program sndrv4 c c Simple program to illustrate the idea of reverse communication c in shift-invert mode for a generalized nonsymmetric eigenvalue c problem. c c We implement example four of ex-nonsym.doc in DOCUMENTS directory c c\Example-4 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 ... The shift sigma is a real number. c c ... OP = inv[A-SIGMA*M]*M and B = M. c c ... Use mode 3 of SNAUPD. c c\BeginLib c c\Routines called: c snaupd ARPACK reverse communication interface routine. c sneupd ARPACK routine that returns Ritz values and (optionally) c Ritz vectors. c sgttrf LAPACK tridiagonal factorization routine. c sgttrs LAPACK tridiagonal linear system solve routine. c slapy2 LAPACK routine to compute sqrt(x**2+y**2) carefully. c saxpy Level 1 BLAS that computes y <- alpha*x+y. c scopy Level 1 BLAS that copies one vector to another. c sdot Level 1 BLAS that computes the dot product of two vectors. c snrm2 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: ndrv4.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) Real & ax(maxn), mx(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, ierr, j, & nconv, maxitr, ishfts, mode Real & tol, h, s, & sigmar, sigmai, s1, s2, s3 logical first, rvec c c %-----------------------------% c | BLAS & LAPACK routines used | c %-----------------------------% c Real & sdot, snrm2, slapy2 external sdot, snrm2, slapy2, sgttrf, sgttrs c c %--------------------% c | Intrinsic function | c %--------------------% c intrinsic abs c c %------------% c | Parameters | c %------------% c Real & one, zero, two, six, rho common /convct/ rho parameter (one = 1.0E+0, zero = 0.0E+0, & two = 2.0E+0, six = 6.0E+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, SIGMAR 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 = 10 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' sigmar = one sigmai = zero c c %--------------------------------------------------% c | Construct C = A - SIGMA*M in real arithmetic, | c | and factor C in real arithmetic (using LAPACK | c | subroutine sgttrf). 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 the mass matrix | c | formed by using piecewise linear elements on | c | [0,1]. | c %--------------------------------------------------% c rho = 1.0E+1 h = one / real(n+1) s = rho / two c s1 = -one/h - s - sigmar*h/six s2 = two/h - 4.0E+0*sigmar*h/six s3 = -one/h + s - sigmar*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 sgttrf(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 SNAUPD 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 SNAUPD 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 SNAUPD is used | c | (IPARAM(7) = 3). All these options can be | c | changed by the user. For details, see the | c | documentation in SNAUPD. | 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 SNAUPD and take | c | actions indicated by parameter IDO until | c | either convergence is indicated or maxitr | c | has been exceeded. | c %---------------------------------------------% c call snaupd ( 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]*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 sgttrs('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 SNAUPD 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 scopy( n, workd(ipntr(3)), 1, workd(ipntr(2)), 1) call sgttrs ('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 SNAUPD 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 SNAUPD again. | c %-----------------------------------------% c go to 20 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 SNAUPD. | 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 SNEUPD. | 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 sneupd ( 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 SNEUPD. | 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 mv(n, v(1,j), mx) call saxpy(n, -d(j,1), mx, 1, ax, 1) d(j,3) = snrm2(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 saxpy(n, -d(j,1), mx, 1, ax, 1) call mv(n, v(1,j+1), mx) call saxpy(n, d(j,2), mx, 1, ax, 1) d(j,3) = snrm2(n, ax, 1) call av(n, v(1,j+1), ax) call mv(n, v(1,j+1), mx) call saxpy(n, -d(j,1), mx, 1, ax, 1) call mv(n, v(1,j), mx) call saxpy(n, -d(j,2), mx, 1, ax, 1) d(j,3) = slapy2( d(j,3), snrm2(n, ax, 1) ) d(j,3) = d(j,3) / slapy2(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 smout(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 *, ' _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 c %---------------------------% c | Done with program sndrv4. | c %---------------------------% c 9000 continue c end c c========================================================================== c c matrix vector multiplication subroutine c subroutine mv (n, v, w) integer n, j Real & v(n), w(n), one, four, six, h parameter (one = 1.0E+0, four = 4.0E+0, six = 6.0E+0) c c Compute the matrix vector multiplication y<---M*x c where M is mass matrix formed by using piecewise linear elements c on [0,1]. c w(1) = ( four*v(1) + one*v(2) ) / six do 10 j = 2,n-1 w(j) = ( one*v(j-1) + four*v(j) + one*v(j+1) ) / six 10 continue w(n) = ( one*v(n-1) + four*v(n) ) / six c h = one / real(n+1) call sscal(n, h, w, 1) return end c------------------------------------------------------------------ subroutine av (n, v, w) integer n, j Real & v(n), w(n), one, two, dd, dl, du, s, h, rho common /convct/ rho parameter (one = 1.0E+0, two = 2.0E+0) c c Compute the matrix vector multiplication y<---A*x c where A is obtained from the finite element discretization of the c 1-dimensional convection diffusion operator c d^u/dx^2 + rho*(du/dx) c on the interval [0,1] with zero Dirichlet boundary condition c using linear elements. c This routine is only used in residual calculation. c h = one / real(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