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# dneupd

Interface for the Implicitly Restarted Arnoldi Iteration, to compute the converged approximations to eigenvalues of A*z = lambda*B*z approximations to a few eigenpairs of a real linear operator

### Calling Sequence

[Dr,Di,Z,RESID,V,IPARAM,IPNTR,WORKD,WORKL,INFO] = dneupd(RVEC,HOWMANY,SELECT,Dr,Di,Z,SIGMAr,SIGMAi,WORKev, BMAT,N,WHICH,NEV,TOL,RESID,NCV,V,IPARAM,IPNTR,WORKD,WORKL,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 the 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 (DR(j), DI(j)), SELECT(j) must be set to 1.

If HOWMANY = 'A' or 'P', SELECT is used as internal workspace.

- DR
Double precision array of dimension NEV+1. (OUTPUT)

If IPARAM(7) = 1,2 or 3 and SIGMAI=0.0 then on exit: DR contains the real part of the Ritz approximations to the eigenvalues of A*z = lambda*B*z.

If IPARAM(7) = 3, 4 and SIGMAI is not equal to zero, then on exit: DR contains the real part of the Ritz values of OP computed by DNAUPD .

A further computation must be performed by the user to transform the Ritz values computed for OP by DNAUPD to those of the original system A*z = lambda*B*z. See remark 3 below

- DI
Double precision array of dimension NEV+1. (OUTPUT)

On exit, DI contains the imaginary part of the Ritz value approximations to the eigenvalues of A*z = lambda*B*z associated with DR.

NOTE: When Ritz values are complex, they will come in complex conjugate pairs. If eigenvectors are requested, the corresponding Ritz vectors will also come in conjugate pairs and the real and imaginary parts of these are represented in two consecutive columns of the array Z (see below).

- Z
Double precision N by NEV+1 array

if RVEC = 1 and HOWMANY = 'A'. (OUTPUT)

On exit, if RVEC = 1 and HOWMANY = 'A', then the columns of Z represent approximate eigenvectors (Ritz vectors) corresponding to the NCONV=IPARAM(5) Ritz values for eigensystem A*z = lambda*B*z. The complex Ritz vector associated with the Ritz value with positive imaginary part is stored in two consecutive columns. The first column holds the real part of the Ritz vector and the second column holds the imaginary part. The Ritz vector associated with the Ritz value with negative imaginary part is simply the complex conjugate of the Ritz vector associated with the positive imaginary part.

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 DNAUPD . In this case the Arnoldi basis will be destroyed and overwritten with the eigenvector basis.

- SIGMAR
Double precision (INPUT)

If IPARAM(7) = 3 or 4, represents the real part of the shift.

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

- SIGMAI
Double precision (INPUT)

If IPARAM(7) = 3 or 4, represents the imaginary part of the shift.

Not referenced if IPARAM(7) = 1 or 2. See remark 3 below.

- WORKEV
Double precision work array of dimension 3*NCV. (WORKSPACE)

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

These arguments MUST NOT BE MODIFIED between the the last call to DNAUPD and the call to DNEUPD .

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

- V
Double precision N by NCV array. (INPUT/OUTPUT)

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

Upon OUTPUT: If RVEC = 1 the first NCONV=IPARAM(5) columns contain approximate Schur vectors that span the desired invariant subspace. See Remark 2 below.

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+3*ncv) contains information obtained in dnaupd . They are not changed by dneupd .

WORKL(ncv*ncv+3*ncv+1:3*ncv*ncv+6*ncv) holds the real and imaginary part of the untransformed Ritz values, the upper quasi-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 dneupd .

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

IPNTR(10): pointer to the imaginary part of the NCV RITZ values of the original system.

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

IPNTR(12): pointer to the NCV by NCV upper quasi-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 dneupd if RVEC = 1 See Remark 2 below.

- INFO
Error flag on output.

0: Normal exit.

1: The Schur form computed by LAPACK routine dlahqr could not be reordered by LAPACK routine dtrsen . Re-enter subroutine dneupd with IPARAM(5)=NCV and increase the size of the arrays DR and DI 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 >= 2 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 calculation of a real Schur form. Informational error from LAPACK routine dlahqr .

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

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

-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 = 1

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

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

### 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 DNAUPD. DNAUPD 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.

