#include "ADMTLM_OPTIONS.h" subroutine admtlm_dsvd ( mythid ) c c This example program is intended to illustrate the c the use of ARPACK to compute the Singular Value Decomposition. c c This code shows how to use ARPACK to find a few of the c largest singular values(sigma) and corresponding right singular c vectors (v) for the the matrix A by solving the symmetric problem: c c (A'*A)*v = sigma*v c c where A is an m by n real matrix. c c This code may be easily modified to estimate the 2-norm c condition number largest(sigma)/smallest(sigma) by setting c which = 'BE' below. This will ask for a few of the smallest c and a few of the largest singular values simultaneously. c The condition number could then be estimated by taking c the ratio of the largest and smallest singular values. c c This formulation is appropriate when m .ge. n. c Reverse the roles of A and A' in the case that m .le. n. c c The main points illustrated here are c c 1) How to declare sufficient memory to find NEV c largest singular values of A . c c 2) Illustration of the reverse communication interface c needed to utilize the top level ARPACK routine DSAUPD c that computes the quantities needed to construct c the desired singular values and vectors(if requested). c c 3) How to extract the desired singular values and vectors c using the ARPACK routine DSEUPD. c c 4) How to construct the left singular vectors U from the c right singular vectors V to obtain the decomposition c c A*V = U*S c c where S = diag(sigma_1, sigma_2, ..., sigma_k). c c The only thing that must be supplied in order to use this c routine on your problem is to change the array dimensions c appropriately, to specify WHICH singular values you want to c compute and to supply a the matrix-vector products c c w <- Ax c y <- A'w c c in place of the calls to AV( ) and ATV( ) respectively below. c c Further documentation is available in the header of DSAUPD c which may be found in the SRC directory. c c This codes implements c c\Example-1 c ... Suppose we want to solve A'A*v = sigma*v in regular mode, c where A is derived from the simplest finite difference c discretization of the 2-dimensional kernel K(s,t)dt where c c K(s,t) = s(t-1) if 0 .le. s .le. t .le. 1, c t(s-1) if 0 .le. t .lt. s .le. 1. c c See subroutines AV and ATV for details. c ... OP = A'*A and B = I. c ... Assume "call av (n,x,y)" computes y = A*x c ... Assume "call atv (n,y,w)" computes w = A'*y c ... Assume exact shifts are used c ... c c\BeginLib 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 dnrm2 Level 1 BLAS that computes the norm of a vector. c daxpy Level 1 BLAS that computes y <- alpha*x+y. c dscal Level 1 BLAS thst computes x <- x*alpha. c dcopy Level 1 BLAS thst computes y <- 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: svd.F SID: 2.4 DATE OF SID: 10/17/00 RELEASE: 2 c c\Remarks c 1. None c c\EndLib c c----------------------------------------------------------------------- c c %------------------------------------------------------% c | Storage Declarations: | c | | c | It is assumed that A is M by N with M .ge. N. | c | | c | The maximum dimensions for all arrays are | c | set here to accommodate a problem size of | c | M .le. MAXM and N .le. MAXN | c | | c | The NEV right singular vectors will be computed in | c | the N by NCV array V. | c | | c | The NEV left singular vectors will be computed in | c | the M by NEV array U. | c | | c | NEV is the number of singular values requested. | c | See specifications for ARPACK usage below. | c | | c | NCV is the largest number of basis vectors that will | c | be used in the Implicitly Restarted Arnoldi | c | Process. Work per major iteration is | c | proportional to N*NCV*NCV. | c | | c | You must set: | c | | c | MAXM: Maximum number of rows of the A allowed. | c | MAXN: Maximum number of columns of the A allowed. | c | MAXNEV: Maximum NEV allowed | c | MAXNCV: Maximum NCV allowed | c %------------------------------------------------------% c C !USES: C == Global variables === #include "SIZE.h" #include "EEPARAMS.h" #include "PARAMS.h" #ifdef ALLOW_ADMTLM # include "tamc.h" # include "ctrl.h" # include "optim.h" # include "cost.h" # include "adcost.h" # include "g_cost.h" #endif C !INPUT/OUTPUT PARAMETERS: C == Routine arguments == INTEGER mythid C == PARAMETERS == integer maxnev, maxncv, ldv, ldu cph( parameter ( maxnev=15, maxncv=30, ldu = maxm, ldv=maxn ) character*(80) fnameGlobal cph) c c %--------------% c | Local Arrays | c %--------------% c Double precision & v(ldv,maxncv), u(ldu, maxnev), & workl(maxncv*(maxncv+8)), workd(3*maxn), & s(maxncv,2), resid(maxn), ax(maxm) logical select(maxncv) integer iparam(11), ipntr(11) c c %---------------% c | Local Scalars | c %---------------% c character bmat*1, which*2 integer ido, m, n, nev, ncv, lworkl, info, ierr, & j, ishfts, maxitr, mode, nconv logical rvec Double precision & tol, sigma, temp c c %------------% c | Parameters | c %------------% c Double precision & one, zero parameter (one = 1.0D+0, zero = 0.0D+0) c c %-----------------------------% c | BLAS & LAPACK routines used | c %-----------------------------% c Double precision & dnrm2 external dnrm2, daxpy, dcopy, dscal cph( integer l, ilinsysinfo, ii, jj integer ipiv(maxn) integer phiwork(maxn) double precision ferr, berr double precision phtmpin(maxn), phtmpout(maxn) double precision phxout(maxn) double precision tmpvec1(maxn), tmpvec2(maxn) double precision phrwork(3*maxn) cph double precision metricLoc(maxn,maxn) cph double precision metricTriag(maxn,maxn) cph double precision metricInv(maxn,maxn) cph DATA (metricInv(ii,1),ii=1,maxn) cph & / 9.9896D+07, 0.0519D+07, -0.0415D+07, cph & 0.0486D+07, -0.2432D+07, 0.1945D+07 / cph DATA (metricInv(ii,2),ii=1,maxn) cph & / 0.0519D+07, 9.7403D+07, 0.2077D+07, cph & -0.2432D+07, 1.2158D+07, -0.9726D+07 / cph DATA (metricInv(ii,3),ii=1,maxn) cph & / -0.0415D+07, 0.2077D+07, 9.8338D+07, cph & 0.1945D+07, -0.9726D+07, 0.7781D+07 / cph DATA (metricInv(ii,4),ii=1,maxn) cph & / 0.0486D+07, -0.2432D+07, 0.1945D+07, cph & 9.7723D+07, 1.1385D+07, -0.9108D+07 / cph DATA (metricInv(ii,5),ii=1,maxn) cph & / -0.2432D+07, 1.2158D+07, -0.9726D+07, cph & 1.1385D+07, 4.3073D+07, 4.5542D+07 / cph DATA (metricInv(ii,6),ii=1,maxn) cph & / 0.1945D+07, -0.9726D+07, 0.7781D+07, cph & -0.9108D+07, 4.5542D+07, 6.3567D+07 / cph) c c %-----------------------% c | Executable Statements | c %-----------------------% c c %-------------------------------------------------% c | The following include statement and assignments | c | initiate trace output from the internal | c | actions of ARPACK. See debug.doc in the | c | DOCUMENTS directory for usage. Initially, the | c | most useful information will be a breakdown of | c | time spent in the various stages of computation | c | given by setting msaupd = 1. | c %-------------------------------------------------% c include 'arpack_debug.h' ndigit = -3 logfil = 6 msgets = 0 msaitr = 0 msapps = 0 msaupd = 1 msaup2 = 0 mseigt = 0 mseupd = 0 c c %-------------------------------------------------% c | The following sets dimensions for this problem. | c %-------------------------------------------------% c cph( m = admtlmrec n = admtlmrec cph) c c %------------------------------------------------% c | Specifications for ARPACK usage are set | c | below: | c | | c | 1) NEV = 4 asks for 4 singular values to be | c | computed. | c | | c | 2) NCV = 20 sets the length of the Arnoldi | c | factorization | c | | c | 3) This is a standard problem | c | (indicated by bmat = 'I') | c | | c | 4) Ask for the NEV singular values of | c | largest magnitude | c | (indicated by which = 'LM') | c | See documentation in DSAUPD for the | c | other options SM, BE. | c | | c | Note: NEV and NCV must satisfy the following | c | conditions: | c | NEV <= MAXNEV, | c | NEV + 1 <= NCV <= MAXNCV | c %------------------------------------------------% c cph( nev = 15 ncv = 30 bmat = 'I' cph) which = 'LM' c if ( n .gt. maxn ) then print *, ' ERROR with _SVD: N is greater than MAXN ' go to 9000 else if ( m .gt. maxm ) then print *, ' ERROR with _SVD: M is greater than MAXM ' go to 9000 else if ( nev .gt. maxnev ) then print *, ' ERROR with _SVD: NEV is greater than MAXNEV ' go to 9000 else if ( ncv .gt. maxncv ) then print *, ' ERROR with _SVD: NCV is greater than MAXNCV ' go to 9000 end if c c %-----------------------------------------------------% c | Specification of stopping rules and initial | c | conditions before calling DSAUPD | c | | c | abs(sigmaC - sigmaT) < TOL*abs(sigmaC) | c | computed true | c | | c | If TOL .le. 0, then TOL <- macheps | c | (machine precision) is used. | c | | c | IDO is the REVERSE COMMUNICATION parameter | c | used to specify actions to be taken on return | c | from DSAUPD. (See usage below.) | c | | c | It MUST initially be set to 0 before the first | c | call to DSAUPD. | c | | c | INFO on entry specifies starting vector information | c | and on return indicates error codes | c | | c | Initially, setting INFO=0 indicates that a | c | random starting vector is requested to | c | start the ARNOLDI iteration. Setting INFO to | c | a nonzero value on the initial call is used | c | if you want to specify your own starting | c | vector (This vector must be placed in RESID.) | c | | c | The work array WORKL is used in DSAUPD as | c | workspace. Its dimension LWORKL is set as | c | illustrated below. | c %-----------------------------------------------------% c lworkl = ncv*(ncv+8) cph( tol = zero cph tol = 1.D-10 cph) info = 0 ido = 0 c c %---------------------------------------------------% c | Specification of Algorithm Mode: | c | | c | This program uses the exact shift strategy | c | (indicated by setting IPARAM(1) = 1.) | c | IPARAM(3) specifies the maximum number of Arnoldi | c | iterations allowed. Mode 1 of DSAUPD is used | c | (IPARAM(7) = 1). All these options can be changed | c | by the user. For details see the documentation in | c | DSAUPD. | c %---------------------------------------------------% c ishfts = 1 cph( maxitr = 10 c maxitr = 5 c mode = 1 cph) iparam(1) = ishfts c iparam(3) = maxitr c iparam(7) = mode c c %------------------------------------------------% c | M A I N L O O P (Reverse communication loop) | c %------------------------------------------------% c C-- Set model configuration (fixed arrays) CALL INITIALISE_FIXED( myThid ) c 10 continue c print *, 'ph----------------------------------------------------' print *, 'ph----------------------------------------------------' print *, 'ph----------------------------------------------------' 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 ctest( CALL DSAUPD ( ido, bmat, n, which, nev, tol, resid, & ncv, v, ldv, iparam, ipntr, workd, workl, & lworkl, info ) ctest ido = -1 ctest) c cph( print *, 'ph-count: optimcycle, ido, info, ipntr(1) ', & optimcycle, ido, info, ipntr(1) cph) if (ido .eq. -1 .or. ido .eq. 1) then c c %---------------------------------------% c | Perform matrix vector multiplications | c | w <--- A*x (av()) | c | y <--- A'*w (atv()) | c | The user should supply his/her own | c | matrix vector multiplication routines | c | here that takes workd(ipntr(1)) as | c | the input, and returns the result in | c | workd(ipntr(2)). | c %---------------------------------------% c c call av (m, n, workd(ipntr(1)), ax) c call atv (m, n, ax, workd(ipntr(2))) c cph( c do l = 1, n c print *, 'ph-test A ', l, c & workd(ipntr(1)+l), workd(ipntr(2)+l), workd(l) c enddo cph) do l = 1, n phtmpadmtlm(l) = workd(ipntr(1)+l-1) enddo IF (optimcycle .GT. 0 ) THEN _BEGIN_MASTER( mythid ) IF ( myProcId .eq. 0 ) THEN call admtlm_dsvd2model( .FALSE., mythid ) ENDIF _END_MASTER( mythid ) cph CALL ADMTLM_UPXX( mythid ) ENDIF c-- MWMWMWMWMWMWMW CALL