c\BeginDoc c c\Name: cnaupd c c\Description: c Reverse communication interface for the Implicitly Restarted Arnoldi c iteration. This is intended to be used to find a few eigenpairs of a c complex linear operator OP with respect to a semi-inner product defined c by a hermitian positive semi-definite real matrix B. B may be the identity c matrix. NOTE: if both OP and B are real, then ssaupd or snaupd should c be used. c c c The computed approximate eigenvalues are called Ritz values and c the corresponding approximate eigenvectors are called Ritz vectors. c c cnaupd is usually called iteratively to solve one of the c following problems: c c Mode 1: A*x = lambda*x. c ===> OP = A and B = I. c c Mode 2: A*x = lambda*M*x, M hermitian positive definite c ===> OP = inv[M]*A and B = M. c ===> (If M can be factored see remark 3 below) c c Mode 3: A*x = lambda*M*x, M hermitian semi-definite c ===> OP = inv[A - sigma*M]*M and B = M. c ===> shift-and-invert mode c If OP*x = amu*x, then lambda = sigma + 1/amu. c c c NOTE: The action of w <- inv[A - sigma*M]*v or w <- inv[M]*v c should be accomplished either by a direct method c using a sparse matrix factorization and solving c c [A - sigma*M]*w = v or M*w = v, c c or through an iterative method for solving these c systems. If an iterative method is used, the c convergence test must be more stringent than c the accuracy requirements for the eigenvalue c approximations. c c\Usage: c call cnaupd c ( IDO, BMAT, N, WHICH, NEV, TOL, RESID, NCV, V, LDV, IPARAM, c IPNTR, WORKD, WORKL, LWORKL, RWORK, INFO ) c c\Arguments c IDO Integer. (INPUT/OUTPUT) c Reverse communication flag. IDO must be zero on the first c call to cnaupd. IDO will be set internally to c indicate the type of operation to be performed. Control is c then given back to the calling routine which has the c responsibility to carry out the requested operation and call c cnaupd with the result. The operand is given in c WORKD(IPNTR(1)), the result must be put in WORKD(IPNTR(2)). c ------------------------------------------------------------- c IDO = 0: first call to the reverse communication interface c IDO = -1: compute Y = OP * X where c IPNTR(1) is the pointer into WORKD for X, c IPNTR(2) is the pointer into WORKD for Y. c This is for the initialization phase to force the c starting vector into the range of OP. c IDO = 1: compute Y = OP * X where c IPNTR(1) is the pointer into WORKD for X, c IPNTR(2) is the pointer into WORKD for Y. c In mode 3, the vector B * X is already c available in WORKD(ipntr(3)). It does not c need to be recomputed in forming OP * X. c IDO = 2: compute Y = M * X where c IPNTR(1) is the pointer into WORKD for X, c IPNTR(2) is the pointer into WORKD for Y. c IDO = 3: compute and return the shifts in the first c NP locations of WORKL. c IDO = 99: done c ------------------------------------------------------------- c After the initialization phase, when the routine is used in c the "shift-and-invert" mode, the vector M * X is already c available and does not need to be recomputed in forming OP*X. c c BMAT Character*1. (INPUT) c BMAT specifies the type of the matrix B that defines the c semi-inner product for the operator OP. c BMAT = 'I' -> standard eigenvalue problem A*x = lambda*x c BMAT = 'G' -> generalized eigenvalue problem A*x = lambda*M*x c c N Integer. (INPUT) c Dimension of the eigenproblem. c c WHICH Character*2. (INPUT) c 'LM' -> want the NEV eigenvalues of largest magnitude. c 'SM' -> want the NEV eigenvalues of smallest magnitude. c 'LR' -> want the NEV eigenvalues of largest real part. c 'SR' -> want the NEV eigenvalues of smallest real part. c 'LI' -> want the NEV eigenvalues of largest imaginary part. c 'SI' -> want the NEV eigenvalues of smallest imaginary part. c c NEV Integer. (INPUT) c Number of eigenvalues of OP to be computed. 