subroutine assst(iv, liv, lv, v) c c *** assess candidate step (***sol version 2.3) *** c integer liv, l integer iv(liv) double precision v(lv) c c *** purpose *** c c this subroutine is called by an unconstrained minimization c routine to assess the next candidate step. it may recommend one c of several courses of action, such as accepting the step, recom- c puting it using the same or a new quadratic model, or halting due c to convergence or false convergence. see the return code listing c below. c c-------------------------- parameter usage -------------------------- c c iv (i/o) integer parameter and scratch vector -- see description c below of iv values referenced. c liv (in) length of iv array. c lv (in) length of v array. c v (i/o) real parameter and scratch vector -- see description c below of v values referenced. c c *** iv values referenced *** c c iv(irc) (i/o) on input for the first step tried in a new iteration, c iv(irc) should be set to 3 or 4 (the value to which it is c set when step is definitely to be accepted). on input c after step has been recomputed, iv(irc) should be c unchanged since the previous return of assst. c on output, iv(irc) is a return code having one of the c following values... c 1 = switch models or try smaller step. c 2 = switch models or accept step. c 3 = accept step and determine v(radfac) by gradient c tests. c 4 = accept step, v(radfac) has been determined. c 5 = recompute step (using the same model). c 6 = recompute step with radius = v(lmaxs) but do not c evaulate the objective function. c 7 = x-convergence (see v(xctol)). c 8 = relative function convergence (see v(rfctol)). c 9 = both x- and relative function convergence. c 10 = absolute function convergence (see v(afctol)). c 11 = singular convergence (see v(lmaxs)). c 12 = false convergence (see v(xftol)). c 13 = iv(irc) was out of range on input. c return code i has precdence over i+1 for i = 9, 10, 11. c iv(mlstgd) (i/o) saved value of iv(model). c iv(model) (i/o) on input, iv(model) should be an integer identifying c the current quadratic model of the objective function. c if a previous step yielded a better function reduction, c then iv(model) will be set to iv(mlstgd) on output. c iv(nfcall) (in) invocation count for the objective function. c iv(nfgcal) (i/o) value of iv(nfcall) at step that gave the biggest c function reduction this iteration. iv(nfgcal) remains c unchanged until a function reduction is obtained. c iv(radinc) (i/o) the number of radius increases (or minus the number c of decreases) so far this iteration. c iv(restor) (out) set to 1 if v(f) has been restored and x should be c restored to its initial value, to 2 if x should be saved, c to 3 if x should be restored from the saved value, and to c 0 otherwise. c iv(stage) (i/o) count of the number of models tried so far in the c current iteration. c iv(stglim) (in) maximum number of models to consider. c iv(switch) (out) set to 0 unless a new model is being tried and it c gives a smaller function value than the previous model, c in which case assst sets iv(switch) = 1. c iv(toobig) (in) is nonzero if step was too big (e.g. if it caused c overflow). c iv(xirc) (i/o) value that iv(irc) would have in the absence of c convergence, false convergence, and oversized steps. c c *** v values referenced *** c c v(afctol) (in) absolute function convergence tolerance. if the c absolute value of the current function value v(f) is less c than v(afctol), then assst returns with iv(irc) = 10. c v(decfac) (in) factor by which to decrease radius when iv(toobig) is c nonzero. c v(dstnrm) (in) the 2-norm of d*step. c v(dstsav) (i/o) value of v(dstnrm) on saved step. c v(dst0) (in) the 2-norm of d times the newton step (when defined, c i.e., for v(nreduc) .ge. 0). c v(f) (i/o) on both input and output, v(f) is the objective func- c tion value at x. if x is restored to a previous value, c then v(f) is restored to the corresponding value. c v(fdif) (out) the function reduction v(f0) - v(f) (for the output c value of v(f) if an earlier step gave a bigger function c decrease, and for the input value of v(f) otherwise). c v(flstgd) (i/o) saved value of v(f). c v(f0) (in) objective function value at start of iteration. c v(gtslst) (i/o) value of v(gtstep) on saved step. c v(gtstep) (in) inner product between step and gradient. c v(incfac) (in) minimum factor by which to increase radius. c v(lmaxs) (in) maximum reasonable step size (and initial step bound). c if the actual function decrease is no more than twice c what was predicted, if a return with iv(irc) = 7, 8, 9, c or 10 does not occur, if v(dstnrm) .gt. v(lmaxs), and if c v(preduc) .le. v(sctol) * abs(v(f0)), then assst re- c turns with iv(irc) = 11. if so doing appears worthwhile, c then assst repeats this test with v(preduc) computed for c a step of length v(lmaxs) (by a return with iv(irc) = 6). c v(nreduc) (i/o) function reduction predicted by quadratic model for c newton step. if assst is called with iv(irc) = 6, i.e., c if v(preduc) has been computed with radius = v(lmaxs) for c use in the singular convervence test, then v(nreduc) is c set to -v(preduc) before the latter is restored. c v(plstgd) (i/o) value of v(preduc) on saved step. c v(preduc) (i/o) function reduction predicted by quadratic model for c current step. c v(radfac) (out) factor to be used in determining the new radius, c which should be v(radfac)*dst, where dst is either the c output value of v(dstnrm) or the 2-norm of c diag(newd)*step for the output value of step and the c updated version, newd, of the scale vector d. for c iv(irc) = 3, v(radfac) = 1.0 is returned. c v(rdfcmn) (in) minimum value for v(radfac) in terms of the input c value of v(dstnrm) -- suggested value = 0.1. c v(rdfcmx) (in) maximum value for v(radfac) -- suggested value = 4.0. c v(reldx) (in) scaled relative change in x caused by step, computed c (e.g.) by function reldst as c max (d(i)*abs(x(i)-x0(i)), 1 .le. i .le. p) / c max (d(i)*(abs(x(i))+abs(x0(i))), 1 .le. i .le. p). c v(rfctol) (in) relative function convergence tolerance. if the c actual function reduction is at most twice what was pre- c dicted and v(nreduc) .le. v(rfctol)*abs(v(f0)), then c assst returns with iv(irc) = 8 or 9. c v(stppar) (in) marquardt parameter -- 0 means full newton step. c v(tuner1) (in) tuning constant used to decide if the function c reduction was much less than expected. suggested c value = 0.1. c v(tuner2) (in) tuning constant used to decide if the function c reduction was large enough to accept step. suggested c value = 10**-4. c v(tuner3) (in) tuning constant used to decide if the radius c should be increased. suggested value = 0.75. c v(xctol) (in) x-convergence criterion. if step is a newton step c (v(stppar) = 0) having v(reldx) .le. v(xctol) and giving c at most twice the predicted function decrease, then c assst returns iv(irc) = 7 or 9. c v(xftol) (in) false convergence tolerance. if step gave no or only c a small function decrease and v(reldx) .le. v(xftol), c then assst returns with iv(irc) = 12. c c------------------------------- notes ------------------------------- c c *** application and usage restrictions *** c c this routine is called as part of the nl2sol (nonlinear c least-squares) package. it may be used in any unconstrained c minimization solver that uses dogleg, goldfeld-quandt-trotter, c or levenberg-marquardt steps. c c *** algorithm notes *** c c see (1) for further discussion of the assessing and model c switching strategies. while nl2sol considers only two models, c assst is designed to handle any number of models. c c *** usage notes *** c c on the first call of an iteration, only the i/o variables c step, x, iv(irc), iv(model), v(f), v(dstnrm), v(gtstep), and c v(preduc) need have been initialized. between calls, no i/o c values execpt step, x, iv(model), v(f) and the stopping toler- c ances should be changed. c after a return for convergence or false convergence, one can c change the stopping tolerances and call assst again, in which c case the stopping tests will be repeated. c c *** references *** c c (1) dennis, j.e., jr., gay, d.m., and welsch, r.e. (1981), c an adaptive nonlinear least-squares algorithm, c acm trans. math. software, vol. 7, no. 3. c c (2) powell, m.j.d. (1970) a fortran subroutine for solving c systems of nonlinear algebraic equations, in numerical c methods for nonlinear algebraic equations, edited by c p. rabinowitz, gordon and breach, london. c c *** history *** c c john dennis designed much of this routine, starting with c ideas in (2). roy welsch suggested the model switching strategy. c david gay and stephen peters cast this subroutine into a more c portable form (winter 1977), and david gay cast it into its c present form (fall 1978). c c *** general *** c c this subroutine was written in connection with research c supported by the national science foundation under grants c mcs-7600324, dcr75-10143, 76-14311dss, mcs76-11989, and c mcs-7906671. c c------------------------ external quantities ------------------------ c c *** no external functions and subroutines *** c c *** intrinsic functions *** c/+ double precision dabs, dmax1 c/ c *** no common blocks *** c c-------------------------- local variables -------------------------- c logical goodx integer i, nfc double precision emax, emaxs, gts, rfac1, xmax double precision half, one, onep2, two, zero c c *** subscripts for iv and v *** c integer afctol, decfac, dstnrm, dstsav, dst0, f, fdif, flstgd, f0, 1 gtslst, gtstep, incfac, irc, lmaxs, mlstgd, model, nfcall, 2 nfgcal, nreduc, plstgd, preduc, radfac, radinc, rdfcmn, 3 rdfcmx, reldx, restor, rfctol, sctol, stage, stglim, 4 stppar, switch, toobig, tuner1, tuner2, tuner3, xctol, 5 xftol, xirc c c *** data initializations *** c c/6 c data half/0.5d+0/, one/1.d+0/, onep2/1.2d+0/, two/2.d+0/, c 1 zero/0.d+0/ c/7 parameter (half=0.5d+0, one=1.d+0, onep2=1.2d+0, two=2.d+0, 1 zero=0.d+0) c/ c c/6 c data irc/29/, mlstgd/32/, model/5/, nfcall/6/, nfgcal/7/, c 1 radinc/8/, restor/9/, stage/10/, stglim/11/, switch/12/, c 2 toobig/2/, xirc/13/ c/7 parameter (irc=29, mlstgd=32, model=5, nfcall=6, nfgcal=7, 1 radinc=8, restor=9, stage=10, stglim=11, switch=12, 2 toobig=2, xirc=13) c/ c/6 c data afctol/31/, decfac/22/, dstnrm/2/, dst0/3/, dstsav/18/, c 1 f/10/, fdif/11/, flstgd/12/, f0/13/, gtslst/14/, gtstep/4/, c 2 incfac/23/, lmaxs/36/, nreduc/6/, plstgd/15/, preduc/7/, c 3 radfac/16/, rdfcmn/24/, rdfcmx/25/, reldx/17/, rfctol/32/, c 4 sctol/37/, stppar/5/, tuner1/26/, tuner2/27/, tuner3/28/, c 5 xctol/33/, xftol/34/ c/7 parameter (afctol=31, decfac=22, dstnrm=2, dst0=3, dstsav=18, 1 f=10, fdif=11, flstgd=12, f0=13, gtslst=14, gtstep=4, 2 incfac=23, lmaxs=36, nreduc=6, plstgd=15, preduc=7, 3 radfac=16, rdfcmn=24, rdfcmx=25, reldx=17, rfctol=32, 4 sctol=37, stppar=5, tuner1=26, tuner2=27, tuner3=28, 