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|
program dndrv6
c
c Simple program to illustrate the idea of reverse communication
c in shift-invert mode for a generalized nonsymmetric eigenvalue problem.
c
c We implement example six of ex-nonsym.doc in DOCUMENTS directory
c
c\Example-6
c
c ... Suppose we want to solve A*x = lambda*B*x in shift-invert mode
c The matrix A is the tridiagonal matrix with 2 on the diagonal,
c -2 on the subdiagonal and 3 on the superdiagonal. The matrix M
c is the tridiagonal matrix with 4 on the diagonal and 1 on the
c off-diagonals.
c ... The shift sigma is a complex number (sigmar, sigmai).
c ... OP = Imaginary_Part{inv[A-(SIGMAR,SIGMAI)*M]*M and B = M.
c ... Use mode 4 of DNAUPD.
c
c\BeginLib
c
c\Routines called:
c dnaupd ARPACK reverse communication interface routine.
c dneupd ARPACK routine that returns Ritz values and (optionally)
c Ritz vectors.
c zgttrf LAPACK complex matrix factorization routine.
c zgttrs LAPACK complex linear system solve routine.
c dlapy2 LAPACK routine to compute sqrt(x**2+y**2) carefully.
c daxpy Level 1 BLAS that computes y <- alpha*x+y.
c ddot Level 1 BLAS that computes the dot product of two vectors.
c dnrm2 Level 1 BLAS that computes the norm of a vector.
c av Matrix vector subroutine that computes A*x.
c mv Matrix vector subroutine 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: ndrv6.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)
Double precision
& ax(maxn), mx(maxn), d(maxncv,3), resid(maxn),
& v(ldv,maxncv), workd(3*maxn),
& workev(3*maxncv),
& workl(3*maxncv*maxncv+6*maxncv)
Complex*16
& cdd(maxn), cdl(maxn), cdu(maxn),
& cdu2(maxn), ctemp(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
Double precision
& tol, numr, numi, denr, deni, sigmar, sigmai
Complex*16
& c1, c2, c3
logical first, rvec
c
c %-----------------------------%
c | BLAS & LAPACK routines used |
c %-----------------------------%
c
external zgttrf, zgttrs
Double precision
& ddot, dnrm2, dlapy2
external ddot, dnrm2, dlapy2
c
c %------------%
c | Parameters |
c %------------%
c
Double precision
& zero
parameter (zero = 0.0D+0)
c
c %--------------------%
c | Intrinsic Function |
c %--------------------%
c
intrinsic dimag, dcmplx, 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 (closest |
c | to the shift (SIGMAR,SIGMAI)) to be approximated. |
c | Since the shift-invert mode is used, WHICH is set |
c | to 'LM'. The user can modify NEV, NCV, SIGMA to |
c | 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 _NDRV6: N is greater than MAXN '
go to 9000
else if ( nev .gt. maxnev ) then
print *, ' ERROR with _NDRV6: NEV is greater than MAXNEV '
go to 9000
else if ( ncv .gt. maxncv ) then
print *, ' ERROR with _NDRV6: NCV is greater than MAXNCV '
go to 9000
end if
bmat = 'G'
which = 'LM'
sigmar = 4.0D-1
sigmai = 6.0D-1
c
c %----------------------------------------------------%
c | Construct C = A - (SIGMAR,SIGMAI)*M in complex |
c | arithmetic, and factor C in complex arithmetic |
c | (using LAPACK subroutine zgttrf). The matrix A is |
c | chosen to be the tridiagonal matrix with -2 on the |
c | subdiagonal, 2 on the diagonal and 3 on the |
c | superdiagonal. The matrix M is chosen to be the |
c | symmetric tridiagonal matrix with 4 on the |
c | diagonal and 1 on the off-diagonals. |
c %----------------------------------------------------%
c
c1 = dcmplx(-2.0D+0-sigmar, -sigmai)
c2 = dcmplx( 2.0D+0-4.0D+0*sigmar, -4.0D+0*sigmai)
c3 = dcmplx( 3.0D+0-sigmar, -sigmai)
c
do 10 j = 1, n-1
cdl(j) = c1
cdd(j) = c2
cdu(j) = c3
10 continue
cdd(n) = c2
c
call zgttrf(n, cdl, cdd, cdu, cdu2, ipiv, ierr)
if ( ierr .ne. 0 ) then
print*, ' '
print*, ' ERROR with _gttrf in _NDRV6.'
