%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: UserManual.dvi %%Pages: 6 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips UserManual %DVIPSParameters: dpi=1200, compressed %DVIPSSource: TeX output 2003.08.28:1743 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 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5978 7 v 900 9210 a Fn(\003)972 9256 y Fm(Corresp)t(onding)46 b(author,)h(e-mail:)60 b(y)l(aro@si.deis.unical.it)900 9422 y(Authors')37 b(addresses:)56 b(M.)38 b(Ga)l(viano)d(and)i(D.)g (Lera,)h(Dipartimen)l(to)c(di)j(Matematica,)f(Univ)l(ersit\022)-71 b(a)35 b(di)h(Cagliari,)900 9589 y(Via)j(Osp)t(edale,)h(72,)h(09100)c (Cagliari)g({)i(Italy;)j(D.E.)e(Kv)-8 b(aso)l(v)40 b(and)g(Y)-12 b(a.D.)40 b(Sergey)l(ev,)i(DEIS,)e(Univ)l(ersit\022)-71 b(a)38 b(della)900 9755 y(Calabria,)45 b(Via)g(P)-12 b(.Bucci,)47 b(Cub)t(o)g(41C,)g(87036)d(Rende)k(\(CS\))f({)f(Italy)-12 b(.)900 9921 y(This)54 b(w)l(ork)i(has)f(b)t(een)g(partially)e(supp)t (orted)j(b)l(y)g(the)f(follo)l(wing)d(pro)8 b(jects:)78 b(FIRB)55 b(RBA)l(U01JYPN,)f(FIRB)900 10087 y(RBNE01WBBB,)45 b(and)j(RFBR)f(01-01-00587.)p eop %%Page: 1 2 1 1 bop 2253 499 a Fl(GKLS-Generato)l(r)53 b(of)e(Classes)h(of)f(T)-13 b(est)52 b(F)l(unctions:)70 b Fk(User)53 b(Manual)317 b Fj(\001)283 b Fl(1)900 931 y Fr(1.)166 b(INTRODUCTION)900 1197 y Fo(This)57 b Fi(User)j(Manual)d Fo(describ)5 b(es)57 b(an)f(implemen)-5 b(tation)57 b(of)e(the)h(GKLS-generator.)76 b(First,)57 b(the)900 1396 y(general)65 b(structure)f(of)h(the)f(pac)-5 b(k)c(age)65 b(is)g(presen)-5 b(ted,)68 b(then)d(instructions)g(for)g (using)h(the)e(test)900 1595 y(classes)78 b(generator)f(are)f(describ)5 b(ed.)139 b(The)77 b Fi(User)j(Manual)d Fo(is)h(strongly)f(connected)f (to)h(the)900 1794 y(pap)5 b(er)77 b(Ga)-5 b(viano)77 b(et)f(al.)h([2003])g(\(called)f(hereafter)f Fi(p)-8 b(ap)g(er)p Fo(\).)137 b(P)-5 b(articularly)-14 b(,)82 b(an)-5 b(y)77 b(form)-5 b(ulae)900 1994 y(referenced)54 b(in)i(the)e Fi(User)60 b(Manual)c Fo(are)f(to)f(b)5 b(e)55 b(found)g(in)h(the)e Fi(p)-8 b(ap)g(er)p Fo(.)73 b(In)55 b(the)g Fi(p)-8 b(ap)g(er)p Fo(,)54 b(the)h(user)900 2193 y(can)g(\014nd)i(an)e(example)g(of)h(a)f(generated)g(test)g (function)g(also.)900 2534 y Fr(2.)166 b(STRUCTURE)57 b(OF)e(THE)g(P)-14 b(A)-5 b(CKA)g(GE)900 2799 y Fo(The)37 b(GKLS-generator)g(pac)-5 b(k)c(age)37 b(has)g(b)5 b(een)37 b(written)f(in)i(ANSI)e(Standard)h(C)g(and)h(successfully)900 2999 y(tested)g(on)h(Windo)-5 b(ws)39 b(and)g(UNIX)e(platforms.)69 b(Our)39 b(implemen)-5 b(tation)39 b(follo)-5 b(ws)40 b(the)e(pro)5 b(cedure)900 3198 y(describ)g(ed)56 b(in)f(Section)h(3)f (of)g(the)g Fi(p)-8 b(ap)g(er)p Fo(.)1066 3397 y(The)55 b(pac)-5 b(k)c(age)55 b(implemen)-5 b(ting)57 b(the)e(generator)g (includes)h(the)f(follo)-5 b(wing)57 b(\014les:)900 3671 y Fp(gkls.c)83 b Fo({)55 b(the)g(main)h(\014le;)900 3898 y Fp(gkls.h)83 b Fo({)62 b(the)h(header)f(\014le)h(that)f(users)i (should)g(include)f(in)g(their)f(application)i(pro)9 b(jects)62 b(in)1066 4097 y(order)56 b(to)e(call)i(subroutines)h(from)e (the)g(\014le)h Fp(gkls.c)p Fo(;)900 4323 y Fp(rnd)p 1202 4323 58 7 v 69 w(gen.c)83 b Fo({)66 b(the)f(\014le)h(con)-5 b(taining)67 b(the)e(uniform)i(random)g(n)-5 b(um)g(b)5 b(er)67 b(generator)f(prop)5 b(osed)1066 4522 y(in)56 b(Kn)-5 b(uth)56 b([1997;)g(2002];)900 4749 y Fp(rnd)p 1202 4749 V 69 w(gen.h)83 b Fo({)55 b(the)g(header)h(\014le)f(for)g (link)-9 b(age)55 b(to)g(the)g(\014le)h Fp(rnd)p 4889 4749 V 68 w(gen.c)p Fo(;)900 4975 y Fp(example.c)81 b Fo({)55 b(an)h(example)f(of)g(the)g(GKLS-generator)h(usage;)900 5201 y Fp(Mak)-5 b(e\014le)81 b Fo({)60 b(an)g(example)g(of)g(a)h(UNIX) d(mak)-5 b(e\014le)61 b(pro)-5 b(vided)61 b(to)e(UNIX)g(users)i(for)f (a)g(simple)1066 5401 y(compilation)c(and)g(link)-9 b(age)55 b(of)g(separate)g(\014les)h(of)g(the)f(application)g(pro)9 b(ject.)1066 5675 y(F)-14 b(or)79 b(implemen)-5 b(tation)79 b(details)g(the)f(user)g(can)h(consult)g(the)e(C)i(co)5 b(des.)142 b(Note)77 b(that)h(the)900 5874 y(random)62 b(n)-5 b(um)g(b)5 b(er)61 b(generator)g(in)f Fp(rnd)p 3369 5874 V 69 w(gen.c)h Fo(uses)g(the)f(logical-and)i(op)5 b(eration)60 b(`&')h(for)g(ef-)900 6073 y(\014ciency)-14 b(,)68 b(so)e(it)f(is)i(not)e(strictly)g(p)5 b(ortable)65 b(unless)i(the)e(computer)h(uses)g(t)-5 b(w)g(o's)67 b(complemen)-5 b(t)900 6273 y(represen)g(tation)79 b(for)f(in)-5 b(teger.)143 b(It)78 