X-Git-Url: http://mmka.chem.univ.gda.pl/gitweb/?p=unres.git;a=blobdiff_plain;f=source%2Funres%2Fsrc_CSA_DiL%2Fsumsld.f;fp=source%2Funres%2Fsrc_CSA_DiL%2Fsumsld.f;h=0000000000000000000000000000000000000000;hp=1ce7b78ad9cc80c8f8ef23c82cdb154f7031d220;hb=2a226bfc86eabc6e4eae0c3ad1cbc3cb5417a05a;hpb=a0e685f844163003749ba91dfbf4644bcc8cfa30 diff --git a/source/unres/src_CSA_DiL/sumsld.f b/source/unres/src_CSA_DiL/sumsld.f deleted file mode 100644 index 1ce7b78..0000000 --- a/source/unres/src_CSA_DiL/sumsld.f +++ /dev/null @@ -1,1446 +0,0 @@ - subroutine sumsl(n, d, x, calcf, calcg, iv, liv, lv, v, - 1 uiparm, urparm, ufparm) -c -c *** minimize general unconstrained objective function using *** -c *** analytic gradient and hessian approx. from secant update *** -c - integer n, liv, lv - integer iv(liv), uiparm(1) - double precision d(n), x(n), v(lv), urparm(1) -c dimension v(71 + n*(n+15)/2), uiparm(*), urparm(*) - external calcf, calcg, ufparm -c -c *** purpose *** -c -c this routine interacts with subroutine sumit in an attempt -c to find an n-vector x* that minimizes the (unconstrained) -c objective function computed by calcf. (often the x* found is -c a local minimizer rather than a global one.) -c -c-------------------------- parameter usage -------------------------- -c -c n........ (input) the number of variables on which f depends, i.e., -c the number of components in x. -c d........ (input/output) a scale vector such that d(i)*x(i), -c i = 1,2,...,n, are all in comparable units. -c d can strongly affect the behavior of sumsl. -c finding the best choice of d is generally a trial- -c and-error process. choosing d so that d(i)*x(i) -c has about the same value for all i often works well. -c the defaults provided by subroutine deflt (see i -c below) require the caller to supply d. -c x........ (input/output) before (initially) calling sumsl, the call- -c er should set x to an initial guess at x*. when -c sumsl returns, x contains the best point so far -c found, i.e., the one that gives the least value so -c far seen for f(x). -c calcf.... (input) a subroutine that, given x, computes f(x). calcf -c must be declared external in the calling program. -c it is invoked by -c call calcf(n, x, nf, f, uiparm, urparm, ufparm) -c when calcf is called, nf is the invocation -c count for calcf. nf is included for possible use -c with calcg. if x is out of bounds (e.g., if it -c would cause overflow in computing f(x)), then calcf -c should set nf to 0. this will cause a shorter step -c to be attempted. (if x is in bounds, then calcf -c should not change nf.) the other parameters are as -c described above and below. calcf should not change -c n, p, or x. -c calcg.... (input) a subroutine that, given x, computes g(x), the gra- -c dient of f at x. calcg must be declared external in -c the calling program. it is invoked by -c call calcg(n, x, nf, g, uiparm, urparm, ufaprm) -c when calcg is called, nf is the invocation -c count for calcf at the time f(x) was evaluated. the -c x passed to calcg is usually the one passed to calcf -c on either its most recent invocation or the one -c prior to it. if calcf saves intermediate results -c for use by calcg, then it is possible to tell from -c nf whether they are valid for the current x (or -c which copy is valid if two copies are kept). if g -c cannot be computed at x, then calcg should set nf to -c 0. in this case, sumsl will return with iv(1) = 65. -c (if g can be computed at x, then calcg should not -c changed nf.) the other parameters to calcg are as -c described above and below. calcg should not change -c n or x. -c iv....... (input/output) an integer value array of length liv (see -c below) that helps control the sumsl algorithm and -c that is used to store various intermediate quanti- -c ties. of particular interest are the initialization/ -c return code iv(1) and the entries in iv that control -c printing and limit the number of iterations and func- -c tion evaluations. see the section on iv input -c values below. -c liv...... (input) length of iv array. must be at least 60. if li -c is too small, then sumsl returns with iv(1) = 15. -c when sumsl returns, the smallest allowed value of -c liv is stored in iv(lastiv) -- see the section on -c iv output values below. (this is intended for use -c with extensions of sumsl that handle constraints.) -c lv....... (input) length of v array. must be at least 71+n*(n+15)/2. -c (at least 77+n*(n+17)/2 for smsno, at least -c 78+n*(n+12) for humsl). if lv is too small, then -c sumsl returns with iv(1) = 16. when sumsl returns, -c the smallest allowed value of lv is stored in -c iv(lastv) -- see the section on iv output values -c below. -c v........ (input/output) a floating-point value array of length l -c (see below) that helps control the sumsl algorithm -c and that is used to store various intermediate -c quantities. of particular interest are the entries -c in v that limit the length of the first step -c attempted (lmax0) and specify convergence tolerances -c (afctol, lmaxs, rfctol, sctol, xctol, xftol). -c uiparm... (input) user integer parameter array passed without change -c to calcf and calcg. -c urparm... (input) user floating-point parameter array passed without -c change to calcf and calcg. -c ufparm... (input) user external subroutine or function passed without -c change to calcf and calcg. -c -c *** iv input values (from subroutine deflt) *** -c -c iv(1)... on input, iv(1) should have a value between 0 and 14...... -c 0 and 12 mean this is a fresh start. 0 means that -c deflt(2, iv, liv, lv, v) -c is to be called to provide all default values to iv and -c v. 12 (the value that deflt assigns to iv(1)) means the -c caller has already called deflt and has possibly changed -c some iv and/or v entries to non-default values. -c 13 means deflt has been called and that sumsl (and -c sumit) should only do their storage allocation. that is, -c they should set the output components of iv that tell -c where various subarrays arrays of v begin, such as iv(g) -c (and, for humsl and humit only, iv(dtol)), and return. -c 14 means that a storage has been allocated (by a call -c with iv(1) = 13) and that the algorithm should be -c started. when called with iv(1) = 13, sumsl returns -c iv(1) = 14 unless liv or lv is too small (or n is not -c positive). default = 12. -c iv(inith).... iv(25) tells whether the hessian approximation h should -c be initialized. 