c-----------------------------------------------------------------------
c\BeginDoc
c
c\Name: dsaupd
c
c\Description:
c
c Reverse communication interface for the Implicitly Restarted Arnoldi
c Iteration. For symmetric problems this reduces to a variant of the Lanczos
c method. This method has been designed to compute approximations to a
c few eigenpairs of a linear operator OP that is real and symmetric
c with respect to a real positive semi-definite symmetric matrix B,
c i.e.
c
c B*OP = (OP')*B.
c
c Another way to express this condition is
c
c < x,OPy > = < OPx,y > where < z,w > = z'Bw .
c
c In the standard eigenproblem B is the identity matrix.
c ( A' denotes transpose of A)
c
c The computed approximate eigenvalues are called Ritz values and
c the corresponding approximate eigenvectors are called Ritz vectors.
c
c dsaupd is usually called iteratively to solve one of the
c following problems:
c
c Mode 1: A*x = lambda*x, A symmetric
c ===> OP = A and B = I.
c
c Mode 2: A*x = lambda*M*x, A symmetric, M symmetric positive definite
c ===> OP = inv[M]*A and B = M.
c ===> (If M can be factored see remark 3 below)
c
c Mode 3: K*x = lambda*M*x, K symmetric, M symmetric positive semi-definite
c ===> OP = (inv[K - sigma*M])*M and B = M.
c ===> Shift-and-Invert mode
c
c Mode 4: K*x = lambda*KG*x, K symmetric positive semi-definite,
c KG symmetric indefinite
c ===> OP = (inv[K - sigma*KG])*K and B = K.
c ===> Buckling mode
c
c Mode 5: A*x = lambda*M*x, A symmetric, M symmetric positive semi-definite
c ===> OP = inv[A - sigma*M]*[A + sigma*M] and B = M.
c ===> Cayley transformed mode
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 dsaupd
c ( IDO, BMAT, N, WHICH, NEV, TOL, RESID, NCV, V, LDV, IPARAM,
c IPNTR, WORKD, WORKL, LWORKL, INFO )
c
c\Arguments
c IDO Integer. (INPUT/OUTPUT)
c Reverse communication flag. IDO must be zero on the first
c call to dsaupd. 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 dsaupd with the result. The operand is given in
c WORKD(IPNTR(1)), the result must be put in WORKD(IPNTR(2)).
c (If Mode = 2 see remark 5 below)
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 * Z and Z = B * X where
c IPNTR(1) is the pointer into WORKD for X,
c IPNTR(2) is the pointer into WORKD for Y,
c IPNTR(3) is the pointer into WORKD for Z.
c IDO = 2: compute Y = B * 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 the IPARAM(8) shifts where
c IPNTR(11) is the pointer into WORKL for
c placing the shifts. See remark 6 below.
c IDO = 99: done
c -------------------------------------------------------------
c After the initialization phase, when the routine is used in
c either the "shift-and-invert" mode or the Cayley transform
c mode, the vector B * X is already available and does not
c 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 B = 'I' -> standard eigenvalue problem A*x = lambda*x
c B = 'G' -> generalized eigenvalue problem A*x = lambda*B*x
c
c N Integer. (INPUT)
c Dimension of the eigenproblem.
c
c WHICH Character*2. (INPUT)
c Specify which of the Ritz values of OP to compute.
c
c 'LA' - compute the NEV largest (algebraic) eigenvalues.
c 'SA' - compute the NEV smallest (algebraic) eigenvalues.
c 'LM' - compute the NEV largest (in magnitude) eigenvalues.
c 'SM' - compute the NEV smallest (in magnitude) eigenvalues.
c 'BE' - compute NEV eigenvalues, half from each end of the
c spectrum. When NEV is odd, compute one more from the
c high end than from the low end.
c (see remark 1 below)
c
c NEV Integer. (INPUT)
c Number of eigenvalues of OP to be computed. 0 < NEV < N.
c
c TOL Double precision scalar. (INPUT)
c Stopping criterion: the relative accuracy of the Ritz value
c is considered acceptable if BOUNDS(I) .LE. TOL*ABS(RITZ(I)).
c If TOL .LE. 0. is passed a default is set:
c DEFAULT = DLAMCH('EPS') (machine precision as computed
c by the LAPACK auxiliary subroutine DLAMCH).
c
c RESID Double precision 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 (less than or equal to N).
c This will indicate how many Lanczos vectors are generated
c at each iteration. After the startup phase in which NEV
c Lanczos vectors are generated, the algorithm generates
c NCV-NEV Lanczos vectors at each subsequent update iteration.
c Most of the cost in generating each Lanczos vector is in the
c matrix-vector product OP*x. (See remark 4 below).
c
c V Double precision N by NCV array. (OUTPUT)
c The NCV columns of V contain the Lanczos basis vectors.
c
c LDV Integer. (INPUT)
c Leading dimension of V exactly as declared in the calling
c 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 restart
c the Arnoldi iteration in an implicit fashion.
c -------------------------------------------------------------
c ISHIFT = 0: the shifts are provided by the user via
c reverse communication. The NCV eigenvalues of
c the current tridiagonal matrix T are returned in
c the part of WORKL array corresponding to RITZ.
c See remark 6 below.
c ISHIFT = 1: exact shifts with respect to the reduced
c tridiagonal matrix T. This is equivalent to
c restarting the iteration with a starting vector
c that is a linear combination of Ritz vectors
c associated with the "wanted" Ritz values.
c -------------------------------------------------------------
c
c IPARAM(2) = LEVEC
c No longer referenced. See remark 2 below.
