program dsdrv6 c c Program to illustrate the idea of reverse communication c in Cayley mode for a generalized symmetric eigenvalue c problem. The following program uses the two LAPACK subroutines c dgttrf.f and dgttrs.f to factor and solve a tridiagonal system of c equations. c c We implement example six of ex-sym.doc in DOCUMENTS directory c c\Example-6 c ... Suppose we want to solve A*x = lambda*M*x in inverse mode, c where A and M are obtained by the finite element of the c 1-dimensional discrete Laplacian c d^2u / dx^2 c on the interval [0,1] with zero Dirichlet boundary condition c using piecewise linear elements. c c ... OP = (inv[A-sigma*M])*(A+sigma*M) and B = M. c c ... Use mode 5 of DSAUPD. c c\BeginLib c c\References: c 1. R.G. Grimes, J.G. Lewis and H.D. Simon, "A Shifted Block Lanczos c Algorithm for Solving Sparse Symmetric Generalized Eigenproblems", c SIAM J. Matr. Anal. Apps., January (1993). c c\Routines called: c dsaupd ARPACK reverse communication interface routine. c dseupd ARPACK routine that returns Ritz values and (optionally) c Ritz vectors. c dgttrf LAPACK tridiagonal factorization routine. c dgttrs LAPACK tridiagonal solve routine. c daxpy Level 1 BLAS that computes y <- alpha*x+y. c dcopy Level 1 BLAS that copies one vector to another. c dscal Level 1 BLAS that scales a vector by a scalar. 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 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: sdrv6.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 Double precision & v(ldv,maxncv), workl(maxncv*(maxncv+8)), & workd(3*maxn), d(maxncv,2), resid(maxn), & ad(maxn), adl(maxn), adu(maxn), adu2(maxn), & temp(maxn), ax(maxn), mx(maxn) logical select(maxncv) integer iparam(11), ipntr(11), ipiv(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 logical rvec Double precision & sigma, r1, r2, tol, h c c %------------% c | Parameters | c %------------% c Double precision & zero, one, two, four, six parameter (zero = 0.0D+0, one = 1.0D+0, & four = 4.0D+0, six = 6.0D+0, & two = 2.0D+0 ) c c %-----------------------------% c | BLAS & LAPACK routines used | c %-----------------------------% c Double precision & dnrm2 external daxpy, dcopy, dscal, dnrm2, dgttrf, dgttrs 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. Since the Cayley mode is used, | c | WHICH is set to 'LM'. The user can modify NEV, | c | NCV, SIGMA to solve problems of different sizes, | c | and to get different parts of the spectrum. | c | However, The following conditions must be | c | satisfied: | c | N <= MAXN, | c | NEV <= MAXNEV, | c | NEV + 1 <= NCV <= MAXNCV | c %--------------------------------------------------% c n = 100 nev = 4 ncv = 20 if ( n .gt. maxn ) then print *, ' ERROR with _SDRV6: N is greater than MAXN ' go to 9000 else if ( nev .gt. maxnev ) then print *, ' ERROR with _SDRV6: NEV is greater than MAXNEV ' go to 9000 else if ( ncv .gt. maxncv ) then print *, ' ERROR with _SDRV6: NCV is greater than MAXNCV ' go to 9000 end if bmat = 'G' which = 'LM' sigma = 1.5D+2 c c %--------------------------------------------------% c | The work array WORKL is used in DSAUPD 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 DSAUPD to start the Arnoldi | c | iteration. | c %--------------------------------------------------% c lworkl = ncv*(ncv+8) 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 5 specified in the | c | documentation of DSAUPD is used (IPARAM(7) = 5). | c | All these options may be changed by the user. For | c | details, see the documentation in DSAUPD. | c %---------------------------------------------------% c ishfts = 1 maxitr = 300 mode = 5 c iparam(1) = ishfts iparam(3) = maxitr iparam(7) = mode c c %------------------------------------------------------% c | Call LAPACK routine to factor (A-sigma*M). The | c | stiffness matrix A is the 1-d discrete Laplacian. | c | The mass matrix M is the associated mass matrix | c | arising from using piecewise linear finite elements | c | on the interval [0, 1]. | c %------------------------------------------------------% c h = one / dble(n+1) r1 = (four / six) * h r2 = (one / six) * h do 20 j=1,n ad(j) = two / h - sigma * r1 adl(j) = -one / h - sigma * r2 20 continue call dcopy (n, adl, 1, adu, 1) call dgttrf (n, adl, ad, adu, adu2, ipiv, ierr) if (ierr .ne. 0) then print *, ' ' print *, ' Error with _gttrf in _SDRV6.' print *, ' ' go to 9000 end if c c %-------------------------------------------% c | M A I N L O O P (Reverse communication) | c %-------------------------------------------% c 10 continue c c %---------------------------------------------% c | Repeatedly call the routine DSAUPD and take | c | actions indicated by parameter IDO until | c | either convergence is indicated or maxitr | c | has been exceeded. | c %---------------------------------------------% c call