1 (the default) means sumit should -c initialize h to the diagonal matrix whose i-th diagonal -c element is d(i)**2. 0 means the caller has supplied a -c cholesky factor l of the initial hessian approximation -c h = l*(l**t) in v, starting at v(iv(lmat)) = v(iv(42)) -c (and stored compactly by rows). note that iv(lmat) may -c be initialized by calling sumsl with iv(1) = 13 (see -c the iv(1) discussion above). default = 1. -c iv(mxfcal)... iv(17) gives the maximum number of function evaluations -c (calls on calcf) allowed. if this number does not suf- -c fice, then sumsl returns with iv(1) = 9. default = 200. -c iv(mxiter)... iv(18) gives the maximum number of iterations allowed. -c it also indirectly limits the number of gradient evalua- -c tions (calls on calcg) to iv(mxiter) + 1. if iv(mxiter) -c iterations do not suffice, then sumsl returns with -c iv(1) = 10. default = 150. -c iv(outlev)... iv(19) controls the number and length of iteration sum- -c mary lines printed (by itsum). iv(outlev) = 0 means do -c not print any summary lines. otherwise, print a summary -c line after each abs(iv(outlev)) iterations. if iv(outlev) -c is positive, then summary lines of length 78 (plus carri- -c age control) are printed, including the following... the -c iteration and function evaluation counts, f = the current -c function value, relative difference in function values -c achieved by the latest step (i.e., reldf = (f0-v(f))/f01, -c where f01 is the maximum of abs(v(f)) and abs(v(f0)) and -c v(f0) is the function value from the previous itera- -c tion), the relative function reduction predicted for the -c step just taken (i.e., preldf = v(preduc) / f01, where -c v(preduc) is described below), the scaled relative change -c in x (see v(reldx) below), the step parameter for the -c step just taken (stppar = 0 means a full newton step, -c between 0 and 1 means a relaxed newton step, between 1 -c and 2 means a double dogleg step, greater than 2 means -c a scaled down cauchy step -- see subroutine dbldog), the -c 2-norm of the scale vector d times the step just taken -c (see v(dstnrm) below), and npreldf, i.e., -c v(nreduc)/f01, where v(nreduc) is described below -- if -c npreldf is positive, then it is the relative function -c reduction predicted for a newton step (one with -c stppar = 0). if npreldf is negative, then it is the -c negative of the relative function reduction predicted -c for a step computed with step bound v(lmaxs) for use in -c testing for singular convergence. -c if iv(outlev) is negative, then lines of length 50 -c are printed, including only the first 6 items listed -c above (through reldx). -c default = 1. -c iv(parprt)... iv(20) = 1 means print any nondefault v values on a -c fresh start or any changed v values on a restart. -c iv(parprt) = 0 means skip this printing. default = 1. -c iv(prunit)... iv(21) is the output unit number on which all printing -c is done. iv(prunit) = 0 means suppress all printing. -c default = standard output unit (unit 6 on most systems). -c iv(solprt)... iv(22) = 1 means print out the value of x returned (as -c well as the gradient and the scale vector d). -c iv(solprt) = 0 means skip this printing. default = 1. -c iv(statpr)... iv(23) = 1 means print summary statistics upon return- -c ing. these consist of the function value, the scaled -c relative change in x caused by the most recent step (see -c v(reldx) below), the number of function and gradient -c evaluations (calls on calcf and calcg), and the relative -c function reductions predicted for the last step taken and -c for a newton step (or perhaps a step bounded by v(lmaxs) -c -- see the descriptions of preldf and npreldf under -c iv(outlev) above). -c iv(statpr) = 0 means skip this printing. -c iv(statpr) = -1 means skip this printing as well as that -c of the one-line termination reason message. default = 1. -c iv(x0prt).... iv(24) = 1 means print the initial x and scale vector d -c (on a fresh start only). iv(x0prt) = 0 means skip this -c printing. default = 1. -c -c *** (selected) iv output values *** -c -c iv(1)........ on output, iv(1) is a return code.... -c 3 = x-convergence. the scaled relative difference (see -c v(reldx)) between the current parameter vector x and -c a locally optimal parameter vector is very likely at -c most v(xctol). -c 4 = relative function convergence. the relative differ- -c ence between the current function value and its lo- -c cally optimal value is very likely at most v(rfctol). -c 5 = both x- and relative function convergence (i.e., the -c conditions for iv(1) = 3 and iv(1) = 4 both hold). -c 6 = absolute function convergence. the current function -c value is at most v(afctol) in absolute value. -c 7 = singular convergence. the hessian near the current -c iterate appears to be singular or nearly so, and a -c step of length at most v(lmaxs) is unlikely to yield -c a relative function decrease of more than v(sctol). -c 8 = false convergence. the iterates appear to be converg- -c ing to a noncritical point. this may mean that the -c convergence tolerances (v(afctol), v(rfctol), -c v(xctol)) are too small for the accuracy to which -c the function and gradient are being computed, that -c there is an error in computing the gradient, or that -c the function or gradient is discontinuous near x. -c 9 = function evaluation limit reached without other con- -c vergence (see iv(mxfcal)). -c 10 = iteration limit reached without other convergence -c (see iv(mxiter)). -c 11 = stopx returned .true. (external interrupt). see the -c usage notes below. -c 14 = storage has been allocated (after a call with -c iv(1) = 13). -c 17 = restart attempted with n changed. -c 18 = d has a negative component and iv(dtype) .le. 0. -c 19...43 = v(iv(1)) is out of range. -c 63 = f(x) cannot be computed at the initial x. -c 64 = bad parameters passed to assess (which should not -c occur). -c 65 = the gradient could not be computed at x (see calcg -c above). -c 67 = bad first parameter to deflt. -c 80 = iv(1) was out of range. -c 81 = n is not positive. -c iv(g)........ iv(28) is the starting subscript in v of the current -c gradient vector (the one corresponding to x). -c iv(lastiv)... iv(44) is the least acceptable value of liv. (it is -c only set if liv is at least 44.) -c iv(lastv).... iv(45) is the least acceptable value of lv. (it is -c only set if liv is large enough, at least iv(lastiv).) -c iv(nfcall)... iv(6) is the number of calls so far made on calcf (i.e., -c function evaluations). -c iv(ngcall)... iv(30) is the number of gradient evaluations (calls on -c calcg). -c iv(niter).... iv(31) is the number of iterations performed. -c -c *** (selected) v input values (from subroutine deflt) *** -c -c v(bias)..... v(43) is the bias parameter used in subroutine dbldog -- -c see that subroutine for details. default = 0.8. -c v(afctol)... v(31) is the absolute function convergence tolerance. -c if sumsl finds a point where the function value is less -c than v(afctol) in absolute value, and if sumsl does not -c return with iv(1) = 3, 4, or 5, then it returns with -c iv(1) = 6. this test can be turned off by setting -c v(afctol) to zero. default = max(10**-20, machep**2), -c where machep is the unit roundoff. -c v(dinit).... v(38), if nonnegative, is the value to which the scale -c vector d is initialized. default = -1. -c v(lmax0).... v(35) gives the maximum 2-norm allowed for d times the -c very first step that sumsl attempts. this parameter can -c markedly affect the performance of sumsl. -c v(lmaxs).... v(36) is used in testing for singular convergence -- if -c the function reduction predicted for a step of length -c bounded by v(lmaxs) is at most v(sctol) * abs(f0), where -c f0 is the function value at the start of the current -c iteration, and if sumsl does not return with iv(1) = 3, -c 4, 5, or 6, then it returns with iv(1) = 7. default = 1. -c v(rfctol)... v(32) is the relative function convergence tolerance. -c if the current model predicts a maximum possible function -c reduction (see v(nreduc)) of at most v(rfctol)*abs(f0) -c at the start of the current iteration, where f0 is the -c then current function value, and if the last step attempt- -c ed achieved no more than twice the predicted function -c decrease, then sumsl returns with iv(1) = 4 (or 5). -c default = max(10**-10, machep**(2/3)), where machep is -c the unit roundoff. -c v(sctol).... v(37) is the singular convergence tolerance -- see the -c description of v(lmaxs) above. -c v(tuner1)... v(26) helps decide when to check for false convergence. -c this is done if the actual function decrease from the -c current step is no more than v(tuner1) times its predict- -c ed value. default = 0.1. -c v(xctol).... v(33) is the x-convergence tolerance. if a newton step -c (see v(nreduc)) is tried that has v(reldx) .le. v(xctol) -c and if this step yields at most twice the predicted func- -c tion decrease, then sumsl returns with iv(1) = 3 (or 5). -c (see the description of v(reldx) below.) -c default = machep**0.5, where machep is the unit roundoff. -c v(xftol).... v(34) is the false convergence tolerance. if a step is -c tried that gives no more than v(tuner1) times the predict- -c ed function decrease and that has v(reldx) .le. v(xftol), -c and if sumsl does not return with iv(1) = 3, 4, 5, 6, or -c 7, then it returns with iv(1) = 8. (see the description -c of v(reldx) below.) default = 100*machep, where -c machep is the unit roundoff. -c v(*)........ deflt supplies to v a number of tuning constants, with -c which it should ordinarily be unnecessary to tinker. see -c section 17 of version 2.2 of the nl2sol usage summary -c (i.e., the appendix to ref. 1) for details on v(i), -c i = decfac, incfac, phmnfc, phmxfc, rdfcmn, rdfcmx, -c tuner2, tuner3, tuner4, tuner5. -c -c *** (selected) v output values *** -c -c v(dgnorm)... v(1) is the 2-norm of (diag(d)**-1)*g, where g is the -c most recently computed gradient. -c v(dstnrm)... v(2) is the 2-norm of diag(d)*step, where step is the -c current step. -c v(f)........ v(10) is the current function value. -c v(f0)....... v(13) is the function value at the start of the current -c iteration. -c v(nreduc)... v(6), if positive, is the maximum function reduction -c possible according to the current model, i.e., the func- -c tion reduction predicted for a newton step (i.e., -c step = -h**-1 * g, where g is the current gradient and -c h is the current hessian approximation). -c if v(nreduc) is negative, then it is the negative of -c the function reduction predicted for a step computed with -c a step bound of v(lmaxs) for use in testing for singular -c convergence. -c v(preduc)... v(7) is the function reduction predicted (by the current -c quadratic model) for the current step. this (divided by -c v(f0)) is used in testing for relative function -c convergence. -c v(reldx).... v(17) is the scaled relative change in x caused by the -c current step, computed as -c max(abs(d(i)*(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 where x = x0 + step. -c -c------------------------------- notes ------------------------------- -c -c *** algorithm notes *** -c -c this routine uses a hessian approximation computed from the -c bfgs update (see ref 3). only a cholesky factor of the hessian -c approximation is stored, and this is updated using ideas from -c ref. 4. steps are computed by the double dogleg scheme described -c in ref. 2. the steps are assessed as in ref. 1. -c -c *** usage notes *** -c -c after a return with iv(1) .le. 11, it is possible to restart, -c i.e., to change some of the iv and v input values described above -c and continue the algorithm from the point where it was interrupt- -c ed. iv(1) should not be changed, nor should any entries of i -c and v other than the input values (those supplied by deflt). -c those who do not wish to write a calcg which computes the -c gradient analytically should call smsno rather than sumsl. -c smsno uses finite differences to compute an approximate gradient. -c those who would prefer to provide f