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

For a brief description, see definitions of IPARAM(7), MODE and WHICH in the documentation of DNAUPD .

### Remarks

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

Let trans(X) denote the transpose of X.

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

are approximately satisfied.

Here T is the leading submatrix of order IPARAM(5) of the real upper quasi-triangular matrix stored workl(ipntr(12)). That is, T is block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block has its diagonal elements equal and its off-diagonal elements of opposite sign. Corresponding to each 2-by-2 diagonal block is a complex conjugate pair of Ritz values. The real Ritz values are stored on the diagonal of T.

If IPARAM(7) = 3 or 4 and SIGMAI is not equal zero, then the user must form the IPARAM(5) Rayleigh quotients in order to transform the Ritz values computed by DNAUPD for OP to those of A*z = lambda*B*z. Set RVEC = 1 and HOWMNY = 'A', and compute

trans(Z(:,I)) * A * Z(:,I) if DI(I) = 0.

If DI(I) is not equal to zero and DI(I+1) = - D(I), then the desired real and imaginary parts of the Ritz value are

trans(Z(:,I)) * A * Z(:,I) + trans(Z(:,I+1)) * A * Z(:,I+1),

trans(Z(:,I)) * A * Z(:,I+1) - trans(Z(:,I+1)) * A * Z(:,I),

respectively.

Another possibility is to set RVEC = 1 and HOWMANY = 'P' and compute

trans(V(:,1:IPARAM(5))) * A * V(:,1:IPARAM(5))

and then an upper quasi-triangular matrix of order IPARAM(5) is computed. See remark 2 above.

### 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); workev = zeros(3*ncv,1); workl = zeros(3*ncv*ncv+6*ncv,1); // Build the 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; sigmar = 0; // the real part of the shift sigmai = 0; // the imaginary part of the shift // M A I N L O O P (Reverse communication) iter = 0; while(iter<maxiter) info_dnaupd = 0; iter = iter + 1; // Repeatedly call the routine DNAUPD 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_dnaupd] = dnaupd(ido,bmat,nx,which,nev,tol,resid,ncv,v,iparam,ipntr,workd,workl,info_dnaupd); if (ido==99) then // BE CAREFUL: DON'T CALL dneupd 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 DNAUPD again. continue end if (info_dnaupd < 0) then printf('\nError with dnaupd, info = %d\n',info_dnaupd); printf('Check the documentation of dnaupd\n\n'); else // Post-Process using DNEUPD. rvec = 1; howmany = 'A'; info_dneupd = 0; [d,d(1,2),v,resid,v,iparam,ipntr,workd,workl,info_dneupd] = dneupd(rvec,howmany,_select,d,d(1,2),v,sigmar,sigmai,workev, ... bmat,nx,which,nev,tol,resid,ncv,v, ... iparam,ipntr,workd,workl,info_dneupd); if (info_dneupd~=0) then printf('\nError with dneupd, info = %d\n', info_dneupd); printf('Check the documentation of dneupd.\n\n'); end // Print additional convergence information. if (info_dneupd==1) then printf('\nMaximum number of iterations reached.\n\n'); elseif (info_dneupd==3) then printf('\nNo shifts could be applied during implicit Arnoldi update, try increasing NCV.\n\n'); end end end printf('\nDNSIMP\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, "The Symmetric Eigenvalue Problem". Prentice-Hall, 1980.

4. B.N. Parlett, B. Nour-Omid, "Towards a Black Box Lanczos Program", Computer Physics Communications, 53 (1989), pp 169-179.

5. B. Nour-Omid, B.N. Parlett, T. Ericson, P.S. Jensen, "How to Implement the Spectral Transformation", Math. Comp., 48 (1987), pp 663-673.

6. R.G. Grimes, J.G. Lewis and H.D. Simon, "A Shifted Block Lanczos Algorithm for Solving Sparse Symmetric Generalized Eigenproblems", SIAM J. Matr. Anal. Apps., January (1993).

7. L. Reichel, W.B. Gragg, "Algorithm 686: FORTRAN Subroutines for Updating the QR decomposition", ACM TOMS, December 1990, Volume 16 Number 4, pp 369-377.

8. R.B. Lehoucq, D.C. Sorensen, "Implementation of Some Spectral Transformations in a k-Step Arnoldi Method". In Preparation.

### Used Functions

Based on ARPACK routine dneupd

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