476 lines
16 KiB
FortranFixed
476 lines
16 KiB
FortranFixed
program dndrv1
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c
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c
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c Example program to illustrate the idea of reverse communication
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c for a standard nonsymmetric eigenvalue problem.
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c
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c We implement example one of ex-nonsym.doc in DOCUMENTS directory
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c
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c\Example-1
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c ... Suppose we want to solve A*x = lambda*x in regular mode,
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c where A is obtained from the standard central difference
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c discretization of the convection-diffusion operator
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c (Laplacian u) + rho*(du / dx)
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c on the unit square [0,1]x[0,1] with zero Dirichlet boundary
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c condition.
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c
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c ... OP = A and B = I.
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c
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c ... Assume "call av (nx,x,y)" computes y = A*x.c
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c
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c ... Use mode 1 of DNAUPD.
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c
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c\BeginLib
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c
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c\Routines called:
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c dnaupd ARPACK reverse communication interface routine.
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c dneupd ARPACK routine that returns Ritz values and (optionally)
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c Ritz vectors.
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c dlapy2 LAPACK routine to compute sqrt(x**2+y**2) carefully.
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c daxpy Level 1 BLAS that computes y <- alpha*x+y.
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c dnrm2 Level 1 BLAS that computes the norm of a vector.
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c av Matrix vector multiplication routine that computes A*x.
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c tv Matrix vector multiplication routine that computes T*x,
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c where T is a tridiagonal matrix. It is used in routine
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c av.
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c
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c\Author
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c Richard Lehoucq
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c Danny Sorensen
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c Chao Yang
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c Dept. of Computational &
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c Applied Mathematics
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c Rice University
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c Houston, Texas
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c
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c\SCCS Information: @(#)
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c FILE: ndrv1.F SID: 2.5 DATE OF SID: 10/17/00 RELEASE: 2
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c
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c\Remarks
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c 1. None
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c
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c\EndLib
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c---------------------------------------------------------------------------
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c
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c %-----------------------------%
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c | Define maximum dimensions |
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c | for all arrays. |
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c | MAXN: Maximum dimension |
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c | of the A allowed. |
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c | MAXNEV: Maximum NEV allowed |
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c | MAXNCV: Maximum NCV allowed |
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c %-----------------------------%
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c
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integer maxn, maxnev, maxncv, ldv
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parameter (maxn=256, maxnev=12, maxncv=30, ldv=maxn)
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c
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c %--------------%
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c | Local Arrays |
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c %--------------%
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c
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integer iparam(11), ipntr(14)
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logical select(maxncv)
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Double precision
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& ax(maxn), d(maxncv,3), resid(maxn),
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& v(ldv,maxncv), workd(3*maxn),
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& workev(3*maxncv),
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& workl(3*maxncv*maxncv+6*maxncv)
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c
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c %---------------%
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c | Local Scalars |
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c %---------------%
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c
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character bmat*1, which*2
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integer ido, n, nx, nev, ncv, lworkl, info, j,
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& ierr, nconv, maxitr, ishfts, mode
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Double precision
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& tol, sigmar, sigmai
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logical first, rvec
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c
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c %------------%
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c | Parameters |
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c %------------%
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c
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Double precision
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& zero
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parameter (zero = 0.0D+0)
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c
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c %-----------------------------%
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c | BLAS & LAPACK routines used |
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c %-----------------------------%
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c
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Double precision
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& dlapy2, dnrm2
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external dlapy2, dnrm2, daxpy
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c
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c %--------------------%
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c | Intrinsic function |
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c %--------------------%
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c
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intrinsic abs
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c
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c %-----------------------%
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c | Executable Statements |
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c %-----------------------%
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c
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c %--------------------------------------------------%
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c | The number NX is the number of interior points |
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c | in the discretization of the 2-dimensional |
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c | convection-diffusion operator on the unit |
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c | square with zero Dirichlet boundary condition. |
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c | The number N(=NX*NX) is the dimension of the |
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c | matrix. A standard eigenvalue problem is |
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c | solved (BMAT = 'I'). NEV is the number of |
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c | eigenvalues to be approximated. The user can |
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c | modify NX, NEV, NCV, WHICH to solve problems of |
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c | different sizes, and to get different parts of |
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c | the spectrum. However, The following |
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c | conditions must be satisfied: |