5 xctol=33, xftol=34) c/ c c+++++++++++++++++++++++++++++++ body ++++++++++++++++++++++++++++++++ c nfc = iv(nfcall) iv(switch) = 0 iv(restor) = 0 rfac1 = one goodx = .true. i = iv(irc) if (i .ge. 1 .and. i .le. 12) 1 go to (20,30,10,10,40,280,220,220,220,220,220,170), i iv(irc) = 13 go to 999 c c *** initialize for new iteration *** c 10 iv(stage) = 1 iv(radinc) = 0 v(flstgd) = v(f0) if (iv(toobig) .eq. 0) go to 110 iv(stage) = -1 iv(xirc) = i go to 60 c c *** step was recomputed with new model or smaller radius *** c *** first decide which *** c 20 if (iv(model) .ne. iv(mlstgd)) go to 30 c *** old model retained, smaller radius tried *** c *** do not consider any more new models this iteration *** iv(stage) = iv(stglim) iv(radinc) = -1 go to 110 c c *** a new model is being tried. decide whether to keep it. *** c 30 iv(stage) = iv(stage) + 1 c c *** now we add the possibiltiy that step was recomputed with *** c *** the same model, perhaps because of an oversized step. *** c 40 if (iv(stage) .gt. 0) go to 50 c c *** step was recomputed because it was too big. *** c if (iv(toobig) .ne. 0) go to 60 c c *** restore iv(stage) and pick up where we left off. *** c iv(stage) = -iv(stage) i = iv(xirc) go to (20, 30, 110, 110, 70), i c 50 if (iv(toobig) .eq. 0) go to 70 c c *** handle oversize step *** c if (iv(radinc) .gt. 0) go to 80 iv(stage) = -iv(stage) iv(xirc) = iv(irc) c 60 v(radfac) = v(decfac) iv(radinc) = iv(radinc) - 1 iv(irc) = 5 iv(restor) = 1 go to 999 c 70 if (v(f) .lt. v(flstgd)) go to 110 c c *** the new step is a loser. restore old model. *** c if (iv(model) .eq. iv(mlstgd)) go to 80 iv(model) = iv(mlstgd) iv(switch) = 1 c c *** restore step, etc. only if a previous step decreased v(f). c 80 if (v(flstgd) .ge. v(f0)) go to 110 iv(restor) = 1 v(f) = v(flstgd) v(preduc) = v(plstgd) v(gtstep) = v(gtslst) if (iv(switch) .eq. 0) rfac1 = v(dstnrm) / v(dstsav) v(dstnrm) = v(dstsav) nfc = iv(nfgcal) goodx = .false. c 110 v(fdif) = v(f0) - v(f) if (v(fdif) .gt. v(tuner2) * v(preduc)) go to 140 if(iv(radinc).gt.0) go to 140 c c *** no (or only a trivial) function decrease c *** -- so try new model or smaller radius c if (v(f) .lt. v(f0)) go to 120 iv(mlstgd) = iv(model) v(flstgd) = v(f) v(f) = v(f0) iv(restor) = 1 go to 130 120 iv(nfgcal) = nfc 130 iv(irc) = 1 if (iv(stage) .lt. iv(stglim)) go to 160 iv(irc) = 5 iv(radinc) = iv(radinc) - 1 go to 160 c c *** nontrivial function decrease achieved *** c 140 iv(nfgcal) = nfc rfac1 = one v(dstsav) = v(dstnrm) if (v(fdif) .gt. v(preduc)*v(tuner1)) go to 190 c c *** decrease was much less than predicted -- either change models c *** or accept step with decreased radius. c if (iv(stage) .ge. iv(stglim)) go to 150 c *** consider switching models *** iv(irc) = 2 go to 160 c c *** accept step with decreased radius *** c 150 iv(irc) = 4 c c *** set v(radfac) to fletcher*s decrease factor *** c 160 iv(xirc) = iv(irc) emax = v(gtstep) + v(fdif) v(radfac) = half * rfac1 if (emax .lt. v(gtstep)) v(radfac) = rfac1 * dmax1(v(rdfcmn), 1 half * v(gtstep)/emax) c c *** do false convergence test *** c 170 if (v(reldx) .le. v(xftol)) go to 180 iv(irc) = iv(xirc) if (v(f) .lt. v(f0)) go to 200 go to 230 c 180 iv(irc) = 12 go to 240 c c *** handle good function decrease *** c 190 if (v(fdif) .lt. (-v(tuner3) * v(gtstep))) go to 210 c c *** increasing radius looks worthwhile. see if we just c *** recomputed step with a decreased radius or restored step c *** after recomputing it with a larger radius. c if (iv(radinc) .lt. 0) go to 210 if (iv(restor) .eq. 1) go to 210 c c *** we did not. try a longer step unless this was a newton c *** step. c v(radfac) = v(rdfcmx) gts = v(gtstep) if (v(fdif) .lt. (half/v(radfac) - one) * gts) 1 v(radfac) = dmax1(v(incfac), half*gts/(gts + v(fdif))) iv(irc) = 4 if (v(stppar) .eq. zero) go to 230 if (v(dst0) .ge. zero .and. (v(dst0) .lt. two*v(dstnrm) 1 .or. v(nreduc) .lt. onep2*v(fdif))) go to 230 c *** step was not a newton step. recompute it with c *** a larger radius. iv(irc) = 5 iv(radinc) = iv(radinc) + 1 c c *** save values corresponding to good step *** c 200 v(flstgd) = v(f) iv(mlstgd) = iv(model) if (iv(restor) .ne. 1) iv(restor) = 2 v(dstsav) = v(dstnrm) iv(nfgcal) = nfc v(plstgd) = v(preduc) v(gtslst) = v(gtstep) go to 230 c c *** accept step with radius unchanged *** c 210 v(radfac) = one iv(irc) = 3 go to 230 c c *** come here for a restart after convergence *** c 220 iv(irc) = iv(xirc) if (v(dstsav) .ge. zero) go to 240 iv(irc) = 12 go to 240 c c *** perform convergence tests *** c 230 iv(xirc) = iv(irc) 240 if (iv(restor) .eq. 1 .and. v(flstgd) .lt. v(f0)) iv(restor) = 3 if (half * v(fdif) .gt. v(preduc)) go to 999 emax = v(rfctol) * dabs(v(f0)) emaxs = v(sctol) * dabs(v(f0)) if (v(dstnrm) .gt. v(lmaxs) .and. v(preduc) .le. emaxs) 1 iv(irc) = 11 if (v(dst0) .lt. zero) go to 250 i = 0 if ((v(nreduc) .gt. zero .and. v(nreduc) .le. emax) .or. 1 (v(nreduc) .eq. zero. and. v(preduc) .eq. zero)) i = 2 if (v(stppar) .eq. zero .and. v(reldx) .le. v(xctol) 1 .and. goodx) i = i + 1 if (i .gt. 0) iv(irc) = i + 6 c c *** consider recomputing step of length v(lmaxs) for singular c *** convergence test. c 250 if (iv(irc) .gt. 5 .and. iv(irc) .ne. 12) go to 999 if (v(dstnrm) .gt. v(lmaxs)) go to 260 if (v(preduc) .ge. emaxs) go to 999 if (v(dst0) .le. zero) go to 270 if (half * v(dst0) .le. v(lmaxs)) go to 999 go to 270 260 if (half * v(dstnrm) .le. v(lmaxs)) go to 999 xmax = v(lmaxs) / v(dstnrm) if (xmax * (two - xmax) * v(preduc) .ge. emaxs) go to 999 270 if (v(nreduc) .lt. zero) go to 290 c c *** recompute v(preduc) for use in singular convergence test *** c v(gtslst) = v(gtstep) v(dstsav) = v(dstnrm) if (iv(irc) .eq. 12) v(dstsav) = -v(dstsav) v(plstgd) = v(preduc) i = iv(restor) iv(restor) = 2 if (i .eq. 3) iv(restor) = 0 iv(irc) = 6 go to 999 c c *** perform singular convergence test with recomputed v(preduc) *** c 280 v(gtstep) = v(gtslst) v(dstnrm) = dabs(v(dstsav)) iv(irc) = iv(xirc) if (v(dstsav) .le. zero) iv(irc) = 12 v(nreduc) = -v(preduc) v(preduc) = v(plstgd) iv(restor) = 3 290 if (-v(nreduc) .le. v(sctol) * dabs(v(f0))) iv(irc) = 11 c 999 return c c *** last card of assst follows *** end subroutine deflt(alg, iv, liv, lv, v) c c *** supply ***sol (version 2.3) default values to iv and v *** c c *** alg = 1 means regression constants. c *** alg = 2 means general unconstrained optimization constants. c integer liv, l integer alg, iv(liv) double precision v(lv) c external imdcon, vdflt integer imdcon c imdcon... returns machine-dependent integer constants. c vdflt.... provides default values to v. c integer miv, m integer miniv(2), minv(2) c c *** subscripts for iv *** c integer algsav, covprt, covreq, dtype, hc, ierr, inith, inits, 1 ipivot, ivneed, lastiv, lastv, lmat, mxfcal, mxiter, 2 nfcov, ngcov, nvdflt, outlev, parprt, parsav, perm, 3 prunit, qrtyp, rdreq, rmat, solprt, statpr, vneed, 4 vsave, x0prt c c *** iv subscript values *** c c/6 c data algsav/51/, covprt/14/, covreq/15/, dtype/16/, hc/71/, c 1 ierr/75/, inith/25/, inits/25/, ipivot/76/, ivneed/3/, c 2 lastiv/44/, lastv/45/, lmat/42/, mxfcal/17/, mxiter/18/, c 3 nfcov/52/, ngcov/53/, nvdflt/50/, outlev/19/, parprt/20/, c 4 parsav/49/, perm/58/, prunit/21/, qrtyp/80/, rdreq/57/, c 5 rmat/78/, solprt/22/, statpr/23/, vneed/4/, vsave/60/, c 6 x0prt/24/ c/7 parameter (algsav=51, covprt=14, covreq=15, dtype=16, hc=71, 1 ierr=75, inith=25, inits=25, ipivot=76, ivneed=3, 2 lastiv=44, lastv=45, lmat=42, mxfcal=17, mxiter=18, 3 nfcov=52, ngcov=53, nvdflt=50, outlev=19, parprt=20, 4 parsav=49, perm=58, prunit=21, qrtyp=80, rdreq=57, 5 rmat=78, solprt=22, statpr=23, vneed=4, vsave=60, 6 x0prt=24) c/ data miniv(1)/80/, miniv(2)/59/, minv(1)/98/, minv(2)/71/ c c------------------------------- body -------------------------------- c if (alg .lt. 1 .or. alg .gt. 2) go to 40 miv = miniv(alg) if (liv .lt. miv) go to 20 mv = minv(alg) if (lv .lt. mv) go to 30 call vdflt(alg, lv, v) iv(1) = 12 iv(algsav) = alg iv(ivneed) = 0 iv(lastiv) = miv iv(lastv) = mv iv(lmat) = mv + 1 iv(mxfcal) = 200 iv(mxiter) = 150 iv(outlev) = 1 iv(parprt) = 1 iv(perm) = miv + 1 iv(prunit) = imdcon(1) iv(solprt) = 1 iv(statpr) = 1 iv(vneed) = 0 iv(x0prt) = 1 c if (alg .ge. 2) go to 10 c c *** regression values c iv(covprt) = 3 iv(covreq) = 1 iv(dtype) = 1 iv(hc) = 0 iv(ierr) = 0 iv(inits) = 0 iv(ipivot) = 0 iv(nvdflt) = 32 iv(parsav) = 67 iv(qrtyp) = 1 iv(rdreq) = 3 iv(rmat) = 0 iv(vsave) = 58 go to 999 c c *** general optimization values c 10 iv(dtype) = 0 iv(inith) = 1 iv(nfcov) = 0 iv(ngcov) = 0 iv(nvdflt) = 25 iv(parsav) = 47 go to 999 c 20 iv(1) = 15 go to 999 c 30 iv(1) = 16 go to 999 c 40 iv(1) = 67 c 999 return c *** last card of deflt follows *** end double precision function dotprd(p, x, y) c c *** return the inner product of the p-vectors x and y. *** c integer p double precision x(p), y(p) c integer i double precision one, sqteta, t, zero c/+ double precision dmax1, dabs c/ external rmdcon double precision rmdcon c c *** rmdcon(2) returns a machine-dependent constant, sqteta, which c *** is slightly larger than the smallest positive number that c *** can be squared without underflowing. c c/6 c data one/1.d+0/, sqteta/0.d+0/, zero/0.d+0/ c/7 parameter (one=1.d+0, zero=0.d+0) data sqteta/0.d+0/ c/ c dotprd = zero if (p .le. 0) go to 999 crc if (sqteta .eq. zero) sqteta = rmdcon(2) do 20 i = 1, p crc t = dmax1(dabs(x(i)), dabs(y(i))) crc if (t .gt. one) go to 10 crc if (t .lt. sqteta) go to 20 crc t = (x(i)/sqteta)*y(i) crc if (dabs(t) .lt. sqteta) go to 20 10 dotprd = dotprd + x(i)*y(i) 20 continue c 999 return c *** last card of dotprd follows *** end subroutine itsum(d, g, iv, liv, lv, p, v, x) c c *** print iteration summary for ***sol (version 2.3) *** c c *** parameter declarations *** c integer liv, lv, p integer iv(liv) double precision d(p), g(p), v(lv), x(p) c c+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ c c *** local variables *** c integer alg, i, iv1, m, nf, ng, ol, pu c/6 c real model1(6), model2(6) c/7 character*4 model1(6), model2(6) c/ double precision nreldf, oldf, preldf, reldf, zero c c *** intrinsic functions *** c/+ integer iabs double precision dabs, dmax1 c/ c *** no external functions or subroutines *** c c *** subscripts for iv and v *** c integer algsav, dstnrm, f, fdif, f0, needhd, nfcall, nfcov, ngcov, 1 ngcall, niter, nreduc, outlev, preduc, prntit, prunit, 2 reldx, solprt, statpr, stppar, sused, x0prt c c *** iv subscript values *** c c/6 c data algsav/51/, needhd/36/, nfcall/6/, nfcov/52/, ngcall/30/, c 1 ngcov/53/, niter/31/, outlev/19/, prntit/39/, prunit/21/, c 2 solprt/22/, statpr/23/, sused/64/, x0prt/24/ c/7 parameter (algsav=51, needhd=36, nfcall=6, nfcov=52, ngcall=30, 1 ngcov=53, niter=31, outlev=19, prntit=39, prunit=21, 2 solprt=22, statpr=23, sused=64, x0prt=24) c/ c c *** v