and g (the function and -c gradient) by reverse communication rather than by writing subrou- -c tines calcf and calcg may call on sumit directly. see the com- -c ments at the beginning of sumit. -c those who use sumsl interactively may wish to supply their -c own stopx function, which should return .true. if the break key -c has been pressed since stopx was last invoked. this makes it -c possible to externally interrupt sumsl (which will return with -c iv(1) = 11 if stopx returns .true.). -c storage for g is allocated at the end of v. thus the caller -c may make v longer than specified above and may allow calcg to use -c elements of g beyond the first n as scratch storage. -c -c *** portability notes *** -c -c the sumsl distribution tape contains both single- and double- -c precision versions of the sumsl source code, so it should be un- -c necessary to change precisions. -c only the functions imdcon and rmdcon contain machine-dependent -c constants. to change from one machine to another, it should -c suffice to change the (few) relevant lines in these functions. -c intrinsic functions are explicitly declared. on certain com- -c puters (e.g. univac), it may be necessary to comment out these -c declarations. so that this may be done automatically by a simple -c program, such declarations are preceded by a comment having c/+ -c in columns 1-3 and blanks in columns 4-72 and are followed by -c a comment having c/ in columns 1 and 2 and blanks in columns 3-72. -c the sumsl source code is expressed in 1966 ansi standard -c fortran. it may be converted to fortran 77 by commenting out all -c lines that fall between a line having c/6 in columns 1-3 and a -c line having c/7 in columns 1-3 and by removing (i.e., replacing -c by a blank) the c in column 1 of the lines that follow the c/7 -c line and precede a line having c/ in columns 1-2 and blanks in -c columns 3-72. these changes convert some data statements into -c parameter statements, convert some variables from real to -c character*4, and make the data statements that initialize these -c variables use character strings delimited by primes instead -c of hollerith constants. (such variables and data statements -c appear only in modules itsum and parck. parameter statements -c appear nearly everywhere.) these changes also add save state- -c ments for variables given machine-dependent constants by rmdcon. -c -c *** references *** -c -c 1. dennis, j.e., gay, d.m., and welsch, r.e. (1981), algorithm 573 -- -c an adaptive nonlinear least-squares algorithm, acm trans. -c math. software 7, pp. 369-383. -c -c 2. dennis, j.e., and mei, h.h.w. (1979), two new unconstrained opti- -c mization algorithms which use function and gradient -c values, j. optim. theory applic. 28, pp. 453-482. -c -c 3. dennis, j.e., and more, j.j. (1977), quasi-newton methods, motiva- -c tion and theory, siam rev. 19, pp. 46-89. -c -c 4. goldfarb, d. (1976), factorized variable metric methods for uncon- -c strained optimization, math. comput. 30, pp. 796-811. -c -c *** general *** -c -c coded by david m. gay (winter 1980). revised summer 1982. -c this subroutine was written in connection with research -c supported in part by the national science foundation under -c grants mcs-7600324, dcr75-10143, 76-14311dss, mcs76-11989, -c and mcs-7906671. -c. -c -c---------------------------- declarations --------------------------- -c - external deflt, sumit -c -c deflt... supplies default iv and v input components. -c sumit... reverse-communication routine that carries out sumsl algo- -c rithm. -c - integer g1, iv1, nf - double precision f -c -c *** subscripts for iv *** -c - integer nextv, nfcall, nfgcal, g, toobig, vneed -c -c/6 -c data nextv/47/, nfcall/6/, nfgcal/7/, g/28/, toobig/2/, vneed/4/ -c/7 - parameter (nextv=47, nfcall=6, nfgcal=7, g=28, toobig=2, vneed=4) -c/ -c -c+++++++++++++++++++++++++++++++ body ++++++++++++++++++++++++++++++++ -c - if (iv(1) .eq. 0) call deflt(2, iv, liv, lv, v) - iv1 = iv(1) - if (iv1 .eq. 12 .or. iv1 .eq. 13) iv(vneed) = iv(vneed) + n - if (iv1 .eq. 14) go to 10 - if (iv1 .gt. 2 .and. iv1 .lt. 12) go to 10 - g1 = 1 - if (iv1 .eq. 12) iv(1) = 13 - go to 20 -c - 10 g1 = iv(g) -c - 20 call sumit(d, f, v(g1), iv, 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 calcg(n, x, iv(nfgcal), v(g1), uiparm, urparm, ufparm) - go to 20 -c - 50 if (iv(1) .ne. 14) go to 999 -c -c *** storage allocation -c - iv(g) = iv(nextv) - iv(nextv) = iv(g) + n - if (iv1 .ne. 13) go to 10 -c - 999 return -c *** last card of sumsl follows *** - end - subroutine sumit(d, fx, g, iv, liv, lv, n, v, x) -c -c *** carry out sumsl (unconstrained minimization) iterations, using -c *** double-dogleg/bfgs steps. -c -c *** parameter declarations *** -c - integer liv, lv, n - integer iv(liv) - double precision d(n), fx, g(n), v(lv), x(n) -c -c-------------------------- parameter usage -------------------------- -c -c d.... scale vector. -c fx... function value. -c g.... gradient vector. -c iv... integer value array. -c liv.. length of iv (at least 60). -c lv... length of v (at least 71 + n*(n+13)/2). -c n.... number of variables (components in x and g). -c v.... floating-point value array. -c x.... vector of parameters to be optimized. -c -c *** discussion *** -c -c parameters iv, n, v, and x are the same as the corresponding -c ones to sumsl (which see), except that v can be shorter (since -c the part of v that sumsl uses for storing g is not needed). -c moreover, compared with sumsl, 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 sumsl (and smsno), is not referenced by -c sumit or the subroutines it calls. -c fx and g need not have been initialized when sumit is called -c with iv(1) = 12, 13, or 14. -c -c iv(1) = 1 means the caller should set fx to f(x), the function value -c at x, and call sumit again, having changed none of the -c other parameters. an exception occurs if f(x) cannot be -c (e.g. if overflow would occur), which may happen because -c of an oversized step. in this case the caller should set -c iv(toobig) = iv(2) to 1, which will cause sumit to ig- -c nore fx and try a smaller step. the parameter nf that -c sumsl passes to calcf (for possible use by calcg) is a -c copy of iv(nfcall) = iv(6). -c iv(1) = 2 means the caller should set g to g(x), the gradient vector -c of f at x, and call sumit again, having changed none of -c the other parameters except possibly the scale vector d -c when iv(dtype) = 0. the parameter nf that sumsl passes -c to calcg is iv(nfgcal) = iv(7). if g(x) cannot be -c evaluated, then the caller may set iv(nfgcal) to 0, in -c which case sumit will return with iv(1) = 65. -c. -c *** general *** -c -c coded by david m. gay (december 1979). 