b(do)5 b(es)78 b(not)h(limit)f(p)5 b(ortabilit)-5 b(y)78 b(of)g(the)g(pac)-5 b(k)c(age)79 b(b)5 b(ecause)900 6472 y(almost)56 b(all)g(mo)5 b(dern)55 b(computers)h(are)f(based)i(on)e(t)-5 b(w)g(o's)56 b(complemen)-5 b(t)56 b(arithmetic.)900 6813 y Fr(3.)166 b(CALLING)36 b(SEQUENCE)i(F)-5 b(OR)37 b(GENERA)-14 b(TION)37 b(AND)f(USA)-5 b(GE)37 b(OF)g(THE)f(TESTS)i(CLASSES)900 7078 y Fo(Here)58 b(w)-5 b(e)59 b(describ)5 b(e)59 b(ho)-5 b(w)59 b(to)f(generate)g(and)i (use)f(classes)h(of)e(non-di\013eren)-5 b(tiable)60 b(\(ND-t)-5 b(yp)5 b(e\),)900 7277 y(di\013eren)-5 b(tiable)65 b(\(D-t)-5 b(yp)5 b(e\),)67 b(and)e(t)-5 b(wice)65 b(con)-5 b(tin)g(uously)67 b(di\013eren)-5 b(tiable)65 b(\(D2-t)-5 b(yp)5 b(e\))64 b(test)g(func-)900 7477 y(tions.)144 b(Again,)84 b(w)-5 b(e)79 b(concen)-5 b(trate)78 b(on)g(the)h(D-t)-5 b(yp)5 b(e)78 b(functions.)143 b(The)79 b(op)5 b(erations)79 b(for)f(the)900 7676 y(remaining)56 b(t)-5 b(w)g(o)56 b(t)-5 b(yp)5 b(es)55 b(are)g(analogous.)1066 7875 y(T)-14 b(o)55 b(utilize)h(the)f(GKLS-generator)g(the)g(user)h(m)-5 b(ust)57 b(p)5 b(erform)55 b(the)g(follo)-5 b(wing)56 b(steps:)900 8149 y Fp(Step)64 b(1.)83 b Fo(Input)56 b(of)f(the)g(parameters)h(de\014ning)g(a)f(sp)5 b(eci\014c)56 b(test)e(class.)900 8376 y Fp(Step)64 b(2.)83 b Fo(Generating)56 b(a)f(sp)5 b(eci\014c)55 b(test)g(function)h(of)f(the)g(de\014ned)h (test)e(class.)900 8602 y Fp(Step)64 b(3.)83 b Fo(Ev)-9 b(aluation)80 b(of)h(the)f(generated)g(test)g(function)g(and,)88 b(if)80 b(necessary)-14 b(,)88 b(its)80 b(partial)1066 8801 y(deriv)-9 b(ativ)k(es.)900 9028 y Fp(Step)64 b(4.)83 b Fo(Memory)56 b(deallo)5 b(cating.)900 9302 y(Let)54 b(us)j(consider)f(these)f(steps)h(in)g(turn.)900 9622 y Fr(3.1)166 b(Input)55 b(of)g(the)g(pa)-5 b(rameters)53 b(de\014ning)i(a)g(sp)5 b(eci\014c)55 b(test)f(class)900 9887 y Fo(This)48 b(step)f(is)h(sub)5 b(divided)48 b(in)-5 b(to:)70 b(\(1\))46 b(de\014ning)i(the)f(parameters)h(of)e(the)h(test)g (class,)i(\(2\))e(de\014n-)900 10087 y(ing)56 b(the)f(admissible)i (region)f(\012,)f(and)h(\(3\))e(c)-5 b(hec)g(king)56 b(\(if)f(necessary\).)p eop %%Page: 2 3 2 2 bop 900 499 a Fl(2)282 b Fj(\001)316 b Fl(M.)51 b(Gaviano,)g(D.)g (E.)g(Kvasov,)f(D.)h(Lera,)g(and)h(Y)-13 b(a.)50 b(D.)h(Sergey)l(ev) 1066 931 y Fo(3.1.1)170 b Fi(De\014ning)50 b(the)f(p)-8 b(ar)g(ameters)47 b(of)h(the)h(test)g(class.)83 b Fo(The)43 b(parameters)h(to)f(b)5 b(e)43 b(de\014ned)g(b)-5 b(y)900 1130 y(the)47 b(user)i(determine)f(a)g(sp)5 b(eci\014c)48 b(class)g(\(of)g(the)f(ND-,)i(D-)f(or)g(D2-t)-5 b(yp)5 b(e\))47 b(of)h(100)g(test)f(functions)900 1329 y(\(a)f(sp)5 b(eci\014c)46 b(function)g(is)h(retriev)-5 b(ed)46 b(b)-5 b(y)46 b(its)h(n)-5 b(um)g(b)5 b(er\).)71 b(There)46 b(are)g(the)g(follo)-5 b(wing)47 b(parameters:)900 1624 y Fi(GKLS)p 1363 1624 51 7 v 62 w(dim)82 b Fo({)73 b(\()p Fp(unsigned)83 b(in)-5 b(t)p Fo(\))72 b(dimension)j Fh(N)91 b Fo(\(from)73 b(\(1\)\))f(of)g(test)h(functions;)82 b Fh(N)94 b Fg(\025)76 b Fo(2)1066 1824 y(\(since)54 b(m)-5 b(ultidimensional)56 b(problems)f(are)e(considered)h(in)g (\(1\)\))f(and)h Fh(N)64 b(<)54 b Fo(NUM)p 6263 1824 50 7 v 59 w(RND)f(in)1066 2023 y Fp(rnd)p 1368 2023 58 7 v 69 w(gen.h)p Fo(;)66 b(this)e(v)-9 b(alue)62 b(is)h(limited)g(b)-5 b(y)63 b(the)g(p)5 b(o)-5 b(w)g(er)63 b(of)f Fp(unsigned)i(in)-5 b(t)o Fo(-represen)g(tation;)1066 2222 y(default)55 b Fh(N)65 b Fo(=)46 b(2;)900 2454 y Fi(GKLS)p 1363 2454 51 7 v 62 w(num)p 1743 2454 V 61 w(minima)83 b Fo({)68 b(\()p Fp(unsigned)77 b(in)-5 b(t)p Fo(\))67 b(n)-5 b(um)g(b)5 b(er)69 b Fh(m)f Fo(\(from)g(\(4\)\))f(of)h(lo)5 b(cal)68 b(minima)h(in-)1066 2653 y(cluding)58 b(the)e(parab)5 b(oloid)58 b Fh(Z)69 b Fo(minim)-5 b(um)59 b(\(from)e(\(3\)\))f(and)h (the)f(global)i(minim)-5 b(um;)60 b Fh(m)49 b Fg(\025)f Fo(2;)1066 2852 y(the)74 b(upp)5 b(er)75 b(b)5 b(ound)75 b(of)f(this)g(parameter)h(is)g(limited)f(b)-5 b(y)75 b(the)f(p)5 b(o)-5 b(w)g(er)75 b(of)f Fp(unsigned)63 b(in)-5 b(t)o Fo(-)1066 3051 y(represen)g(tation;)56 b(default)f Fh(m)46 b Fo(=)g(10;)900 3283 y Fi(GKLS)p 1363 3283 V 62 w(glob)-8 b(al)p 1823 3283 V 62 w(value)84 b Fo({)70 b(\()p Fp(double)p Fo(\))f(global)j(minim)-5 b(um)72 b(v)-9 b(alue)71 b Fh(f)5040 3223 y Fq(\003)5187 3283 y Fo(of)f Fh(f)18 b Fo(\()p Fh(x)p Fo(\);)77 b(condition)71 b(\(16\))1066 3482 y(m)-5 b(ust)62 b(b)5 b(e)60 b(satis\014ed;)65 b(the)60 b(default)h(v)-9 b(alue)60 