X-Git-Url: http://mmka.chem.univ.gda.pl/gitweb/?a=blobdiff_plain;f=source%2Funres%2Fsrc_Eshel%2Fcored.f;fp=source%2Funres%2Fsrc_Eshel%2Fcored.f;h=1cf25e5a6a1ce35ec0b5428d39e99fe3ebe0e0d2;hb=d101c97dea752458d76055fdbae49c26fff03c1f;hp=0000000000000000000000000000000000000000;hpb=325eda160c9ad2982501e091ca40606a29043712;p=unres.git diff --git a/source/unres/src_Eshel/cored.f b/source/unres/src_Eshel/cored.f new file mode 100644 index 0000000..1cf25e5 --- /dev/null +++ b/source/unres/src_Eshel/cored.f @@ -0,0 +1,3151 @@ + 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