print*, ' '
go to 9000
end if
c
c %-----------------------------------------------------%
c | The work array WORKL is used in DNAUPD 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 DNAUPD 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 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 3 of DNAUPD is used |
c | (IPARAM(7) = 3). All these options can be |
c | changed by the user. For details, see the |
c | documentation in DNAUPD. |
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 DNAUPD and take |
c | actions indicated by parameter IDO until |
c | either convergence is indicated or maxitr |
c | has been exceeded. |
c %---------------------------------------------%
c
call dnaupd ( 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 |
c | y <--- OP*x = Imaginary_Part{inv[A-(SIGMAR,SIGMAI)*M]*M*x} |
c | to force starting vector into the range of OP. The user |
c | should supply his/her own matrix vector multiplication |
c | routine and a complex linear system solver. The matrix |
c | vector multiplication routine should take workd(ipntr(1)) |
c | as the input. The final result (a real vector) should be |
c | returned to workd(ipntr(2)). |
c %------------------------------------------------------------%
c
call mv (n, workd(ipntr(1)), workd(ipntr(2)))
do 30 j = 1, n
ctemp(j) = dcmplx(workd(ipntr(2)+j-1))
30 continue
c
call zgttrs('N', n, 1, cdl, cdd, cdu, cdu2, ipiv,
& ctemp, maxn, ierr)
if ( ierr .ne. 0 ) then
print*, ' '
print*, ' ERROR with _gttrs in _NDRV6.'
print*, ' '
go to 9000
end if
do 40 j = 1, n
workd(ipntr(2)+j-1) = dimag(ctemp(j))
40 continue
c
c %-----------------------------------------%
c | L O O P B A C K to call DNAUPD again. |
c %-----------------------------------------%
c
go to 20
c
else if ( ido .eq. 1) then
c
c %------------------------------------------------------------%
c | Perform |
c | y <--- OP*x = Imaginary_Part{inv[A-(SIGMAR,SIGMAI)*M]*M*x} |
c | M*x has been saved in workd(ipntr(3)). The user only need |
c | the complex linear system solver here that takes |
c | complex[workd(ipntr(3))] as input, and returns the result |
c | to workd(ipntr(2)). |
c %------------------------------------------------------------%
c
do 50 j = 1,n
ctemp(j) = dcmplx(workd(ipntr(3)+j-1))
50 continue
call zgttrs ('N', n, 1, cdl, cdd, cdu, cdu2, ipiv,
& ctemp, maxn, ierr)
if ( ierr .ne. 0 ) then
print*, ' '
print*, ' ERROR with _gttrs in _NDRV6.'