subscript values *** c c/6 c data dstnrm/2/, f/10/, f0/13/, fdif/11/, nreduc/6/, preduc/7/, c 1 reldx/17/, stppar/5/ c/7 parameter (dstnrm=2, f=10, f0=13, fdif=11, nreduc=6, preduc=7, 1 reldx=17, stppar=5) c/ c c/6 c data zero/0.d+0/ c/7 parameter (zero=0.d+0) c/ c/6 c data model1(1)/4h /, model1(2)/4h /, model1(3)/4h /, c 1 model1(4)/4h /, model1(5)/4h g /, model1(6)/4h s /, c 2 model2(1)/4h g /, model2(2)/4h s /, model2(3)/4hg-s /, c 3 model2(4)/4hs-g /, model2(5)/4h-s-g/, model2(6)/4h-g-s/ c/7 data model1/' ',' ',' ',' ',' g ',' s '/, 1 model2/' g ',' s ','g-s ','s-g ','-s-g','-g-s'/ c/ c c------------------------------- body -------------------------------- c pu = iv(prunit) if (pu .eq. 0) go to 999 iv1 = iv(1) if (iv1 .gt. 62) iv1 = iv1 - 51 ol = iv(outlev) alg = iv(algsav) if (iv1 .lt. 2 .or. iv1 .gt. 15) go to 370 if (iv1 .ge. 12) go to 120 if (iv1 .eq. 2 .and. iv(niter) .eq. 0) go to 390 if (ol .eq. 0) go to 120 if (iv1 .ge. 10 .and. iv(prntit) .eq. 0) go to 120 if (iv1 .gt. 2) go to 10 iv(prntit) = iv(prntit) + 1 if (iv(prntit) .lt. iabs(ol)) go to 999 10 nf = iv(nfcall) - iabs(iv(nfcov)) iv(prntit) = 0 reldf = zero preldf = zero oldf = dmax1(dabs(v(f0)), dabs(v(f))) if (oldf .le. zero) go to 20 reldf = v(fdif) / oldf preldf = v(preduc) / oldf 20 if (ol .gt. 0) go to 60 c c *** print short summary line *** c if (iv(needhd) .eq. 1 .and. alg .eq. 1) write(pu,30) 30 format(/10h it nf,6x,1hf,7x,5hreldf,3x,6hpreldf,3x,5hreldx, 1 2x,13hmodel stppar) if (iv(needhd) .eq. 1 .and. alg .eq. 2) write(pu,40) 40 format(/11h it nf,7x,1hf,8x,5hreldf,4x,6hpreldf,4x,5hreldx, 1 3x,6hstppar) iv(needhd) = 0 if (alg .eq. 2) go to 50 m = iv(sused) write(pu,100) iv(niter), nf, v(f), reldf, preldf, v(reldx), 1 model1(m), model2(m), v(stppar) go to 120 c 50 write(pu,110) iv(niter), nf, v(f), reldf, preldf, v(reldx), 1 v(stppar) go to 120 c c *** print long summary line *** c 60 if (iv(needhd) .eq. 1 .and. alg .eq. 1) write(pu,70) 70 format(/11h it nf,6x,1hf,7x,5hreldf,3x,6hpreldf,3x,5hreldx, 1 2x,13hmodel stppar,2x,6hd*step,2x,7hnpreldf) if (iv(needhd) .eq. 1 .and. alg .eq. 2) write(pu,80) 80 format(/11h it nf,7x,1hf,8x,5hreldf,4x,6hpreldf,4x,5hreldx, 1 3x,6hstppar,3x,6hd*step,3x,7hnpreldf) iv(needhd) = 0 nreldf = zero if (oldf .gt. zero) nreldf = v(nreduc) / oldf if (alg .eq. 2) go to 90 m = iv(sused) write(pu,100) iv(niter), nf, v(f), reldf, preldf, v(reldx), 1 model1(m), model2(m), v(stppar), v(dstnrm), nreldf go to 120 c 90 write(pu,110) iv(niter), nf, v(f), reldf, preldf, 1 v(reldx), v(stppar), v(dstnrm), nreldf 100 format(i6,i5,d10.3,2d9.2,d8.1,a3,a4,2d8.1,d9.2) 110 format(i6,i5,d11.3,2d10.2,3d9.1,d10.2) c 120 if (iv(statpr) .lt. 0) go to 430 go to (999, 999, 130, 150, 170, 190, 210, 230, 250, 270, 290, 310, 1 330, 350, 520), iv1 c 130 write(pu,140) 140 format(/26h ***** x-convergence *****) go to 430 c 150 write(pu,160) 160 format(/42h ***** relative function convergence *****) go to 430 c 170 write(pu,180) 180 format(/49h ***** x- and relative function convergence *****) go to 430 c 190 write(pu,200) 200 format(/42h ***** absolute function convergence *****) go to 430 c 210 write(pu,220) 220 format(/33h ***** singular convergence *****) go to 430 c 230 write(pu,240) 240 format(/30h ***** false convergence *****) go to 430 c 250 write(pu,260) 260 format(/38h ***** function evaluation limit *****) go to 430 c 270 write(pu,280) 280 format(/28h ***** iteration limit *****) go to 430 c 290 write(pu,300) 300 format(/18h ***** stopx *****) go to 430 c 310 write(pu,320) 320 format(/44h ***** initial f(x) cannot be computed *****) c go to 390 c 330 write(pu,340) 340 format(/37h ***** bad parameters to assess *****) go to 999 c 350 write(pu,360) 360 format(/43h ***** gradient could not be computed *****) if (iv(niter) .gt. 0) go to 480 go to 390 c 370 write(pu,380) iv(1) 380 format(/14h ***** iv(1) =,i5,6h *****) go to 999 c c *** initial call on itsum *** c 390 if (iv(x0prt) .ne. 0) write(pu,400) (i, x(i), d(i), i = 1, p) 400 format(/23h i initial x(i),8x,4hd(i)//(1x,i5,d17.6,d14.3)) c *** the following are to avoid undefined variables when the c *** function evaluation limit is 1... v(dstnrm) = zero v(fdif) = zero v(nreduc) = zero v(preduc) = zero v(reldx) = zero if (iv1 .ge. 12) go to 999 iv(needhd) = 0 iv(prntit) = 0 if (ol .eq. 0) go to 999 if (ol .lt. 0 .and. alg .eq. 1) write(pu,30) if (ol .lt. 0 .and. alg .eq. 2) write(pu,40) if (ol .gt. 0 .and. alg .eq. 1) write(pu,70) if (ol .gt. 0 .and. alg .eq. 2) write(pu,80) if (alg .eq. 1) write(pu,410) v(f) if (alg .eq. 2) write(pu,420) v(f) 410 format(/11h 0 1,d10.3) c365 format(/11h 0 1,e11.3) 420 format(/11h 0 1,d11.3) go to 999 c c *** print various information requested on solution *** c 430 iv(needhd) = 1 if (iv(statpr) .eq. 0) go to 480 oldf = dmax1(dabs(v(f0)), dabs(v(f))) preldf = zero nreldf = zero if (oldf .le. zero) go to 440 preldf = v(preduc) / oldf nreldf = v(nreduc) / oldf 440 nf = iv(nfcall) - iv(nfcov) ng = iv(ngcall) - iv(ngcov) write(pu,450) v(f), v(reldx), nf, ng, preldf, nreldf 450 format(/9h function,d17.6,8h reldx,d17.3/12h func. evals, 1 i8,9x,11hgrad. evals,i8/7h preldf,d16.3,6x,7hnpreldf,d15.3) c if (iv(nfcov) .gt. 0) write(pu,460) iv(nfcov) 460 format(/1x,i4,50h extra func. evals for covariance and diagnost 1ics.) if (iv(ngcov) .gt. 0) write(pu,470) iv(ngcov) 470 format(1x,i4,50h extra grad. evals for covariance and diagnosti 1cs.) c 480 if (iv(solprt) .eq. 0) go to 999 iv(needhd) = 1 write(pu,490) 490 format(/22h i final x(i),8x,4hd(i),10x,4hg(i)/) do 500 i = 1, p write(pu,510) i, x(i), d(i), g(i) 500 continue 510 format(1x,i5,d16.6,2d14.3) go to 999 c 520 write(pu,530) 530 format(/24h inconsistent dimensions) 999 return c *** last card of itsum follows *** end subroutine litvmu(n, x, l, y) c c *** solve (l**t)*x = y, where l is an n x n lower triangular c *** matrix stored compactly by rows. x and y may occupy the same c *** storage. *** c integer n cal double precision x(n), l(1), y(n) double precision x(n), l(n*(n+1)/2), y(n) integer i, ii, ij, im1, i0, j, np1 double precision xi, zero c/6 c data zero/0.d+0/ c/7 parameter (zero=0.d+0) c/ c do 10 i = 1, n 10 x(i) = y(i) np1 = n + 1 i0 = n*(n+1)/2 do 30 ii = 1, n i = np1 - ii xi = x(i)/l(i0) x(i) = xi if (i .le. 1) go to 999 i0 = i0 - i if (xi .eq. zero) go to 30 im1 = i - 1 do 20 j = 1, im1 ij = i0 + j x(j) = x(j) - xi*l(ij) 20 continue 30 continue 999 return c *** last card of litvmu follows *** end subroutine livmul(n, x, l, y) c c *** solve l*x = y, where l is an n x n lower triangular c *** matrix stored compactly by rows. x and y may occupy the same c *** storage. *** c integer n cal double precision x(n), l(1), y(n) double precision x(n), l(n*(n+1)/2), y(n) external dotprd double precision dotprd integer i, j, k double precision t, zero c/6 c data zero/0.d+0/ c/7 parameter (zero=0.d+0) c/ c do 10 k = 1, n if (y(k) .ne. zero) go to 20 x(k) = zero 10 continue go to 999 20 j = k*(k+1)/2 x(k) = y(k) / l(j) if (k .ge. n) go to 999 k = k + 1 do 30 i = k, n t = dotprd(i-1, l(j+1), x) j = j + i x(i) = (y(i) - t)/l(j) 30 continue 999 return c *** last card of livmul follows *** end subroutine parck(alg, d, iv, liv, lv, n, v) c c *** check ***sol (version 2.3) parameters, print changed values *** c c *** alg = 1 for regression, alg = 2 for general unconstrained opt. c integer alg, liv, lv, n integer iv(liv) double precision d(n), v(lv) c external rmdcon, vcopy, vdflt double precision rmdcon c rmdcon -- returns machine-dependent constants. c vcopy -- copies one vector to another. c vdflt -- supplies default parameter values to v alone. c/+ integer max0 c/ c c *** local variables *** c integer i, ii, iv1, j, k, l, m, miv1, miv2, ndfalt, parsv1, pu integer ijmp, jlim(2), miniv(2), ndflt(2) c/6 c integer varnm(2), sh(2) c real cngd(3), dflt(3), vn(2,34), which(3) c/7 character*1 varnm(2), sh(2) character*4 cngd(3), dflt(3), vn(2,34), which(3) c/ double precision big, machep, tiny, vk, vm(34), vx(34), zero c c *** iv and v subscripts *** c integer algsav, dinit, dtype, dtype0, epslon, inits, ivneed, 1 lastiv, lastv, lmat, nextiv, nextv, nvdflt, oldn, 2 parprt, parsav, perm, prunit, vneed c c c/6 c data algsav/51/, dinit/38/, dtype/16/, dtype0/54/, epslon/19/, c 1 inits/25/, ivneed/3/, lastiv/44/, lastv/45/, lmat/42/, c 2 nextiv/46/, nextv/47/, nvdflt/50/, oldn/38/, parprt/20/, c 3 parsav/49/, perm/58/, prunit/21/, vneed/4/ c/7 parameter (algsav=51, dinit=38, dtype=16, dtype0=54, epslon=19, 1 inits=25, ivneed=3, lastiv=44, lastv=45, lmat=42, 2 nextiv=46, nextv=47, nvdflt=50, oldn=38, parprt=20, 3 parsav=49, perm=58, prunit=21, vneed=4) save big, machep, tiny c/ c data big/0.d+0/, machep/-1.d+0/, tiny/1.d+0/, zero/0.d+0/ c/6 c data vn(1,1),vn(2,1)/4hepsl,4hon../ c data vn(1,2),vn(2,2)/4hphmn,4hfc../ c data vn(1,3),vn(2,3)/4hphmx,4hfc../ c data vn(1,4),vn(2,4)/4hdecf,4hac../ c data vn(1,5),vn(2,5)/4hincf,4hac../ c data vn(1,6),vn(2,6)/4hrdfc,4hmn../ c data vn(1,7),vn(2,7)/4hrdfc,4hmx../ c data vn(1,8),vn(2,8)/4htune,4hr1../ c data vn(1,9),vn(2,9)/4htune,4hr2../ c data vn(1,10),vn(2,10)/4htune,4hr3../ c data vn(1,11),vn(2,11)/4htune,4hr4../ c data vn(1,12),vn(2,12)/4htune,4hr5../ c data vn(1,13),vn(2,13)/4hafct,4hol../ c data vn(1,14),vn(2,14)/4hrfct,4hol../ c data vn(1,15),vn(2,15)/4hxcto,4hl.../ c data vn(1,16),vn(2,16)/4hxfto,4hl.../ c data vn(1,17),vn(2,17)/4hlmax,4h0.../ c data vn(1,18),vn(2,18)/4hlmax,4hs.../ c data vn(1,19),vn(2,19)/4hscto,4hl.../ c data vn(1,20),vn(2,20)/4hdini,4ht.../ c data vn(1,21),vn(2,21)/4hdtin,4hit../ c data vn(1,22),vn(2,22)/4hd0in,4hit../ c data vn(1,23),vn(2,23)/4hdfac,4h..../ c data vn(1,24),vn(2,24)/4hdltf,4hdc../ c data vn(1,25),vn(2,25)/4hdltf,4hdj../ c data vn(1,26),vn(2,26)/4hdelt,4ha0../ c data vn(1,27),vn(2,27)/4hfuzz,4h..../ c data vn(1,28),vn(2,28)/4hrlim,4hit../ c data vn(1,29),vn(2,29)/4hcosm,4hin../ c data vn(1,30),vn(2,30)/4hhube,4hrc../ c data vn(1,31),vn(2,31)/4hrspt,4hol../ c data vn(1,32),vn(2,32)/4hsigm,4hin../ c data vn(1,33),vn(2,33)/4heta0,4h..../ c data vn(1,34),vn(2,34)/4hbias,4h..../ c/7 data vn(1,1),vn(2,1)/'epsl','on..'/ data vn(1,2),vn(2,2)/'phmn','fc..'/ data vn(1,3),vn(2,3)/'phmx','fc..'/ data vn(1,4),vn(2,4)/'decf','ac..'/ data vn(1,5),vn(2,5)/'incf','ac..'/ data vn(1,6),vn(2,6)/'rdfc','mn..'/ data vn(1,7),vn(2,7)/'rdfc','mx..'/ data vn(1,8),vn(2,8)/'tune','r1..'/ data vn(1,9),vn(2,9)/'tune','r2..'/ data vn(1,10),vn(2,10)/'tune','r3..'/ data vn(1,11),vn(2,11)/'tune','r4..'/ data vn(1,12),vn(2,12)/'tune','r5..'/ data vn(1,13),vn(2,13)/'afct','ol..'/ data vn(1,14),vn(2,14)/'rfct','ol..'/ data vn(1,15),vn(2,15)/'xcto','l...'/ data vn(1,16),vn(2,16)/'xfto','l...'/ data vn(1,17),vn(2,17)/'lmax','0...'/ data vn(1,18),vn(2,18)/'lmax','s...'/ data vn(1,19),vn(2,19)/'scto','l...'/ data vn(1,20),vn(2,20)/'dini','t...'/ data vn(1,21),vn(2,21)/'dtin','it..'/ data vn(1,22),vn(2,22)/'d0in','it..'