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# DNAUPD

```c\BeginDoc
c
c\Name: dnaupd
c
c\Description:
c  Reverse communication interface for the Implicitly Restarted Arnoldi
c  iteration. This subroutine computes approximations to a few eigenpairs
c  of a linear operator "OP" with respect to a semi-inner product defined by
c  a symmetric positive semi-definite real matrix B. B may be the identity
c  matrix. NOTE: If the linear operator "OP" is real and symmetric
c  with respect to the real positive semi-definite symmetric matrix B,
c  i.e. B*OP = (OP')*B, then subroutine ssaupd should be used instead.
c
c  The computed approximate eigenvalues are called Ritz values and
c  the corresponding approximate eigenvectors are called Ritz vectors.
c
c  dnaupd 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 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:  A*x = lambda*M*x, M symmetric semi-definite
c           ===> OP = Real_Part{ inv[A - sigma*M]*M }  and  B = M.
c           ===> shift-and-invert mode (in real arithmetic)
c           If OP*x = amu*x, then
c           amu = 1/2 * [ 1/(lambda-sigma) + 1/(lambda-conjg(sigma)) ].
c           Note: If sigma is real, i.e. imaginary part of sigma is zero;
c                 Real_Part{ inv[A - sigma*M]*M } == inv[A - sigma*M]*M
c                 amu == 1/(lambda-sigma).
c
c  Mode 4:  A*x = lambda*M*x, M symmetric semi-definite
c           ===> OP = Imaginary_Part{ inv[A - sigma*M]*M }  and  B = M.
c           ===> shift-and-invert mode (in real arithmetic)
c           If OP*x = amu*x, then
c           amu = 1/2i * [ 1/(lambda-sigma) - 1/(lambda-conjg(sigma)) ].
c
c  Both mode 3 and 4 give the same enhancement to eigenvalues close to
c  the (complex) shift sigma.  However, as lambda goes to infinity,
c  the operator OP in mode 4 dampens the eigenvalues more strongly than
c  does OP defined in mode 3.
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 dnaupd
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 dnaupd.  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          dnaupd 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 * 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) real and imaginary parts
c                    of the shifts where INPTR(14) is the pointer
c                    into WORKL for placing the shifts. See Remark
c                    5 below.
c          IDO =  4: compute Z = OP * X
c          IDO = 99: done
c          -------------------------------------------------------------
c          After the initialization phase, when the routine is used in
c          the "shift-and-invert" mode, the vector B * 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*B*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     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          where ABS(RITZ(I)) is the magnitude when RITZ(I) is complex.
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. NCV must satisfy the two
c          inequalities 2 <= 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.
c          NOTE: 2 <= NCV-NEV in order that complex conjugate pairs of Ritz
c          values are kept together. (See remark 4 below)
c
c  V       Double precision 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 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 real and imaginary
c                      parts of the NCV eigenvalues of the Hessenberg
c                      matrix H are returned in the part of the WORKL
c                      array corresponding to RITZR and RITZI. See remark
c                      5 below.
c          ISHIFT = 1: exact shifts with respect to the current
c                      Hessenberg matrix H.  This is equivalent to
c                      restarting the iteration with a starting vector
c                      that is a linear combination of approximate Schur
c                      vectors associated with the "wanted" Ritz values.
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 1,2,3,4; See under \Description of dnaupd 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), dnaupd returns NP, the number
c          of shifts the user is to provide. 0 < NP <=NCV-NEV. See Remark
c          5 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 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 matrix
c                    H in WORKL.
c          IPNTR(6): pointer to the real part of the ritz value array
c                    RITZR in WORKL.
c          IPNTR(7): pointer to the imaginary part of the ritz value array
c                    RITZI in WORKL.
c          IPNTR(8): pointer to the Ritz estimates in array WORKL associated
c                    with the Ritz values located in RITZR and RITZI in WORKL.
c
c          Note: IPNTR(9:13) is only referenced by dneupd. See Remark 2 below.
c
c          IPNTR(9): pointer to the real part of the NCV RITZ values of the
c                    original system.
c          IPNTR(10): pointer to the imaginary part of the NCV RITZ values of
c                     the original system.
c          IPNTR(11): pointer to the NCV corresponding error bounds.
c          IPNTR(12): pointer to the NCV by NCV upper quasi-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                     dneupd if RVEC = .TRUE. See Remark 2 below.
c          Note: IPNTR(9:13) is only referenced by dneupd. See Remark 2 below.
c          IPNTR(14): pointer to the NP shifts in WORKL. See Remark 5 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 an invariant subspace
c          associated with the converged Ritz values is desired, see remark
c          2 below, subroutine dneupd 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 3*NCV**2 + 6*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-NEV >= 2 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 1,2,3,4.
c          = -11: IPARAM(7) = 1 and BMAT = 'G' are incompatable.
c          = -12: IPARAM(1) must be equal to 0 or 1.
c          = -9999: Could not build an Arnoldi factorization.
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
c     Mode = 3 and 4.  After convergence, approximate eigenvalues of the
c     original problem may be obtained with the ARPACK subroutine dneupd.
c
c  2. If a basis for the invariant subspace corresponding to the converged Ritz
c     values is needed, the user must call dneupd immediately following
c     completion of dnaupd. 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 of NCV
c     relative to NEV.  The only formal requirement is that NCV > NEV + 2.
c     However, it is recommended that NCV .ge. 2*NEV+1.  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) real and imaginary parts of the shifts in locations
c         real part                  imaginary part
c         -----------------------    --------------
c     1   WORKL(IPNTR(14))           WORKL(IPNTR(14)+NP)
c     2   WORKL(IPNTR(14)+1)         WORKL(IPNTR(14)+NP+1)
c                        .                          .
c                        .                          .
c                        .                          .
c     NP  WORKL(IPNTR(14)+NP-1)      WORKL(IPNTR(14)+2*NP-1).
c
c     Only complex conjugate pairs of shifts may be applied and the pairs
c     must be placed in consecutive locations. The real part of the
c     eigenvalues of the current upper Hessenberg matrix are located in
c     WORKL(IPNTR(6)) through WORKL(IPNTR(6)+NCV-1) and the imaginary part
c     in WORKL(IPNTR(7)) through WORKL(IPNTR(7)+NCV-1). They are ordered
c     according to the order defined by WHICH. The complex conjugate
c     pairs are kept together and the associated Ritz estimates are located in
c     WORKL(IPNTR(8)), WORKL(IPNTR(8)+1), ... , WORKL(IPNTR(8)+NCV-1).
c
c-----------------------------------------------------------------------
```

Chao Yang
11/7/1997