b(is)h Fg(\000)p Fo(1)p Fh(:)p Fo(0)g(\(de\014ned)g(in)g(the)g(\014le)g Fp(gkls.h)f Fo(as)h(a)g(con-)1066 3681 y(stan)-5 b(t)56 b(GKLS)p 1952 3681 50 7 v 60 w(GLOBAL)p 2722 3681 V 59 w(MIN)p 3118 3681 V 59 w(V)-18 b(ALUE\);)900 3913 y Fi(GKLS)p 1363 3913 51 7 v 62 w(glob)-8 b(al)p 1823 3913 V 62 w(dist)83 b Fo({)67 b(\()p Fp(double)p Fo(\))f(distance)i Fh(r)3857 3853 y Fq(\003)4000 3913 y Fo(from)g(the)f(parab)5 b(oloid)68 b(v)-5 b(ertex)67 b Fh(T)90 b Fo(in)68 b(\(3\))f(to)1066 4112 y(the)47 b(global)h(minimizer)g Fh(x)2687 4052 y Fq(\003)2809 4112 y Fg(2)e Fo(\012)h(of)g Fh(f)18 b Fo(\()p Fh(x)p Fo(\);)50 b(condition)d(\(17\))g(m)-5 b(ust)48 b(b)5 b(e)47 b(satis\014ed;)k(the)c(default)1066 4311 y(v)-9 b(alue)55 b(is)2346 4604 y(GKLS)p 2811 4604 50 7 v 60 w(global)p 3304 4604 V 61 w(dist)3724 4510 y Ff(def)3751 4604 y Fo(=)217 b(min)4009 4712 y Ff(1)p Fq(\024)p Fe(j)7 b Fq(\024)p Fe(N)4490 4604 y Fg(j)p Fh(b)p Fo(\()p Fh(j)j Fo(\))36 b Fg(\000)h Fh(a)p Fo(\()p Fh(j)10 b Fo(\))p Fg(j)p Fh(=)28 b Fo(3)p Fh(;)1066 4986 y Fo(where)55 b(the)g(v)-5 b(ectors)55 b Fh(a)g Fo(and)h Fh(b)g Fo(determine)f(the)g (admissible)j(region)d(\012)g(in)h(\(2\);)900 5217 y Fi(GKLS)p 1363 5217 51 7 v 62 w(glob)-8 b(al)p 1823 5217 V 62 w(r)g(adius)82 b Fo({)c(\()p Fp(double)p Fo(\))f(radius)i Fh(\032)3928 5157 y Fq(\003)4083 5217 y Fo(of)f(the)f(attraction)h (region)g(of)g(the)g(global)1066 5416 y(minimizer)56 b Fh(x)1936 5356 y Fq(\003)2059 5416 y Fg(2)46 b Fo(\012)55 b(of)g Fh(f)18 b Fo(\()p Fh(x)p Fo(\);)55 b(condition)g(\(18\))g(m)-5 b(ust)56 b(b)5 b(e)55 b(satis\014ed;)i(the)e(default)g(v)-9 b(alue)55 b(is)2258 5763 y(GKLS)p 2723 5763 50 7 v 60 w(global)p 3216 5763 V 61 w(radius)3812 5669 y Ff(def)3839 5763 y Fo(=)217 b(min)4097 5871 y Ff(1)p Fq(\024)p Fe(j)7 b Fq(\024)p Fe(N)4577 5763 y Fg(j)p Fh(b)p Fo(\()p Fh(j)j Fo(\))37 b Fg(\000)g Fh(a)p Fo(\()p Fh(j)10 b Fo(\))p Fg(j)p Fh(=)28 b Fo(6)p Fh(:)900 6149 y Fo(The)55 b(user)g(ma)-5 b(y)56 b(call)f(subroutine)h Fi(GKLS)p 3541 6149 51 7 v 61 w(set)p 3801 6149 V 62 w(default)p Fo(\(\))f(to)f(set)h(the)f (default)h(v)-9 b(alues)55 b(of)g(these)900 6348 y(\014v)-5 b(e)55 b(v)-9 b(ariables.)1066 6647 y(3.1.2)170 b