      program zndrv2
c
c     Simple program to illustrate the idea of reverse communication
c     in shift-invert mode for a standard complex nonsymmetric eigenvalue 
c     problem.
c
c     We implement example two of ex-complex.doc in DOCUMENTS directory
c
c\Example-2
c     ... Suppose we want to solve A*x = lambda*x in shift-invert mode,
c         where A is derived from the central difference discretization
c         of the 1-dimensional convection-diffusion operator
c                   (d^2u/dx^2) + rho*(du/dx)
c         on the interval [0,1] with zero Dirichlet boundary condition.
c     ... The shift sigma is a complex number.
c
c     ... OP = inv[A-sigma*I] and  B = I.
c
c     ... Use mode 3 of ZNAUPD.
c
c\BeginLib
c
c\Routines called:
c     znaupd  ARPACK reverse communication interface routine.
c     zneupd  ARPACK routine that returns Ritz values and (optionally)
c             Ritz vectors.
c     zgttrf  LAPACK tridiagonal factorization routine.
c     zgttrs  LAPACK tridiagonal solve routine.
c     dlapy2  LAPACK routine to compute sqrt(x**2+y**2) carefully.
c     zaxpy   Level 1 BLAS that computes y <- alpha*x+y.
c     zcopy   Level 1 BLAS that copies one vector to another.
c     dznrm2  Level 1 BLAS that computes the norm of a vector.
c     av      Matrix vector multiplication routine that computes A*x.
c
c\Author
c     Richard Lehoucq
c     Danny Sorensen
c     Chao Yang
c     Dept. of Computational &
c     Applied Mathematics
c     Rice University
c     Houston, Texas
c
c\SCCS Information: @(#)
c FILE: ndrv2.F   SID: 2.2   DATE OF SID: 4/22/96   RELEASE: 2
c
c\Remarks
c     1. None
c
c\EndLib
c--------------------------------------------------------------------------
c
c     %-----------------------------%
c     | Define leading dimensions   |
c     | for all arrays.             |
c     | MAXN:   Maximum dimension   |
c     |         of the A allowed.   |
c     | MAXNEV: Maximum NEV allowed |
c     | MAXNCV: Maximum NCV allowed |
c     %-----------------------------%
c
      integer           maxn, maxnev, maxncv, ldv
      parameter         (maxn=256, maxnev=10, maxncv=25, 
     &                   ldv=maxn )
c
c     %--------------%
c     | Local Arrays |
c     %--------------%
c
      integer           iparam(11), ipntr(14), ipiv(maxn)
      logical           select(maxncv)
      Complex*16
     &                  ax(maxn), d(maxncv), resid(maxn),
     &                  v(ldv, maxncv), workd(3*maxn),
     &                  workev(2*maxncv),
     &                  workl(3*maxncv*maxncv+5*maxncv),
     &                  dd(maxn), dl(maxn), du(maxn),
     &                  du2(maxn)
      Double precision
     &                  rwork(maxn), rd(maxncv,3)
c
c     %---------------%
c     | Local Scalars |
c     %---------------%
c
      character         bmat*1, which*2
      integer           ido, n, nev, ncv, lworkl, info, j, ierr,
     &                  nconv, maxitr, ishfts, mode
      Complex*16 
     &                  h, h2, s, sigma, s1, s2, s3, rho
      common            /convct/ rho
c
      Double precision
     &                  tol
      logical           rvec
c
c     %------------%
c     | Parameters |
c     %------------%
c
      Complex*16 
     &                   one, zero, two        
      parameter         (one = (1.0D+0, 0.0D+0), 
     &                   zero = (0.0D+0, 0.0D+0),
     &                   two = (2.0D+0, 0.0D+0))
c
c     %-----------------------------%
c     | BLAS & LAPACK routines used |
c     %-----------------------------%
c
      Double precision
     &                  dznrm2, dlapy2
      external          zgttrf, zgttrs, zaxpy, zcopy, dznrm2, dlapy2
c
c     %-----------------------%
c     | Executable statements |
c     %-----------------------%
c
c     %--------------------------------------------------%
c     | The number N is the dimension of the matrix.  A  |
c     | standard eigenvalue problem is solved (BMAT =    |
c     | 'I').  NEV is the number of eigenvalues (closest |
c     | to the shift SIGMA) to be approximated.  Since   |
c     | the shift-invert mode is used, WHICH is set to   |
c     | 'LM'.  The user can modify NEV, NCV, SIGMA to    |
c     | solve problems of different sizes, and to get    |
c     | different parts of the spectrum.  However, The   |
c     | following conditions must be satisfied:          |
c     |                 N <= MAXN,                       |
c     |               NEV <= MAXNEV,                     |
c     |           NEV + 2 <= NCV <= MAXNCV               |
c     %--------------------------------------------------%

c
      n     = 100
      nev   = 4
      ncv   = 20
      if ( n .gt. maxn ) then
         print *, ' ERROR with _NDRV2: N is greater than MAXN '
         go to 9000
      else if ( nev .gt. maxnev ) then
         print *, ' ERROR with _NDRV2: NEV is greater than MAXNEV '