print*, ' '
go to 9000
end if
do 60 j = 1, n
workd(ipntr(2)+j-1) = dimag(ctemp(j))
60 continue
c
c %-----------------------------------------%
c | L O O P B A C K to call DNAUPD 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 DNAUPD again. |
c %-----------------------------------------%
c
go to 20
c
end if
c
c %-----------------------------------------%
c | Either we have convergence, or there is |
c | an error. |
c %-----------------------------------------%
c
c
if ( info .lt. 0 ) then
c
c %--------------------------%
c | Error message, check the |
c | documentation in DNAUPD. |
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 DNEUPD. |
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 dneupd ( 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 DNEUPD. |
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 70 j=1, nconv
c
c %-------------------------------------%
c | Use Rayleigh Quotient to recover |
c | eigenvalues of the original problem.|
c %-------------------------------------%
c
if ( d(j,2) .eq. zero) then
c
c %----------------------------%
c | Eigenvalue is real. |
c | Compute d = x'(Ax)/x'(Mx). |
c %----------------------------%
c
call av(n, v(1,j), ax )
numr = ddot(n, v(1,j), 1, ax, 1)
call mv(n, v(1,j), ax )
denr = ddot(n, v(1,j), 1, ax, 1)
d(j,1) = numr / denr
c
else if (first) then
c
c %------------------------%
c | Eigenvalue is complex. |
c | Compute the first one |
c | of the conjugate pair. |
c %------------------------%
c
c %----------------%
c | Compute x'(Ax) |
c %----------------%
c
call av(n, v(1,j), ax )
numr = ddot(n, v(1,j), 1, ax, 1)
numi = ddot(n, v(1,j+1), 1, ax, 1)
call av(n, v(1,j+1), ax)
numr = numr + ddot(n,v(1,j+1),1,ax,1)
numi = -numi + ddot(n,v(1,j),1,ax,1)
c
c %----------------%
c | Compute x'(Mx) |
c %----------------%
c
call mv(n, v(1,j), ax )
denr = ddot(n, v(1,j), 1, ax, 1)
deni = ddot(n, v(1,j+1), 1, ax, 1)
call mv(n, v(1,j+1), ax)
denr = denr + ddot(n,v(1,j+1),1,ax,1)
deni = -deni + ddot(n,v(1,j),1, ax,1)
c
c %----------------%
c | d=x'(Ax)/x'(Mx)|
c %----------------%
c
d(j,1) = (numr*denr+numi*deni) /
& dlapy2(denr, deni)
d(j,2) = (numi*denr-numr*deni) /
& dlapy2(denr, deni)
first = .false.
c
else
c
c %------------------------------%
c | Get the second eigenvalue of |
c | the conjugate pair by taking |
c | the conjugate of the last |
c | eigenvalue computed. |
c %------------------------------%
c
d(j,1) = d(j-1,1)
d(j,2) = -d(j-1,2)
first = .true.
c
end if
c
70 continue
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
first = .true.
nconv = iparam(5)
do 80 j=1, nconv
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 daxpy(n, -d(j,1), mx, 1, ax, 1)
d(j,3) = dnrm2(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 daxpy(n, -d(j,1), mx, 1, ax, 1)
call mv(n, v(1,j+1), mx)
call daxpy(n, d(j,2), mx, 1, ax, 1)
d(j,3) = dnrm2(n, ax, 1)
call av(n, v(1,j+1), ax)
call mv(n, v(1,j+1), mx)
call daxpy(n, -d(j,1), mx, 1, ax, 1)
call mv(n, v(1,j), mx)
call daxpy(n, -d(j,2), mx, 1, ax, 1)
d(j,3) = dlapy2( d(j,3), dnrm2(n, ax, 1) )
d(j,3) = d(j,3) / dlapy2(d(j,1),d(j,2))
d(j+1,3) = d(j,3)
first = .false.
else
first = .true.
end if
c
80 continue
c
c %-----------------------------%
c | Display computed residuals. |
c %-----------------------------%
c
call dmout(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 *, ' _NDRV6 '
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 dndrv6. |
c %---------------------------%
c
9000 continue
c
end
c
c==========================================================================
c
c matrix vector multiplication subroutine
c
subroutine mv (n, v, w)
integer n, j
Double precision
& v(n), w(n), one, four
parameter (one = 1.0D+0, four = 4.0D+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)
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)
return
end
c------------------------------------------------------------------
subroutine av (n, v, w)
integer n, j
Double precision
& v(n), w(n), three, two
parameter (three = 3.0D+0, two = 2.0D+0)
c
c Compute the matrix vector multiplication y<---A*x
c where M is a n by n symmetric tridiagonal matrix with 2 on the
c diagonal, -2 on the subdiagonal and 3 on the superdiagonal.
c
w(1) = two*v(1) + three*v(2)
do 10 j = 2,n-1
w(j) = -two*v(j-1) + two*v(j) + three*v(j+1)
10 continue
w(n) = -two*v(n-1) + two*v(n)
return
end
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