Fi(De\014ning)83 b(the)e(admissible)f(r)-8 b(e)g(gion.)83 b Fo(With)78 b Fh(N)97 b Fo(determined,)84 b(the)78 b(user)h(m)-5 b(ust)79 b(al-)900 6846 y(lo)5 b(cate)79 b(dynamic)h(arra)-5 b(ys)80 b Fi(GKLS)p 3089 6846 V 62 w(domain)p 3686 6846 V 60 w(left)h Fo(and)f Fi(GKLS)p 4851 6846 V 62 w(domain)p 5448 6846 V 61 w(right)g Fo(to)f(de\014ne)h(the)900 7045 y(b)5 b(oundary)56 b(of)f(the)g(h)-5 b(yp)5 b(errectangle)54 b(\012.)74 b(This)56 b(is)g(done)g(b)-5 b(y)56 b(calling)f(subroutine) 1066 7245 y Fp(in)-5 b(t)54 b Fi(GKLS)p 1812 7245 V 62 w(domain)p 2409 7245 V 61 w(al)8 b(lo)-8 b(c)56 b Fo(\(\);)900 7444 y(whic)-5 b(h)56 b(has)g(no)g(parameters)g(and)f(returns)h(the)f (follo)-5 b(wing)56 b(error)f(co)5 b(des)56 b(de\014ned)g(in)f Fp(gkls.h)p Fo(:)900 7739 y Fp(GKLS)p 1432 7739 58 7 v 68 w(OK)84 b Fo({)55 b(no)h(errors;)900 7970 y Fp(GKLS)p 1432 7970 V 68 w(DIM)p 1899 7970 V 70 w(ERR)-5 b(OR)85 b Fo({)39 b(the)f(problem)i(dimension)h(is)e(out)g(of)g(range;)45 b(it)39 b(m)-5 b(ust)40 b(b)5 b(e)38 b(greater)1066 8169 y(than)55 b(or)h(equal)f(to)g(2)g(and)h(less)g(than)g(NUM)p 3865 8169 50 7 v 59 w(RND)f(de\014ned)h(in)g Fp(rnd)p 5411 8169 58 7 v 68 w(gen.h)p Fo(;)900 8401 y Fp(GKLS)p 1432 8401 V 68 w(MEMOR)-16 b(Y)p 2401 8401 V 71 w(ERR)-5 b(OR)85 b Fo({)55 b(there)g(is)h(not)f(enough)h(memory)f(to)g(allo)5 b(cate.)1066 8696 y(The)71 b(same)h(subroutine)h(de\014nes)f(the)f (admissible)i(region)f(\012.)121 b(The)72 b(default)f(v)-9 b(alue)71 b(\012)h(=)900 8895 y([)p Fg(\000)p Fo(1)p Fh(;)28 b Fo(1])1361 8835 y Fe(N)1543 8895 y Fo(is)55 b(set)h(b)-5 b(y)55 b Fi(GKLS)p 2662 8895 51 7 v 62 w(set)p 2923 8895 V 62 w(default)p Fo(\(\))o(.)1066 9194 y(3.1.3)170 b Fi(Che)-8 b(cking.)84 b Fo(The)71 b(follo)-5 b(wing)73 b(subroutine)f(allo)-5 b(ws)73 b(the)e(user)h(to)f(c)-5 b(hec)g(k)72 b(v)-9 b(alidit)k(y)71 b(of)900 9393 y(the)55 b(input)h(parameters:)1066 9592 y Fp(in)-5 b(t)54 b Fi(GKLS)p 1812 9592 V 62 w(p)-8 b(ar)g(ameters)p 2664 9592 V 61 w(che)g(ck)56 b Fo(\(\).)900 9792 y(It)f(has)h(no)g(parameters)f(and)h (returns)g(the)f(follo)-5 b(wing)56 b(error)f(co)5 b(des)56 b(\(see)f Fp(gkls.h)p Fo(\):)900 10087 y Fp(GKLS)p 1432 10087 58 7 v 68 w(OK)84 b