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|
program zndrv1
c
c Example program to illustrate the idea of reverse communication
c for a standard complex nonsymmetric eigenvalue problem.
c
c We implement example one of ex-complex.doc in DOCUMENTS directory
c
c\Example-1
c ... Suppose we want to solve A*x = lambda*x in regular mode,
c where A is obtained from the standard central difference
c discretization of the convection-diffusion operator
c (Laplacian u) + rho*(du / dx)
c on the unit squre [0,1]x[0,1] with zero Dirichlet boundary
c condition.
c
c ... OP = A and B = I.
c
c ... Assume "call av (nx,x,y)" computes y = A*x
c
c ... Use mode 1 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 dlapy2 LAPACK routine to compute sqrt(x**2+y**2) carefully.
c dznrm2 Level 1 BLAS that computes the norm of a complex vector.
c zaxpy Level 1 BLAS that computes y <- alpha*x+y.
c av Matrix vector multiplication routine that computes A*x.
c tv Matrix vector multiplication routine that computes T*x,
c where T is a tridiagonal matrix. It is used in routine
c av.
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: ndrv1.F SID: 2.4 DATE OF SID: 10/17/00 RELEASE: 2
c
c\Remarks
c 1. None
c
c\EndLib
c---------------------------------------------------------------------------
c
c %-----------------------------%
c | Define maximum 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=12, maxncv=30, ldv=maxn)
c
c %--------------%
c | Local Arrays |
c %--------------%
c
integer iparam(11), ipntr(14)
logical select(maxncv)
Complex*16
& ax(maxn), d(maxncv),
& v(ldv,maxncv), workd(3*maxn),
& workev(3*maxncv), resid(maxn),
& workl(3*maxncv*maxncv+5*maxncv)
Double precision
& rwork(maxncv), rd(maxncv,3)
c
c %---------------%
c | Local Scalars |
c %---------------%
c
character bmat*1, which*2
integer ido, n, nx, nev, ncv, lworkl, info, j,
& ierr, nconv, maxitr, ishfts, mode
Complex*16
& sigma
Double precision
& tol
logical rvec
c
c %-----------------------------%
c | BLAS & LAPACK routines used |
c %-----------------------------%
c
Double precision
& dznrm2 , dlapy2
external dznrm2 , zaxpy , dlapy2
c
c %-----------------------%
c | Executable Statements |
c %-----------------------%
c
c %--------------------------------------------------%
c | The number NX is the number of interior points |
c | in the discretization of the 2-dimensional |
c | convection-diffusion operator on the unit |
c | square with zero Dirichlet boundary condition. |
c | The number N(=NX*NX) is the dimension of the |
c | matrix. A standard eigenvalue problem is |
c | solved (BMAT = 'I'). NEV is the number of |
c | eigenvalues to be approximated. The user can |
c | modify NX, NEV, NCV, WHICH to solve problems of |
c | different sizes, and to get different parts of |
c | the spectrum. However, The following |
c | conditions must be satisfied: |
c | N <= MAXN |
c | NEV <= MAXNEV |
c | NEV + 2 <= NCV <= MAXNCV |
c %--------------------------------------------------%
c
nx = 10
n = nx*nx
nev = 4
ncv = 20
if ( n .gt. maxn ) then
print *, ' ERROR with _NDRV1: N is greater than MAXN '
go to 9000
else if ( nev .gt. maxnev ) then
print *, ' ERROR with _NDRV1: NEV is greater than MAXNEV '
go to 9000
else if ( ncv .gt. maxncv ) then
print *, ' ERROR with _NDRV1: NCV is greater than MAXNCV '
go to 9000
end if
bmat = 'I'
which = 'LM'
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 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 shift with respect to |
c | the current Hessenberg matrix (IPARAM(1) = 1). |
c | IPARAM(3) specifies the maximum number of Arnoldi |
c | iterations allowed. Mode 1 of ZNAUPD is used |
c | (IPARAM(7) = 1). All these options can be changed |
c | by the user. For details see the documentation in |
c | ZNAUPD . |
c %---------------------------------------------------%
c
ishfts = 1
maxitr = 300
mode = 1
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
10 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 matrix vector multiplication |
c | y <--- OP*x |
c | The user should supply his/her own |
c | matrix vector multiplication routine here |
c | that takes workd(ipntr(1)) as the input |
c | vector, and return the matrix vector |
c | product to workd(ipntr(2)). |
c %-------------------------------------------%
c
call av (nx, workd(ipntr(1)), workd(ipntr(2)))
c
c %-----------------------------------------%
c | L O O P B A C K to call ZNAUPD again. |
c %-----------------------------------------%
c
go to 10
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 of _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 *, ' '
c
else
c
nconv = iparam(5)
do 20 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(nx, 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))
20 continue
c
c %-----------------------------%
c | Display computed residuals. |
c %-----------------------------%
c
call dmout (6, nconv, 3, rd, maxncv, -6,
& 'Ritz values (Real, Imag) and relative residuals')
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 *, '_NDRV1'
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 zndrv1 . |
c %---------------------------%
c
9000 continue
c
end
c
c==========================================================================
c
c matrix vector subroutine
c
c The matrix used is the convection-diffusion operator
c discretized using centered difference.
c
subroutine av (nx, v, w)
integer nx, j, lo
Complex*16
& v(nx*nx), w(nx*nx), one, h2
parameter (one = (1.0D+0, 0.0D+0) )
external zaxpy , tv
c
c Computes w <--- OP*v, where OP is the nx*nx by nx*nx block
c tridiagonal matrix
c
c | T -I |
c |-I T -I |
c OP = | -I T |
c | ... -I|
c | -I T|
c
c derived from the standard central difference discretization
c of the convection-diffusion operator (Laplacian u) + rho*(du/dx)
c with zero boundary condition.
c
c The subroutine TV is called to computed y<---T*x.
c
c
h2 = one / dcmplx ((nx+1)*(nx+1))
c
call tv(nx,v(1),w(1))
call zaxpy (nx, -one/h2, v(nx+1), 1, w(1), 1)
c
do 10 j = 2, nx-1
lo = (j-1)*nx
call tv(nx, v(lo+1), w(lo+1))
call zaxpy (nx, -one/h2, v(lo-nx+1), 1, w(lo+1), 1)
call zaxpy (nx, -one/h2, v(lo+nx+1), 1, w(lo+1), 1)
10 continue
c
lo = (nx-1)*nx
call tv(nx, v(lo+1), w(lo+1))
call zaxpy (nx, -one/h2, v(lo-nx+1), 1, w(lo+1), 1)
c
return
end
c=========================================================================
subroutine tv (nx, x, y)
c
integer nx, j
Complex*16
& x(nx), y(nx), h, h2, dd, dl, du
c
Complex*16
& one, rho
parameter (one = (1.0D+0, 0.0D+0) ,
& rho = (1.0D+2, 0.0D+0) )
c
c Compute the matrix vector multiplication y<---T*x
c where T is a nx by nx tridiagonal matrix with DD on the
c diagonal, DL on the subdiagonal, and DU on the superdiagonal
c
h = one / dcmplx (nx+1)
h2 = h*h
dd = (4.0D+0, 0.0D+0) / h2
dl = -one/h2 - (5.0D-1, 0.0D+0) *rho/h
du = -one/h2 + (5.0D-1, 0.0D+0) *rho/h
c
y(1) = dd*x(1) + du*x(2)
do 10 j = 2,nx-1
y(j) = dl*x(j-1) + dd*x(j) + du*x(j+1)
10 continue
y(nx) = dl*x(nx-1) + dd*x(nx)
return
end
|
