dnaupd - Interface for the Implicitly Restarted Arnoldi Iteration, to compute approximations to a few eigenpairs of a real linear operator
: 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
B = 'I' -> standard eigenvalue problem A*x = lambda*x
B = 'G' -> generalized eigenvalue problem A*x = lambda*B*x
'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.
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.
NOTE: 2 <= NCV-NEV in order that complex conjugate pairs of Ritz values are kept together. (See remark 4 below)
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 dsaupd for the five modes available.
IPARAM(8) = NP When ido = 3 and the user provides shifts through reverse communication (IPARAM(1)=0), dsaupd 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, 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(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 dseupd . 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(11): pointer to the NCV by NCV matrix of eigenvectors of the upper Hessenberg matrix H. Only referenced by dneupd if RVEC = .TRUE. See Remark 2 below.
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.
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)) ] .
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.
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).
dsaupd ,
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.