         go to 9000
      else if ( ncv .gt. maxncv ) then
         print *, ' ERROR with _NDRV2: NCV is greater than MAXNCV '
         go to 9000
      end if
      bmat  = 'I'
      which = 'LM'
      sigma = zero
c
c     %----------------------------------------------------%
c     | Construct C = A - SIGMA*I, factor C in complex     |
c     | arithmetic (using LAPACK subroutine zgttrf). The   |
c     | matrix A is chosen to be the tridiagonal matrix    |
c     | derived from standard central difference of the    |
c     | 1-d convection diffusion operator - u" + rho*u' on |
c     | the interval [0, 1] with zero Dirichlet boundary   |
c     | condition.                                         |
c     %----------------------------------------------------%
c
      rho = (1.0D+1, 0.0D+0)
      h = one / dcmplx(n+1)
      h2 = h*h
      s = rho / two
c
      s1 = -one/h2 - s/h 
      s2 = two/h2 - sigma
      s3 = -one/h2 + s/h  
c
      do 10 j = 1, n-1
         dl(j) = s1 
         dd(j) = s2
         du(j) = s3
  10  continue 
      dd(n) = s2 
c 
      call zgttrf(n, dl, dd, du, du2, ipiv, ierr)
      if ( ierr .ne. 0 ) then
         print*, ' '
         print*, ' ERROR with _gttrf in _NDRV2.'
         print*, ' '
         go to 9000
      end if
c
c     %-----------------------------------------------------%
c     | The work array WORKL is used in ZNAUPD as           |
c     | workspace.  Its dimension LWORKL is set as          |
c     | illustrated below.  The parameter TOL determines    |
c     | the stopping criterion. If TOL<=0, machine          |
c     | precision is used.  The variable IDO is used for    |
c     | reverse communication, and is initially set to 0.   |
c     | Setting INFO=0 indicates that a random vector is    |
c     | generated in ZNAUPD to start the Arnoldi iteration. |
c     %-----------------------------------------------------%
c
      lworkl = 3*ncv**2+5*ncv 
      tol    = 0.0 
      ido    = 0
      info   = 0
c
c     %---------------------------------------------------%
c     | This program uses exact shifts with respect to    |
c     | the current Hessenberg matrix (IPARAM(1) = 1).    |
c     | IPARAM(3) specifies the maximum number of Arnoldi |
c     | iterations allowed. Mode 3 of ZNAUPD is used      |
c     | (IPARAM(7) = 3).  All these options can be        |
c     | changed by the user. For details see the          |
c     | documentation in ZNAUPD.                          |
c     %---------------------------------------------------%
c
      ishfts = 1
      maxitr = 300
      mode   = 3
c
      iparam(1) = ishfts 
      iparam(3) = maxitr 
      iparam(7) = mode 
c
c     %-------------------------------------------%
c     | M A I N   L O O P (Reverse communication) | 
c     %-------------------------------------------%
c
 20   continue
c
c        %---------------------------------------------%
c        | Repeatedly call the routine ZNAUPD and take | 
c        | actions indicated by parameter IDO until    |
c        | either convergence is indicated or maxitr   |
c        | has been exceeded.                          |
c        %---------------------------------------------%
c
         call znaupd ( ido, bmat, n, which, nev, tol, resid, ncv,
     &        v, ldv, iparam, ipntr, workd, workl, lworkl,
     &        rwork,info )
c
         if (ido .eq. -1 .or. ido .eq. 1 ) then
c
c           %-------------------------------------------%
c           | Perform  y <--- OP*x = inv[A-SIGMA*I]*x   |
c           | The user should supply his/her own linear |
c           | system solver here that takes             |
c           | workd(ipntr(1)) as the input, and returns |
c           | the result to workd(ipntr(2)).            |
c           %-------------------------------------------%
c
            call zcopy( n, workd(ipntr(1)),1, workd(ipntr(2)), 1)
c
            call zgttrs('N', n, 1, dl, dd, du, du2, ipiv, 
     &                  workd(ipntr(2)), n, ierr) 
            if ( ierr .ne. 0 ) then
               print*, ' '
               print*, ' ERROR with _gttrs in _NDRV2.'
               print*, ' '
               go to 9000
            end if
c
c           %-----------------------------------------%
c           | L O O P   B A C K to call ZNAUPD again. |
c           %-----------------------------------------%
c
            go to 20
c
         end if
c
c     %-----------------------------------------%
c     | Either we have convergence, or there is |
c     | an error.                               |
c     %-----------------------------------------%
c
      if ( info .lt. 0 ) then
c
c        %--------------------------%
c        | Error message, check the |
c        | documentation in ZNAUPD  |
c        %--------------------------%
c
         print *, ' '
         print *, ' Error with _naupd, info = ',info
         print *, ' Check the documentation in _naupd.'