/ data vn(1,23),vn(2,23)/'dfac','....'/ data vn(1,24),vn(2,24)/'dltf','dc..'/ data vn(1,25),vn(2,25)/'dltf','dj..'/ data vn(1,26),vn(2,26)/'delt','a0..'/ data vn(1,27),vn(2,27)/'fuzz','....'/ data vn(1,28),vn(2,28)/'rlim','it..'/ data vn(1,29),vn(2,29)/'cosm','in..'/ data vn(1,30),vn(2,30)/'hube','rc..'/ data vn(1,31),vn(2,31)/'rspt','ol..'/ data vn(1,32),vn(2,32)/'sigm','in..'/ data vn(1,33),vn(2,33)/'eta0','....'/ data vn(1,34),vn(2,34)/'bias','....'/ c/ c data vm(1)/1.0d-3/, vm(2)/-0.99d+0/, vm(3)/1.0d-3/, vm(4)/1.0d-2/, 1 vm(5)/1.2d+0/, vm(6)/1.d-2/, vm(7)/1.2d+0/, vm(8)/0.d+0/, 2 vm(9)/0.d+0/, vm(10)/1.d-3/, vm(11)/-1.d+0/, vm(13)/0.d+0/, 3 vm(15)/0.d+0/, vm(16)/0.d+0/, vm(19)/0.d+0/, vm(20)/-10.d+0/, 4 vm(21)/0.d+0/, vm(22)/0.d+0/, vm(23)/0.d+0/, vm(27)/1.01d+0/, 5 vm(28)/1.d+10/, vm(30)/0.d+0/, vm(31)/0.d+0/, vm(32)/0.d+0/, 6 vm(34)/0.d+0/ data vx(1)/0.9d+0/, vx(2)/-1.d-3/, vx(3)/1.d+1/, vx(4)/0.8d+0/, 1 vx(5)/1.d+2/, vx(6)/0.8d+0/, vx(7)/1.d+2/, vx(8)/0.5d+0/, 2 vx(9)/0.5d+0/, vx(10)/1.d+0/, vx(11)/1.d+0/, vx(14)/0.1d+0/, 3 vx(15)/1.d+0/, vx(16)/1.d+0/, vx(19)/1.d+0/, vx(23)/1.d+0/, 4 vx(24)/1.d+0/, vx(25)/1.d+0/, vx(26)/1.d+0/, vx(27)/1.d+10/, 5 vx(29)/1.d+0/, vx(31)/1.d+0/, vx(32)/1.d+0/, vx(33)/1.d+0/, 6 vx(34)/1.d+0/ c c/6 c data varnm(1)/1hp/, varnm(2)/1hn/, sh(1)/1hs/, sh(2)/1hh/ c data cngd(1),cngd(2),cngd(3)/4h---c,4hhang,4hed v/, c 1 dflt(1),dflt(2),dflt(3)/4hnond,4hefau,4hlt v/ c/7 data varnm(1)/'p'/, varnm(2)/'n'/, sh(1)/'s'/, sh(2)/'h'/ data cngd(1),cngd(2),cngd(3)/'---c','hang','ed v'/, 1 dflt(1),dflt(2),dflt(3)/'nond','efau','lt v'/ c/ data ijmp/33/, jlim(1)/0/, jlim(2)/24/, ndflt(1)/32/, ndflt(2)/25/ data miniv(1)/80/, miniv(2)/59/ c c............................... body ................................ c pu = 0 if (prunit .le. liv) pu = iv(prunit) if (alg .lt. 1 .or. alg .gt. 2) go to 340 if (iv(1) .eq. 0) call deflt(alg, iv, liv, lv, v) iv1 = iv(1) if (iv1 .ne. 13 .and. iv1 .ne. 12) go to 10 miv1 = miniv(alg) if (perm .le. liv) miv1 = max0(miv1, iv(perm) - 1) if (ivneed .le. liv) miv2 = miv1 + max0(iv(ivneed), 0) if (lastiv .le. liv) iv(lastiv) = miv2 if (liv .lt. miv1) go to 300 iv(ivneed) = 0 iv(lastv) = max0(iv(vneed), 0) + iv(lmat) - 1 iv(vneed) = 0 if (liv .lt. miv2) go to 300 if (lv .lt. iv(lastv)) go to 320 10 if (alg .eq. iv(algsav)) go to 30 if (pu .ne. 0) write(pu,20) alg, iv(algsav) 20 format(/39h the first parameter to deflt should be,i3, 1 12h rather than,i3) iv(1) = 82 go to 999 30 if (iv1 .lt. 12 .or. iv1 .gt. 14) go to 60 if (n .ge. 1) go to 50 iv(1) = 81 if (pu .eq. 0) go to 999 write(pu,40) varnm(alg), n 40 format(/8h /// bad,a1,2h =,i5) go to 999 50 if (iv1 .ne. 14) iv(nextiv) = iv(perm) if (iv1 .ne. 14) iv(nextv) = iv(lmat) if (iv1 .eq. 13) go to 999 k = iv(parsav) - epslon call vdflt(alg, lv-k, v(k+1)) iv(dtype0) = 2 - alg iv(oldn) = n which(1) = dflt(1) which(2) = dflt(2) which(3) = dflt(3) go to 110 60 if (n .eq. iv(oldn)) go to 80 iv(1) = 17 if (pu .eq. 0) go to 999 write(pu,70) varnm(alg), iv(oldn), n 70 format(/5h /// ,1a1,14h changed from ,i5,4h to ,i5) go to 999 c 80 if (iv1 .le. 11 .and. iv1 .ge. 1) go to 100 iv(1) = 80 if (pu .ne. 0) write(pu,90) iv1 90 format(/13h /// iv(1) =,i5,28h should be between 0 and 14.) go to 999 c 100 which(1) = cngd(1) which(2) = cngd(2) which(3) = cngd(3) c 110 if (iv1 .eq. 14) iv1 = 12 if (big .gt. tiny) go to 120 tiny = rmdcon(1) machep = rmdcon(3) big = rmdcon(6) vm(12) = machep vx(12) = big vx(13) = big vm(14) = machep vm(17) = tiny vx(17) = big vm(18) = tiny vx(18) = big vx(20) = big vx(21) = big vx(22) = big vm(24) = machep vm(25) = machep vm(26) = machep vx(28) = rmdcon(5) vm(29) = machep vx(30) = big vm(33) = machep 120 m = 0 i = 1 j = jlim(alg) k = epslon ndfalt = ndflt(alg) do 150 l = 1, ndfalt vk = v(k) if (vk .ge. vm(i) .and. vk .le. vx(i)) go to 140 m = k if (pu .ne. 0) write(pu,130) vn(1,i), vn(2,i), k, vk, 1 vm(i), vx(i) 130 format(/6h /// ,2a4,5h.. v(,i2,3h) =,d11.3,7h should, 1 11h be between,d11.3,4h and,d11.3) 140 k = k + 1 i = i + 1 if (i .eq. j) i = ijmp 150 continue c if (iv(nvdflt) .eq. ndfalt) go to 170 iv(1) = 51 if (pu .eq. 0) go to 999 write(pu,160) iv(nvdflt), ndfalt 160 format(/13h iv(nvdflt) =,i5,13h rather than ,i5) go to 999 170 if ((iv(dtype) .gt. 0 .or. v(dinit) .gt. zero) .and. iv1 .eq. 12) 1 go to 200 do 190 i = 1, n if (d(i) .gt. zero) go to 190 m = 18 if (pu .ne. 0) write(pu,180) i, d(i) 180 format(/8h /// d(,i3,3h) =,d11.3,19h should be positive) 190 continue 200 if (m .eq. 0) go to 210 iv(1) = m go to 999 c 210 if (pu .eq. 0 .or. iv(parprt) .eq. 0) go to 999 if (iv1 .ne. 12 .or. iv(inits) .eq. alg-1) go to 230 m = 1 write(pu,220) sh(alg), iv(inits) 220 format(/22h nondefault values..../5h init,a1,14h..... iv(25) =, 1 i3) 230 if (iv(dtype) .eq. iv(dtype0)) go to 250 if (m .eq. 0) write(pu,260) which m = 1 write(pu,240) iv(dtype) 240 format(20h dtype..... iv(16) =,i3) 250 i = 1 j = jlim(alg) k = epslon l = iv(parsav) ndfalt = ndflt(alg) do 290 ii = 1, ndfalt if (v(k) .eq. v(l)) go to 280 if (m .eq. 0) write(pu,260) which 260 format(/1h ,3a4,9halues..../) m = 1 write(pu,270) vn(1,i), vn(2,i), k, v(k) 270 format(1x,2a4,5h.. v(,i2,3h) =,d15.7) 280 k = k + 1 l = l + 1 i = i + 1 if (i .eq. j) i = ijmp 290 continue c iv(dtype0) = iv(dtype) parsv1 = iv(parsav) call vcopy(iv(nvdflt), v(parsv1), v(epslon)) go to 999 c 300 iv(1) = 15 if (pu .eq. 0) go to 999 write(pu,310) liv, miv2 310 format(/10h /// liv =,i5,17h must be at least,i5) if (liv .lt. miv1) go to 999 if (lv .lt. iv(lastv)) go to 320 go to 999 c 320 iv(1) = 16 if (pu .eq. 0) go to 999 write(pu,330) lv, iv(lastv) 330 format(/9h /// lv =,i5,17h must be at least,i5) go to 999 c 340 iv(1) = 67 if (pu .eq. 0) go to 999 write(pu,350) alg 350 format(/10h /// alg =,i5,15h must be 1 or 2) c 999 return c *** last card of parck follows *** end double precision function reldst(p, d, x, x0) c c *** compute and return relative difference between x and x0 *** c *** nl2sol version 2.2 *** c integer p double precision d(p), x(p), x0(p) c/+ double precision dabs c/ integer i double precision emax, t, xmax, zero c/6 c data zero/0.d+0/ c/7 parameter (zero=0.d+0) c/ c emax = zero xmax = zero do 10 i = 1, p t = dabs(d(i) * (x(i) - x0(i))) if (emax .lt. t) emax = t t = d(i) * (dabs(x(i)) + dabs(x0(i))) if (xmax .lt. t) xmax = t 10 continue reldst = zero if (xmax .gt. zero) reldst = emax / xmax 999 return c *** last card of reldst follows *** end c logical function stopx(idummy) c *****parameters... c integer idummy c c .................................................................. c c *****purpose... c this function may serve as the stopx (asynchronous interruption) c function for the nl2sol (nonlinear least-squares) package at c those installations which do not wish to implement a c dynamic stopx. c c *****algorithm notes... c at installations where the nl2sol system is used c interactively, this dummy stopx should be replaced by a c function that returns .true. if and only if the interrupt c (break) key has been pressed since the last call on stopx. c c .................................................................. c c stopx = .false. c return c end subroutine vaxpy(p, w, a, x, y) c c *** set w = a*x + y -- w, x, y = p-vectors, a = scalar *** c integer p double precision a, w(p), x(p), y(p) c integer i c do 10 i = 1, p 10 w(i) = a*x(i) + y(i) return end subroutine vcopy(p, y, x) c c *** set y = x, where x and y are p-vectors *** c integer p double precision x(p), y(p) c integer i c do 10 i = 1, p 10 y(i) = x(i) return end subroutine vdflt(alg, lv, v) c c *** supply ***sol (version 2.3) default values to v *** c c *** alg = 1 means regression constants. c *** alg = 2 means general unconstrained optimization constants. c integer alg, l double precision v(lv) c/+ double precision dmax1 c/ external rmdcon double precision rmdcon c rmdcon... returns machine-dependent constants c double precision machep, mepcrt, one, sqteps, three c c *** subscripts for v *** c integer afctol, bias, cosmin, decfac, delta0, dfac, dinit, dltfdc, 1 dltfdj, dtinit, d0init, epslon, eta0, fuzz, huberc, 2 incfac, lmax0, lmaxs, phmnfc, phmxfc, rdfcmn, rdfcmx, 3 rfctol, rlimit, rsptol, sctol, sigmin, tuner1, tuner2, 4 tuner3, tuner4, tuner5, xctol, xftol c c/6 c data one/1.d+0/, three/3.d+0/ c/7 parameter (one=1.d+0, three=3.d+0) c/ c c *** v subscript values *** c c/6 c data afctol/31/, bias/43/, cosmin/47/, decfac/22/, delta0/44/, c 1 dfac/41/, dinit/38/, dltfdc/42/, dltfdj/43/, dtinit/39/, c 2 d0init/40/, epslon/19/, eta0/42/, fuzz/45/, huberc/48/, c 3 incfac/23/, lmax0/35/, lmaxs/36/, phmnfc/20/, phmxfc/21/, c 4 rdfcmn/24/, rdfcmx/25/, rfctol/32/, rlimit/46/, rsptol/49/, c 5 sctol/37/, sigmin/50/, tuner1/26/, tuner2/27/, tuner3/28/, c 6 tuner4/29/, tuner5/30/, xctol/33/, xftol/34/ c/7 parameter (afctol=31, bias=43, cosmin=47, decfac=22, delta0=44, 1 dfac=41, dinit=38, dltfdc=42, dltfdj=43, dtinit=39, 2 d0init=40, epslon=19, eta0=42, fuzz=45, huberc=48, 3 incfac=23, lmax0=35, lmaxs=36, phmnfc=20, phmxfc=21, 4 rdfcmn=24, rdfcmx=25, rfctol=32, rlimit=46, rsptol=49, 5 sctol=37, sigmin=50, tuner1=26, tuner2=27, tuner3=28, 6 tuner4=29, tuner5=30, xctol=33, xftol=34) c/ c c------------------------------- body -------------------------------- c machep = rmdcon(3) v(afctol) = 1.d-20 if (machep .gt. 1.d-10) v(afctol) = machep**2 v(decfac) = 0.5d+0 sqteps = rmdcon(4) v(dfac) = 0.6d+0 v(delta0) = sqteps v(dtinit) = 1.d-6 mepcrt = machep ** (one/three) v(d0init) = 1.d+0 v(epslon) = 0.1d+0 v(incfac) = 2.d+0 v(lmax0) = 1.d+0 v(lmaxs) = 1.d+0 v(phmnfc) = -0.1d+0 v(phmxfc) = 0.1d+0 v(rdfcmn) = 0.1d+0 v(rdfcmx) = 4.d+0 v(rfctol) = dmax1(1.d-10, mepcrt**2) v(sctol) = v(rfctol) v(tuner1) = 0.1d+0 v(tuner2) = 1.d-4 v(tuner3) = 0.75d+0 v(tuner4) = 0.5d+0 v(tuner5) = 0.75d+0 v(xctol) = sqteps v(xftol) = 1.d+2 * machep c if (alg .ge. 2) go to 10 c c *** regression values c v(cosmin) = dmax1(1.d-6, 1.d+2 * machep) v(dinit) = 0.d+0 v(dltfdc) = mepcrt v(dltfdj) = sqteps v(fuzz) = 1.5d+0 v(huberc) = 0.7d+0 v(rlimit) = rmdcon(5) v(rsptol) = 1.d-3 v(sigmin) = 1.d-4 go to 999 c c *** general optimization values c 10 v(bias) = 0.8d+0 v(dinit) = -1.0d+0 v(eta0) = 1.0d+3 * machep c 999 return c *** last card of vdflt follows *** end subroutine vscopy(p, y, s) c c *** set p-vector y to scalar s *** c integer p double precision s, y(p) c integer i c do 10 i = 1, p 10 y(i) = s return end double precision function v2norm(p, x) c c *** return the 2-norm of the p-vector x, taking *** c *** care to avoid the most likely underflows. *** c integer p double precision x(p) c integer i, j double precision one, r, scale, sqteta, t, xi, zero c/+ double precision dabs, dsqrt c/ external rmdcon double precision rmdcon c c/6 c data one/1.d+0/, zero/0.d+0/ c/7 parameter (one=1.d+0, zero=0.d+0) save sqteta c/ data sqteta/0.d+0/ c if (p .gt. 0) go to 10 v2norm = zero go to 999 10 do 