program dndrv3 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 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 dpttrf LAPACK symmetric positive definite tridiagonal factorization c routine. c dpttrs LAPACK symmetric positive definite 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 dnrm2 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) 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), & 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 Double precision & tol, sigmar, sigmai, h logical first, rvec c c %------------% c | Parameters | c %------------% c Double precision & zero, one parameter (zero = 0.0D+0, one = 1.0D+0) c c %-----------------------------% c | BLAS & LAPACK routines used | c %-----------------------------% Double precision & dnrm2, dlapy2 external daxpy, dnrm2, dpttrf, dpttrs, 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 | 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 / dble(n+1) do 20 j = 1, n-1 md(j) = 4.0D+0*h me(j) = one*h 20 continue md(n) = 4.0D+0*h c call dpttrf(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 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 = 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 DNAUPD is used | c | (IPARAM(7) = 2). 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 = 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 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[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 dpttrs(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 DNAUPD 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 DNAUPD 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 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 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 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)**2 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 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 *, ' _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 dndrv3. | 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), one, two, dd, dl, du, s, h, rho parameter ( rho = 1.0D+1, one = 1.0D+0, & two = 2.0D+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 / dble(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 Double precision & v(n), w(n), one, four, h parameter ( one = 1.0D+0, four = 4.0D+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 / dble(n+1) call dscal(n, h, w, 1) return end