program sndrv3
c
c Simple program to illustrate the idea of reverse communication
c in inverse mode for a generalized nonsymmetric eigenvalue problem.
c
c We implement example three of ex-nonsym.doc in DOCUMENTS directory
c
c\Example-3
c ... Suppose we want to solve A*x = lambda*B*x in inverse mode,
c where A and B are derived from the finite element 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 using linear elements.
c
c ... So OP = inv[M]*A and B = M.
c
c ... Use mode 2 of SNAUPD.
c
c\BeginLib
c
c\Routines called:
c snaupd ARPACK reverse communication interface routine.
c sneupd ARPACK routine that returns Ritz values and (optionally)
c Ritz vectors.
c spttrf LAPACK symmetric positive definite tridiagonal factorization
c routine.
c spttrs LAPACK symmetric positive definite tridiagonal solve routine.
c slapy2 LAPACK routine to compute sqrt(x**2+y**2) carefully.
c saxpy Level 1 BLAS that computes y <- alpha*x+y.
c snrm2 Level 1 BLAS that computes the norm of a 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 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: ndrv3.F SID: 2.5 DATE OF SID: 10/17/00 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)
logical select(maxncv)
Real
& ax(maxn), mx(maxn), d(maxncv, 3), resid(maxn),
& v(ldv,maxncv), workd(3*maxn),
& workev(3*maxncv),
& workl(3*maxncv*maxncv+6*maxncv),
& md(maxn), me(maxn-1)
c
c %---------------%
c | Local Scalars |
c %---------------%
c
character bmat*1, which*2
integer ido, n, nev, ncv, lworkl, info, ierr, j,
& nconv, maxitr, ishfts, mode
Real
& tol, sigmar, sigmai, h
logical first, rvec
c
c %------------%
c | Parameters |
c %------------%
c
Real
& zero, one
parameter (zero = 0.0E+0, one = 1.0E+0)
c
c %-----------------------------%
c | BLAS & LAPACK routines used |
c %-----------------------------%
Real
& snrm2, slapy2
external saxpy, snrm2, spttrf, spttrs, slapy2
c
c %--------------------%
c | Intrinsic function |
c %--------------------%
c
intrinsic abs
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 to be |
c | approximated. The user can modify NEV, NCV, WHICH |
c | to 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 _NDRV3: N is greater than MAXN '
go to 9000
else if ( nev .gt. maxnev ) then
print *, ' ERROR with _NDRV3: NEV is greater than MAXNEV '
go to 9000
else if ( ncv .gt. maxncv ) then
print *, ' ERROR with _NDRV3: NCV is greater than MAXNCV '
go to 9000
end if
bmat = 'G'
which = 'LM'
c
c %------------------------------------------------%
c | M is the mass matrix formed by using piecewise |
c | linear elements on [0,1]. |
c %------------------------------------------------%
c
h = one / real(n+1)
do 20 j = 1, n-1
md(j) = 4.0E+0*h
me(j) = one*h
20 continue
md(n) = 4.0E+0*h
c
call spttrf(n, md, me, ierr)
if ( ierr .ne. 0 ) then
print*, ' '
print*, ' ERROR with _pttrf. '
print*, ' '
go to 9000
end if
c
c %-----------------------------------------------------%
c | The work array WORKL is used in SNAUPD 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 SNAUPD to start the Arnoldi iteration. |
c %-----------------------------------------------------%
c
lworkl = 3*ncv**2+6*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 2 of SNAUPD is used |
c | (IPARAM(7) = 2). All these options can be |
c | changed by the user. For details, see the |
c | documentation in SNAUPD. |
c %---------------------------------------------------%
c
ishfts = 1
maxitr = 300
mode = 2
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 SNAUPD and take |
c | actions indicated by parameter IDO until |
c | either convergence is indicated or maxitr |
c | has been exceeded. |
c %---------------------------------------------%
c
call snaupd ( ido, bmat, n, which, nev, tol, resid,
& ncv, v, ldv, iparam, ipntr, workd,
& workl, lworkl, info )
c
if (ido .eq. -1 .or. ido .eq. 1) then
c
c %----------------------------------------%
c | Perform y <--- OP*x = inv[M]*A*x |
c | The user should supply his/her own |
c | matrix vector routine and a linear |
c | system solver. The matrix-vector |
c | subroutine should take workd(ipntr(1)) |
c | as input, and the final result should |
c | be returned to workd(ipntr(2)). |
c %----------------------------------------%
c
call av (n, workd(ipntr(1)), workd(ipntr(2)))
call spttrs(n, 1, md, me, workd(ipntr(2)), n,
& ierr)
if ( ierr .ne. 0 ) then
print*, ' '
print*, ' ERROR with _pttrs. '
print*, ' '
go to 9000
end if
c
c %-----------------------------------------%
c | L O O P B A C K to call SNAUPD again. |
c %-----------------------------------------%
c
go to 10
c
else if ( ido .eq. 2) then
c
c %-------------------------------------%
c | Perform y <--- M*x |
c | The matrix vector multiplication |
c | routine should take workd(ipntr(1)) |
c | as input and return the result to |
c | 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 SNAUPD again. |
c %-----------------------------------------%
c
go to 10
c
end if
c
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 SNAUPD. |
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 SNEUPD. |
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.