20 i = 1, p if (x(i) .ne. zero) go to 30 20 continue v2norm = zero go to 999 c 30 scale = dabs(x(i)) if (i .lt. p) go to 40 v2norm = scale go to 999 40 t = one if (sqteta .eq. zero) sqteta = rmdcon(2) c c *** sqteta is (slightly larger than) the square root of the c *** smallest positive floating point number on the machine. c *** the tests involving sqteta are done to prevent underflows. c j = i + 1 do 60 i = j, p xi = dabs(x(i)) if (xi .gt. scale) go to 50 r = xi / scale if (r .gt. sqteta) t = t + r*r go to 60 50 r = scale / xi if (r .le. sqteta) r = zero t = one + t * r*r scale = xi 60 continue c v2norm = scale * dsqrt(t) 999 return c *** last card of v2norm follows *** end subroutine humsl(n, d, x, calcf, calcgh, iv, liv, lv, v, 1 uiparm, urparm, ufparm) c c *** minimize general unconstrained objective function using *** c *** (analytic) gradient and hessian provided by the caller. *** c integer liv, lv, n integer iv(liv), uiparm(1) double precision d(n), x(n), v(lv), urparm(1) c dimension v(78 + n*(n+12)), uiparm(*), urparm(*) external calcf, calcgh, ufparm c c------------------------------ discussion --------------------------- c c this routine is like sumsl, except that the subroutine para- c meter calcg of sumsl (which computes the gradient of the objec- c tive function) is replaced by the subroutine parameter calcgh, c which computes both the gradient and (lower triangle of the) c hessian of the objective function. the calling sequence is... c call calcgh(n, x, nf, g, h, uiparm, urparm, ufparm) c parameters n, x, nf, g, uiparm, urparm, and ufparm are the same c as for sumsl, while h is an array of length n*(n+1)/2 in which c calcgh must store the lower triangle of the hessian at x. start- c ing at h(1), calcgh must store the hessian entries in the order c (1,1), (2,1), (2,2), (3,1), (3,2), (3,3), ... c the value printed (by itsum) in the column labelled stppar c is the levenberg-marquardt used in computing the current step. c zero means a full newton step. if the special case described in c ref. 1 is detected, then stppar is negated. the value printed c in the column labelled npreldf is zero if the current hessian c is not positive definite. c it sometimes proves worthwhile to let d be determined from the c diagonal of the hessian matrix by setting iv(dtype) = 1 and c v(dinit) = 0. the following iv and v components are relevant... c c iv(dtol)..... iv(59) gives the starting subscript in v of the dtol c array used when d is updated. (iv(dtol) can be c initialized by calling humsl with iv(1) = 13.) c iv(dtype).... iv(16) tells how the scale vector d should be chosen. c iv(dtype) .le. 0 means that d should not be updated, and c iv(dtype) .ge. 1 means that d should be updated as c described below with v(dfac). default = 0. c v(dfac)..... v(41) and the dtol and d0 arrays (see v(dtinit) and c v(d0init)) are used in updating the scale vector d when c iv(dtype) .gt. 0. (d is initialized according to c v(dinit), described in sumsl.) let c d1(i) = max(sqrt(abs(h(i,i))), v(dfac)*d(i)), c where h(i,i) is the i-th diagonal element of the current c hessian. if iv(dtype) = 1, then d(i) is set to d1(i) c unless d1(i) .lt. dtol(i), in which case d(i) is set to c max(d0(i), dtol(i)). c if iv(dtype) .ge. 2, then d is updated during the first c iteration as for iv(dtype) = 1 (after any initialization c due to v(dinit)) and is left unchanged thereafter. c default = 0.6. c v(dtinit)... v(39), if positive, is the value to which all components c of the dtol array (see v(dfac)) are initialized. if c v(dtinit) = 0, then it is assumed that the caller has c stored dtol in v starting at v(iv(dtol)). c default = 10**-6. c v(d0init)... v(40), if positive, is the value to which all components c of the d0 vector (see v(dfac)) are initialized. if c v(dfac) = 0, then it is assumed that the caller has c stored d0 in v starting at v(iv(dtol)+n). default = 1.0. c c *** reference *** c c 1. gay, d.m. (1981), computing optimal locally constrained steps, c siam j. sci. statist. comput. 2, pp. 186-197. c. c *** general *** c c coded by david m. gay (winter 1980). revised sept. 1982. c this subroutine was written in connection with research supported c in part by the national science foundation under grants c mcs-7600324 and mcs-7906671. c c---------------------------- declarations --------------------------- c external deflt, humit c c deflt... provides default input values for iv and v. c humit... reverse-communication routine that does humsl algorithm. c integer g1, h1, iv1, lh, nf double precision f c c *** subscripts for iv *** c integer g, h, nextv, nfcall, nfgcal, toobig, vneed c c/6 c data nextv/47/, nfcall/6/, nfgcal/7/, g/28/, h/56/, toobig/2/, c 1 vneed/4/ c/7 parameter (nextv=47, nfcall=6, nfgcal=7, g=28, h=56, toobig=2, 1 vneed=4) c/ c c+++++++++++++++++++++++++++++++ body ++++++++++++++++++++++++++++++++ c lh = n * (n + 1) / 2 if (iv(1) .eq. 0) call deflt(2, iv, liv, lv, v) if (iv(1) .eq. 12 .or. iv(1) .eq. 13) 1 iv(vneed) = iv(vneed) + n*(n+3)/2 iv1 = iv(1) if (iv1 .eq. 14) go to 10 if (iv1 .gt. 2 .and. iv1 .lt. 12) go to 10 g1 = 1 h1 = 1 if (iv1 .eq. 12) iv(1) = 13 go to 20 c 10 g1 = iv(g) h1 = iv(h) c 20 call humit(d, f, v(g1), v(h1), iv, lh, liv, lv, n, v, x) if (iv(1) - 2) 30, 40, 50 c 30 nf = iv(nfcall) call calcf(n, x, nf, f, uiparm, urparm, ufparm) if (nf .le. 0) iv(toobig) = 1 go to 20 c 40 call calcgh(n, x, iv(nfgcal), v(g1), v(h1), uiparm, urparm, 1 ufparm) go to 20 c 50 if (iv(1) .ne. 14) go to 999 c c *** storage allocation c iv(g) = iv(nextv) iv(h) = iv(g) + n iv(nextv) = iv(h) + n*(n+1)/2 if (iv1 .ne. 13) go to 10 c 999 return c *** last card of humsl follows *** end subroutine humit(d, fx, g, h, iv, lh, liv, lv, n, v, x) c c *** carry out humsl (unconstrained minimization) iterations, using c *** hessian matrix provided by the caller. c c *** parameter declarations *** c integer lh, liv, lv, n integer iv(liv) double precision d(n), fx, g(n), h(lh), v(lv), x(n) c c-------------------------- parameter usage -------------------------- c c d.... scale vector. c fx... function value. c g.... gradient vector. c h.... lower triangle of the hessian, stored rowwise. c iv... integer value array. c lh... length of h = p*(p+1)/2. c liv.. length of iv (at least 60). c lv... length of v (at least 78 + n*(n+21)/2). c n.... number of variables (components in x and g). c v.... floating-point value array. c x.... parameter vector. c c *** discussion *** c c parameters iv, n, v, and x are the same as the corresponding c ones to humsl (which see), except that v can be shorter (since c the part of v that humsl uses for storing g and h is not needed). c moreover, compared with humsl, iv(1) may have the two additional c output values 1 and 2, which are explained below, as is the use c of iv(toobig) and iv(nfgcal). the value iv(g), which is an c output value from humsl, is not referenced by humit or the c subroutines it calls. c c iv(1) = 1 means the caller should set fx to f(x), the function value c at x, and call humit again, having changed none of the c other parameters. an exception occurs if f(x) cannot be c computed (e.g. if overflow would occur), which may happen c because of an oversized step. in this case the caller c should set iv(toobig) = iv(2) to 1, which will cause c humit to ignore fx and try a smaller step. the para- c meter nf that humsl passes to calcf (for possible use by c calcgh) is a copy of iv(nfcall) = iv(6). c iv(1) = 2 means the caller should set g to g(x), the gradient of f at c x, and h to the lower triangle of h(x), the hessian of f c at x, and call humit again, having changed none of the c other parameters except perhaps the scale vector d. c the parameter nf that humsl passes to calcg is c iv(nfgcal) = iv(7). if g(x) and h(x) cannot be evaluated, c then the caller may set iv(nfgcal) to 0, in which case c humit will return with iv(1) = 65. c note -- humit overwrites h with the lower triangle c of diag(d)**-1 * h(x) * diag(d)**-1. c. c *** general *** c c coded by david m. gay (winter 1980). revised sept. 1982. c this subroutine was written in connection with research supported c in part by the national science foundation under grants c mcs-7600324 and mcs-7906671. c c (see sumsl and humsl for references.) c c+++++++++++++++++++++++++++ declarations ++++++++++++++++++++++++++++ c c *** local variables *** c integer dg1, dummy, i, j, k, l, lstgst, nn1o2, step1, 1 temp1, w1, x01 double precision t c c *** constants *** c double precision one, onep2, zero c c *** no intrinsic functions *** c c *** external functions and subroutines *** c external assst, deflt, dotprd, dupdu, gqtst, itsum, parck, 1 reldst, slvmul, stopx, vaxpy, vcopy, vscopy, v2norm logical stopx double precision dotprd, reldst, v2norm c c assst.... assesses candidate step. c deflt.... provides default iv and v input values. c dotprd... returns inner product of two vectors. c dupdu.... updates scale vector d. c gqtst.... computes optimally locally constrained step. c itsum.... prints iteration summary and info on initial and final x. c parck.... checks validity of input iv and v values. c reldst... computes v(reldx) = relative step size. c slvmul... multiplies symmetric matrix times vector, given the lower c triangle of the matrix. c stopx.... returns .true. if the break key has been pressed. c vaxpy.... computes scalar times one vector plus another. c vcopy.... copies one vector to another. c vscopy... sets all elements of a vector to a scalar. c v2norm... returns the 2-norm of a vector. c c *** subscripts for iv and v *** c integer cnvcod, dg, dgnorm, dinit, dstnrm, dtinit, dtol, 1 dtype, d0init, f, f0, fdif, gtstep, incfac, irc, kagqt, 2 lmat, lmax0, lmaxs, mode, model, mxfcal, mxiter, nextv, 3 nfcall, nfgcal, ngcall, niter, preduc, radfac, radinc, 4 radius, rad0, reldx, restor, step, stglim, stlstg, stppar, 5 toobig, tuner4, tuner5, vneed, w, xirc, x0 c c *** iv subscript values *** c c/6 c data cnvcod/55/, dg/37/, dtol/59/, dtype/16/, irc/29/, kagqt/33/, c 1 lmat/42/, mode/35/, model/5/, mxfcal/17/, mxiter/18/, c 2 nextv/47/, nfcall/6/, nfgcal/7/, ngcall/30/, niter/31/, c 3 radinc/8/, restor/9/, step/40/, stglim/11/, stlstg/41/, c 4 toobig/2/, vneed/4/, w/34/, xirc/13/, x0/43/ c/7 parameter (cnvcod=55, dg=37, dtol=59, dtype=16, irc=29, kagqt=33, 1 lmat=42, mode=35, model=5, mxfcal=17, mxiter=18, 2 nextv=47, nfcall=6, nfgcal=7, ngcall=30, niter=31, 3 radinc=8, restor=9, step=40, stglim=11, stlstg=41, 4 toobig=2, vneed=4, w=34, xirc=13, x0=43) c/ c c *** v subscript values *** c c/6 c data dgnorm/1/, dinit/38/, dstnrm/2/, dtinit/39/, d0init/40/, c 1 f/10/, f0/13/, fdif/11/, gtstep/4/, incfac/23/, lmax0/35/, c 2 lmaxs/36/, preduc/7/, radfac/16/, radius/8/, rad0/9/, c 3 reldx/17/, stppar/5/, tuner4/29/, tuner5/30/ c/7 parameter (dgnorm=1, dinit=38, dstnrm=2, dtinit=39, d0init=40, 1 