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 for references.) -c -c+++++++++++++++++++++++++++ declarations ++++++++++++++++++++++++++++ -c -c *** local variables *** -c - integer dg1, dummy, g01, i, k, l, lstgst, nwtst1, step1, - 1 temp1, w, x01, z - double precision t -c -c *** constants *** -c - double precision half, negone, one, onep2, zero -c -c *** no intrinsic functions *** -c -c *** external functions and subroutines *** -c - external assst, dbdog, deflt, dotprd, itsum, litvmu, livmul, - 1 ltvmul, lupdat, lvmul, parck, reldst, stopx, vaxpy, - 2 vcopy, vscopy, vvmulp, v2norm, wzbfgs - logical stopx - double precision dotprd, reldst, v2norm -c -c assst.... assesses candidate step. -c dbdog.... computes double-dogleg (candidate) step. -c deflt.... supplies default iv and v input components. -c dotprd... returns inner product of two vectors. -c itsum.... prints iteration summary and info on initial and final x. -c litvmu... multiplies inverse transpose of lower triangle times vector. -c livmul... multiplies inverse of lower triangle times vector. -c ltvmul... multiplies transpose of lower triangle times vector. -c lupdt.... updates cholesky factor of hessian approximation. -c lvmul.... multiplies lower triangle times vector. -c parck.... checks validity of input iv and v values. -c reldst... computes v(reldx) = relative step size. -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 vvmulp... multiplies vector by vector raised to power (componentwise). -c v2norm... returns the 2-norm of a vector. -c wzbfgs... computes w and z for lupdat corresponding to bfgs update. -c -c *** subscripts for iv and v *** -c - integer afctol - integer cnvcod, dg, dgnorm, dinit, dstnrm, dst0, f, f0, fdif, - 1 gthg, gtstep, g0, incfac, inith, irc, kagqt, lmat, lmax0, - 2 lmaxs, mode, model, mxfcal, mxiter, nextv, nfcall, nfgcal, - 3 ngcall, niter, nreduc, nwtstp, preduc, radfac, radinc, - 4 radius, rad0, reldx, restor, step, stglim, stlstg, toobig, - 5 tuner4, tuner5, vneed, xirc, x0 -c -c *** iv subscript values *** -c -c/6 -c data cnvcod/55/, dg/37/, g0/48/, inith/25/, irc/29/, kagqt/33/, -c 1 mode/35/, model/5/, mxfcal/17/, mxiter/18/, nfcall/6/, -c 2 nfgcal/7/, ngcall/30/, niter/31/, nwtstp/34/, radinc/8/, -c 3 restor/9/, step/40/, stglim/11/, stlstg/41/, toobig/2/, -c 4 vneed/4/, xirc/13/, x0/43/ -c/7 - parameter (cnvcod=55, dg=37, g0=48, inith=25, irc=29, kagqt=33, - 1 mode=35, model=5, mxfcal=17, mxiter=18, nfcall=6, - 2 nfgcal=7, ngcall=30, niter=31, nwtstp=34, radinc=8, - 3 restor=9, step=40, stglim=11, stlstg=41, toobig=2, - 4 vneed=4, xirc=13, x0=43) -c/ -c -c *** v subscript values *** -c -c/6 -c data afctol/31/ -c data dgnorm/1/, dinit/38/, dstnrm/2/, dst0/3/, f/10/, f0/13/, -c 1 fdif/11/, gthg/44/, gtstep/4/, incfac/23/, lmat/42/, -c 2 lmax0/35/, lmaxs/36/, nextv/47/, nreduc/6/, preduc/7/, -c 3 radfac/16/, radius/8/, rad0/9/, reldx/17/, tuner4/29/, -c 4 tuner5/30/ -c/7 - parameter (afctol=31) - parameter (dgnorm=1, dinit=38, dstnrm=2, dst0=3, f=10, f0=13, - 1 fdif=11, gthg=44, gtstep=4, incfac=23, lmat=42, - 2 lmax0=35, lmaxs=36, nextv=47, nreduc=6, preduc=7, - 3 radfac=16, radius=8, rad0=9, reldx=17, tuner4=29, - 4 tuner5=30) -c/ -c -c/6 -c data half/0.5d+0/, negone/-1.d+0/, one/1.d+0/, onep2/1.2d+0/, -c 1 zero/0.d+0/ -c/7 - parameter (half=0.5d+0, negone=-1.d+0, one=1.d+0, onep2=1.2d+0, - 1 zero=0.d+0) -c/ -c -c+++++++++++++++++++++++++++++++ body ++++++++++++++++++++++++++++++++ -c -C Following SAVE statement inserted. - save l - i = iv(1) - if (i .eq. 1) go to 50 - if (i .eq. 2) go to 60 -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+13)/2 - call parck(2, d, iv, liv, lv, n, v) - i = iv(1) - 2 - if (i .gt. 12) go to 999 - go to (180, 180, 180, 180, 180, 180, 120, 90, 120, 10, 10, 20), i -c -c *** storage allocation *** -c -10 l = iv(lmat) - iv(x0) = l + n*(n+1)/2 - iv(step) = iv(x0) + n - iv(stlstg) = iv(step) + n - iv(g0) = iv(stlstg) + n - iv(nwtstp) = iv(g0) + n - iv(dg) = iv(nwtstp) + n - iv(nextv) = iv(dg) + n - 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 - if (v(dinit) .ge. zero) call vscopy(n, d, v(dinit)) - if (iv(inith) .ne. 1) go to 40 -c -c *** set the initial hessian approximation to diag(d)**-2 *** -c - l = iv(lmat) - call vscopy(n*(n+1)/2, v(l), zero) - k = l - 1 - do 30 i = 1, n - k = k + i - t = d(i) - if (t .le. zero) t = one - v(k) = t - 30 continue -c -c *** compute initial function value *** -c - 40 iv(1) = 1 - go to 999 -c - 50 v(f) = fx - if (iv(mode) .ge. 0) go to 180 - iv(1) = 2 - if (iv(toobig) .eq. 0) go to 999 - iv(1) = 63 - go to 300 -c -c *** make sure gradient could be computed *** -c - 60 if (iv(nfgcal) .ne. 0) go to 70 - iv(1) = 65 - go to 300 -c - 70 dg1 = iv(dg) - call vvmulp(n, v(dg1), g, d, -1) - v(dgnorm) = v2norm(n, v(dg1)) -c -c *** test norm of gradient *** -c - if (v(dgnorm) .gt. v(afctol)) go to 75 - iv(irc) = 10 - iv(cnvcod) = iv(irc) - 4 -c - 75 if (iv(cnvcod) .ne. 0) go to 290 - if (iv(mode) .eq. 0) go to 250 -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 - 80 call itsum(d, g, iv, liv, lv, n, v, x) - 90 k = iv(niter) - if (k .lt. iv(mxiter)) go to 100 - iv(1) = 10 - go to 300 -c -c *** update radius *** -c - 100 iv(niter) = k + 1 - if(k.gt.0)v(radius) = v(radfac) * v(dstnrm) -c -c *** initialize for start of next iteration *** -c - g01 = iv(g0) - x01 = iv(x0) - v(f0) = v(f) - iv(irc) = 4 - iv(kagqt) = -1 -c -c *** copy x to x0, g to g0 *** -c - call vcopy(n, v(x01), x) - call vcopy(n, v(g01), g) -c -c *** check stopx and function evaluation limit *** -c -C AL 4/30/95 - dummy=iv(nfcall) - 110 if (.not. stopx(dummy)) go to 130 - iv(1) = 