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c | N <= MAXN |
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c | NEV <= MAXNEV |
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c | NEV + 2 <= NCV <= MAXNCV |
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c %--------------------------------------------------%
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c
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nx = 10
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n = nx*nx
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nev = 4
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ncv = 20
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if ( n .gt. maxn ) then
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print *, ' ERROR with _NDRV1: N is greater than MAXN '
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go to 9000
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else if ( nev .gt. maxnev ) then
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print *, ' ERROR with _NDRV1: NEV is greater than MAXNEV '
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go to 9000
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else if ( ncv .gt. maxncv ) then
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print *, ' ERROR with _NDRV1: NCV is greater than MAXNCV '
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go to 9000
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end if
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bmat = 'I'
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which = 'SM'
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c
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c %-----------------------------------------------------%
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c | The work array WORKL is used in DNAUPD as |
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c | workspace. Its dimension LWORKL is set as |
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c | illustrated below. The parameter TOL determines |
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c | the stopping criterion. If TOL<=0, machine |
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c | precision is used. The variable IDO is used for |
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c | reverse communication, and is initially set to 0. |
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c | Setting INFO=0 indicates that a random vector is |
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c | generated in DNAUPD to start the Arnoldi iteration. |
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c %-----------------------------------------------------%
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c
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lworkl = 3*ncv**2+6*ncv
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tol = zero
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ido = 0
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info = 0
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c
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c %---------------------------------------------------%
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c | This program uses exact shifts with respect to |
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c | the current Hessenberg matrix (IPARAM(1) = 1). |
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c | IPARAM(3) specifies the maximum number of Arnoldi |
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c | iterations allowed. Mode 1 of DNAUPD is used |
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c | (IPARAM(7) = 1). All these options can be changed |
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c | by the user. For details see the documentation in |
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c | DNAUPD. |
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c %---------------------------------------------------%
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c
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ishfts = 1
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maxitr = 300
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mode = 1
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c
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iparam(1) = ishfts
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iparam(3) = maxitr
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iparam(7) = mode
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c
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c %-------------------------------------------%
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c | M A I N L O O P (Reverse communication) |
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c %-------------------------------------------%
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c
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10 continue
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c
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c %---------------------------------------------%
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c | Repeatedly call the routine DNAUPD and take |
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c | actions indicated by parameter IDO until |
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c | either convergence is indicated or maxitr |
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c | has been exceeded. |
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c %---------------------------------------------%
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c
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call dnaupd ( ido, bmat, n, which, nev, tol, resid,
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& ncv, v, ldv, iparam, ipntr, workd, workl, lworkl,
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& info )
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c
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if (ido .eq. -1 .or. ido .eq. 1) then
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c
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c %-------------------------------------------%
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c | Perform matrix vector multiplication |
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c | y <--- OP*x |
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c | The user should supply his/her own |
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c | matrix vector multiplication routine here |
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c | that takes workd(ipntr(1)) as the input |
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c | vector, and return the matrix vector |
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c | product to workd(ipntr(2)). |
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c %-------------------------------------------%
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c
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call av (nx, workd(ipntr(1)), workd(ipntr(2)))
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c
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c %-----------------------------------------%
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c | L O O P B A C K to call DNAUPD again. |
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c %-----------------------------------------%
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c
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go to 10
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c
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end if
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c
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c %----------------------------------------%
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c | Either we have convergence or there is |
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c | an error. |
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c %----------------------------------------%
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c
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if ( info .lt. 0 ) then
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c
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c %--------------------------%
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c | Error message, check the |
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c | documentation in DNAUPD. |
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c %--------------------------%
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c
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print *, ' '
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print *, ' Error with _naupd, info = ', info
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print *, ' Check the documentation of _naupd'
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print *, ' '
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c
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else
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c
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c %-------------------------------------------%
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c | No fatal errors occurred. |
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c | Post-Process using DNEUPD. |
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c | |
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c | Computed eigenvalues may be extracted. |
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c | |
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c | Eigenvectors may also be computed now if |
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c | desired. (indicated by rvec = .true.) |
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c %-------------------------------------------%
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c
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rvec = .true.
