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