11 - go to 140 -c -c *** come here when restarting after func. eval. limit or stopx. -c - 120 if (v(f) .ge. v(f0)) go to 130 - v(radfac) = one - k = iv(niter) - go to 100 -c - 130 if (iv(nfcall) .lt. iv(mxfcal)) go to 150 - iv(1) = 9 - 140 if (v(f) .ge. v(f0)) go to 300 -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 240 -c -c. . . . . . . . . . . . . compute candidate step . . . . . . . . . . -c - 150 step1 = iv(step) - dg1 = iv(dg) - nwtst1 = iv(nwtstp) - if (iv(kagqt) .ge. 0) go to 160 - l = iv(lmat) - call livmul(n, v(nwtst1), v(l), g) - v(nreduc) = half * dotprd(n, v(nwtst1), v(nwtst1)) - call litvmu(n, v(nwtst1), v(l), v(nwtst1)) - call vvmulp(n, v(step1), v(nwtst1), d, 1) - v(dst0) = v2norm(n, v(step1)) - call vvmulp(n, v(dg1), v(dg1), d, -1) - call ltvmul(n, v(step1), v(l), v(dg1)) - v(gthg) = v2norm(n, v(step1)) - iv(kagqt) = 0 - 160 call dbdog(v(dg1), lv, n, v(nwtst1), v(step1), v) - if (iv(irc) .eq. 6) go to 180 -c -c *** check whether evaluating f(x0 + step) looks worthwhile *** -c - if (v(dstnrm) .le. zero) go to 180 - if (iv(irc) .ne. 5) go to 170 - if (v(radfac) .le. one) go to 170 - if (v(preduc) .le. onep2 * v(fdif)) go to 180 -c -c *** compute f(x0 + step) *** -c - 170 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 - 180 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 190 - 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 - 190 k = iv(irc) - go to (200,230,230,230,200,210,220,220,220,220,220,220,280,250), k -c -c *** recompute step with changed radius *** -c - 200 v(radius) = v(radfac) * v(dstnrm) - go to 110 -c -c *** compute step of length v(lmaxs) for singular convergence test. -c - 210 v(radius) = v(lmaxs) - go to 150 -c -c *** convergence or false convergence *** -c - 220 iv(cnvcod) = k - 4 - if (v(f) .ge. v(f0)) go to 290 - if (iv(xirc) .eq. 14) go to 290 - iv(xirc) = 14 -c -c. . . . . . . . . . . . process acceptable step . . . . . . . . . . . -c - 230 if (iv(irc) .ne. 3) go to 240 - step1 = iv(step) - temp1 = iv(stlstg) -c -c *** set temp1 = hessian * step for use in gradient tests *** -c - l = iv(lmat) - call ltvmul(n, v(temp1), v(l), v(step1)) - call lvmul(n, v(temp1), v(l), v(temp1)) -c -c *** compute gradient *** -c - 240 iv(ngcall) = iv(ngcall) + 1 - iv(1) = 2 - go to 999 -c -c *** initializations -- g0 = g - g0, etc. *** -c - 250 g01 = iv(g0) - call vaxpy(n, v(g01), negone, v(g01), g) - step1 = iv(step) - temp1 = iv(stlstg) - if (iv(irc) .ne. 3) go to 270 -c -c *** set v(radfac) by gradient tests *** -c -c *** set temp1 = diag(d)**-1 * (hessian*step + (g(x0)-g(x))) *** -c - call vaxpy(n, v(temp1), negone, v(g01), v(temp1)) - call vvmulp(n, v(temp1), v(temp1), d, -1) -c -c *** do gradient tests *** -c - if (v2norm(n, v(temp1)) .le. v(dgnorm) * v(tuner4)) - 1 go to 260 - if (dotprd(n, g, v(step1)) - 1 .ge. v(gtstep) * v(tuner5)) go to 270 - 260 v(radfac) = v(incfac) -c -c *** update h, loop *** -c - 270 w = iv(nwtstp) - z = iv(x0) - l = iv(lmat) - call wzbfgs(v(l), n, v(step1), v(w), v(g01), v(z)) -c -c ** use the n-vectors starting at v(step1) and v(g01) for scratch.. - call lupdat(v(temp1), v(step1), v(l), v(g01), v(l), n, v(w), v(z)) - iv(1) = 2 - go to 80 -c -c. . . . . . . . . . . . . . misc. details . . . . . . . . . . . . . . -c -c *** bad parameters to assess *** -c - 280 iv(1) = 64 - go to 300 -c -c *** print summary of final iteration and other requested items *** -c - 290 iv(1) = iv(cnvcod) - iv(cnvcod) = 0 - 300 call itsum(d, g, iv, liv, lv, n, v, x) -c - 999 return -c -c *** last line of sumit follows *** - end - subroutine dbdog(dig, lv, n, nwtstp, step, v) -c -c *** compute double dogleg step *** -c -c *** parameter declarations *** -c - integer lv, n - double precision dig(n), nwtstp(n), step(n), v(lv) -c -c *** purpose *** -c -c this subroutine computes a candidate step (for use in an uncon- -c strained minimization code) by the double dogleg algorithm of -c dennis and mei (ref. 1), which is a variation on powell*s dogleg -c scheme (ref. 2, p. 95). -c -c-------------------------- parameter usage -------------------------- -c -c dig (input) diag(d)**-2 * g -- see algorithm notes. -c g (input) the current gradient vector. -c lv (input) length of v. -c n (input) number of components in dig, g, nwtstp, and step. -c nwtstp (input) negative newton step -- see algorithm notes. -c step (output) the computed step. -c v (i/o) values array, the following components of which are -c used here... -c v(bias) (input) bias for relaxed newton step, which is v(bias) of -c the way from the full newton to the fully relaxed newton -c step. recommended value = 0.8 . -c v(dgnorm) (input) 2-norm of diag(d)**-1 * g -- see algorithm notes. -c v(dstnrm) (output) 2-norm of diag(d) * step, which is v(radius) -c unless v(stppar) = 0 -- see algorithm notes. -c v(dst0) (input) 2-norm of diag(d) * nwtstp -- see algorithm notes. -c v(grdfac) (output) the coefficient of dig in the step returned -- -c step(i) = v(grdfac)*dig(i) + v(nwtfac)*nwtstp(i). -c v(gthg) (input) square-root of (dig**t) * (hessian) * dig -- see -c algorithm notes. -c v(gtstep) (output) inner product between g and step. -c v(nreduc) (output) function reduction predicted for the full newton -c step. -c v(nwtfac) (output) the coefficient of nwtstp in the step returned -- -c see v(grdfac) above. -c v(preduc) (output) function reduction predicted for the step returned. -c v(radius) (input) the trust region radius. d times the step returned -c has 2-norm v(radius) unless v(stppar) = 0. -c v(stppar) (output) code telling how step was computed... 0 means a -c full newton step. between 0 and 1 means v(stppar) of the -c way from the newton to the relaxed newton step. between -c 1 and 2 means a true double dogleg step, v(stppar) - 1 of -c the way from the relaxed newton to the cauchy step. -c greater than 2 means 1 / (v(stppar) - 1) times the cauchy -c step. -c -c------------------------------- notes ------------------------------- -c -c *** algorithm notes *** -c -c let g and h be the current gradient and hessian approxima- -c tion respectively and let d be the current scale vector. this -c routine assumes dig = diag(d)**-2 * g and nwtstp = h**-1 * g. -c the step computed is the same one would get by replacing g and h -c by diag(d)**-1 * g and diag(d)**-1 * h * diag(d)**-1, -c computing step, and translating step back to the original -c variables, i.e., premultiplying it by diag(d)**-1. -c -c *** references *** -c -c 1. dennis, j.e., and mei, h.h.w. (1979), two new unconstrained opti- -c mization algorithms which use function and gradient -c values, j. optim. theory applic. 