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# znaupd

Interface for the Implicitly Restarted Arnoldi Iteration, to compute a few eigenpairs of a complex linear operator OP with respect to a semi-inner product defined by a hermitian positive semi-definite real matrix B.

### Calling Sequence

[IDO,RESID,V,IPARAM,IPNTR,WORKD,WORKL,RWORKINFO] = znaupd(ID0,BMAT,N,WHICH,NEV,TOL,RESID,NCV,V,IPARAM,IPNTR,WORKD,WORKL,RWORK,INFO)

### Arguments

- IDO
Integer. (INPUT/OUTPUT)

Reverse communication flag. IDO must be zero on the first call to znaupd. IDO will be set internally to indicate the type of operation to be performed. Control is then given back to the calling routine which has the responsibility to carry out the requested operation and call znaupd with the result.

The operand is given in WORKD(IPNTR(1)), the result must be put in WORKD(IPNTR(2)).

IDO = 0: first call to the reverse communication interface

IDO = -1: compute Y = OP * X where IPNTR(1) is the pointer into WORKD for X, IPNTR(2) is the pointer into WORKD for Y.

This is for the initialization phase to force the starting vector into the range of OP.

IDO = 1: compute Y = OP * X where IPNTR(1) is the pointer into WORKD for X, IPNTR(2) is the pointer into WORKD for Y.

In mode 3, the vector B * X is already available in WORKD(ipntr(3)). It does not need to be recomputed in forming OP * X.

IDO = 2: compute Y = M * X where IPNTR(1) is the pointer into WORKD for X, IPNTR(2) is the pointer into WORKD for Y.

IDO = 3: compute and return the shifts in the first NP locations of WORKL.

IDO = 99: done

After the initialization phase, when the routine is used in the "shift-and-invert" mode, the vector M * X is already available and does not need to be recomputed in forming OP*X.

- BMAT
Character. (INPUT)

BMAT specifies the type of the matrix B that defines the semi-inner product for the operator OP.

'I': standard eigenvalue problem A*x = lambda*x

'G': generalized eigenvalue problem A*x = lambda*M*x

- N
Integer. (INPUT)

Dimension of the eigenproblem.

- WHICH
Characters. (INPUT)

'LM': want the NEV eigenvalues of largest magnitude.

'SM': want the NEV eigenvalues of smallest magnitude.

'LR': want the NEV eigenvalues of largest real part.

'SR': want the NEV eigenvalues of smallest real part.

'LI': want the NEV eigenvalues of largest imaginary part.

'SI': want the NEV eigenvalues of smallest imaginary part.

- NEV
Integer. (INPUT)

Number of eigenvalues of OP to be computed. 0 < NEV < N-1.

- TOL
Double precision scalar. (INPUT)

Stopping criteria: the relative accuracy of the Ritz value is considered acceptable if BOUNDS(I) .LE. TOL*ABS(RITZ(I)) where ABS(RITZ(I)) is the magnitude when RITZ(I) is complex. DEFAULT = dlamch('EPS') (machine precision as computed by the LAPACK auxiliary subroutine dlamch).

- RESID
Complex*16 array of length N. (INPUT/OUTPUT)

On INPUT: If INFO .EQ. 0, a random initial residual vector is used. If INFO .NE. 0, RESID contains the initial residual vector, possibly from a previous run.

On OUTPUT: RESID contains the final residual vector.

- NCV
Integer. (INPUT)

Number of columns of the matrix V. NCV must satisfy the two inequalities 1 <= NCV-NEV and NCV <= N.

This will indicate how many Arnoldi vectors are generated at each iteration. After the startup phase in which NEV Arnoldi vectors are generated, the algorithm generates approximately NCV-NEV Arnoldi vectors at each subsequent update iteration. Most of the cost in generating each Arnoldi vector is in the matrix-vector operation OP*x. (See remark 4 below.)

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

Contains the final set of Arnoldi basis vectors.

