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# dnaupd

Interface for the Implicitly Restarted Arnoldi Iteration, to compute approximations to a few eigenpairs of a real linear operator

### Calling Sequence

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

### Arguments

- ID0
Integer. (INPUT/OUTPUT) Reverse communication flag. IDO must be zero on the first call to dnaupd. 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 dnaupd 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 and 4, 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 = B * X where IPNTR(1) is the pointer into WORKD for X, IPNTR(2) is the pointer into WORKD for Y.

IDO = 3: compute the IPARAM(8) real and imaginary parts of the shifts where INPTR(14) is the pointer into WORKL for placing the shifts. See Remark 5 below.

IDO = 99: done

- BMAT
Character, 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*B*x

- N
Integer, dimension of the eigenproblem.

- WHICH
string of length 2, Specify which of the Ritz values of OP to compute.

'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, number of eigenvalues of OP to be computed. 0 < NEV < N-1.

- TOL
scalar. Stopping criterion: the relative accuracy of the Ritz value is considered acceptable if BOUNDS(I) <= TOL*ABS(RITZ(I)). If TOL <= 0. is passed the machine precision is set.

- RESID
array of length N (INPUT/OUTPUT)

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

On OUTPUT: RESID contains the final residual vector.

- NCV
Integer, number of columns of the matrix V. NCV must satisfy the two inequalities 2 <= 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.

NOTE: 2 <= NCV-NEV in order that complex conjugate pairs of Ritz values are kept together. (See remark 4 below)

- V
N by NCV array. Contains the final set of Arnoldi basis vectors.

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

IPARAM(1) = ISHIFT: method for selecting the implicit shifts. The shifts selected at each iteration are used to restart the Arnoldi iteration in an implicit fashion.

ISHIFT = 0: the shifts are provided by the user via reverse communication. The real and imaginary parts of the NCV eigenvalues of the Hessenberg matrix H are returned in the part of the WORKL array corresponding to RITZR and RITZI. See remark 5 below.

ISHIFT = 1: exact shifts with respect to the current Hessenberg matrix H. This is equivalent to restarting the iteration with a starting vector that is a linear combination of approximate Schur vectors associated with the "wanted" Ritz values.

IPARAM(2) = LEVEC 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 1,2,3,4; See under Description of dnaupd for the five modes available.

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

0 < NP <= NCV-NEV. See Remark 5 below.

IPARAM(9) = NUMOP,

IPARAM(10) = NUMOPB,

IPARAM(11) = NUMREO,

On 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
array of length 14. 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 real part of the ritz value array RITZR in WORKL.

IPNTR(7): pointer to the imaginary part of the ritz value array RITZI in WORKL.

IPNTR(8): pointer to the Ritz estimates in array WORKL associated with the Ritz values located in RITZR and RITZI in WORKL.

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

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

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.

- WORKD
Double precision 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. Upon termination WORKD(1:N) contains B*RESID(1:N). If an invariant subspace associated with the converged Ritz values is desired, see remark 2 below, subroutine dneupd uses this output. See Data Distribution Note below.

- WORKL
work array of length at least 3*NCV**2 + 6*NCV. (OUTPUT/WORKSPACE) Private (replicated) array on each PE or array allocated on the front end. See Data Distribution Note below.

- INFO
Integer. (INPUT/OUTPUT)

If INFO == 0, a randomly initial residual vector is used, else 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 >= 2 and less than or equal to N.

-4: The maximum number of Arnoldi update iterations allowed 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 WORKL is not sufficient.

-8: Error return from LAPACK eigenvalue calculation;

-9: Starting vector is zero.

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

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

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

-9999: Could not build an Arnoldi factorization. IPARAM(5) returns the size of the current Arnoldi factorization. The user is advised to check that enough workspace and array storage has been allocated.

### Description

Reverse communication interface for the Implicitly Restarted Arnoldi iteration. This subroutine computes approximations to a few eigenpairs of a linear operator "OP" with respect to a semi-inner product defined by a symmetric positive semi-definite real matrix B. B may be the identity matrix. NOTE: If the linear operator "OP" is real and symmetric with respect to the real positive semi-definite symmetric matrix B, i.e. B*OP = (OP`)*B, then subroutine dsaupd should be used instead.

The computed approximate eigenvalues are called Ritz values and the corresponding approximate eigenvectors are called Ritz vectors.

dnaupd is usually called iteratively to solve one of the following problems:

Mode 1: A*x = lambda*x.

`OP = A , B = I`

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

`OP = inv[M]*A, B = M`

. (If M can be factored see remark 3 below)Mode 3: A*x = lambda*M*x, M symmetric positive semi-definite.

`OP = Real_Part{ inv[A - sigma*M]*M }, B = M`

. shift-and-invert mode (in real arithmetic)If

`OP*x = amu*x`

, then`amu = 1/2 * [ 1/(lambda-sigma) + 1/(lambda-conjg(sigma))]`

.Note: If sigma is real, i.e. imaginary part of sigma is zero;

`Real_Part{ inv[A - sigma*M]*M } == inv[A - sigma*M]*M`

`amu == 1/(lambda-sigma)`

.Mode 4: A*x = lambda*M*x, M symmetric semi-definite

`OP = Imaginary_Part{ inv[A - sigma*M]*M } , B = M`

. shift-and-invert mode (in real arithmetic)If

`OP*x = amu*x`

, then`amu = 1/2i * [ 1/(lambda-sigma) - 1/(lambda-conjg(sigma)) ]`

.

Both mode 3 and 4 give the same enhancement to eigenvalues close to the (complex) shift sigma. However, as lambda goes to infinity, the operator OP in mode 4 dampens the eigenvalues more strongly than does OP defined in mode 3.

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.

### 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);

### Remarks

1. The computed Ritz values are approximate eigenvalues of OP. The selection of WHICH should be made with this in mind when Mode = 3 and 4. After convergence, approximate eigenvalues of the original problem may be obtained with the ARPACK subroutine dneupd.

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

3. 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.

4. At present there is no a-priori analysis to guide the selection of NCV relative to NEV. The only formal requrement is that NCV > NEV + 2. However, it is recommended that NCV >= 2*NEV+1. 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.

5. When IPARAM(1) = 0, and IDO = 3, the user needs to provide the NP = IPARAM(8) real and imaginary parts of the shifts in locations

real part imaginary part ----------------------- -------------- 1 WORKL(IPNTR(14)) WORKL(IPNTR(14)+NP) 2 WORKL(IPNTR(14)+1) WORKL(IPNTR(14)+NP+1) . . . . . . NP WORKL(IPNTR(14)+NP-1) WORKL(IPNTR(14)+2*NP-1).

Only complex conjugate pairs of shifts may be applied and the pairs must be placed in consecutive locations. The real part of the eigenvalues of the current upper Hessenberg matrix are located in WORKL(IPNTR(6)) through WORKL(IPNTR(6)+NCV-1) and the imaginary part in WORKL(IPNTR(7)) through WORKL(IPNTR(7)+NCV-1). They are ordered according to the order defined by WHICH. The complex conjugate pairs are kept together and the associated Ritz estimates are located in WORKL(IPNTR(8)), WORKL(IPNTR(8)+1), ... , WORKL(IPNTR(8)+NCV-1).

### See Also

- dsaupd — Interface for the Implicitly Restarted Arnoldi Iteration, to compute approximations to a few eigenpairs of a real and symmetric linear operator

### 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 dnaupd

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