28, pp. 453-482. -c 2. powell, m.j.d. (1970), a hybrid method for non-linear equations, -c in numerical methods for non-linear equations, edited by -c p. rabinowitz, gordon and breach, london. -c -c *** general *** -c -c coded by david m. gay. -c this subroutine was written in connection with research supported -c by the national science foundation under grants mcs-7600324 and -c mcs-7906671. -c -c------------------------ external quantities ------------------------ -c -c *** functions and subroutines called *** -c - external dotprd, v2norm - double precision dotprd, v2norm -c -c dotprd... returns inner product of two vectors. -c v2norm... returns 2-norm of a vector. -c -c *** intrinsic functions *** -c/+ - double precision dsqrt -c/ -c-------------------------- local variables -------------------------- -c - integer i - double precision cfact, cnorm, ctrnwt, ghinvg, femnsq, gnorm, - 1 nwtnrm, relax, rlambd, t, t1, t2 - double precision half, one, two, zero -c -c *** v subscripts *** -c - integer bias, dgnorm, dstnrm, dst0, grdfac, gthg, gtstep, - 1 nreduc, nwtfac, preduc, radius, stppar -c -c *** data initializations *** -c -c/6 -c data half/0.5d+0/, one/1.d+0/, two/2.d+0/, zero/0.d+0/ -c/7 - parameter (half=0.5d+0, one=1.d+0, two=2.d+0, zero=0.d+0) -c/ -c -c/6 -c data bias/43/, dgnorm/1/, dstnrm/2/, dst0/3/, grdfac/45/, -c 1 gthg/44/, gtstep/4/, nreduc/6/, nwtfac/46/, preduc/7/, -c 2 radius/8/, stppar/5/ -c/7 - parameter (bias=43, dgnorm=1, dstnrm=2, dst0=3, grdfac=45, - 1 gthg=44, gtstep=4, nreduc=6, nwtfac=46, preduc=7, - 2 radius=8, stppar=5) -c/ -c -c+++++++++++++++++++++++++++++++ body ++++++++++++++++++++++++++++++++ -c - nwtnrm = v(dst0) - rlambd = one - if (nwtnrm .gt. zero) rlambd = v(radius) / nwtnrm - gnorm = v(dgnorm) - ghinvg = two * v(nreduc) - v(grdfac) = zero - v(nwtfac) = zero - if (rlambd .lt. one) go to 30 -c -c *** the newton step is inside the trust region *** -c - v(stppar) = zero - v(dstnrm) = nwtnrm - v(gtstep) = -ghinvg - v(preduc) = v(nreduc) - v(nwtfac) = -one - do 20 i = 1, n - 20 step(i) = -nwtstp(i) - go to 999 -c - 30 v(dstnrm) = v(radius) - cfact = (gnorm / v(gthg))**2 -c *** cauchy step = -cfact * g. - cnorm = gnorm * cfact - relax = one - v(bias) * (one - gnorm*cnorm/ghinvg) - if (rlambd .lt. relax) go to 50 -c -c *** step is between relaxed newton and full newton steps *** -c - v(stppar) = one - (rlambd - relax) / (one - relax) - t = -rlambd - v(gtstep) = t * ghinvg - v(preduc) = rlambd * (one - half*rlambd) * ghinvg - v(nwtfac) = t - do 40 i = 1, n - 40 step(i) = t * nwtstp(i) - go to 999 -c - 50 if (cnorm .lt. v(radius)) go to 70 -c -c *** the cauchy step lies outside the trust region -- -c *** step = scaled cauchy step *** -c - t = -v(radius) / gnorm - v(grdfac) = t - v(stppar) = one + cnorm / v(radius) - v(gtstep) = -v(radius) * gnorm - v(preduc) = v(radius)*(gnorm - half*v(radius)*(v(gthg)/gnorm)**2) - do 60 i = 1, n - 60 step(i) = t * dig(i) - go to 999 -c -c *** compute dogleg step between cauchy and relaxed newton *** -c *** femur = relaxed newton step minus cauchy step *** -c - 70 ctrnwt = cfact * relax * ghinvg / gnorm -c *** ctrnwt = inner prod. of cauchy and relaxed newton steps, -c *** scaled by gnorm**-1. - t1 = ctrnwt - gnorm*cfact**2 -c *** t1 = inner prod. of femur and cauchy step, scaled by -c *** gnorm**-1. - t2 = v(radius)*(v(radius)/gnorm) - gnorm*cfact**2 - t = relax * nwtnrm - femnsq = (t/gnorm)*t - ctrnwt - t1 -c *** femnsq = square of 2-norm of femur, scaled by gnorm**-1. - t = t2 / (t1 + dsqrt(t1**2 + femnsq*t2)) -c *** dogleg step = cauchy step + t * femur. - t1 = (t - one) * cfact - v(grdfac) = t1 - t2 = -t * relax - v(nwtfac) = t2 - v(stppar) = two - t - v(gtstep) = t1*gnorm**2 + t2*ghinvg - v(preduc) = -t1*gnorm * ((t2 + one)*gnorm) - 1 - t2 * (one + half*t2)*ghinvg - 2 - half * (v(gthg)*t1)**2 - do 80 i = 1, n - 80 step(i) = t1*dig(i) + t2*nwtstp(i) -c - 999 return -c *** last line of dbdog follows *** - end - subroutine ltvmul(n, x, l, y) -c -c *** compute x = (l**t)*y, where l is an n x n lower -c *** triangular matrix stored compactly by rows. x and y may -c *** occupy the same storage. *** -c - integer n -cal double precision x(n), l(1), y(n) - double precision x(n), l(n*(n+1)/2), y(n) -c dimension l(n*(n+1)/2) - integer i, ij, i0, j - double precision yi, zero -c/6 -c data zero/0.d+0/ -c/7 - parameter (zero=0.d+0) -c/ -c - i0 = 0 - do 20 i = 1, n - yi = y(i) - x(i) = zero - do 10 j = 1, i - ij = i0 + j - x(j) = x(j) + yi*l(ij) - 10 continue - i0 = i0 + i - 20 continue - 999 return -c *** last card of ltvmul follows *** - end - subroutine lupdat(beta, gamma, l, lambda, lplus, n, w, z) -c -c *** compute lplus = secant update of l *** -c -c *** parameter declarations *** -c - integer n -cal double precision beta(n), gamma(n), l(1), lambda(n), lplus(1), - double precision beta(n), gamma(n), l(n*(n+1)/2), lambda(n), - 1 lplus(n*(n+1)/2),w(n), z(n) -c dimension l(n*(n+1)/2), lplus(n*(n+1)/2) -c -c-------------------------- parameter usage -------------------------- -c -c beta = scratch vector. -c gamma = scratch vector. -c l (input) lower triangular matrix, stored rowwise. -c lambda = scratch vector. -c lplus (output) lower triangular matrix, stored rowwise, which may -c occupy the same storage as l. -c n (input) length of vector parameters and order of matrices. -c w (input, destroyed on output) right singular vector of rank 1 -c correction to l. -c z (input, destroyed on output) left singular vector of rank 1 -c correction to l. -c -c------------------------------- notes ------------------------------- -c -c *** application and usage restrictions *** -c -c this routine updates the cholesky factor l of a symmetric -c positive definite matrix to which a secant update is being -c applied -- it computes a cholesky factor lplus of -c l * (i + z*w**t) * (i + w*z**t) * l**t. it is assumed that w -c and z have been chosen so that the updated matrix is strictly -c positive definite. -c -c *** algorithm notes *** -c -c this code uses recurrence 3 of ref. 1 (with d(j) = 1 for all j) -c to compute lplus of the form l * (i + z*w**t) * q, where q -c is an orthogonal matrix that makes the result lower triangular. -c lplus may have some negative diagonal elements. -c -c *** references *** -c -c 1. goldfarb, d. (1976), factorized variable metric methods for uncon- -c strained optimization, math. comput. 