Fo({)55 b(no)h(errors;)p eop %%Page: 3 4 3 3 bop 2253 499 a Fl(GKLS-Generato)l(r)53 b(of)e(Classes)h(of)f(T)-13 b(est)52 b(F)l(unctions:)70 b Fk(User)53 b(Manual)317 b Fj(\001)283 b Fl(3)900 931 y Fp(GKLS)p 1432 931 58 7 v 68 w(DIM)p 1899 931 V 70 w(ERR)-5 b(OR)85 b Fo({)55 b(problem)h(dimension)h(error;)900 1156 y Fp(GKLS)p 1432 1156 V 68 w(NUM)p 1977 1156 V 70 w(MINIMA)p 2846 1156 V 71 w(ERR)-5 b(OR)84 b Fo({)55 b(n)-5 b(um)g(b)5 b(er)57 b(of)e(lo)5 b(cal)55 b(minima)i(error;)900 1380 y Fp(GKLS)p 1432 1380 V 68 w(BOUND)-5 b(AR)-16 b(Y)p 2631 1380 V 71 w(ERR)-5 b(OR)84 b Fo({)d(the)e(admissible)k(region)d(b)5 b(oundary)81 b(v)-5 b(ectors)80 b(are)1066 1580 y(ill-de\014ned;)900 1804 y Fp(GKLS)p 1432 1804 V 68 w(GLOBAL)p 2303 1804 V 70 w(MIN)p 2775 1804 V 70 w(V)-21 b(ALUE)p 3499 1804 V 69 w(ERR)-5 b(OR)84 b Fo({)64 b(the)g(global)h(minim)-5 b(um)66 b(v)-9 b(alue)64 b(is)h(not)1066 2004 y(less)71 b(than)f(the)g(parab)5 b(oloid)70 b(\(3\))f(minim)-5 b(um)73 b(v)-9 b(alue)69 b Fh(t)h Fo(de\014ned)h(in)f Fp(gkls.h)g Fo(as)g(a)g(constan)-5 b(t)1066 2203 y(GKLS)p 1531 2203 50 7 v 60 w(P)-14 b(ARABOLOID)p 2729 2203 V 59 w(MIN;)900 2427 y Fp(GKLS)p 1432 2427 58 7 v 68 w(GLOBAL)p 2303 2427 V 70 w(DIST)p 2830 2427 V 69 w(ERR)-5 b(OR)84 b Fo({)56 b(the)f(parameter)g Fh(r)4963 2367 y Fq(\003)5094 2427 y Fo(do)5 b(es)56 b(not)f(satisfy)g(\(17\);)900 2652 y Fp(GKLS)p 1432 2652 V 68 w(GLOBAL)p 2303 2652 V 70 w(RADIUS)p 3131 2652 V 70 w(ERR)-5 b(OR)84 b Fo({)48 b(the)f(parameter)g Fh(\032)5247 2592 y Fq(\003)5371 2652 y Fo(do)5 b(es)48 b(not)g(satisfy)f(\(18\).)900 2969 y Fr(3.2)166 b(Generating)55 b(a)g(sp)5 b(eci\014c)54 b(test)h(function)g(of)g(the)g(de\014ned)f(test)h(class)900 3234 y Fo(After)70 b(a)h(sp)5 b(eci\014c)71 b(test)g(class)h(has)g(b)5 b(een)70 b(c)-5 b(hosen)73 b(\(i.e.,)h(the)d(input)h(parameters)f(ha)-5 b(v)g(e)72 b(b)5 b(een)900 3434 y(determined\))68 b(the)f(user)i(can)f (generate)f(a)h(sp)5 b(eci\014c)68 b(function)g(that)g(b)5 b(elongs)68 b(to)g(the)g(c)-5 b(hosen)900 3633 y(class)56 b(of)f(100)h(test)f(functions.)74 b(This)56 b(is)g(done)g(b)-5 b(y)55 b(calling)h(subroutine)1066 3832 y Fp(in)-5 b(t)54 b Fi(GKLS)p 1812 3832 51 7 v 62 