- IPARAM
Integer array of length 11. (INPUT/OUTPUT)

IPARAM(1) = ISHIFT: method for selecting the implicit shifts. The shifts selected at each iteration are used to filter out the components of the unwanted eigenvector.

ISHIFT = 0: the shifts are to be provided by the user via reverse communication. The NCV eigenvalues of the Hessenberg matrix H are returned in the part of WORKL array corresponding to RITZ.

ISHIFT = 1: exact shifts with respect to the current Hessenberg matrix H. This is equivalent to restarting the iteration from the beginning after updating the starting vector with a linear combination of Ritz vectors associated with the "wanted" eigenvalues.

ISHIFT = 2: other choice of internal shift to be defined.

IPARAM(2) = No longer referenced

IPARAM(3) = MXITER

On INPUT: maximum number of Arnoldi update iterations allowed.

On OUTPUT: actual number of Arnoldi update iterations taken.

IPARAM(4) = NB: blocksize to be used in the recurrence. The code currently works only for NB = 1.

IPARAM(5) = NCONV: number of "converged" Ritz values. This represents the number of Ritz values that satisfy the convergence criterion.

IPARAM(6) = IUPD No longer referenced. Implicit restarting is ALWAYS used.

IPARAM(7) = MODE On INPUT determines what type of eigenproblem is being solved. Must be 1,2,3; See under Description of znaupd for the four modes available.

IPARAM(8) = NP When ido = 3 and the user provides shifts through reverse communication (IPARAM(1)=0), _naupd returns NP, the number of shifts the user is to provide. 0 < NP < NCV-NEV.

IPARAM(9) = NUMOP,

IPARAM(10) = NUMOPB,

IPARAM(11) = NUMREO, OUTPUT: NUMOP = total number of OP*x operations, NUMOPB = total number of B*x operations if BMAT='G', NUMREO = total number of steps of re-orthogonalization.

- IPNTR
Integer array of length 14. (OUTPUT)

Pointer to mark the starting locations in the WORKD and WORKL arrays for matrices/vectors used by the Arnoldi iteration.

IPNTR(1): pointer to the current operand vector X in WORKD.

IPNTR(2): pointer to the current result vector Y in WORKD.

IPNTR(3): pointer to the vector B * X in WORKD when used in the shift-and-invert mode.

IPNTR(4): pointer to the next available location in WORKL that is untouched by the program.

IPNTR(5): pointer to the NCV by NCV upper Hessenberg matrix H in WORKL.

IPNTR(6): pointer to the ritz value array RITZ

IPNTR(7): pointer to the (projected) ritz vector array Q

IPNTR(8): pointer to the error BOUNDS array in WORKL.

IPNTR(14): pointer to the NP shifts in WORKL. See Remark 5 below.

Note: IPNTR(9:13) is only referenced by zneupd. See Remark 2 below.

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

IPNTR(10): Not Used

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

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.

- WORKD
Complex*16 work array of length 3*N. (REVERSE COMMUNICATION)

Distributed array to be used in the basic Arnoldi iteration for reverse communication.

The user should not use WORKD as temporary workspace during the iteration !!!!!!!!!!

See Data Distribution Note below.

- WORKL
Complex*16 work array of length 3*NCV**2 + 5*NCV. (OUTPUT/WORKSPACE)

Private (replicated) array on each PE or array allocated on the front end. See Data Distribution Note below.

- RWORK
Double precision work array of length NCV (WORKSPACE) Private (replicated) array on each PE or array allocated on the front end.

- INFO
Integer. (INPUT/OUTPUT)

If INFO == 0, a randomly initial residual vector is used.

If INFO ~= 0, RESID contains the initial residual vector, possibly from a previous run.

Error flag on output.

0: Normal exit.

1: Maximum number of iterations taken. All possible eigenvalues of OP has been found. IPARAM(5) returns the number of wanted converged Ritz values.

2: No longer an informational error. Deprecated starting with release 2 of ARPACK.

