program znbdr1 c c ... Construct the matrix A in LAPACK-style band form. c The matrix A is derived from the discretization of c the 2-d convection-diffusion operator c c -Laplacian(u) + rho*partial(u)/partial(x). c c on the unit square with zero Dirichlet boundary condition c using standard central difference. c c ... Call ZNBAND to find eigenvalues LAMBDA such that c A*x = x*LAMBDA. c c ... Use mode 1 of ZNAUPD . c c\BeginLib c c znband ARPACK banded eigenproblem solver. c dlapy2 LAPACK routine to compute sqrt(x**2+y**2) carefully. c zlaset LAPACK routine to initialize a matrix to zero. c zaxpy Level 1 BLAS that computes y <- alpha*x+y. c dznrm2 Level 1 BLAS that computes the norm of a vector. c zgbmv Level 2 BLAS that computes the band matrix vector product 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: nbdr1.F SID: 2.3 DATE OF SID: 08/26/96 RELEASE: 2 c c\Remarks c 1. None c c\EndLib c c---------------------------------------------------------------------- c c %-------------------------------------% c | Define leading dimensions for all | c | arrays. | c | MAXN - Maximum size of the matrix | c | MAXNEV - Maximum number of | c | eigenvalues to be computed | c | MAXNCV - Maximum number of Arnoldi | c | vectors stored | c | MAXBDW - Maximum bandwidth | c %-------------------------------------% c integer maxn, maxnev, maxncv, maxbdw, lda, & lworkl, ldv parameter ( maxn = 1000, maxnev = 25, maxncv=50, & maxbdw=50, lda = maxbdw, ldv = maxn ) c c %--------------% c | Local Arrays | c %--------------% c integer iparam(11), iwork(maxn) logical select(maxncv) Complex*16 & a(lda,maxn), m(lda,maxn), fac(lda,maxn), & workl(3*maxncv*maxncv+5*maxncv), workd(3*maxn), & workev(2*maxncv), v(ldv, maxncv), & resid(maxn), d(maxncv), ax(maxn) Double precision & rwork(maxn), rd(maxncv,3) c c %---------------% c | Local Scalars | c %---------------% c character which*2, bmat integer nev, ncv, kl, ku, info, i, j, & n, nx, lo, isub, isup, idiag, maxitr, mode, & nconv logical rvec Double precision & tol Complex*16 & rho, h, h2, sigma c c %------------% c | Parameters | c %------------% c Complex*16 & one, zero, two parameter ( one = (1.0D+0, 0.0D+0) , & zero = (0.0D+0, 0.0D+0) , & two = (2.0D+0, 0.0D+0) ) c c %-----------------------------% c | BLAS & LAPACK routines used | c %-----------------------------% c Double precision & dznrm2 , dlapy2 external dznrm2 , zgbmv , zaxpy , dlapy2 , zlaset c c %-----------------------% c | Executable Statements | c %-----------------------% c c %-------------------------------------------------% c | The number NX is the number of interior points | c | in the discretization of the 2-dimensional | c | convection-diffusion operator on the unit | c | square with zero Dirichlet boundary condition. | c | The number N(=NX*NX) is the dimension of the | c | matrix. A standard eigenvalue problem is | c | solved (BMAT = 'I'). NEV is the number of | c | eigenvalues to be approximated. The user can | c | modify NX, NEV, NCV and WHICH to solve problems | c | of different sizes, and to get different parts | c | the spectrum. However, the following | c | conditions must be satisfied: | c | N <= MAXN | c | NEV <= MAXNEV | c | NEV + 2 <= NCV <= MAXNCV | c %-------------------------------------------------% c nx = 10 n = nx*nx nev = 4 ncv = 10 if ( n .gt. maxn ) then print *, ' ERROR with _NBDR1: N is greater than MAXN ' go to 9000 else if ( nev .gt. maxnev ) then print *, ' ERROR with _NBDR1: NEV is greater than MAXNEV ' go to 9000 else if ( ncv .gt. maxncv ) then print *, ' ERROR with _NBDR1: NCV is greater than MAXNCV ' go to 9000 end if bmat = 'I' which = 'LM' 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. Setting INFO=0 indicates that a | c | random vector is generated in ZNAUPD to start the | c | Arnoldi iteration. | c %-----------------------------------------------------% c lworkl = 3*ncv**2+5*ncv tol = 0.0 info = 0 c c %---------------------------------------------------% c | IPARAM(3) specifies the maximum number of Arnoldi | c | iterations allowed. Mode 1 of ZNAUPD is used | c | (IPARAM(7) = 1). All these options can be changed | c | by the user. For details, see the documentation | c | in znband . | c %---------------------------------------------------% c maxitr = 300 mode = 1 c iparam(3) = maxitr iparam(7) = mode c c %----------------------------------------% c | Construct the matrix A in LAPACK-style | c | banded form. | c %----------------------------------------% c c %---------------------------------------------% c | Zero out the workspace for banded matrices. | c %---------------------------------------------% c call zlaset ('A', lda, n, zero, zero, a, lda) call zlaset ('A', lda, n, zero, zero, m, lda) call zlaset ('A', lda, n, zero, zero, fac, lda) c c %-------------------------------------% c | KU, KL are number of superdiagonals | c | and subdiagonals within the band of | c | matrices A and M. | c %-------------------------------------% c kl = nx ku = nx c c %---------------% c | Main diagonal | c %---------------% c h = one / dcmplx (nx+1) h2 = h*h c idiag = kl+ku+1 do 30 j = 1, n a(idiag,j) = (4.0D+0, 0.0D+0) / h2 30 continue c c %-------------------------------------% c | First subdiagonal and superdiagonal | c %-------------------------------------% c rho = (1.0D+2, 0.0D+0) isup = kl+ku isub = kl+ku+2 do 50 i = 1, nx lo = (i-1)*nx do 40 j = lo+1, lo+nx-1 a(isup,j+1) = -one/h2 + rho/two/h a(isub,j) = -one/h2 - rho/two/h 40 continue 50 continue c c %------------------------------------% c | KL-th subdiagonal and KU-th super- | c | diagonal. | c %------------------------------------% c isup = kl+1 isub = 2*kl+ku+1 do 80 i = 1, nx-1 lo = (i-1)*nx do 70 j = lo+1, lo+nx a(isup,nx+j) = -one / h2 a(isub,j) = -one / h2 70 continue 80 continue c c %-----------------------------------------------% c | Call ARPACK banded solver to find eigenvalues | c | and eigenvectors. Eigenvalues are returned in | c | the one dimensional array D. Eigenvectors | c | are returned in the first NCONV (=IPARAM(5)) | c | columns of V. | c %-----------------------------------------------% c rvec = .true. call znband (rvec, 'A', select, d, v, ldv, sigma, & workev, n, a, m, lda, fac, kl, ku, which, & bmat, nev, tol, resid, ncv, v, ldv, iparam, & workd, workl, lworkl, rwork, iwork, info) c if ( info .eq. 0) then c nconv = iparam(5) c c %-----------------------------------% c | Print out convergence information | c %-----------------------------------% c print *, ' ' print *, '_NBDR1 ' print *, '====== ' print *, ' ' print *, ' The size of the matrix is ', n print *, ' Number of eigenvalue requested is ', nev print *, ' The number of Arnoldi vectors generated', & ' (NCV) is ', ncv print *, ' The number of converged Ritz values is ', & nconv print *, ' What portion of the spectrum ', which print *, ' The number of Implicit Arnoldi ', & ' update taken is ', iparam(3) print *, ' The number of OP*x is ', iparam(9) print *, ' The convergence tolerance is ', tol print *, ' ' c c %----------------------------% c | Compute the residual norm. | c | || A*x - lambda*x || | c %----------------------------% c do 90 j = 1, nconv c c %---------------------------% c | Compute the residual norm | c | || A*x - lambda*x || | c %---------------------------% c call zgbmv ('Notranspose', n, n, kl, ku, one, & a(kl+1,1), lda, v(1,j), 1, zero, & ax, 1) call zaxpy (n, -d(j), v(1,j), 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)) 90 continue call dmout (6, nconv, 3, rd, maxncv, -6, & 'Ritz values (Real,Imag) and relative residuals') else c c %-------------------------------------% c | Either convergence failed, or there | c | is error. Check the documentation | c | for znband . | c %-------------------------------------% c print *, ' ' print *, ' Error with _nband, info= ', info print *, ' Check the documentation of _nband ' print *, ' ' c end if c 9000 end