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 1,2,3,4,5; See under \Description of dsaupd for the
c five modes available.
c
c IPARAM(8) = NP
c When ido = 3 and the user provides shifts through reverse
c communication (IPARAM(1)=0), dsaupd returns NP, the number
c of shifts the user is to provide. 0 < NP <=NCV-NEV. See Remark
c 6 below.
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 11. (OUTPUT)
c Pointer to mark the starting locations in the WORKD and WORKL
c arrays for matrices/vectors used by the Lanczos 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 2 tridiagonal matrix T in WORKL.
c IPNTR(6): pointer to the NCV RITZ values array in WORKL.
c IPNTR(7): pointer to the Ritz estimates in array WORKL associated
c with the Ritz values located in RITZ in WORKL.
c Note: IPNTR(8:10) is only referenced by dseupd. See Remark 2.
c IPNTR(8): pointer to the NCV RITZ values of the original system.
c IPNTR(9): pointer to the NCV corresponding error bounds.
c IPNTR(10): pointer to the NCV by NCV matrix of eigenvectors
c of the tridiagonal matrix T. Only referenced by
c dseupd if RVEC = .TRUE. See Remarks.
c Note: IPNTR(8:10) is only referenced by dseupd. See Remark 2.
c IPNTR(11): pointer to the NP shifts in WORKL. See Remark 6 below.
c -------------------------------------------------------------
c
c WORKD Double precision 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. Upon termination
c WORKD(1:N) contains B*RESID(1:N). If the Ritz vectors are desired
c subroutine dseupd uses this output.
c See Data Distribution Note below.
c
c WORKL Double precision 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 NCV**2 + 8*NCV .
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 must be greater than NEV and less than or equal to N.
c = -4: The maximum number of Arnoldi update iterations allowed
c must be greater than zero.
c = -5: WHICH must be one of 'LM', 'SM', 'LA', 'SA' or 'BE'.
c = -6: BMAT must be one of 'I' or 'G'.
c = -7: Length of private work array WORKL is not sufficient.
c = -8: Error return from trid. eigenvalue calculation;
c Informational error from LAPACK routine dsteqr.
c = -9: Starting vector is zero.
c = -10: IPARAM(7) must be 1,2,3,4,5.
c = -11: IPARAM(7) = 1 and BMAT = 'G' are incompatable.
c = -12: IPARAM(1) must be equal to 0 or 1.
c = -13: NEV and WHICH = 'BE' are incompatable.
c = -9999: Could not build an Arnoldi factorization.
c IPARAM(5) returns the size of the current Arnoldi
c factorization. The user is advised to check that
c enough workspace and array storage has been allocated.
c
c
c\Remarks
c 1. The converged Ritz values are always returned in ascending
c algebraic order. The computed Ritz values are approximate
c eigenvalues of OP. The selection of WHICH should be made
c with this in mind when Mode = 3,4,5. After convergence,
c approximate eigenvalues of the original problem may be obtained
c with the ARPACK subroutine dseupd.
c
c 2. If the Ritz vectors corresponding to the converged Ritz values
c are needed, the user must call dseupd immediately following completion
c of dsaupd. This is new starting with version 2.1 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.
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
c 5. If IPARAM(7) = 2 then in the Reverse communication interface the user
c must do the following. When IDO = 1, Y = OP * X is to be computed.
c When IPARAM(7) = 2 OP = inv(B)*A. After computing A*X the user
c must overwrite X with A*X. Y is then the solution to the linear set
c of equations B*Y = A*X.
c
c 6. When IPARAM(1) = 0, and IDO = 3, the user needs to provide the
c NP = IPARAM(8) shifts in locations:
c 1 WORKL(IPNTR(11))
c 2 WORKL(IPNTR(11)+1)
c .
c .
c .
c NP WORKL(IPNTR(11)+NP-1).
c
c The eigenvalues of the current tridiagonal matrix are located in
c WORKL(IPNTR(6)) through WORKL(IPNTR(6)+NCV-1). They are in the
c order defined by WHICH. The associated Ritz estimates are located in
c WORKL(IPNTR(8)), WORKL(IPNTR(8)+1), ... , WORKL(IPNTR(8)+NCV-1).
c
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