call sneupd ( rvec, 'A', select, d, d(1,2), v, ldv,
& sigmar, sigmai, workev, bmat, n, which, nev, tol,
& resid, ncv, v, ldv, iparam, ipntr, workd,
& workl, lworkl, ierr )
c
c %-----------------------------------------------%
c | The real part of the eigenvalue is returned |
c | in the first column of the two dimensional |
c | array D, and the IMAGINARY part is returned |
c | in the second column of D. The corresponding |
c | eigenvectors are returned in the first NEV |
c | columns of the two dimensional array V if |
c | requested. Otherwise, an orthogonal basis |
c | for the invariant subspace corresponding to |
c | the eigenvalues in D is returned in V. |
c %-----------------------------------------------%
c
if ( ierr .ne. 0 ) then
c
c %------------------------------------%
c | Error condition: |
c | Check the documentation of SNEUPD. |
c %------------------------------------%
c
print *, ' '
print *, ' Error with _neupd, info = ', ierr
print *, ' Check the documentation of _neupd'
print *, ' '
c
else
c
first = .true.
nconv = iparam(5)
do 30 j=1, iparam(5)
c
c %---------------------------%
c | Compute the residual norm |
c | |
c | || A*x - lambda*M*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
if (d(j,2) .eq. zero) then
c
c %--------------------%
c | Ritz value is real |
c %--------------------%
c
call av(n, v(1,j), ax)
call mv(n, v(1,j), mx)
call saxpy(n, -d(j,1), mx, 1, ax, 1)
d(j,3) = snrm2(n, ax, 1)
d(j,3) = d(j,3) / abs(d(j,1))
c
else if (first) then
c
c %------------------------%
c | Ritz value is complex |
c | Residual of one Ritz |
c | value of the conjugate |
c | pair is computed. |
c %------------------------%
c
call av(n, v(1,j), ax)
call mv(n, v(1,j), mx)
call saxpy(n, -d(j,1), mx, 1, ax, 1)
call mv(n, v(1,j+1), mx)
call saxpy(n, d(j,2), mx, 1, ax, 1)
d(j,3) = snrm2(n, ax, 1)**2
call av(n, v(1,j+1), ax)
call mv(n, v(1,j+1), mx)
call saxpy(n, -d(j,1), mx, 1, ax, 1)
call mv(n, v(1,j), mx)
call saxpy(n, -d(j,2), mx, 1, ax, 1)
d(j,3) = slapy2( d(j,3), snrm2(n, ax, 1) )
d(j,3) = d(j,3) / slapy2(d(j,1),d(j,2))
d(j+1,3) = d(j,3)
first = .false.
else
first = .true.
end if
c
30 continue
c
c %-----------------------------%
c | Display computed residuals. |
c %-----------------------------%
c
call smout(6, nconv, 3, d, 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 *, ' _NDRV3 '
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 sndrv3. |
c %---------------------------%
c
9000 continue
c
end
c
c==========================================================================
c
c matrix vector multiplication subroutine
c
subroutine av (n, v, w)
integer n, j
Real
& v(n), w(n), one, two, dd, dl, du, s, h, rho
parameter ( rho = 1.0E+1, one = 1.0E+0,
& two = 2.0E+0)
c
c Compute the matrix vector multiplication y<---A*x
c where A is stiffness matrix obtained from the finite element
c discretization 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 using
c linear elements.
c
h = one / real(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 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
c------------------------------------------------------------------------
subroutine mv (n, v, w)
integer n, j
Real
& v(n), w(n), one, four, h
parameter ( one = 1.0E+0, four = 4.0E+0)
c
c Compute the matrix vector multiplication y<---M*x
c where M is the mass matrix formed by using piecewise linear
c elements on [0,1].
c
w(1) = four*v(1) + one*v(2)
do 10 j = 2,n-1
w(j) = one*v(j-1) + four*v(j) + one*v(j+1)
10 continue
w(n) = one*v(n-1) + four*v(n)
c
h = one / real(n+1)
call sscal(n, h, w, 1)
return
end