dsaupd ( 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])*(A+SIGMA*M)*x | c | to force starting vector into the range of OP. The | c | user should provide his/her matrix vector (A*x, M*x) | c | multiplication routines and a linear system solver | c | here. The matrix vector multiplication routine takes | c | workd(ipntr(1)) as the input vector. The final | c | result is returned to workd(ipntr(2)). | c %-------------------------------------------------------% c call av (n, workd(ipntr(1)), workd(ipntr(2))) call mv (n, workd(ipntr(1)), temp) call daxpy(n, sigma, temp, 1, workd(ipntr(2)), 1) c call dgttrs ('Notranspose', n, 1, adl, ad, adu, adu2, ipiv, & workd(ipntr(2)), n, ierr) if (ierr .ne. 0) then print *, ' ' print *, ' Error with _gttrs in _SDRV6.' print *, ' ' go to 9000 end if c c %-----------------------------------------% c | L O O P B A C K to call DSAUPD again. | c %-----------------------------------------% c go to 10 c else if (ido .eq. 1) then c c %----------------------------------------------------% c | Perform y <-- OP*x = inv[A-SIGMA*M]*(A+SIGMA*M)*x. | c | M*x has been saved in workd(ipntr(3)). The user | c | need the matrix vector multiplication (A*x) | c | routine and a linear system solver here. The | c | matrix vector multiplication routine takes | c | workd(ipntr(1)) as the input, and the result is | c | combined with workd(ipntr(3)) to form the input | c | for the linear system solver. The final result is | c | returned to workd(ipntr(2)). | c %----------------------------------------------------% c call av (n, workd(ipntr(1)), workd(ipntr(2))) call daxpy(n, sigma, workd(ipntr(3)), 1, workd(ipntr(2)), 1) call dgttrs ('Notranspose', n, 1, adl, ad, adu, adu2, ipiv, & workd(ipntr(2)), n, ierr) if (ierr .ne. 0) then print *, ' ' print *, ' Error with _gttrs in _SDRV6. ' print *, ' ' go to 9000 end if c c %-----------------------------------------% c | L O O P B A C K to call DSAUPD again. | c %-----------------------------------------% c go to 10 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 DSAUPD again. | c %-----------------------------------------% c go to 10 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 DSAUPD | c %--------------------------% c print *, ' ' print *, ' Error with _saupd, info = ',info print *, ' Check the documentation of _saupd. ' print *, ' ' c else c c %-------------------------------------------% c | No fatal errors occurred. | c | Post-Process using DSEUPD. | 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 dseupd ( rvec, 'All', select, d, v, ldv, sigma, & bmat, n, which, nev, tol, resid, ncv, v, ldv, & iparam, ipntr, workd, workl, lworkl, ierr ) c c %----------------------------------------------% c | Eigenvalues are returned in the first column | c | of the two dimensional array D and the | c | corresponding eigenvectors are returned in | c | the first NEV 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 DSEUPD. | c %------------------------------------% c print *, ' ' print *, ' Error with _seupd, info = ', ierr print *, ' Check the documentation of _seupd ' print *, ' ' c else 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 nconv = iparam(5) do 30 j=1, nconv 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,2) = dnrm2(n, ax, 1) d(j,2) = d(j,2) / abs(d(j,1)) 30 continue c call dmout(6, nconv, 2, d, maxncv, -6, & 'Ritz values 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 *, ' _SDRV6 ' 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 dsdrv6. | c %---------------------------% c 9000 continue c end c c------------------------------------------------------------------------ c Matrix vector subroutine c where the matrix used is the 1 dimensional mass matrix c arising from using the piecewise linear finite element c on the interval [0,1]. c subroutine mv (n, v, w) integer n, j Double precision & v(n), w(n), one, four, six, h parameter (one = 1.0D+0, four = 4.0D+0, & six = 6.0D+0) c w(1) = four*v(1) + v(2) do 100 j = 2,n-1 w(j) = v(j-1) + four*v(j) + v(j+1) 100 continue j = n w(j) = v(j-1) + four*v(j) c c Scale the vector w by h. c h = one / (six*dble(n+1)) call dscal(n, h, w, 1) return end c c------------------------------------------------------------------------ c Matrix vector subroutine c where the matrix is the stiffness matrix obtained from the c finite element discretization of the 1-dimensional discrete Laplacian c on the interval [0,1] with zero Dirichlet boundary condition c using piecewise linear elements. c subroutine av (n, v, w) integer n, j Double precision & v(n), w(n), one, two, h parameter (one = 1.0D+0, two = 2.0D+0) c w(1) = two*v(1) - v(2) do 100 j = 2,n-1 w(j) = - v(j-1) + two*v(j) - v(j+1) 100 continue j = n w(j) = - v(j-1) + two*v(j) c c Scale the vector w by (1/h). c h = one / dble(n+1) call dscal(n, one/h, w, 1) return end