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|
program dndrv2
c
c Simple program to illustrate the idea of reverse communication
c in shift-invert mode for a standard nonsymmetric eigenvalue problem.
c
c We implement example two of ex-nonsym.doc in DOCUMENTS directory
c
c\Example-2
c ... Suppose we want to solve A*x = lambda*x in shift-invert mode,
c where A is derived from the centered difference 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
c ... The shift sigma is a real number.
c
c ... OP = inv[A-sigma*I] and B = I.
c
c ... Use mode 3 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 dgttrf LAPACK tridiagonal factorization routine.
c dgttrs LAPACK tridiagonal 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 dcopy Level 1 BLAS that copies one vector to another.
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 multiplication routine that computes A*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: ndrv2.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), 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, j,
& ierr, nconv, maxitr, ishfts, mode
Double precision
& tol, h, s,
& sigmar, sigmai, s1, s2, s3
logical first, rvec
c
c %------------%
c | Parameters |
c %------------%
c
Double precision
& one, zero, two, rho
common /convct/ rho
parameter (one = 1.0D+0, zero = 0.0D+0,
& two = 2.0D+0)
c
c %-----------------------------%
c | BLAS & LAPACK routines used |
c %-----------------------------%
c
Double precision
& ddot, dnrm2, dlapy2
external dgttrf, dgttrs, ddot, dnrm2, dlapy2
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 | standard eigenvalue problem is solved (BMAT = |
c | 'I'). NEV is the number of eigenvalues (closest |
c | to the shift SIGMAR) to be approximated. Since |
c | the shift-invert mode is used, WHICH is set to |
c | 'LM'. The user can modify NEV, NCV, SIGMAR 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 _NDRV2: N is greater than MAXN '
go to 9000
else if ( nev .gt. maxnev ) then
print *, ' ERROR with _NDRV2: NEV is greater than MAXNEV '
go to 9000
else if ( ncv .gt. maxncv ) then
print *, ' ERROR with _NDRV2: NCV is greater than MAXNCV '
go to 9000
end if
bmat = 'I'
which = 'LM'
sigmar = 1.0D+0
sigmai = 0.0D+0
c
c %----------------------------------------------------%
c | Construct C = A - SIGMA*I in real arithmetic, and |
c | factor C in real arithmetic using LAPACK |
c | subroutine dgttrf. The matrix A is chosen to be |
c | the tridiagonal matrix derived from standard |
c | central difference of the 1-d convection diffusion |
c | operator u" + rho*u' on the interval [0, 1] with |
c | zero Dirichlet boundary condition. |
c %----------------------------------------------------%
c
rho = 1.0D+1
h = one / dble(n+1)
s = rho*h / two
c
s1 = -one-s
s2 = two - sigmar
s3 = -one+s
c
do 10 j = 1, n-1
dl(j) = s1
dd(j) = s2
du(j) = s3
10 continue
dd(n) = s2
c
call dgttrf(n, dl, dd, du, du2, ipiv, ierr)
if ( ierr .ne. 0 ) then
print*, ' '
print*, ' ERROR with _gttrf in _NDRV2.'
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 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 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
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 .or. ido .eq. 1) then
c
c %-------------------------------------------%
c | Perform y <--- OP*x = inv[A-SIGMA*I]*x |
c | The user should supply his/her own linear |
c | system solver here that takes |
c | workd(ipntr(1)) as the input, and returns |
c | the result to workd(ipntr(2)). |
c %-------------------------------------------%
c
call dcopy( n, workd(ipntr(1)), 1, workd(ipntr(2)), 1)
c
call dgttrs('N', n, 1, dl, dd, du, du2, ipiv,
& workd(ipntr(2)), n, ierr)
if ( ierr .ne. 0 ) then
print*, ' '
print*, ' ERROR with _gttrs in _NDRV2.'
print*, ' '
go to 9000
end if
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
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 in _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.
c
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 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 daxpy(n, -d(j,1), v(1,j), 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 daxpy(n, -d(j,1), v(1,j), 1, ax, 1)
call daxpy(n, d(j,2), v(1,j+1), 1, ax, 1)
d(j,3) = dnrm2(n, ax, 1)
call av(n, v(1,j+1), ax)
call daxpy(n, -d(j,2), v(1,j), 1, ax, 1)
call daxpy(n, -d(j,1), v(1,j+1), 1, ax, 1)
d(j,3) = dlapy2( d(j,3), dnrm2(n, ax, 1) )
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 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 *, ' _NDRV2 '
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 dndrv2. |
c %---------------------------%
c
9000 continue
c
end
c
c-------------------------------------------------------------------
c
c matrix vector multiplication subroutine
c
subroutine av (n, v, w)
integer n, j
Double precision
& v(n), w(n), rho, two, one, dd, dl, du, s, h
common /convct/ rho
parameter (one = 1.0D+0, two = 2.0D+0 )
c
c Compute the matrix vector multiplication y<---A*x
c where A is a n by n nonsymmetric tridiagonal matrix derived from
c the central difference discretization of the 1-dimensional
c convection diffusion operator on the interval [0,1] with
c zero Dirichlet boundary condition.
c
c
h = one / dble(n+1)
s = rho *h / two
dd = two
dl = -one - s
du = -one + 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
|