         print *, ' ' 
c
      else 
c
c        %-------------------------------------------%
c        | No fatal errors occurred.                 |
c        | Post-Process using ZNEUPD.                |
c        |                                           |
c        | Computed eigenvalues may be extracted.    |  
c        |                                           |
c        | Eigenvectors may also be computed now if  |
c        | desired.  (indicated by rvec = .true.)    | 
c        %-------------------------------------------%
c 
         rvec = .true.
c
         call zneupd (rvec, 'A', select, d, v, ldv, sigma, 
     &                workev, bmat, n, which, nev, tol,
     &                resid, ncv, v, ldv, iparam, ipntr, workd,
     &                workl, lworkl, rwork, ierr)
c
c        %----------------------------------------------%
c        | Eigenvalues are returned in the one          |
c        | dimensional array D.  The corresponding      |
c        | eigenvectors are returned in the first NCONV |
c        | (=IPARAM(5)) columns of the two dimensional  |
c        | array V if requested.  Otherwise, an         |
c        | orthogonal basis for the invariant subspace  |
c        | corresponding to the eigenvalues in D is     |
c        | returned in V.                               |
c        %----------------------------------------------%
c
         if ( ierr .ne. 0) then
c 
c           %------------------------------------%
c           | Error condition:                   | 
c           | Check the documentation of ZNEUPD. |
c           %------------------------------------%
c 
            print *, ' '
            print *, ' Error with _neupd, info = ', ierr
            print *, ' Check the documentation of _neupd. '
            print *, ' '
         else
c
            nconv = iparam(5) 
            do 60 j=1, nconv
c
c              %---------------------------%
c              | Compute the residual norm |
c              |                           |
c              |   ||  A*x - lambda*x ||   |
c              |                           |
c              | for the NCONV accurately  |
c              | computed eigenvalues and  |
c              | eigenvectors.  (iparam(5) |
c              | indicates how many are    |
c              | accurate to the requested |
c              | tolerance)                |
c              %---------------------------%
c
               call av(n, v(1,j), ax)
               call zaxpy(n, -d(j), v(1,j), 1, ax, 1)
               rd(j,1) = dble(d(j))
               rd(j,2) = dimag(d(j))
               rd(j,3) = dznrm2(n, ax, 1)
               rd(j,3) = rd(j,3) / dlapy2(rd(j,1),rd(j,2))
 60         continue
c
c           %-----------------------------%
c           | Display computed residuals. |
c           %-----------------------------%
c
            call dmout(6, nconv, 3, rd, maxncv, -6,
     &           'Ritz values (Real, Imag) and relative residuals')
c
         end if
c
c        %-------------------------------------------%
c        | Print additional convergence information. |
c        %-------------------------------------------%
c
         if ( info .eq. 1) then
             print *, ' '
             print *, ' Maximum number of iterations reached.'
             print *, ' '
         else if ( info .eq. 3) then
             print *, ' ' 
             print *, ' No shifts could be applied during implicit
     &                  Arnoldi update, try increasing NCV.'
             print *, ' '
         end if      
c
         print *, ' '
         print *, '_NDRV2 '
         print *, '====== '
         print *, ' '
         print *, ' Size of the matrix is ', n
         print *, ' The number of Ritz values requested is ', nev
         print *, ' The number of Arnoldi vectors generated',
     &            ' (NCV) is ', ncv
         print *, ' What portion of the spectrum: ', which
         print *, ' The number of converged Ritz values is ', 
     &              nconv 
         print *, ' The number of Implicit Arnoldi update',
     &            ' iterations taken is ', iparam(3)
         print *, ' The number of OP*x is ', iparam(9)
         print *, ' The convergence criterion is ', tol
         print *, ' '
c
      end if
c
c     %---------------------------%
c     | Done with program zndrv2. |
c     %---------------------------%
c
 9000 continue
c
      end
c
c-------------------------------------------------------------------
c
c     matrix vector multiplication subroutine
c
      subroutine av (n, v, w)
      integer           n, j
      Complex*16 
     &                   v(n), w(n), rho, two, one, dd, dl, du, s, h,
     &                   h2
      parameter         (one = (1.0D+0, 0.0D+0), two = (2.0D+0, 0.0D+0))
      common            /convct/ rho
c
      h = one / dcmplx(n+1)
      h2 = h*h
      s = rho / two
      dd = two / h2
      dl = -one/h2 - s/h
      du = -one/h2 + s/h
c
      w(1) =  dd*v(1) + du*v(2)
      do 10 j = 2,n-1
         w(j) = dl*v(j-1) + dd*v(j) + du*v(j+1)
 10   continue
      w(n) =  dl*v(n-1) + dd*v(n)
      return
      end
