545 lines
19 KiB
FortranFixed
545 lines
19 KiB
FortranFixed
program sndrv4
|
|
c
|
|
c Simple program to illustrate the idea of reverse communication
|
|
c in shift-invert mode for a generalized nonsymmetric eigenvalue
|
|
c problem.
|
|
c
|
|
c We implement example four of ex-nonsym.doc in DOCUMENTS directory
|
|
c
|
|
c\Example-4
|
|
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 ... The shift sigma is a real number.
|
|
c
|
|
c ... OP = inv[A-SIGMA*M]*M and B = M.
|
|
c
|
|
c ... Use mode 3 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 sgttrf LAPACK tridiagonal factorization routine.
|
|
c sgttrs LAPACK tridiagonal linear system 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 scopy Level 1 BLAS that copies one vector to another.
|
|
c sdot Level 1 BLAS that computes the dot product of two vectors.
|
|
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: ndrv4.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), ipiv(maxn)
|
|
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),
|
|
& dd(maxn), dl(maxn), du(maxn),
|
|
& du2(maxn)
|
|
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, h, s,
|
|
& sigmar, sigmai, s1, s2, s3
|
|
logical first, rvec
|
|
c
|
|
c %-----------------------------%
|
|
c | BLAS & LAPACK routines used |
|
|
c %-----------------------------%
|
|
c
|
|
Real
|
|
& sdot, snrm2, slapy2
|
|
external sdot, snrm2, slapy2, sgttrf, sgttrs
|
|
c
|
|
c %--------------------%
|
|
c | Intrinsic function |
|
|
c %--------------------%
|
|
c
|
|
intrinsic abs
|
|
c
|
|
c %------------%
|
|
c | Parameters |
|
|
c %------------%
|
|
c
|
|
Real
|
|
& one, zero, two, six, rho
|
|
common /convct/ rho
|
|
parameter (one = 1.0E+0, zero = 0.0E+0,
|
|
& two = 2.0E+0, six = 6.0E+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, SIGMAR 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 = 10
|
|
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'
|
|
sigmar = one
|
|
sigmai = zero
|
|
c
|
|
c %--------------------------------------------------%
|
|
c | Construct C = A - SIGMA*M in real arithmetic, |
|
|
c | and factor C in real arithmetic (using LAPACK |
|
|
c | subroutine sgttrf). 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 the mass matrix |
|
|
c | formed by using piecewise linear elements on |
|
|
c | [0,1]. |
|
|
c %--------------------------------------------------%
|
|
c
|
|
rho = 1.0E+1
|
|
h = one / real(n+1)
|
|
s = rho / two
|
|
c
|
|
s1 = -one/h - s - sigmar*h/six
|
|
s2 = two/h - 4.0E+0*sigmar*h/six
|
|
s3 = -one/h + s - sigmar*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 sgttrf(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 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 = zero
|
|
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 SNAUPD is used |
|
|
c | (IPARAM(7) = 3). 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 = 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 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) 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 sgttrs('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 SNAUPD 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 scopy( n, workd(ipntr(3)), 1, workd(ipntr(2)), 1)
|
|
call sgttrs ('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 SNAUPD 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 SNAUPD again. |
|
|
c %-----------------------------------------%
|
|
c
|
|
go to 20
|
|
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 in _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, 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
|
|
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)
|
|
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 *, ' _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
|
|
c %---------------------------%
|
|
c | Done with program sndrv4. |
|
|
c %---------------------------%
|
|
c
|
|
9000 continue
|
|
c
|
|
end
|
|
c
|
|
c==========================================================================
|
|
c
|
|
c matrix vector multiplication subroutine
|
|
c
|
|
subroutine mv (n, v, w)
|
|
integer n, j
|
|
Real
|
|
& v(n), w(n), one, four, six, h
|
|
parameter (one = 1.0E+0, four = 4.0E+0, six = 6.0E+0)
|
|
c
|
|
c Compute the matrix vector multiplication y<---M*x
|
|
c where M is mass matrix formed by using piecewise linear elements
|
|
c on [0,1].
|
|
c
|
|
w(1) = ( four*v(1) + one*v(2) ) / six
|
|
do 10 j = 2,n-1
|
|
w(j) = ( one*v(j-1) + four*v(j) + one*v(j+1) ) / six
|
|
10 continue
|
|
w(n) = ( one*v(n-1) + four*v(n) ) / six
|
|
c
|
|
h = one / real(n+1)
|
|
call sscal(n, h, w, 1)
|
|
return
|
|
end
|
|
c------------------------------------------------------------------
|
|
subroutine av (n, v, w)
|
|
integer n, j
|
|
Real
|
|
& v(n), w(n), one, two, dd, dl, du, s, h, rho
|
|
common /convct/ rho
|
|
parameter (one = 1.0E+0, two = 2.0E+0)
|
|
c
|
|
c Compute the matrix vector multiplication y<---A*x
|
|
c where A is obtained from the finite element discretization of the
|
|
c 1-dimensional convection diffusion operator
|
|
c d^u/dx^2 + rho*(du/dx)
|
|
c on the interval [0,1] with zero Dirichlet boundary condition
|
|
c using linear elements.
|
|
c This routine is only used in residual calculation.
|
|
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
|
|
|