ADMTLM_DRIVER( mythid ) c-- MWMWMWMWMWMWMW _BEGIN_MASTER( mythid ) IF ( myProcId .eq. 0 ) THEN call admtlm_model2dsvd( .FALSE., mythid ) ENDIF _END_MASTER( mythid ) do l = 1, n workd(ipntr(2)+l-1) = phtmpadmtlm(l) enddo c if (optimcycle .EQ. 4) & STOP 'in ADMTLM_DSVD after the_model_main' cph( cph Since we solve for a generalized EV problem, we have to solve cph M*y=A*x for y with known matrix M and vector A*x c do ii = 1, n c phxout(ii) = 0. c do jj = 1, n c phxout(ii) = phxout(ii) + c & metricInv(ii,jj)*phtmpout(jj) c enddo c print *, 'ph-test C ', ii, phxout(ii) c enddo c c call dposvx ( c & 'E', 'U', maxn, 1, metricLoc, 6, metricTriag, 6, c & 'N', tmpvec1, phtmpout, 6, phxout, 6, tmpvec2, c & ferr, berr, phrwork, phiwork, ilinsysinfo) cph c call dgesv ( maxn, 1, metricLoc, maxn, c & ipiv, phtmpout, maxn, ilinsysinfo ) cph c cph Finally schift result y -> workd(ipntr(2)) cph call dcopy ( maxn, tmpvec1, 1, workd(ipntr(2)), 1 ) cph We have restored orig. standard EV problem cph OP*x = lambda*x for OP = INV(M)*A cph) c c %-----------------------------------------% c | L O O P B A C K to call DSAUPD again. | c %-----------------------------------------% c optimcycle = optimcycle + 1 go to 10 c end if c cph( 1001 continue print *, 'ph-continue ', info, ierr cph) 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 documentation in _saupd ' print *, ' ' c else c c %--------------------------------------------% c | No fatal errors occurred. | c | Post-Process using DSEUPD. | c | | c | Computed singular values may be extracted. | c | | c | Singular vectors may also be computed now | c | if desired. (indicated by rvec = .true.) | c | | c | The routine DSEUPD now called to do this | c | post processing | c %--------------------------------------------% c rvec = .true. c call dseupd ( rvec, 'All', select, s, v, ldv, sigma, & bmat, n, which, nev, tol, resid, ncv, v, ldv, & iparam, ipntr, workd, workl, lworkl, ierr ) c c %-----------------------------------------------% c | Singular values are returned in the first | c | column of the two dimensional array S | c | and the corresponding right singular vectors | c | are returned in the first NEV columns of the | c | two dimensional array V as requested here. | 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 nconv = iparam(5) cph( do j=1, nconv print *, 'ph-ev ', j, s(j,1), s(j,2) enddo STOP 'TEST AFTER dneupd' cph) do 20 j=1, nconv c s(j,1) = sqrt(s(j,1)) c c %-----------------------------% c | Compute the left singular | c | vectors from the formula | c | | c | u = Av/sigma | c | | c | u should have norm 1 so | c | divide by norm(Av) instead. | c %-----------------------------% c do l = 1, n tmpvec1(l) = v(l,j) enddo c c-- MWMWMWMWMWMWMW cph call box_main( tmpvec1, tmpvec2, metricLoc, ldoadjoint ) c-- MWMWMWMWMWMWMW c call dcopy( m, tmpvec2, 1, u(l,j), 1 ) temp = one / dnrm2( m, u(l,j), 1 ) cph( print *, 'ph-print ', j, dnrm2( m, u(l,j), 1 ) cph) call dscal(m , temp, u(l,j), 1 ) cph do l = 1, n cph u(l,j) = tmpvec2(l) cph enddo cph temp = one/dnrm2(m, tmpvec2, 1) cph call dscal(m, temp, tmpvec2, 1) c c %---------------------------% c | | c | Compute the residual norm | c | | c | || A*v - sigma*u || | c | | c | for the NCONV accurately | c | computed singular values | c | and vectors. (iparam(5) | c | indicates how many are | c | accurate to the requested | c | tolerance). | c | Store the result in 2nd | c | column of array S. | c %---------------------------% c call daxpy(m, -s(j,1), u(1,j), 1, tmpvec2, 1) s(j,2) = dnrm2(m, tmpvec2, 1) c 20 continue c c %-------------------------------% c | Display computed residuals | c %-------------------------------% c call dmout(6, nconv, 2, s, maxncv, -6, & 'Singular values and direct residuals') 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 *, ' _SVD ' 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 dsvd. | c %-------------------------% c 9000 continue c end