f=10, f0=13, fdif=11, gtstep=4, incfac=23, lmax0=35, 2 lmaxs=36, preduc=7, radfac=16, radius=8, rad0=9, 3 reldx=17, stppar=5, tuner4=29, tuner5=30) c/ c c/6 c data one/1.d+0/, onep2/1.2d+0/, zero/0.d+0/ c/7 parameter (one=1.d+0, onep2=1.2d+0, zero=0.d+0) c/ c c+++++++++++++++++++++++++++++++ body ++++++++++++++++++++++++++++++++ c i = iv(1) if (i .eq. 1) go to 30 if (i .eq. 2) go to 40 c c *** check validity of iv and v input values *** c if (iv(1) .eq. 0) call deflt(2, iv, liv, lv, v) if (iv(1) .eq. 12 .or. iv(1) .eq. 13) 1 iv(vneed) = iv(vneed) + n*(n+21)/2 + 7 call parck(2, d, iv, liv, lv, n, v) i = iv(1) - 2 if (i .gt. 12) go to 999 nn1o2 = n * (n + 1) / 2 if (lh .ge. nn1o2) go to (210,210,210,210,210,210,160,120,160, 1 10,10,20), i iv(1) = 66 go to 350 c c *** storage allocation *** c 10 iv(dtol) = iv(lmat) + nn1o2 iv(x0) = iv(dtol) + 2*n iv(step) = iv(x0) + n iv(stlstg) = iv(step) + n iv(dg) = iv(stlstg) + n iv(w) = iv(dg) + n iv(nextv) = iv(w) + 4*n + 7 if (iv(1) .ne. 13) go to 20 iv(1) = 14 go to 999 c c *** initialization *** c 20 iv(niter) = 0 iv(nfcall) = 1 iv(ngcall) = 1 iv(nfgcal) = 1 iv(mode) = -1 iv(model) = 1 iv(stglim) = 1 iv(toobig) = 0 iv(cnvcod) = 0 iv(radinc) = 0 v(rad0) = zero v(stppar) = zero if (v(dinit) .ge. zero) call vscopy(n, d, v(dinit)) k = iv(dtol) if (v(dtinit) .gt. zero) call vscopy(n, v(k), v(dtinit)) k = k + n if (v(d0init) .gt. zero) call vscopy(n, v(k), v(d0init)) iv(1) = 1 go to 999 c 30 v(f) = fx if (iv(mode) .ge. 0) go to 210 iv(1) = 2 if (iv(toobig) .eq. 0) go to 999 iv(1) = 63 go to 350 c c *** make sure gradient could be computed *** c 40 if (iv(nfgcal) .ne. 0) go to 50 iv(1) = 65 go to 350 c c *** update the scale vector d *** c 50 dg1 = iv(dg) if (iv(dtype) .le. 0) go to 70 k = dg1 j = 0 do 60 i = 1, n j = j + i v(k) = h(j) k = k + 1 60 continue call dupdu(d, v(dg1), iv, liv, lv, n, v) c c *** compute scaled gradient and its norm *** c 70 dg1 = iv(dg) k = dg1 do 80 i = 1, n v(k) = g(i) / d(i) k = k + 1 80 continue v(dgnorm) = v2norm(n, v(dg1)) c c *** compute scaled hessian *** c k = 1 do 100 i = 1, n t = one / d(i) do 90 j = 1, i h(k) = t * h(k) / d(j) k = k + 1 90 continue 100 continue c if (iv(cnvcod) .ne. 0) go to 340 if (iv(mode) .eq. 0) go to 300 c c *** allow first step to have scaled 2-norm at most v(lmax0) *** c v(radius) = v(lmax0) c iv(mode) = 0 c c c----------------------------- main loop ----------------------------- c c c *** print iteration summary, check iteration limit *** c 110 call itsum(d, g, iv, liv, lv, n, v, x) 120 k = iv(niter) if (k .lt. iv(mxiter)) go to 130 iv(1) = 10 go to 350 c 130 iv(niter) = k + 1 c c *** initialize for start of next iteration *** c dg1 = iv(dg) x01 = iv(x0) v(f0) = v(f) iv(irc) = 4 iv(kagqt) = -1 c c *** copy x to x0 *** c call vcopy(n, v(x01), x) c c *** update radius *** c if (k .eq. 0) go to 150 step1 = iv(step) k = step1 do 140 i = 1, n v(k) = d(i) * v(k) k = k + 1 140 continue v(radius) = v(radfac) * v2norm(n, v(step1)) c c *** check stopx and function evaluation limit *** c C AL 4/30/95 dummy=iv(nfcall) 150 if (.not. stopx(dummy)) go to 170 iv(1) = 11 go to 180 c c *** come here when restarting after func. eval. limit or stopx. c 160 if (v(f) .ge. v(f0)) go to 170 v(radfac) = one k = iv(niter) go to 130 c 170 if (iv(nfcall) .lt. iv(mxfcal)) go to 190 iv(1) = 9 180 if (v(f) .ge. v(f0)) go to 350 c c *** in case of stopx or function evaluation limit with c *** improved v(f), evaluate the gradient at x. c iv(cnvcod) = iv(1) go to 290 c c. . . . . . . . . . . . . compute candidate step . . . . . . . . . . c 190 step1 = iv(step) dg1 = iv(dg) l = iv(lmat) w1 = iv(w) call gqtst(d, v(dg1), h, iv(kagqt), v(l), n, v(step1), v, v(w1)) if (iv(irc) .eq. 6) go to 210 c c *** check whether evaluating f(x0 + step) looks worthwhile *** c if (v(dstnrm) .le. zero) go to 210 if (iv(irc) .ne. 5) go to 200 if (v(radfac) .le. one) go to 200 if (v(preduc) .le. onep2 * v(fdif)) go to 210 c c *** compute f(x0 + step) *** c 200 x01 = iv(x0) step1 = iv(step) call vaxpy(n, x, one, v(step1), v(x01)) iv(nfcall) = iv(nfcall) + 1 iv(1) = 1 iv(toobig) = 0 go to 999 c c. . . . . . . . . . . . . assess candidate step . . . . . . . . . . . c 210 x01 = iv(x0) v(reldx) = reldst(n, d, x, v(x01)) call assst(iv, liv, lv, v) step1 = iv(step) lstgst = iv(stlstg) if (iv(restor) .eq. 1) call vcopy(n, x, v(x01)) if (iv(restor) .eq. 2) call vcopy(n, v(lstgst), v(step1)) if (iv(restor) .ne. 3) go to 220 call vcopy(n, v(step1), v(lstgst)) call vaxpy(n, x, one, v(step1), v(x01)) v(reldx) = reldst(n, d, x, v(x01)) c 220 k = iv(irc) go to (230,260,260,260,230,240,250,250,250,250,250,250,330,300), k c c *** recompute step with new radius *** c 230 v(radius) = v(radfac) * v(dstnrm) go to 150 c c *** compute step of length v(lmaxs) for singular convergence test. c 240 v(radius) = v(lmaxs) go to 190 c c *** convergence or false convergence *** c 250 iv(cnvcod) = k - 4 if (v(f) .ge. v(f0)) go to 340 if (iv(xirc) .eq. 14) go to 340 iv(xirc) = 14 c c. . . . . . . . . . . . process acceptable step . . . . . . . . . . . c 260 if (iv(irc) .ne. 3) go to 290 temp1 = lstgst c c *** prepare for gradient tests *** c *** set temp1 = hessian * step + g(x0) c *** = diag(d) * (h * step + g(x0)) c c use x0 vector as temporary. k = x01 do 270 i = 1, n v(k) = d(i) * v(step1) k = k + 1 step1 = step1 + 1 270 continue call slvmul(n, v(temp1), h, v(x01)) do 280 i = 1, n v(temp1) = d(i) * v(temp1) + g(i) temp1 = temp1 + 1 280 continue c c *** compute gradient and hessian *** c 290 iv(ngcall) = iv(ngcall) + 1 iv(1) = 2 go to 999 c 300 iv(1) = 2 if (iv(irc) .ne. 3) go to 110 c c *** set v(radfac) by gradient tests *** c temp1 = iv(stlstg) step1 = iv(step) c c *** set temp1 = diag(d)**-1 * (hessian*step + (g(x0)-g(x))) *** c k = temp1 do 310 i = 1, n v(k) = (v(k) - g(i)) / d(i) k = k + 1 310 continue c c *** do gradient tests *** c if (v2norm(n, v(temp1)) .le. v(dgnorm) * v(tuner4)) go to 320 if (dotprd(n, g, v(step1)) 1 .ge. v(gtstep) * v(tuner5)) go to 110 320 v(radfac) = v(incfac) go to 110 c c. . . . . . . . . . . . . . misc. details . . . . . . . . . . . . . . c c *** bad parameters to assess *** c 330 iv(1) = 64 go to 350 c c *** print summary of final iteration and other requested items *** c 340 iv(1) = iv(cnvcod) iv(cnvcod) = 0 350 call itsum(d, g, iv, liv, lv, n, v, x) c 999 return c c *** last card of humit follows *** end subroutine dupdu(d, hdiag, iv, liv, lv, n, v) c c *** update scale vector d for humsl *** c c *** parameter declarations *** c integer liv, lv, n integer iv(liv) double precision d(n), hdiag(n), v(lv) c c *** local variables *** c integer dtoli, d0i, i double precision t, vdfac c c *** intrinsic functions *** c/+ double precision dabs, dmax1, dsqrt c/ c *** subscripts for iv and v *** c integer dfac, dtol, dtype, niter c/6 c data dfac/41/, dtol/59/, dtype/16/, niter/31/ c/7 parameter (dfac=41, dtol=59, dtype=16, niter=31) c/ c c------------------------------- body -------------------------------- c i = iv(dtype) if (i .eq. 1) go to 10 if (iv(niter) .gt. 0) go to 999 c 10 dtoli = iv(dtol) d0i = dtoli + n vdfac = v(dfac) do 20 i = 1, n t = dmax1(dsqrt(dabs(hdiag(i))), vdfac*d(i)) if (t .lt. v(dtoli)) t = dmax1(v(dtoli), v(d0i)) d(i) = t dtoli = dtoli + 1 d0i = d0i + 1 20 continue c 999 return c *** last card of dupdu follows *** end subroutine gqtst(d, dig, dihdi, ka, l, p, step, v, w) c c *** compute goldfeld-quandt-trotter step by more-hebden technique *** c *** (nl2sol version 2.2), modified a la more and sorensen *** c c *** parameter declarations *** c integer ka, p cal double precision d(p), dig(p), dihdi(1), l(1), v(21), step(p), cal 1 w(1) double precision d(p), dig(p), dihdi(p*(p+1)/2), l(p*(p+1)/2), 1 v(21), step(p),w(4*p+7) c dimension dihdi(p*(p+1)/2), l(p*(p+1)/2), w(4*p+7) c c+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ c c *** purpose *** c c given the (compactly stored) lower triangle of a scaled c hessian (approximation) and a nonzero scaled gradient vector, c this subroutine computes a goldfeld-quandt-trotter step of c approximate length v(radius) by the more-hebden technique. in c other words, step is computed to (approximately) minimize c psi(step) = (g**t)*step + 0.5*(step**t)*h*step such that the c 2-norm of d*step is at most (approximately) v(radius), where c g is the gradient, h is the hessian, and d is a diagonal c scale matrix whose diagonal is stored in the parameter d. c (gqtst assumes dig = d**-1 * g and dihdi = d**-1 * h * d**-1.) c c *** parameter description *** c c d (in) = the scale vector, i.e. the diagonal of the scale c matrix d mentioned above under purpose. c dig (in) = the scaled gradient vector, d**-1 * g. if g = 0, then c step = 0 and v(stppar) = 0 are returned. c dihdi (in) = lower triangle of the scaled hessian (approximation), c i.e., d**-1 * h * d**-1, stored compactly by rows., i.e., c in the order (1,1), (2,1), (2,2), (3,1), (3,2), etc. c ka (i/o) = the number of hebden iterations (so far) taken to deter- c mine step. ka .lt. 0 on input means this is the first c attempt to determine step (for the present dig and dihdi) c -- ka is initialized to 0 in this case. output with c ka = 0 (or v(stppar) = 0) means step = -(h**-1)*g. c l (i/o) = workspace of length p*(p+1)/2 for cholesky factors. c p (in) = number of parameters -- the hessian is a p x p matrix. c step (i/o) = the step computed. c v (i/o) contains various constants and variables described below. c w (i/o) = workspace of length 4*p + 6. c c *** entries in v *** c c v(dgnorm) (i/o) = 2-norm of (d**-1)*g. c v(dstnrm) (output) = 2-norm of d*step. c v(dst0) (i/o) = 2-norm of d*(h**-1)*g (for pos. def. h only), or c overestimate of smallest eigenvalue of (d**-1)*h*(d**-1). c v(epslon) (in) = max. rel. error allowed for psi(step). for the c step returned, psi(step) will exceed its optimal value c by less than -v(epslon)*psi(step). suggested value = 0.1. c v(gtstep) (out) = inner product between g and step. c v(nreduc) (out) = psi(-(h**-1)*g) = psi(newton step) (for pos. def. c h only -- v(nreduc) is set to zero otherwise). c v(phmnfc) (in) = tol. (together with v(phmxfc)) for accepting step c (more*s sigma). the error v(dstnrm) - v(radius) must lie c between v(phmnfc)*v(radius) and v(phmxfc)*v(radius). c v(phmxfc) (in) (see v(phmnfc).) c suggested values -- v(phmnfc) = -0.25, v(phmxfc) = 0.5. c v(preduc) (out) = psi(step) = predicted obj. func. reduction for step. c v(radius) (in) = radius of current (scaled) trust region. c v(rad0) (i/o) = value of v(radius) from previous call. c v(stppar) (i/o) is normally the marquardt parameter, i.e. the alpha c described below under algorithm notes. if h + alpha*d**2 c (see algorithm notes) is (nearly) singular, however, c then v(stppar) = -alpha. c c *** usage notes *** c c if it is desired to recompute step using a different value of c v(radius), then this routine may