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c
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call dneupd ( rvec, 'A', select, d, d(1,2), v, ldv,
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& sigmar, sigmai, workev, bmat, n, which, nev, tol,
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& resid, ncv, v, ldv, iparam, ipntr, workd, workl,
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& lworkl, ierr )
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c
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c %-----------------------------------------------%
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c | The real part of the eigenvalue is returned |
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c | in the first column of the two dimensional |
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c | array D, and the imaginary part is returned |
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c | in the second column of D. The corresponding |
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c | eigenvectors are returned in the first NEV |
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c | columns of the two dimensional array V if |
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c | requested. Otherwise, an orthogonal basis |
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c | for the invariant subspace corresponding to |
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c | the eigenvalues in D is returned in V. |
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c %-----------------------------------------------%
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c
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if ( ierr .ne. 0) then
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c
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c %------------------------------------%
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c | Error condition: |
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c | Check the documentation of DNEUPD. |
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c %------------------------------------%
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c
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print *, ' '
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print *, ' Error with _neupd, info = ', ierr
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print *, ' Check the documentation of _neupd. '
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print *, ' '
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c
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else
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c
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first = .true.
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nconv = iparam(5)
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do 20 j=1, nconv
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c
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c %---------------------------%
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c | Compute the residual norm |
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c | |
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c | || A*x - lambda*x || |
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c | |
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c | for the NCONV accurately |
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c | computed eigenvalues and |
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c | eigenvectors. (iparam(5) |
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c | indicates how many are |
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c | accurate to the requested |
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c | tolerance) |
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c %---------------------------%
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c
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if (d(j,2) .eq. zero) then
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c
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c %--------------------%
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c | Ritz value is real |
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c %--------------------%
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c
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call av(nx, v(1,j), ax)
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call daxpy(n, -d(j,1), v(1,j), 1, ax, 1)
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d(j,3) = dnrm2(n, ax, 1)
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d(j,3) = d(j,3) / abs(d(j,1))
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c
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else if (first) then
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c
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c %------------------------%
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c | Ritz value is complex. |
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c | Residual of one Ritz |
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c | value of the conjugate |
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c | pair is computed. |
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c %------------------------%
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c
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call av(nx, v(1,j), ax)
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call daxpy(n, -d(j,1), v(1,j), 1, ax, 1)
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call daxpy(n, d(j,2), v(1,j+1), 1, ax, 1)
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d(j,3) = dnrm2(n, ax, 1)
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call av(nx, v(1,j+1), ax)
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call daxpy(n, -d(j,2), v(1,j), 1, ax, 1)
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call daxpy(n, -d(j,1), v(1,j+1), 1, ax, 1)
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d(j,3) = dlapy2( d(j,3), dnrm2(n, ax, 1) )
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d(j,3) = d(j,3) / dlapy2(d(j,1),d(j,2))
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d(j+1,3) = d(j,3)
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first = .false.
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else
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first = .true.
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end if
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c
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20 continue
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c
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c %-----------------------------%
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c | Display computed residuals. |
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c %-----------------------------%
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c
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call dmout(6, nconv, 3, d, maxncv, -6,
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& 'Ritz values (Real,Imag) and relative residuals')
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end if
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c
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c %-------------------------------------------%
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c | Print additional convergence information. |
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c %-------------------------------------------%
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c
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if ( info .eq. 1) then
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print *, ' '
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print *, ' Maximum number of iterations reached.'
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print *, ' '
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else if ( info .eq. 3) then
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print *, ' '
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print *, ' No shifts could be applied during implicit',
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& ' Arnoldi update, try increasing NCV.'