0 < NEV < N-1. c c TOL Real scalar. (INPUT) c Stopping criteria: the relative accuracy of the Ritz value c is considered acceptable if BOUNDS(I) .LE. TOL*ABS(RITZ(I)) c where ABS(RITZ(I)) is the magnitude when RITZ(I) is complex. c DEFAULT = slamch('EPS') (machine precision as computed c by the LAPACK auxiliary subroutine slamch). c c RESID Complex array of length N. (INPUT/OUTPUT) c On INPUT: c If INFO .EQ. 0, a random initial residual vector is used. c If INFO .NE. 0, RESID contains the initial residual vector, c possibly from a previous run. c On OUTPUT: c RESID contains the final residual vector. c c NCV Integer. (INPUT) c Number of columns of the matrix V. NCV must satisfy the two c inequalities 1 <= NCV-NEV and NCV <= N. c This will indicate how many Arnoldi vectors are generated c at each iteration. After the startup phase in which NEV c Arnoldi vectors are generated, the algorithm generates c approximately NCV-NEV Arnoldi vectors at each subsequent update c iteration. Most of the cost in generating each Arnoldi vector is c in the matrix-vector operation OP*x. (See remark 4 below.) c c V Complex array N by NCV. (OUTPUT) c Contains the final set of Arnoldi basis vectors. c c LDV Integer. (INPUT) c Leading dimension of V exactly as declared in the calling program. c c IPARAM Integer array of length 11. (INPUT/OUTPUT) c IPARAM(1) = ISHIFT: method for selecting the implicit shifts. c The shifts selected at each iteration are used to filter out c the components of the unwanted eigenvector. c ------------------------------------------------------------- c ISHIFT = 0: the shifts are to be provided by the user via c reverse communication. The NCV eigenvalues of c the Hessenberg matrix H are returned in the part c of WORKL array corresponding to RITZ. c ISHIFT = 1: exact shifts with respect to the current c Hessenberg matrix H. This is equivalent to c restarting the iteration from the beginning c after updating the starting vector with a linear c combination of Ritz vectors associated with the c "wanted" eigenvalues. c ISHIFT = 2: other choice of internal shift to be defined. c ------------------------------------------------------------- c c IPARAM(2) = No longer referenced c c IPARAM(3) = MXITER c On INPUT: maximum number of Arnoldi update iterations allowed. c On OUTPUT: actual number of Arnoldi update iterations taken. c c IPARAM(4) = NB: blocksize to be used in the recurrence. c The code currently works only for NB = 1. c c IPARAM(5) = NCONV: number of "converged" Ritz values. c This represents the number of Ritz values that satisfy c the convergence criterion. c c IPARAM(6) = IUPD c No longer referenced. Implicit restarting is ALWAYS used. c c IPARAM(7) = MODE c On INPUT determines what type of eigenproblem is being solved. c Must be 1,2,3; See under \Description of cnaupd for the c four modes available. c c IPARAM(8) = NP c When ido = 3 and the user provides shifts through reverse c communication (IPARAM(1)=0), _naupd returns NP, the number c of shifts the user is to provide. 0 < NP < NCV-NEV. c c IPARAM(9) = NUMOP, IPARAM(10) = NUMOPB, IPARAM(11) = NUMREO, c OUTPUT: NUMOP = total number of OP*x operations, c NUMOPB = total number of B*x operations if BMAT='G', c NUMREO = total number of steps of re-orthogonalization. c c IPNTR Integer array of length 14. (OUTPUT) c Pointer to mark the starting locations in the WORKD and WORKL c arrays for matrices/vectors used by the Arnoldi iteration. c ------------------------------------------------------------- c IPNTR(1): pointer to the current operand vector X in WORKD. c IPNTR(2): pointer to the current result vector Y in WORKD. c IPNTR(3): pointer to the vector B * X in WORKD when used in c the