be restarted by calling it c with all parameters unchanged except v(radius). (this explains c why step and w are listed as i/o). on an initial call (one with c ka .lt. 0), step and w need not be initialized and only compo- c nents v(epslon), v(stppar), v(phmnfc), v(phmxfc), v(radius), and c v(rad0) of v must be initialized. c c *** algorithm notes *** c c the desired g-q-t step (ref. 2, 3, 4, 6) satisfies c (h + alpha*d**2)*step = -g for some nonnegative alpha such that c h + alpha*d**2 is positive semidefinite. alpha and step are c computed by a scheme analogous to the one described in ref. 5. c estimates of the smallest and largest eigenvalues of the hessian c are obtained from the gerschgorin circle theorem enhanced by a c simple form of the scaling described in ref. 7. cases in which c h + alpha*d**2 is nearly (or exactly) singular are handled by c the technique discussed in ref. 2. in these cases, a step of c (exact) length v(radius) is returned for which psi(step) exceeds c its optimal value by less than -v(epslon)*psi(step). the test c suggested in ref. 6 for detecting the special case is performed c once two matrix factorizations have been done -- doing so sooner c seems to degrade the performance of optimization routines that c call this routine. c c *** functions and subroutines called *** c c dotprd - returns inner product of two vectors. c litvmu - applies inverse-transpose of compact lower triang. matrix. c livmul - applies inverse of compact lower triang. matrix. c lsqrt - finds cholesky factor (of compactly stored lower triang.). c lsvmin - returns approx. to min. sing. value of lower triang. matrix. c rmdcon - returns machine-dependent constants. c v2norm - returns 2-norm of a vector. c c *** references *** c c 1. dennis, j.e., gay, d.m., and welsch, r.e. (1981), an adaptive c nonlinear least-squares algorithm, acm trans. math. c software, vol. 7, no. 3. c 2. gay, d.m. (1981), computing optimal locally constrained steps, c siam j. sci. statist. computing, vol. 2, no. 2, pp. c 186-197. c 3. goldfeld, s.m., quandt, r.e., and trotter, h.f. (1966), c maximization by quadratic hill-climbing, econometrica 34, c pp. 541-551. c 4. hebden, m.d. (1973), an algorithm for minimization using exact c second derivatives, report t.p. 515, theoretical physics c div., a.e.r.e. harwell, oxon., england. c 5. more, j.j. (1978), the levenberg-marquardt algorithm, implemen- c tation and theory, pp.105-116 of springer lecture notes c in mathematics no. 630, edited by g.a. watson, springer- c verlag, berlin and new york. c 6. more, j.j., and sorensen, d.c. (1981), computing a trust region c step, technical report anl-81-83, argonne national lab. c 7. varga, r.s. (1965), minimal gerschgorin sets, pacific j. math. 15, c pp. 719-729. c c *** general *** c c coded by david m. gay. c this subroutine was written in connection with research c supported by the national science foundation under grants c mcs-7600324, dcr75-10143, 76-14311dss, mcs76-11989, and c mcs-7906671. c c+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ c c *** local variables *** c logical restrt integer dggdmx, diag, diag0, dstsav, emax, emin, i, im1, inc, irc, 1 j, k, kalim, kamin, k1, lk0, phipin, q, q0, uk0, x double precision alphak, aki, akk, delta, dst, eps, gtsta, lk, 1 oldphi, phi, phimax, phimin, psifac, rad, radsq, 2 root, si, sk, sw, t, twopsi, t1, t2, uk, wi c c *** constants *** double precision big, dgxfac, epsfac, four, half, kappa, negone, 1 one, p001, six, three, two, zero c c *** intrinsic functions *** c/+ double precision dabs, dmax1, dmin1, dsqrt c/ c *** external functions and subroutines *** c external dotprd, litvmu, livmul, lsqrt, lsvmin, rmdcon, v2norm double precision dotprd, lsvmin, rmdcon, v2norm c c *** subscripts for v *** c integer dgnorm, dstnrm, dst0, epslon, gtstep, stppar, nreduc, 1 phmnfc, phmxfc, preduc, radius, rad0 c/6 c data dgnorm/1/, dstnrm/2/, dst0/3/, epslon/19/, gtstep/4/, c 1 nreduc/6/, phmnfc/20/, phmxfc/21/, preduc/7/, radius/8/, c 2 rad0/9/, stppar/5/ c/7 parameter (dgnorm=1, dstnrm=2, dst0=3, epslon=19, gtstep=4, 1 nreduc=6, phmnfc=20, phmxfc=21, preduc=7, radius=8, 2 rad0=9, stppar=5) c/ c c/6 c data epsfac/50.0d+0/, four/4.0d+0/, half/0.5d+0/, c 1 kappa/2.0d+0/, negone/-1.0d+0/, one/1.0d+0/, p001/1.0d-3/, c 2 six/6.0d+0/, three/3.0d+0/, two/2.0d+0/, zero/0.0d+0/ c/7 parameter (epsfac=50.0d+0, four=4.0d+0, half=0.5d+0, 1 kappa=2.0d+0, negone=-1.0d+0, one=1.0d+0, p001=1.0d-3, 2 six=6.0d+0, three=3.0d+0, two=2.0d+0, zero=0.0d+0) save dgxfac c/ data big/0.d+0/, dgxfac/0.d+0/ c c *** body *** c c *** store largest abs. entry in (d**-1)*h*(d**-1) at w(dggdmx). dggdmx = p + 1 c *** store gerschgorin over- and underestimates of the largest c *** and smallest eigenvalues of (d**-1)*h*(d**-1) at w(emax) c *** and w(emin) respectively. emax = dggdmx + 1 emin = emax + 1 c *** for use in recomputing step, the final values of lk, uk, dst, c *** and the inverse derivative of more*s phi at 0 (for pos. def. c *** h) are stored in w(lk0), w(uk0), w(dstsav), and w(phipin) c *** respectively. lk0 = emin + 1 phipin = lk0 + 1 uk0 = phipin + 1 dstsav = uk0 + 1 c *** store diag of (d**-1)*h*(d**-1) in w(diag),...,w(diag0+p). diag0 = dstsav diag = diag0 + 1 c *** store -d*step in w(q),...,w(q0+p). q0 = diag0 + p q = q0 + 1 c *** allocate storage for scratch vector x *** x = q + p rad = v(radius) radsq = rad**2 c *** phitol = max. error allowed in dst = v(dstnrm) = 2-norm of c *** d*step. phimax = v(phmxfc) * rad phimin = v(phmnfc) * rad psifac = two * v(epslon) / (three * (four * (v(phmnfc) + one) * 1 (kappa + one) + kappa + two) * rad**2) c *** oldphi is used to detect limits of numerical accuracy. if c *** we recompute step and it does not change, then we accept it. oldphi = zero eps = v(epslon) irc = 0 restrt = .false. kalim = ka + 50 c c *** start or restart, depending on ka *** c if (ka .ge. 0) go to 290 c c *** fresh start *** c k = 0 uk = negone ka = 0 kalim = 50 v(dgnorm) = v2norm(p, dig) v(nreduc) = zero v(dst0) = zero kamin = 3 if (v(dgnorm) .eq. zero) kamin = 0 c c *** store diag(dihdi) in w(diag0+1),...,w(diag0+p) *** c j = 0 do 10 i = 1, p j = j + i k1 = diag0 + i w(k1) = dihdi(j) 10 continue c c *** determine w(dggdmx), the largest element of dihdi *** c t1 = zero j = p * (p + 1) / 2 do 20 i = 1, j t = dabs(dihdi(i)) if (t1 .lt. t) t1 = t 20 continue w(dggdmx) = t1 c c *** try alpha = 0 *** c 30 call lsqrt(1, p, l, dihdi, irc) if (irc .eq. 0) go to 50 c *** indef. h -- underestimate smallest eigenvalue, use this c *** estimate to initialize lower bound lk on alpha. j = irc*(irc+1)/2 t = l(j) l(j) = one do 40 i = 1, irc 40 w(i) = zero w(irc) = one call litvmu(irc, w, l, w) t1 = v2norm(irc, w) lk = -t / t1 / t1 v(dst0) = -lk if (restrt) go to 210 go to 70 c c *** positive definite h -- compute unmodified newton step. *** 50 lk = zero t = lsvmin(p, l, w(q), w(q)) if (t .ge. one) go to 60 if (big .le. zero) big = rmdcon(6) if (v(dgnorm) .ge. t*t*big) go to 70 60 call livmul(p, w(q), l, dig) gtsta = dotprd(p, w(q), w(q)) v(nreduc) = half * gtsta call litvmu(p, w(q), l, w(q)) dst = v2norm(p, w(q)) v(dst0) = dst phi = dst - rad if (phi .le. phimax) go to 260 if (restrt) go to 210 c c *** prepare to compute gerschgorin estimates of largest (and c *** smallest) eigenvalues. *** c 70 k = 0 do 100 i = 1, p wi = zero if (i .eq. 1) go to 90 im1 = i - 1 do 80 j = 1, im1 k = k + 1 t = dabs(dihdi(k)) wi = wi + t w(j) = w(j) + t 80 continue 90 w(i) = wi k = k + 1 100 continue c c *** (under-)estimate smallest eigenvalue of (d**-1)*h*(d**-1) *** c k = 1 t1 = w(diag) - w(1) if (p .le. 1) go to 120 do 110 i = 2, p j = diag0 + i t = w(j) - w(i) if (t .ge. t1) go to 110 t1 = t k = i 110 continue c 120 sk = w(k) j = diag0 + k akk = w(j) k1 = k*(k-1)/2 + 1 inc = 1 t = zero do 150 i = 1, p if (i .eq. k) go to 130 aki = dabs(dihdi(k1)) si = w(i) j = diag0 + i t1 = half * (akk - w(j) + si - aki) t1 = t1 + dsqrt(t1*t1 + sk*aki) if (t .lt. t1) t = t1 if (i .lt. k) go to 140 130 inc = i 140 k1 = k1 + inc 150 continue c w(emin) = akk - t uk = v(dgnorm)/rad - w(emin) if (v(dgnorm) .eq. zero) uk = uk + p001 + p001*uk if (uk .le. zero) uk = p001 c c *** compute gerschgorin (over-)estimate of largest eigenvalue *** c k = 1 t1 = w(diag) + w(1) if (p .le. 1) go to 170 do 160 i = 2, p j = diag0 + i t = w(j) + w(i) if (t .le. t1) go to 160 t1 = t k = i 160 continue c 170 sk = w(k) j = diag0 + k akk = w(j) k1 = k*(k-1)/2 + 1 inc = 1 t = zero do 200 i = 1, p if (i .eq. k) go to 180 aki = dabs(dihdi(k1)) si = w(i) j = diag0 + i t1 = half * (w(j) + si - aki - akk) t1 = t1 + dsqrt(t1*t1 + sk*aki) if (t .lt. t1) t = t1 if (i .lt. k) go to 190 180 inc = i 190 k1 = k1 + inc 200 continue c w(emax) = akk + t lk = dmax1(lk, v(dgnorm)/rad - w(emax)) c c *** alphak = current value of alpha (see alg. notes above). we c *** use more*s scheme for initializing it. alphak = dabs(v(stppar)) * v(rad0)/rad c if (irc .ne. 0) go to 210 c c *** compute l0 for positive definite h *** c call livmul(p, w, l, w(q)) t = v2norm(p, w) w(phipin) = dst / t / t lk = dmax1(lk, phi*w(phipin)) c c *** safeguard alphak and add alphak*i to (d**-1)*h*(d**-1) *** c 210 ka = ka + 1 if (-v(dst0) .ge. alphak .or. alphak .lt. lk .or. alphak .ge. uk) 1 alphak = uk * dmax1(p001, dsqrt(lk/uk)) if (alphak .le. zero) alphak = half * uk if (alphak .le. zero) alphak = uk k = 0 do 220 i = 1, p k = k + i j = diag0 + i dihdi(k) = w(j) + alphak 220 continue c c *** try computing cholesky decomposition *** c call lsqrt(1, p, l, dihdi, irc) if (irc .eq. 0) go to 240 c c *** (d**-1)*h*(d**-1) + alphak*i is indefinite -- overestimate c *** smallest eigenvalue for use in updating lk *** c j = (irc*(irc+1))/2 t = l(j) l(j) = one do 230 i = 1, irc 230 w(i) = zero w(irc) = one call litvmu(irc, w, l, w) t1 = v2norm(irc, w) lk = alphak - t/t1/t1 v(dst0) = -lk go to 210 c c *** alphak makes (d**-1)*h*(d**-1) positive definite. c *** compute q = -d*step, check for convergence. *** c 240 call livmul(p, w(q), l, dig) gtsta = dotprd(p, w(q), w(q)) call litvmu(p, w(q), l, w(q)) dst = v2norm(p, w(q)) phi = dst - rad if (phi .le. phimax .and. phi .ge. phimin) go to 270 if (phi .eq. oldphi) go to 270 oldphi = phi if (phi .lt. zero) go to 330 c c *** unacceptable alphak -- update lk, uk, alphak *** c 250 if (ka .ge. kalim) go to 270 c *** the following dmin1 is necessary because of restarts *** if (phi .lt. zero) uk = dmin1(uk, alphak) c *** kamin = 0 only iff the gradient vanishes *** if (kamin .eq. 0) go to 210 call livmul(p, w, l, w(q)) t1 = v2norm(p, w) alphak = alphak + (phi/t1) * (dst/t1) * (dst/rad) lk = dmax1(lk, alphak) go to 210 c c *** acceptable step on first try *** c 260 alphak = zero c c *** successful step in general. compute step = -(d**-1)*q *** c 270 do 280 i = 1, p j = q0 + i step(i) = -w(j)/d(i) 280 continue v(gtstep) = -gtsta v(preduc) = half * (dabs(alphak)*dst*dst + gtsta) go to 410 c c c *** restart with new radius *** c 290 if (v(dst0) .le. zero .or. v(dst0) - rad .gt. phimax) go to 310 c c *** prepare to return newton step *** c restrt = .true. ka = ka + 1 k = 0 do 300 i = 1, p k = k + i j = diag0 + i dihdi(k) = w(j) 300 continue uk = negone go to 30 c 310 kamin = ka + 3 if (v(dgnorm) .eq. zero) kamin = 0 if (ka .eq. 0) go to 50 c dst = w(dstsav) alphak = dabs(v(stppar)) phi = dst - rad t = v(dgnorm)/rad uk = t - w(emin) if (v(dgnorm) .eq. zero) uk = uk + p001 + p001*uk if (uk .le. zero) uk = p001 if (rad .gt. v(rad0)) go to 320 c c *** smaller radius *** lk = zero if (alphak .gt. zero) lk = w(lk0) lk = dmax1(lk, t - w(emax)) if (v(dst0) .gt. zero) lk = dmax1(lk, (v(dst0)-rad)*w(phipin)) go to 250 c c *** bigger radius *** 320 if (alphak .gt. zero) uk = dmin1(uk, w(uk0)) lk = dmax1(zero, -v(dst0), t - w(emax)) if (v(dst0) .gt. zero) lk = dmax1(lk, (v(dst0)-rad)*w(phipin)) go to 250 c c *** decide whether to check for special case... in practice (from c *** the standpoint of the calling optimization code) it seems best c *** not to check until a few iterations have failed -- hence the c *** test on kamin below. c 330 delta = alphak + dmin1(zero, v(dst0)) twopsi = alphak*dst*dst + gtsta if (ka .ge. kamin) go to 340 c *** if the test in ref. 2 is satisfied, fall through to handle c *** the special case (as soon as the more-sorensen test detects c *** it). if (delta .ge. psifac*twopsi) go to 370 c c *** check for the special case of h + alpha*d**2 (nearly) c *** singular. use one step of inverse power method with start c *** from lsvmin to obtain approximate eigenvector corresponding c *** to smallest eigenvalue of (d**-1)*h*(d**-1). lsvmin returns c *** x and w with l*w = x. c 340 t = lsvmin(p, l, w(x), w) c c *** normalize w *** do 350 i = 1, p 350 w(i) = t*w(i) c *** complete current inv. power iter. -- replace w by (l**-t)*w. call litvmu(p, w, l, w) t2 = one/v2norm(p, w) do 360 i = 1, p 360 w(i) = t2*w(i) t = t2 * t c c *** now w is the desired approximate (unit) eigenvector and c *** t*x = ((d**-1)*h*(d**-1) + alphak*i)*w. c sw = dotprd(p, w(q), w) t1 = (rad + dst) * (rad - dst) root = dsqrt(sw*sw + t1) if (sw .lt. zero) root = -root si = t1 / (sw + root) c c *** the actual test for the special case... c if ((t2*si)**2 .le. eps*(dst**2 + alphak*radsq)) go to 380 c c *** update upper bound on smallest eigenvalue (when not positive) c *** (as recommended by more and sorensen) and continue... c if (v(dst0) .le. zero) v(dst0) = dmin1(v(dst0), t2**2 - alphak) lk = dmax1(lk, -v(dst0)) c c *** check whether we can hope to detect the special case in c *** the available arithmetic. accept step as it is if not. c c *** if not yet available, obtain machine dependent value dgxfac. 370 if (dgxfac .eq. zero) dgxfac = epsfac * rmdcon(3) c if (delta .gt. dgxfac*w(dggdmx)) go to 250 go to 270 c c *** special case detected... negate alphak to indicate special case c 380 alphak = -alphak v(preduc) = half * twopsi c c *** accept current step if adding si*w would lead to a c *** further relative reduction in psi of less than v(epslon)/3. c t1 = zero t = si*(alphak*sw - half*si*(alphak + t*dotprd(p,w(x),w))) if (t .lt. eps*twopsi/six) go to 390 v(preduc) = v(preduc) + t dst = rad t1 = -si 390 do 400 i = 1, p j = q0 + i w(j) = t1*w(i) - w(j) step(i) = w(j) / d(i) 400 continue v(gtstep) = dotprd(p, dig, w(q)) c c *** save values for use in a possible restart *** c 410 v(dstnrm) = dst v(stppar) = alphak w(lk0) = lk w(uk0) = uk v(rad0) = rad w(dstsav) = dst c c *** restore diagonal of dihdi *** c j = 0 do 420 i = 1, p j = j + i k = diag0 + i dihdi(j) = w(k) 420 continue c 999 return c c *** last card of gqtst follows *** end subroutine lsqrt(n1, n, l, a, irc) c c *** compute rows n1 through n of the cholesky factor l of c *** a = l*(l**t), where l and the lower triangle of a are both c *** stored compactly by rows (and may occupy the same storage). c *** irc = 0 means all went well. irc = j means the leading c *** principal j x j submatrix of a is not positive definite -- c *** and l(j*(j+1)/2) contains the (nonpos.) reduced j-th diagonal. c c *** parameters *** c integer n1, n, irc cal double precision l(1), a(1) double precision l(n*(n+1)/2), a(n*(n+1)/2) c dimension l(n*(n+1)/2), a(n*(n+1)/2) c c *** local variables *** c integer i, ij, ik, im1, i0, j, jk, jm1, j0, k double precision t, td, zero c c *** intrinsic functions *** c/+ double precision dsqrt c/ c/6 c data zero/0.d+0/ c/7 parameter (zero=0.d+0) c/ c c *** body *** c i0 = n1 * (n1 - 1) / 2 do 50 i = n1, n td = zero if (i .eq. 1) go to 40 j0 = 0 im1 = i - 1 do 30 j = 1, im1 t = zero if (j .eq. 1) go to 20 jm1 = j - 1 do 10 k = 1, jm1 ik = i0 + k jk = j0 + k t = t + l(ik)*l(jk) 10 continue 20 ij = i0 + j j0 = j0 + j t = (a(ij) - t) / l(j0) l(ij) = t td = td + t*t 30 continue 40 i0 = i0 + i t = a(i0) - td if (t .le. zero) go to 60 l(i0) = dsqrt(t) 50 continue c irc = 0 go to 999 c 60 l(i0) = t irc = i c 999 return c c *** last card of lsqrt *** end double precision function lsvmin(p, l, x, y) c c *** estimate smallest sing. value of packed lower triang. matrix l c c *** parameter declarations *** c integer p cal double precision l(1), x(p), y(p) double precision l(p*(p+1)/2), x(p), y(p) c dimension l(p*(p+1)/2) c c+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ c c *** purpose *** c c this function returns a good over-estimate of the smallest c singular value of the packed lower triangular matrix l. c c *** parameter description *** c c p (in) = the order of l. l is a p x p lower triangular matrix. c l (in) = array holding the elements of l in row order, i.e. c l(1,1), l(2,1), l(2,2), l(3,1), l(3,2), l(3,3), etc. c x (out) if lsvmin returns a positive value, then x is a normalized c approximate left singular vector corresponding to the c smallest singular value. this approximation may be very c crude. if lsvmin returns zero, then some components of x c are zero and the rest retain their input values. c y (out) if lsvmin returns a positive value, then y = (l**-1)*x is an c unnormalized approximate right singular vector correspond- c ing to the smallest singular value. this approximation c may be crude. if lsvmin returns zero, then y retains its c input value. the caller may pass the same vector for x c and y (nonstandard fortran usage), in which case y over- c writes x (for nonzero lsvmin returns). c c *** algorithm notes *** c c the algorithm is based on (1), with the additional provision that c lsvmin = 0 is returned if the smallest diagonal element of l c (in magnitude) is not more than the unit roundoff times the c largest. the algorithm uses a random number generator proposed c in (4), which passes the spectral test with flying colors -- see c (2) and (3). c c *** subroutines and functions called *** c c v2norm - function, returns the 2-norm of a vector. c c *** references *** c c (1) cline, a., moler, c., stewart, g., and wilkinson, j.h.(1977), c an estimate for the condition number of a matrix, report c tm-310, applied math. div., argonne national laboratory. c c (2) hoaglin, d.c. (1976), theoretical properties of congruential c random-number generators -- an empirical view, c memorandum ns-340, dept. of statistics, harvard univ. c c (3) knuth, d.e. (1969), the art of computer programming, vol. 2 c (seminumerical algorithms), addison-wesley, reading, mass. c c (4) smith, c.s. (1971), multiplicative pseudo-random number c generators with prime modulus, j. assoc. comput. mach. 18, c pp. 586-593. c c *** history *** c c designed and coded by david m. gay (winter 1977/summer 1978). c c *** general *** c c this subroutine was written in connection with research c supported by the national science foundation under grants c mcs-7600324, dcr75-10143, 76-14311dss, and mcs76-11989. c c+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ c c *** local variables *** c integer i, ii, ix, j, ji, jj, jjj, jm1, j0, pm1 double precision b, sminus, splus, t, xminus, xplus c c *** constants *** c double precision half, one, r9973, zero c c *** intrinsic functions *** c/+ integer mod real float double precision dabs c/ c *** external functions and subroutines *** c external dotprd, v2norm, vaxpy double precision dotprd, v2norm c c/6 c data half/0.5d+0/, one/1.d+0/, r9973/9973.d+0/, zero/0.d+0/ c/7 parameter (half=0.5d+0, one=1.d+0, r9973=9973.d+0, zero=0.d+0) c/ c c *** body *** c ix = 2 pm1 = p - 1 c c *** first check whether to return lsvmin = 0 and initialize x *** c ii = 0 j0 = p*pm1/2 jj = j0 + p if (l(jj) .eq. zero) go to 110 ix = mod(3432*ix, 9973) b = half*(one + float(ix)/r9973) xplus = b / l(jj) x(p) = xplus if (p .le. 1) go to 60 do 10 i = 1, pm1 ii = ii + i if (l(ii) .eq. zero) go to 110 ji = j0 + i x(i) = xplus * l(ji) 10 continue c c *** solve (l**t)*x = b, where the components of b have randomly c *** chosen magnitudes in (.5,1) with signs chosen to make x large. c c do j = p-1 to 1 by -1... do 50 jjj = 1, pm1 j = p - jjj c *** determine x(j) in this iteration. note for i = 1,2,...,j c *** that x(i) holds the current partial sum for row i. ix = mod(3432*ix, 9973) b = half*(one + float(ix)/r9973) xplus = (b - x(j)) xminus = (-b - x(j)) splus = dabs(xplus) sminus = dabs(xminus) jm1 = j - 1 j0 = j*jm1/2 jj = j0 + j xplus = xplus/l(jj) xminus = xminus/l(jj) if (jm1 .eq. 0) go to 30 do 20 i = 1, jm1 ji = j0 + i splus = splus + dabs(x(i) + l(ji)*xplus) sminus = sminus + dabs(x(i) + l(ji)*xminus) 20 continue 30 if (sminus .gt. splus) xplus = xminus x(j) = xplus c *** update partial sums *** if (jm1 .gt. 0) call vaxpy(jm1, x, xplus, l(j0+1), x) 50 continue c c *** normalize x *** c 60 t = one/v2norm(p, x) do 70 i = 1, p 70 x(i) = t*x(i) c c *** solve l*y = x and return lsvmin = 1/twonorm(y) *** c do 100 j = 1, p jm1 = j - 1 j0 = j*jm1/2 jj = j0 + j t = zero if (jm1 .gt. 0) t = dotprd(jm1, l(j0+1), y) y(j) = (x(j) - t) / l(jj) 100 continue c lsvmin = one/v2norm(p, y) go to 999 c 110 lsvmin = zero 999 return c *** last card of lsvmin follows *** end subroutine slvmul(p, y, s, x) c c *** set y = s * x, s = p x p symmetric matrix. *** c *** lower triangle of s stored rowwise. *** c c *** parameter declarations *** c integer p cal double precision s(1), x(p), y(p) double precision s(p*(p+1)/2), x(p), y(p) c dimension s(p*(p+1)/2) c c *** local variables *** c integer i, im1, j, k double precision xi c c *** no intrinsic functions *** c c *** external function *** c external dotprd double precision dotprd c c----------------------------------------------------------------------- c j = 1 do 10 i = 1, p y(i) = dotprd(i, s(j), x) j = j + i 10 continue c if (p .le. 1) go to 999 j = 1 do 40 i = 2, p xi = x(i) im1 = i - 1 j = j + 1 do 30 k = 1, im1 y(k) = y(k) + s(j)*xi j = j + 1 30 continue 40 continue c 999 return c *** last card of slvmul follows *** end