program ssvd 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 SSAUPD 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 SSEUPD. 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 ssaupd ARPACK reverse communication interface routine. c sseupd ARPACK routine that returns Ritz values and (optionally) c Ritz vectors. c snrm2 Level 1 BLAS that computes the norm of a vector. c saxpy Level 1 BLAS that computes y <- alpha*x+y. c sscal Level 1 BLAS thst computes x <- x*alpha. c scopy 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 integer maxm, maxn, maxnev, maxncv, ldv, ldu parameter (maxm = 500, maxn=250, maxnev=10, maxncv=25, & ldu = maxm, ldv=maxn ) c c %--------------% c | Local Arrays | c %--------------% c Real & 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, mode1, nconv logical rvec Real & tol, sigma, temp c c %------------% c | Parameters | c %------------% c Real & one, zero parameter (one = 1.0E+0, zero = 0.0E+0) c c %-----------------------------% c | BLAS & LAPACK routines used | c %-----------------------------% c Real & snrm2 external snrm2, saxpy, scopy, sscal 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 '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 m = 500 n = 100 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 SSAUPD 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 nev = 4 ncv = 10 bmat = 'I' 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 SSAUPD | 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 SSAUPD. (See usage below.) | c | | c | It MUST initially be set to 0 before the first | c | call to SSAUPD. | 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 SSAUPD as | c | workspace. Its dimension LWORKL is set as | c | illustrated below. | c %-----------------------------------------------------% c lworkl = ncv*(ncv+8) tol = zero 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 SSAUPD is used | c | (IPARAM(7) = 1). All these options can be changed | c | by the user. For details see the documentation in | c | SSAUPD. | c %---------------------------------------------------% c ishfts = 1 maxitr = n mode1 = 1 c iparam(1) = ishfts c iparam(3) = maxitr c iparam(7) = mode1 c c %------------------------------------------------% c | M A I N L O O P (Reverse communication loop) | c %------------------------------------------------% c 10 continue c c %---------------------------------------------% c | Repeatedly call the routine SSAUPD and take | c | actions indicated by parameter IDO until | c | either convergence is indicated or maxitr | c | has been exceeded. | c %---------------------------------------------% c call ssaupd ( ido, bmat, n, which, nev, tol, resid, & ncv, v, ldv, iparam, ipntr, workd, workl, & lworkl, info ) c 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 call av (m, n, workd(ipntr(1)), ax) call atv (m, n, ax, workd(ipntr(2))) c c %-----------------------------------------% c | L O O P B A C K to call SSAUPD 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 SSAUPD. | 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 SSEUPD. | 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 SSEUPD now called to do this | c | post processing | c %--------------------------------------------% c rvec = .true. c call sseupd ( 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 SSEUPD. | c %------------------------------------% c print *, ' ' print *, ' Error with _seupd, info = ', ierr print *, ' Check the documentation of _seupd. ' print *, ' ' c else c nconv = iparam(5) 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 call av(m, n, v(1,j), ax) call scopy(m, ax, 1, u(1,j), 1) temp = one/snrm2(m, u(1,j), 1) call sscal(m, temp, u(1,j), 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 saxpy(m, -s(j,1), u(1,j), 1, ax, 1) s(j,2) = snrm2(m, ax, 1) c 20 continue c c %-------------------------------% c | Display computed residuals | c %-------------------------------% c call smout(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 ssvd. | c %-------------------------% c 9000 continue c end c c ------------------------------------------------------------------ c matrix vector subroutines c c The matrix A is derived from the simplest finite difference c discretization of the integral operator c c f(s) = integral(K(s,t)x(t)dt). c c Thus, the matrix A is a discretization of the 2-dimensional kernel c 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 Thus A is an m by n matrix with entries c c A(i,j) = k*(si)*(tj - 1) if i .le. j, c k*(tj)*(si - 1) if i .gt. j c c where si = i/(m+1) and tj = j/(n+1) and k = 1/(n+1). c c------------------------------------------------------------------- c subroutine av (m, n, x, w) c c computes w <- A*x c integer m, n, i, j Real & x(n), w(m), one, zero, h, k, s, t parameter ( one = 1.0E+0, zero = 0.0E+0 ) c h = one / real(m+1) k = one / real(n+1) do 5 i = 1,m w(i) = zero 5 continue t = zero c do 30 j = 1,n t = t+k s = zero do 10 i = 1,j s = s+h w(i) = w(i) + k*s*(t-one)*x(j) 10 continue do 20 i = j+1,m s = s+h w(i) = w(i) + k*t*(s-one)*x(j) 20 continue 30 continue c return end c c------------------------------------------------------------------- c subroutine atv (m, n, w, y) c c computes y <- A'*w c integer m, n, i, j Real & w(m), y(n), one, zero, h, k, s, t parameter ( one = 1.0E+0, zero = 0.0E+0 ) c h = one / real(m+1) k = one / real(n+1) do 5 i = 1,n y(i) = zero 5 continue t = zero c do 30 j = 1,n t = t+k s = zero do 10 i = 1,j s = s+h y(j) = y(j) + k*s*(t-one)*w(i) 10 continue do 20 i = j+1,m s = s+h y(j) = y(j) + k*t*(s-one)*w(i) 20 continue 30 continue c return end c