shift-and-invert mode. c IPNTR(4): pointer to the next available location in WORKL c that is untouched by the program. c IPNTR(5): pointer to the NCV by NCV upper Hessenberg c matrix H in WORKL. c IPNTR(6): pointer to the ritz value array RITZ c IPNTR(7): pointer to the (projected) ritz vector array Q c IPNTR(8): pointer to the error BOUNDS array in WORKL. c IPNTR(14): pointer to the NP shifts in WORKL. See Remark 5 below. c c Note: IPNTR(9:13) is only referenced by cneupd. See Remark 2 below. c c IPNTR(9): pointer to the NCV RITZ values of the c original system. c IPNTR(10): Not Used c IPNTR(11): pointer to the NCV corresponding error bounds. c IPNTR(12): pointer to the NCV by NCV upper triangular c Schur matrix for H. c IPNTR(13): pointer to the NCV by NCV matrix of eigenvectors c of the upper Hessenberg matrix H. Only referenced by c cneupd if RVEC = .TRUE. See Remark 2 below. c c ------------------------------------------------------------- c c WORKD Complex work array of length 3*N. (REVERSE COMMUNICATION) c Distributed array to be used in the basic Arnoldi iteration c for reverse communication. The user should not use WORKD c as temporary workspace during the iteration !!!!!!!!!! c See Data Distribution Note below. c c WORKL Complex work array of length LWORKL. (OUTPUT/WORKSPACE) c Private (replicated) array on each PE or array allocated on c the front end. See Data Distribution Note below. c c LWORKL Integer. (INPUT) c LWORKL must be at least 3*NCV**2 + 5*NCV. c c RWORK Real work array of length NCV (WORKSPACE) c Private (replicated) array on each PE or array allocated on c the front end. c c c INFO Integer. (INPUT/OUTPUT) c If INFO .EQ. 0, a randomly initial residual vector is used. c If INFO .NE. 0, RESID contains the initial residual vector, c possibly from a previous run. c Error flag on output. c = 0: Normal exit. c = 1: Maximum number of iterations taken. c All possible eigenvalues of OP has been found. IPARAM(5) c returns the number of wanted converged Ritz values. c = 2: No longer an informational error. Deprecated starting c with release 2 of ARPACK. c = 3: No shifts could be applied during a cycle of the c Implicitly restarted Arnoldi iteration. One possibility c is to increase the size of NCV relative to NEV. c See remark 4 below. c = -1: N must be positive. c = -2: NEV must be positive. c = -3: NCV-NEV >= 1 and less than or equal to N. c = -4: The maximum number of Arnoldi update iteration c must be greater than zero. c = -5: WHICH must be one of 'LM', 'SM', 'LR', 'SR', 'LI', 'SI' c = -6: BMAT must be one of 'I' or 'G'. c = -7: Length of private work array is not sufficient. c = -8: Error return from LAPACK eigenvalue calculation; c = -9: Starting vector is zero. c = -10: IPARAM(7) must be 1,2,3. c = -11: IPARAM(7) = 1 and BMAT = 'G' are incompatible. c = -12: IPARAM(1) must be equal to 0 or 1. c = -9999: Could not build an Arnoldi factorization. c User input error highly likely. Please c check actual array dimensions and layout. c IPARAM(5) returns the size of the current Arnoldi c factorization. c c\Remarks c 1. The computed Ritz values are approximate eigenvalues of OP. The c selection of WHICH should be made with this in mind when using c Mode = 3. When operating in Mode = 3 setting WHICH = 'LM' will c compute the NEV eigenvalues of the original problem that are c closest to the shift SIGMA . After convergence, approximate eigenvalues c of the original problem may be obtained with the ARPACK subroutine cneupd. c c 2. If a basis for the invariant subspace corresponding to the converged Ritz c values is needed, the user must call cneupd immediately following c completion of cnaupd. This is new starting with release 2 of ARPACK. c c 3. If M can be factored into a Cholesky factorization M = LL` c then Mode = 2 should not be selected. Instead one should use c Mode = 1 with OP = inv(L)*A*inv(L`). Appropriate triangular c linear systems should be solved with L and L` rather c than computing inverses. After convergence, an approximate c eigenvector z of the original problem is recovered by solving c L`z = x where x is a Ritz vector of OP. c c 4. At present there is no a-priori analysis to guide the selection c of NCV relative to NEV. The only formal requirement is that NCV > NEV + 1. c However, it is recommended that NCV .ge. 2*NEV. If many problems of c the same type are to be solved, one should experiment with increasing c NCV while keeping NEV fixed for a given test problem. This will c usually decrease the required number of OP*x operations but it c also increases the work and storage required to maintain the orthogonal c basis vectors. The optimal "cross-over" with respect to CPU time c is problem dependent and must be determined empirically. c See Chapter 8 of Reference 2 for further information. c c 5. When IPARAM(1) = 0, and IDO = 3, the user needs to provide the c NP = IPARAM(8) complex shifts in locations c WORKL(IPNTR(14)), WORKL(IPNTR(14)+1), ... , WORKL(IPNTR(14)+NP). c Eigenvalues of the current upper Hessenberg matrix are located in c WORKL(IPNTR(6)) through WORKL(IPNTR(6)+NCV-1). They are ordered c according to the order defined by WHICH. The associated Ritz estimates c are located in WORKL(IPNTR(8)), WORKL(IPNTR(8)+1), ... , c WORKL(IPNTR(8)+NCV-1). c c----------------------------------------------------------------------- c c\Data Distribution Note: c c Fortran-D syntax: c ================ c Complex resid(n), v(ldv,ncv), workd(3*n), workl(lworkl) c decompose d1(n), d2(n,ncv) c align resid(i) with d1(i) c align v(i,j) with d2(i,j) c align workd(i) with d1(i) range (1:n) c align workd(i) with d1(i-n) range (n+1:2*n) c align workd(i) with d1(i-2*n) range (2*n+1:3*n) c distribute d1(block), d2(block,:) c replicated workl(lworkl) c c Cray MPP syntax: c =============== c Complex resid(n), v(ldv,ncv), workd(n,3), workl(lworkl) c shared resid(block), v(block,:), workd(block,:) c replicated workl(lworkl) c c CM2/CM5 syntax: c ============== c c----------------------------------------------------------------------- c c include 'ex-nonsym.doc' c c----------------------------------------------------------------------- c c\BeginLib c c\Local variables: c xxxxxx Complex c c\References: c 1. D.C. Sorensen, "Implicit Application of Polynomial Filters in c a k-Step Arnoldi Method", SIAM J. Matr. Anal. Apps., 13 (1992), c pp 357-385. c 2. R.B. Lehoucq, "Analysis and Implementation of an Implicitly c Restarted Arnoldi Iteration", Rice University Technical Report c TR95-13, Department of Computational and Applied Mathematics. c 3. B.N. Parlett & Y. Saad, "_Complex_ Shift and Invert Strategies for c Real Matrices", Linear Algebra and its Applications, vol 88/89, c pp 575-595, (1987). c c\Routines called: c cnaup2 ARPACK routine that implements the Implicitly Restarted c Arnoldi Iteration. c cstatn ARPACK routine that initializes the timing variables. c ivout ARPACK utility routine that prints integers. c cvout ARPACK utility routine that prints vectors. c second ARPACK utility routine for timing. c slamch LAPACK routine that determines machine constants. c c\Author c Danny Sorensen Phuong Vu c Richard Lehoucq CRPC / Rice University c Dept. of Computational & Houston, Texas c Applied Mathematics c Rice University c Houston, Texas c c\SCCS Information: @(#) c FILE: naupd.F SID: 2.9 DATE OF SID: 07/21/02 RELEASE: 2 c c\Remarks c c\EndLib c c----------------------------------------------------------------------- c subroutine cnaupd & ( ido, bmat, n, which, nev, tol, resid, ncv, v, ldv, iparam, & ipntr, workd, workl, lworkl, rwork, info ) c c %----------------------------------------------------% c | Include files for debugging and timing information | c %----------------------------------------------------% c include 'debug.h' include 'stat.h' c c %------------------% c | Scalar Arguments | c %------------------% c character bmat*1, which*2 integer ido, info, ldv, lworkl, n, ncv, nev Real & tol c c %-----------------% c | Array Arguments | c %-----------------% c integer iparam(11), ipntr(14) Complex & resid(n), v(ldv,ncv), workd(3*n), workl(lworkl) Real & rwork(ncv) c c %------------% c | Parameters | c %------------% c Complex & one, zero parameter (one = (1.0E+0, 0.0E+0), zero = (0.0E+0, 0.0E+0)) c c %---------------% c | Local Scalars | c %---------------% c integer bounds, ierr, ih, iq, ishift, iupd, iw, & ldh, ldq, levec, mode, msglvl, mxiter, nb, & nev0, next, np, ritz, j save bounds, ih, iq, ishift, iupd, iw, & ldh, ldq, levec, mode, msglvl, mxiter, nb, & nev0, next, np, ritz c c %----------------------% c | External Subroutines | c %----------------------% c external cnaup2, cvout, ivout, second, cstatn c c %--------------------% c | External Functions | c %--------------------% c Real & slamch external slamch c c %-----------------------% c | Executable Statements | c %-----------------------% c if (ido .eq. 0) then c c %-------------------------------% c | Initialize timing statistics | c | & message level for debugging | c %-------------------------------% c call cstatn call second (t0) msglvl = mcaupd c c %----------------% c | Error checking | c %----------------% c ierr = 0 ishift = iparam(1) c levec = iparam(2) mxiter = iparam(3) c nb = iparam(4) nb = 1 c c %--------------------------------------------% c | Revision 2 performs only implicit restart. | c %--------------------------------------------% c iupd = 1 mode = iparam(7) c if (n .le. 0) then ierr = -1 else if (nev .le. 0) then ierr = -2 else if (ncv .le. nev .or. ncv .gt. n) then ierr = -3 else if (mxiter .le. 0) then ierr = -4 else if (which .ne. 'LM' .and. & which .ne. 'SM' .and. & which .ne. 'LR' .and. & which .ne. 'SR' .and. & which .ne. 'LI' .and. & which .ne. 'SI') then ierr = -5 else if (bmat .ne. 'I' .and. bmat .ne. 'G') then ierr = -6 else if (lworkl .lt. 3*ncv**2 + 5*ncv) then ierr = -7 else if (mode .lt. 1 .or. mode .gt. 3) then ierr = -10 else if (mode .eq. 1 .and. bmat .eq. 'G') then ierr = -11 end if c c %------------% c | Error Exit | c %------------% c if (ierr .ne. 0) then info = ierr ido = 99 go to 9000 end if c c %------------------------% c | Set default parameters | c %------------------------% c if (nb .le. 0) nb = 1 if (tol .le. 0.0E+0 ) tol = slamch('EpsMach') if (ishift .ne. 0 .and. & ishift .ne. 1 .and. & ishift .ne. 2) ishift = 1 c c %----------------------------------------------% c | NP is the number of additional steps to | c | extend the length NEV Lanczos factorization. | c | NEV0 is the local variable designating the | c | size of the invariant subspace desired. | c %----------------------------------------------% c np = ncv - nev nev0 = nev c c %-----------------------------% c | Zero out internal workspace | c %-----------------------------% c do 10 j = 1, 3*ncv**2 + 5*ncv workl(j) = zero 10 continue c c %-------------------------------------------------------------% c | Pointer into WORKL for address of H, RITZ, BOUNDS, Q | c | etc... and the remaining workspace. | c | Also update pointer to be used on output. | c | Memory is laid out as follows: | c | workl(1:ncv*ncv) := generated Hessenberg matrix | c | workl(ncv*ncv+1:ncv*ncv+ncv) := the ritz values | c | workl(ncv*ncv+ncv+1:ncv*ncv+2*ncv) := error bounds | c | workl(ncv*ncv+2*ncv+1:2*ncv*ncv+2*ncv) := rotation matrix Q | c | workl(2*ncv*ncv+2*ncv+1:3*ncv*ncv+5*ncv) := workspace | c | The final workspace is needed by subroutine cneigh called | c | by cnaup2. Subroutine cneigh calls LAPACK routines for | c | calculating eigenvalues and the last row of the eigenvector | c | matrix. | c %-------------------------------------------------------------% c ldh = ncv ldq = ncv ih = 1 ritz = ih + ldh*ncv bounds = ritz + ncv iq = bounds + ncv iw = iq + ldq*ncv next = iw + ncv**2 + 3*ncv c ipntr(4) = next ipntr(5) = ih ipntr(6) = ritz ipntr(7) = iq ipntr(8) = bounds ipntr(14) = iw end if c c %-------------------------------------------------------% c | Carry out the Implicitly restarted Arnoldi Iteration. | c %-------------------------------------------------------% c call cnaup2 & ( ido, bmat, n, which, nev0, np, tol, resid, mode, iupd, & ishift, mxiter, v, ldv, workl(ih), ldh, workl(ritz), & workl(bounds), workl(iq), ldq, workl(iw), & ipntr, workd, rwork, info ) c c %--------------------------------------------------% c | ido .ne. 99 implies use of reverse communication | c | to compute operations involving OP. | c %--------------------------------------------------% c if (ido .eq. 3) iparam(8) = np if (ido .ne. 99) go to 9000 c iparam(3) = mxiter iparam(5) = np iparam(9) = nopx iparam(10) = nbx iparam(11) = nrorth c c %------------------------------------% c | Exit if there was an informational | c | error within cnaup2. | c %------------------------------------% c if (info .lt. 0) go to 9000 if (info .eq. 2) info = 3 c if (msglvl .gt. 0) then call ivout (logfil, 1, mxiter, ndigit, & '_naupd: Number of update iterations taken') call ivout (logfil, 1, np, ndigit, & '_naupd: Number of wanted "converged" Ritz values') call cvout (logfil, np, workl(ritz), ndigit, & '_naupd: The final Ritz values') call cvout (logfil, np, workl(bounds), ndigit, & '_naupd: Associated Ritz estimates') end if c call second (t1) tcaupd = t1 - t0 c if (msglvl .gt. 0) then c c %--------------------------------------------------------% c | Version Number & Version Date are defined in version.h | c %--------------------------------------------------------% c write (6,1000) write (6,1100) mxiter, nopx, nbx, nrorth, nitref, nrstrt, & tmvopx, tmvbx, tcaupd, tcaup2, tcaitr, titref, & tgetv0, tceigh, tcgets, tcapps, tcconv, trvec 1000 format (//, & 5x, '=============================================',/ & 5x, '= Complex implicit Arnoldi update code =',/ & 5x, '= Version Number: ', ' 2.3', 21x, ' =',/ & 5x, '= Version Date: ', ' 07/31/96', 16x, ' =',/ & 5x, '=============================================',/ & 5x, '= Summary of timing statistics =',/ & 5x, '=============================================',//) 1100 format ( & 5x, 'Total number update iterations = ', i5,/ & 5x, 'Total number of OP*x operations = ', i5,/ & 5x, 'Total number of B*x operations = ', i5,/ & 5x, 'Total number of reorthogonalization steps = ', i5,/ & 5x, 'Total number of iterative refinement steps = ', i5,/ & 5x, 'Total number of restart steps = ', i5,/ & 5x, 'Total time in user OP*x operation = ', f12.6,/ & 5x, 'Total time in user B*x operation = ', f12.6,/ & 5x, 'Total time in Arnoldi update routine = ', f12.6,/ & 5x, 'Total time in naup2 routine = ', f12.6,/ & 5x, 'Total time in basic Arnoldi iteration loop = ', f12.6,/ & 5x, 'Total time in reorthogonalization phase = ', f12.6,/ & 5x, 'Total time in (re)start vector generation = ', f12.6,/ & 5x, 'Total time in Hessenberg eig. subproblem = ', f12.6,/ & 5x, 'Total time in getting the shifts = ', f12.6,/ & 5x, 'Total time in applying the shifts = ', f12.6,/ & 5x, 'Total time in convergence testing = ', f12.6,/ & 5x, 'Total time in computing final Ritz vectors = ', f12.6/) end if c 9000 continue c return c c %---------------% c | End of cnaupd | c %---------------% c end