1 columns of the array V and RVEC=1 and HOWMANY= 'A', then the Arnoldi basis held by V has been overwritten by the desired Ritz vectors. If a separate array Z has been passed then the first NCONV=IPARAM(5) columns of V will contain approximate Schur vectors that span the desired invariant subspace.

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

WORKL(1:ncv*ncv+2*ncv) contains information obtained in znaupd. They are not changed by zneupd.

WORKL(ncv*ncv+2*ncv+1:3*ncv*ncv+4*ncv) holds the untransformed Ritz values, the untransformed error estimates of the Ritz values, the upper triangular matrix for H, and the associated matrix representation of the invariant subspace for H.

Note: IPNTR(9:13) contains the pointer into WORKL for addresses of the above information computed by zneupd.

IPNTR(9): pointer to the NCV RITZ values of the original system.

IPNTR(10): Not used

IPNTR(11): pointer to the NCV corresponding error estimates.

IPNTR(12): pointer to the NCV by NCV upper triangular Schur matrix for H.

IPNTR(13): pointer to the NCV by NCV matrix of eigenvectors of the upper Hessenberg matrix H. Only referenced by zneupd if RVEC = 1 See Remark 2 below.

- INFO
Integer. (OUTPUT)

Error flag on output.

0: Normal exit.

1: The Schur form computed by LAPACK routine csheqr could not be reordered by LAPACK routine ztrsen. Re-enter subroutine zneupd with IPARAM(5)=NCV and increase the size of the array D to have dimension at least dimension NCV and allocate at least NCV columns for Z.

NOTE: Not necessary if Z and V share the same space. Please notify the authors if this error occurs.

-1: N must be positive.

-2: NEV must be positive.

-3: NCV-NEV >= 1 and less than or equal to N.

-5: WHICH must be one of 'LM', 'SM', 'LR', 'SR', 'LI', 'SI'

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

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

-8: Error return from LAPACK eigenvalue calculation. This should never happened.

-9: Error return from calculation of eigenvectors. Informational error from LAPACK routine ztrevc

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

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

-12: HOWMANY = 'S' not yet implemented

-13: HOWMANY must be one of 'A' or 'P' if RVEC = .true.

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

-15: ZNEUPD got a different count of the number of converged Ritz values than ZNAUPD got. This indicates the user probably made an error in passing data from ZNAUPD to ZNEUPD or that the data was modified before entering ZNEUPD

### Description

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

The corresponding approximate eigenvectors;

An orthonormal basis for the associated approximate invariant subspace;

Both.

There is negligible additional cost to obtain eigenvectors. An orthonormal 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).

The approximate eigenvalues and eigenvectors of A*z = lambda*B*z are derived from approximate eigenvalues and eigenvectors of of the linear operator OP prescribed by the MODE selection in the call to ZNAUPD.

ZNAUPD 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 Schur basis.

The 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 may be found in the header of ZNAUPD. For a brief description, see definitions of IPARAM(7), MODE and WHICH in the documentation of ZNAUPD.

### Remarks

Currently only HOWMNY = 'A' and 'P' are implemented.

Schur vectors are an orthogonal representation for the basis of Ritz vectors. Thus, their numerical properties are often superior.

If RVEC = 1 then the relationship

A * V(:,1:IPARAM(5)) = V(:,1:IPARAM(5)) * T,

and

transpose( V(:,1:IPARAM(5)) ) * V(:,1:IPARAM(5)) = I

are approximately satisfied. Here T is the leading submatrix of order IPARAM(5) of the upper triangular matrix stored workl(ipntr(12)).

### 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) + 0*%i; v = zeros(nx,ncv) + 0*%i; workd = zeros(3*nx+1,1) + 0*%i; workev = zeros(3*ncv,1); rwork = zeros(ncv,1); workl = zeros(3*ncv*ncv+5*ncv,1) + 0*%i; // Build the complex test matrix A = diag(10*ones(nx,1)+%i*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_znaupd = 0; iter = iter + 1; // Repeatedly call the routine ZNAUPD 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_znaupd] = znaupd(ido,bmat,nx,which,nev,tol,resid,ncv,v,iparam,ipntr,workd,workl,rwork,info_znaupd); if (ido==99) then // BE CAREFUL: DON'T CALL zneupd 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 ZNAUPD again. continue end if (info_znaupd < 0) then printf('\nError with znaupd, info = %d\n',info_znaupd); printf('Check the documentation of znaupd\n\n'); else // Post-Process using ZNEUPD. rvec = 1; howmany = 'A'; info_zneupd = 0; [d,d(1,2),v,resid,v,iparam,ipntr,workd,workl,info_zneupd] = zneupd(rvec,howmany,_select,d,d(1,2),v,sigma,workev, ... bmat,nx,which,nev,tol,resid,ncv,v, ... iparam,ipntr,workd,workl,info_zneupd); if (info_zneupd~=0) then printf('\nError with zneupd, info = %d\n', info_zneupd); printf('Check the documentation of zneupd.\n\n'); end if (info_zneupd==1) then printf('\nMaximum number of iterations reached.\n\n'); elseif (info_zneupd==3) then printf('\nNo shifts could be applied during implicit Arnoldi update, try increasing NCV.\n\n'); end end end // Done with program znsimp. printf('\nZNSIMP\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 a 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 zneupd

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