30, pp. 796-811. -c -c *** general *** -c -c coded by david m. gay (fall 1979). -c this subroutine was written in connection with research supported -c by the national science foundation under grants mcs-7600324 and -c mcs-7906671. -c -c------------------------ external quantities ------------------------ -c -c *** intrinsic functions *** -c/+ - double precision dsqrt -c/ -c-------------------------- local variables -------------------------- -c - integer i, ij, j, jj, jp1, k, nm1, np1 - double precision a, b, bj, eta, gj, lj, lij, ljj, nu, s, theta, - 1 wj, zj - double precision one, zero -c -c *** data initializations *** -c -c/6 -c data one/1.d+0/, zero/0.d+0/ -c/7 - parameter (one=1.d+0, zero=0.d+0) -c/ -c -c+++++++++++++++++++++++++++++++ body ++++++++++++++++++++++++++++++++ -c - nu = one - eta = zero - if (n .le. 1) go to 30 - nm1 = n - 1 -c -c *** temporarily store s(j) = sum over k = j+1 to n of w(k)**2 in -c *** lambda(j). -c - s = zero - do 10 i = 1, nm1 - j = n - i - s = s + w(j+1)**2 - lambda(j) = s - 10 continue -c -c *** compute lambda, gamma, and beta by goldfarb*s recurrence 3. -c - do 20 j = 1, nm1 - wj = w(j) - a = nu*z(j) - eta*wj - theta = one + a*wj - s = a*lambda(j) - lj = dsqrt(theta**2 + a*s) - if (theta .gt. zero) lj = -lj - lambda(j) = lj - b = theta*wj + s - gamma(j) = b * nu / lj - beta(j) = (a - b*eta) / lj - nu = -nu / lj - eta = -(eta + (a**2)/(theta - lj)) / lj - 20 continue - 30 lambda(n) = one + (nu*z(n) - eta*w(n))*w(n) -c -c *** update l, gradually overwriting w and z with l*w and l*z. -c - np1 = n + 1 - jj = n * (n + 1) / 2 - do 60 k = 1, n - j = np1 - k - lj = lambda(j) - ljj = l(jj) - lplus(jj) = lj * ljj - wj = w(j) - w(j) = ljj * wj - zj = z(j) - z(j) = ljj * zj - if (k .eq. 1) go to 50 - bj = beta(j) - gj = gamma(j) - ij = jj + j - jp1 = j + 1 - do 40 i = jp1, n - lij = l(ij) - lplus(ij) = lj*lij + bj*w(i) + gj*z(i) - w(i) = w(i) + lij*wj - z(i) = z(i) + lij*zj - ij = ij + i - 40 continue - 50 jj = jj - j - 60 continue -c - 999 return -c *** last card of lupdat follows *** - end - subroutine lvmul(n, x, l, y) -c -c *** compute x = l*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) -c dimension l(n*(n+1)/2) - integer i, ii, ij, i0, j, np1 - double precision t, zero -c/6 -c data zero/0.d+0/ -c/7 - parameter (zero=0.d+0) -c/ -c - np1 = n + 1 - i0 = n*(n+1)/2 - do 20 ii = 1, n - i = np1 - ii - i0 = i0 - i - t = zero - do 10 j = 1, i - ij = i0 + j - t = t + l(ij)*y(j) - 10 continue - x(i) = t - 20 continue - 999 return -c *** last card of lvmul follows *** - end - subroutine vvmulp(n, x, y, z, k) -c -c *** set x(i) = y(i) * z(i)**k, 1 .le. i .le. n (for k = 1 or -1) *** -c - integer n, k - double precision x(n), y(n), z(n) - integer i -c - if (k .ge. 0) go to 20 - do 10 i = 1, n - 10 x(i) = y(i) / z(i) - go to 999 -c - 20 do 30 i = 1, n - 30 x(i) = y(i) * z(i) - 999 return -c *** last card of vvmulp follows *** - end - subroutine wzbfgs (l, n, s, w, y, z) -c -c *** compute y and z for lupdat corresponding to bfgs update. -c - integer n -cal double precision l(1), s(n), w(n), y(n), z(n) - double precision l(n*(n+1)/2), s(n), w(n), y(n), z(n) -c dimension l(n*(n+1)/2) -c -c-------------------------- parameter usage -------------------------- -c -c l (i/o) cholesky factor of hessian, a lower triang. matrix stored -c compactly by rows. -c n (input) order of l and length of s, w, y, z. -c s (input) the step just taken. -c w (output) right singular vector of rank 1 correction to l. -c y (input) change in gradients corresponding to s. -c z (output) left singular vector of rank 1 correction to l. -c -c------------------------------- notes ------------------------------- -c -c *** algorithm notes *** -c -c when s is computed in certain ways, e.g. by gqtstp or -c dbldog, it is possible to save n**2/2 operations since (l**t)*s -c or l*(l**t)*s is then known. -c if the bfgs update to l*(l**t) would reduce its determinant to -c less than eps times its old value, then this routine in effect -c replaces y by theta*y + (1 - theta)*l*(l**t)*s, where theta -c (between 0 and 1) is chosen to make the reduction factor = eps. -c -c *** general *** -c -c coded by david m. gay (fall 1979). -c this subroutine was written in connection with research supported -c by the national science foundation under grants mcs-7600324 and -c mcs-7906671. -c -c------------------------ external quantities ------------------------ -c -c *** functions and subroutines called *** -c - external dotprd, livmul, ltvmul - double precision dotprd -c dotprd returns inner product of two vectors. -c livmul multiplies l**-1 times a vector. -c ltvmul multiplies l**t times a vector. -c -c *** intrinsic functions *** -c/+ - double precision dsqrt -c/ -c-------------------------- local variables -------------------------- -c - integer i - double precision cs, cy, eps, epsrt, one, shs, ys, theta -c -c *** data initializations *** -c -c/6 -c data eps/0.1d+0/, one/1.d+0/ -c/7 - parameter (eps=0.1d+0, one=1.d+0) -c/ -c -c+++++++++++++++++++++++++++++++ body ++++++++++++++++++++++++++++++++ -c - call ltvmul(n, w, l, s) - shs = dotprd(n, w, w) - ys = dotprd(n, y, s) - if (ys .ge. eps*shs) go to 10 - theta = (one - eps) * shs / (shs - ys) - epsrt = dsqrt(eps) - cy = theta / (shs * epsrt) - cs = (one + (theta-one)/epsrt) / shs - go to 20 - 10 cy = one / (dsqrt(ys) * dsqrt(shs)) - cs = one / shs - 20 call livmul(n, z, l, y) - do 30 i = 1, n - 30 z(i) = cy * z(i) - cs * w(i) -c - 999 return -c *** last card of wzbfgs follows *** - end