3: 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 remark 4 below.

-1: N must be positive.

-2: NEV must be positive.

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

-4: The maximum number of Arnoldi update iteration must be greater than zero.

-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 array is not sufficient.

-8: Error return from LAPACK eigenvalue calculation;

-9: Starting vector is zero.

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

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

-12: IPARAM(1) must be equal to 0 or 1.

-9999: Could not build an Arnoldi factorization. User input error highly likely. Please check actual array dimensions and layout. IPARAM(5) returns the size of the current Arnoldi factorization.

### Description

Reverse communication interface for the Implicitly Restarted Arnoldi iteration. This is intended to be used to find a few eigenpairs of a complex linear operator OP with respect to a semi-inner product defined by a hermitian positive semi-definite real matrix B. B may be the identity matrix.

NOTE: if both OP and B are real, then dsaupd or dnaupd should be used.

The computed approximate eigenvalues are called Ritz values and the corresponding approximate eigenvectors are called Ritz vectors. znaupd is usually called iteratively to solve one of the following problems:

Mode 1: A*x = lambda*x.

===> OP = A and B = I.

Mode 2: A*x = lambda*M*x, M hermitian positive definite

===> OP = inv[M]*A and B = M.

===> (If M can be factored see remark 3 below)

Mode 3: A*x = lambda*M*x, M hermitian semi-definite

===> OP = inv[A - sigma*M]*M and B = M.

===> shift-and-invert mode If OP*x = amu*x, then lambda = sigma + 1/amu.

NOTE: The action of w <- inv[A - sigma*M]*v or w <- inv[M]*v should be accomplished either by a direct method using a sparse matrix factorization and solving

[A - sigma*M]*w = v or M*w = v,

or through an iterative method for solving these systems. If an iterative method is used, the convergence test must be more stringent than the accuracy requirements for the eigenvalue approximations.

### Remarks

The computed Ritz values are approximate eigenvalues of OP. The selection of WHICH should be made with this in mind when using Mode = 3. When operating in Mode = 3 setting WHICH = 'LM' will compute the NEV eigenvalues of the original problem that are closest to the shift SIGMA . After convergence, approximate eigenvalues of the original problem may be obtained with the ARPACK subroutine zneupd.

If a basis for the invariant subspace corresponding to the converged Ritz values is needed, the user must call zneupd immediately following completion of znaupd. This is new starting with release 2 of ARPACK.

If M can be factored into a Cholesky factorization M = LL` then Mode = 2 should not be selected. Instead one should use Mode = 1 with OP = inv(L)*A*inv(L`). Appropriate triangular linear systems should be solved with L and L` rather than computing inverses. After convergence, an approximate eigenvector z of the original problem is recovered by solving L`z = x where x is a Ritz vector of OP.

At present there is no a-priori analysis to guide the selection of NCV relative to NEV. The only formal requirement is that NCV > NEV + 1. However, it is recommended that NCV .ge. 2*NEV. If many problems of the same type are to be solved, one should experiment with increasing NCV while keeping NEV fixed for a given test problem. This will usually decrease the required number of OP*x operations but it also increases the work and storage required to maintain the orthogonal basis vectors. The optimal "cross-over" with respect to CPU time is problem dependent and must be determined empirically. See Chapter 8 of Reference 2 for further information.

When IPARAM(1) = 0, and IDO = 3, the user needs to provide the NP = IPARAM(8) complex shifts in locations

WORKL(IPNTR(14)), WORKL(IPNTR(14)+1), ... , WORKL(IPNTR(14)+NP).

Eigenvalues of the current upper Hessenberg matrix are located in WORKL(IPNTR(6)) through WORKL(IPNTR(6)+NCV-1). They are ordered according to the order defined by WHICH. The associated Ritz estimates are located in

WORKL(IPNTR(8)), WORKL(IPNTR(8)+1), ... , WORKL(IPNTR(8)+NCV-1).

### 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 znaupd

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