program zndrv4
c
c Simple program to illustrate the idea of reverse communication
c in shift and invert mode for a generalized complex nonsymmetric
c eigenvalue problem.
c
c We implement example four of ex-complex.doc in DOCUMENTS directory
c
c\Example-4
c ... Suppose we want to solve A*x = lambda*B*x in shift-invert mode,
c where A and B are derived from a finite element discretization
c of a 1-dimensional convection-diffusion operator
c (d^2u/dx^2) + rho*(du/dx)
c on the interval [0,1] with zero boundary condition using
c piecewise linear elements.
c
c ... where the shift sigma is a complex number.
c
c ... OP = inv[A-SIGMA*M]*M and B = M.
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 complex vector.
c av Matrix vector multiplication routine that computes A*x.
c mv Matrix vector multiplication routine that computes M*x.
c
c\Author
c Danny Sorensen
c Richard Lehoucq
c Chao Yang
c Dept. of Computational &
c Applied Mathematics
c Rice University
c Houston, Texas
c
c\SCCS Information: @(#)
c FILE: ndrv4.F SID: 2.4 DATE OF SID: 10/18/00 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), mx(maxn), d(maxncv),
& v(ldv,maxncv), workd(3*maxn), resid(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
& rho, h, s,
& sigma, s1, s2, s3
common /convct/ rho
c
Double precision
& tol
logical rvec
c
c %-----------------------------%
c | BLAS & LAPACK routines used |
c %-----------------------------%
c
Double precision
& dznrm2 , dlapy2
external dznrm2 , zaxpy , zcopy , zgttrf , zgttrs ,
& dlapy2
c
c %------------%
c | Parameters |
c %------------%
c
Complex*16
& one, zero, two, four, six
parameter (one = (1.0D+0, 0.0D+0) ,
& zero = (0.0D+0, 0.0D+0) ,
& two = (2.0D+0, 0.0D+0) ,
& four = (4.0D+0, 0.0D+0) ,
& six = (6.0D+0, 0.0D+0) )
c
c %-----------------------%
c | Executable statements |
c %-----------------------%
c
c %----------------------------------------------------%
c | The number N is the dimension of the matrix. A |
c | generalized eigenvalue problem is solved (BMAT = |
c | 'G'). NEV is the number of eigenvalues (closest |
c | to SIGMAR) to be approximated. Since the |
c | shift-invert mode is used, WHICH is set to 'LM'. |
c | The user can modify NEV, NCV, SIGMA to solve |
c | problems of different sizes, and to get different |
c | parts of the spectrum. However, The following |
c | 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 _NDRV4: N is greater than MAXN '
go to 9000
else if ( nev .gt. maxnev ) then
print *, ' ERROR with _NDRV4: NEV is greater than MAXNEV '
go to 9000
else if ( ncv .gt. maxncv ) then
print *, ' ERROR with _NDRV4: NCV is greater than MAXNCV '
go to 9000
end if
bmat = 'G'
which = 'LM'
sigma = one
c
c %--------------------------------------------------%
c | Construct C = A - SIGMA*M in COMPLEX arithmetic. |
c | Factor C in COMPLEX arithmetic (using LAPACK |
c | subroutine zgttrf ). The matrix A is chosen to be |
c | the tridiagonal matrix derived from the standard |
c | central difference discretization of the 1-d |
c | convection-diffusion operator u``+ rho*u` on the |
c | interval [0, 1] with zero Dirichlet boundary |
c | condition. The matrix M is chosen to be the |
c | symmetric tridiagonal matrix with 4.0 on the |
c | diagonal and 1.0 on the off-diagonals. |
c %--------------------------------------------------%
c
rho = (1.0D+1, 0.0D+0)
h = one / dcmplx (n+1)
s = rho / two
c
s1 = -one/h - s - sigma*h/six
s2 = two/h - four*sigma*h/six
s3 = -one/h + s - sigma*h/six
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 _NDRV4.'
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) then
c
c %-------------------------------------------%
c | Perform y <--- OP*x = inv[A-SIGMA*M]*M*x |
c | to force starting vector into the range |
c | of OP. The user should supply his/her |
c | own matrix vector multiplication routine |
c | and a linear system solver. The matrix |
c | vector multiplication routine should take |
c | workd(ipntr(1)) as the input. The final |
c | result should be returned to |
c | workd(ipntr(2)). |
c %-------------------------------------------%
c
call mv (n, workd(ipntr(1)), workd(ipntr(2)))
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 _NDRV4.'
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
else if ( ido .eq. 1) then
c
c %-----------------------------------------%
c | Perform y <-- OP*x = inv[A-sigma*M]*M*x |
c | M*x has been saved in workd(ipntr(3)). |
c | The user only need the linear system |
c | solver here that takes workd(ipntr(3)) |
c | as input, and returns the result to |
c | workd(ipntr(2)). |
c %-----------------------------------------%
c
call zcopy ( n, workd(ipntr(3)), 1, workd(ipntr(2)), 1)
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 _NDRV4.'
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
else if ( ido .eq. 2) then
c
c %---------------------------------------------%
c | Perform y <--- M*x |
c | Need matrix vector multiplication routine |
c | here that takes workd(ipntr(1)) as input |
c | and returns the result to workd(ipntr(2)). |
c %---------------------------------------------%
c
call mv (n, 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 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 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 80 j=1, nconv
c
call av(n, v(1,j), ax)
call mv(n, v(1,j), mx)
call zaxpy (n, -d(j), mx, 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))
80 continue
c
c %-----------------------------%
c | Display computed residuals. |
c %-----------------------------%
c
call dmout (6, nconv, 3, rd, maxncv, -6,
& 'Ritz values (Real, Imag) and direct 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 *, '_NDRV4 '
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
9000 continue
c
end
c
c==========================================================================
c
c matrix vector multiplication subroutine
c
subroutine mv (n, v, w)
integer n, j
Complex*16
& v(n), w(n), one, four, six, h
parameter (one = (1.0D+0, 0.0D+0) ,
& four = (4.0D+0, 0.0D+0) ,
& six = (6.0D+0, 0.0D+0) )
c
c Compute the matrix vector multiplication y<---M*x
c where M is a n by n symmetric tridiagonal matrix with 4 on the
c diagonal, 1 on the subdiagonal and superdiagonal.
c
w(1) = ( four*v(1) + one*v(2) ) / six
do 40 j = 2,n-1
w(j) = ( one*v(j-1) + four*v(j) + one*v(j+1) ) / six
40 continue
w(n) = ( one*v(n-1) + four*v(n) ) / six
c
h = one / dcmplx (n+1)
call zscal (n, h, w, 1)
return
end
c------------------------------------------------------------------
subroutine av (n, v, w)
integer n, j
Complex*16
& v(n), w(n), one, two, dd, dl, du, s, h, rho
parameter (one = (1.0D+0, 0.0D+0) ,
& two = (2.0D+0, 0.0D+0) )
common /convct/ rho
c
h = one / dcmplx (n+1)
s = rho / two
dd = two / h
dl = -one/h - s
du = -one/h + s
c
w(1) = dd*v(1) + du*v(2)
do 40 j = 2,n-1
w(j) = dl*v(j-1) + dd*v(j) + du*v(j+1)
40 continue
w(n) = dl*v(n-1) + dd*v(n)
return
end
