program dnbdr2 c c ... Construct matrices 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 ... Define the shift SIGMA = (SIGMAR, SIGMAI). c c ... Call DNBAND to find eigenvalues LAMBDA closest to SIGMA c such that c A*x = LAMBDA*x. c c ... Use mode 3 of DNAUPD . c c\BeginLib c c\Routines called: c dnband ARPACK banded eigenproblem solver. c dlapy2 LAPACK routine to compute sqrt(x**2+y**2) carefully. c dlaset LAPACK routine to initialize a matrix to zero. c daxpy Level 1 BLAS that computes y <- alpha*x+y. c dnrm2 Level 1 BLAS that computes the norm of a vector. c dgbmv 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: nbdr2.F SID: 2.5 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) Double precision & a(lda,maxn), m(lda,maxn), rfac(lda,maxn), & workl(3*maxncv*maxncv+6*maxncv), workd(3*maxn), & workev(3*maxncv), v(ldv, maxncv), & resid(maxn), d(maxncv, 3), ax(maxn) Complex*16 & cfac(lda, maxn), workc(maxn) c c %---------------% c | Local Scalars | c %---------------% c character which*2, bmat integer nev, ncv, ku, kl, info, i, j, ido, & n, nx, lo, idiag, isub, isup, mode, maxitr, & nconv logical rvec, first Double precision & tol, rho, h2, h, sigmar, sigmai c c %------------% c | Parameters | c %------------% c Double precision & one, zero, two parameter (one = 1.0D+0 , zero = 0.0D+0 , & two = 2.0D+0 ) c c %-----------------------------% c | BLAS & LAPACK routines used | c %-----------------------------% c Double precision & dlapy2 , dnrm2 external dlapy2 , dnrm2 , daxpy , dgbmv c c %--------------------% c | Intrinsic function | c %--------------------% c intrinsic abs 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 (closest to (SIGMAR,SIGMAI)) to be | c | approximated. Since the shift-invert moded is | c | used, WHICH is set to 'LM'. The user can modify | c | NX, NEV, NCV, SIGMAR, SIGMAI to solve problems | c | of different sizes, and to get different parts | c | the spectrum. However, The following conditions | c | must be satisfied: | c | N <= MAXN | c | NEV <= MAXNEV | c | NEV + 2 <= NCV <= MAXNCV | c %-------------------------------------------------% c nx = 10 n = nx*nx nev = 4 ncv = 20 if ( n .gt. maxn ) then print *, ' ERROR with _NBDR2: N is greater than MAXN ' go to 9000 else if ( nev .gt. maxnev ) then print *, ' ERROR with _NBDR2: NEV is greater than MAXNEV ' go to 9000 else if ( ncv .gt. maxncv ) then print *, ' ERROR with _NBDR2: NCV is greater than MAXNCV ' go to 9000 end if bmat = 'I' which = 'LM' sigmar = 1.0D+4 sigmai = 0.0D+0 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 = zero ido = 0 info = 0 c c %---------------------------------------------------% c | IPARAM(3) specifies the maximum number of Arnoldi | c | iterations allowed. Mode 3 of DNAUPD is used | c | (IPARAM(7) = 3). All these options can be changed | c | by the user. For details, see the documentation | c | in DNBAND . | c %---------------------------------------------------% c maxitr = 300 mode = 3 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 dlaset ('A', lda, n, zero, zero, a, lda) call dlaset ('A', lda, n, zero, zero, m, lda) call dlaset ('A', lda, n, zero, zero, rfac, lda) c c %-------------------------------------% c | KU, KL are number of superdiagonals | c | and subdiagonals within the band of | c | matrices A. | c %-------------------------------------% c kl = nx ku = nx c c %---------------% c | Main diagonal | c %---------------% c h = one / dble (nx+1) h2 = h*h c idiag = kl+ku+1 do 30 j = 1, n a(idiag,j) = 4.0D+0 / h2 30 continue c c %-------------------------------------% c | First subdiagonal and superdiagonal | c %-------------------------------------% c isup = kl+ku isub = kl+ku+2 rho = 1.0D+1 do 50 i = 1, nx lo = (i-1)*nx do 40 j = lo+1, lo+nx-1 a(isub,j+1) = -one/h2 + rho/two/h a(isup,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. The real parts of the | c | eigenvalues are returned in the first column | c | of D, the imaginary parts are returned in the | c | second column of D. Eigenvectors are returned | c | in the first NCONV (=IPARAM(5)) columns of V. | c %------------------------------------------------% c rvec = .true. call dnband (rvec, 'A', select, d, d(1,2), v, ldv, sigmar, & sigmai, workev, n, a, m, lda, rfac, cfac, kl, ku, & which, bmat, nev, tol, resid, ncv, v, ldv, iparam, & workd, workl, lworkl, workc, iwork, info) c if ( info .eq. 0) then c c %-----------------------------------% c | Print out convergence information | c %-----------------------------------% c nconv = iparam(5) c print *, ' ' print *, ' _NBDR2 ' 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 first = .true. do 90 j = 1, nconv c if ( d(j,2) .eq. zero ) then c c %--------------------% c | Ritz value is real | c %--------------------% c call dgbmv ('Notranspose', n, n, kl, ku, one, & a(kl+1,1), lda, v(1,j), 1, zero, & ax, 1) call daxpy (n, -d(j,1), v(1,j), 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 dgbmv ('Notranspose', n, n, kl, ku, one, & a(kl+1,1), lda, v(1,j), 1, zero, & ax, 1) call daxpy (n, -d(j,1), v(1,j), 1, ax, 1) call daxpy (n, d(j,2), v(1,j+1), 1, ax, 1) d(j,3) = dnrm2 (n, ax, 1) call dgbmv ('Notranspose', n, n, kl, ku, one, & a(kl+1,1), lda, v(1,j+1), 1, zero, & ax, 1) call daxpy (n, -d(j,1), v(1,j+1), 1, ax, 1) call daxpy (n, -d(j,2), v(1,j), 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 90 continue call dmout (6, nconv, 3, d, 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 DNBAND . | c %-------------------------------------% c print *, ' ' print *, ' Error with _nband, info= ', info print *, ' Check the documentation of _nband ' print *, ' ' c end if c 9000 end