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print *, ' '
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end if
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c
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print *, ' '
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print *, ' _NDRV1 '
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print *, ' ====== '
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print *, ' '
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print *, ' Size of the matrix is ', n
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print *, ' The number of Ritz values requested is ', nev
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print *, ' The number of Arnoldi vectors generated',
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& ' (NCV) is ', ncv
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print *, ' What portion of the spectrum: ', which
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print *, ' The number of converged Ritz values is ',
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& nconv
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print *, ' The number of Implicit Arnoldi update',
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& ' iterations taken is ', iparam(3)
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print *, ' The number of OP*x is ', iparam(9)
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print *, ' The convergence criterion is ', tol
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print *, ' '
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c
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end if
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c
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c %---------------------------%
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c | Done with program dndrv1. |
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c %---------------------------%
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c
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9000 continue
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c
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end
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c
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c==========================================================================
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c
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c matrix vector subroutine
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c
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c The matrix used is the 2 dimensional convection-diffusion
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c operator discretized using central difference.
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c
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subroutine av (nx, v, w)
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integer nx, j, lo
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Double precision
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& v(nx*nx), w(nx*nx), one, h2
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parameter (one = 1.0D+0)
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external daxpy
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c
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c Computes w <--- OP*v, where OP is the nx*nx by nx*nx block
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c tridiagonal matrix
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c
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c | T -I |
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c |-I T -I |
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c OP = | -I T |
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c | ... -I|
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c | -I T|
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c
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c derived from the standard central difference discretization
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c of the 2 dimensional convection-diffusion operator
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c (Laplacian u) + rho*(du/dx) on a unit square with zero boundary
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c condition.
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c
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c When rho*h/2 <= 1, the discrete convection-diffusion operator
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c has real eigenvalues. When rho*h/2 > 1, it has COMPLEX
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c eigenvalues.
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c
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c The subroutine TV is called to compute y<---T*x.
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c
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c
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h2 = one / dble((nx+1)*(nx+1))
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c
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call tv(nx,v(1),w(1))
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call daxpy(nx, -one/h2, v(nx+1), 1, w(1), 1)
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c
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do 10 j = 2, nx-1
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lo = (j-1)*nx
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call tv(nx, v(lo+1), w(lo+1))
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call daxpy(nx, -one/h2, v(lo-nx+1), 1, w(lo+1), 1)
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call daxpy(nx, -one/h2, v(lo+nx+1), 1, w(lo+1), 1)
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10 continue
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c
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lo = (nx-1)*nx
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call tv(nx, v(lo+1), w(lo+1))
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call daxpy(nx, -one/h2, v(lo-nx+1), 1, w(lo+1), 1)
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c
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return
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end
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c=========================================================================
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subroutine tv (nx, x, y)
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c
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integer nx, j
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Double precision
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& x(nx), y(nx), h, h2, dd, dl, du
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c
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Double precision
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& one, zero, rho
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parameter (one = 1.0D+0, zero = 0.0D+0,
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& rho = 0.0D+0)
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c
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c Compute the matrix vector multiplication y<---T*x
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c where T is a nx by nx tridiagonal matrix with DD on the
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c diagonal, DL on the subdiagonal, and DU on the superdiagonal.
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c
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c When rho*h/2 <= 1, the discrete convection-diffusion operator
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c has real eigenvalues. When rho*h/2 > 1, it has COMPLEX
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c eigenvalues.
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c
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h = one / dble(nx+1)
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h2 = h*h
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dd = 4.0D+0 / h2
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dl = -one / h2 - 5.0D-1*rho / h
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du = -one / h2 + 5.0D-1*rho / h
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c
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y(1) = dd*x(1) + du*x(2)
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do 10 j = 2,nx-1
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y(j) = dl*x(j-1) + dd*x(j) + du*x(j+1)
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10 continue
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y(nx) = dl*x(nx-1) + dd*x(nx)
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return
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end
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