w(ar)-8 b(g)p 2097 3832 V 61 w(gener)g(ate)57 b Fo(\()p Fp(unsigned)63 b(in)-5 b(t)54 b Fi(nf)p Fo(\);)900 4031 y(where)900 4299 y Fi(nf)83 b Fo({)56 b(the)f(n)-5 b(um)g(b)5 b(er)56 b(of)f(a)h(function)f(from)h (the)f(test)f(class)j(\(from)e(1)g(to)g(100\).)900 4567 y(This)48 b(subroutine)g(initializes)g(the)f(random)h(n)-5 b(um)g(b)5 b(er)48 b(generator,)h(c)-5 b(hec)g(ks)47 b(the)g(input)h(parame-)900 4766 y(ters,)c(allo)5 b(cates)40 b(dynamic)h(arra)-5 b(ys,)45 b(and)c(generates)f(a)h(test)f(function)h (follo)-5 b(wing)41 b(the)g(pro)5 b(cedure)900 4965 y(of)53 b(Section)g(3)g(of)g(the)f Fi(p)-8 b(ap)g(er)p Fo(.)72 b(It)53 b(returns)g(an)h(error)f(co)5 b(de)52 b(that)h(can)g(b)5 b(e)52 b(the)h(same)h(as)f(for)g(sub-)900 5165 y(routines)j Fi(GKLS)p 2001 5165 V 62 w(p)-8 b(ar)g(ameters)p 2853 5165 V 60 w(che)g(ck)p Fo(\(\))55 b(and)h Fi(GKLS)p 4255 5165 V 62 w(domain)p 4852 5165 V 61 w(al)8 b(lo)-8 b(c)p Fo(\(\),)55 b(or)h(additionally:)900 5432 y Fp(GKLS)p 1432 5432 58 7 v 68 w(FUNC)p 2054 5432 V 70 w(NUMBER)p 3005 5432 V 69 w(ERR)-5 b(OR)85 b Fo({)42 b(the)g(n)-5 b(um)g(b)5 b(er)43 b(of)f(a)h(test)e(function)h(to)g(generate)1066 5632 y(exceeds)55 b(100)g(or)h(it)f(is)h(less)g(than)f(1.)1066 5900 y Fi(GKLS)p 1529 5900 51 7 v 62 w(ar)-8 b(g)p 1814 5900 V 61 w(gener)g(ate)p Fo(\(\))74 b(generates)f(the)g(list)h(of)f (all)g(lo)5 b(cal)73 b(minima)i(and)f(the)f(list)h(of)f(the)900 6099 y(global)f(minima)g(as)g(parts)f(of)g(the)g(structures)g Fi(GKLS)p 4444 6099 V 62 w(minima)g Fo(and)g Fi(GKLS)p 5920 6099 V 62 w(glob)p Fo(,)76 b(resp)5 b(ec-)900 6298 y(tiv)-5 b(ely)-14 b(.)72 b(The)52 b(\014rst)g(structure)f(gathers)h (the)g(follo)-5 b(wing)52 b(information)g(ab)5 b(out)52 b(all)g(lo)5 b(cal)51 b(minima)900 6497 y(\(including)57 b(the)f(parab)5 b(oloid)57 b(minim)-5 b(um)59 b(and)e(the)f(global)g (one\):)76 b(co)5 b(ordinates)56 b(of)g(lo)5 b(cal)56 b(min-)900 6697 y(imizers,)75 b(lo)5 b(cal)70 b(minima)h(v)-9 b(alues,)75 b(and)70 b(attraction)g(regions)h(radii.)119 b(The)70 b(second)h(structure)900 6896 y(con)-5 b(tains)69 b(information)f(ab)5 b(out)68 b(the)f(n)-5 b(um)g(b)5 b(er)70 b(of)d(global)i(minimizers)g(and)g(their)e(indices)i(in)900 7095 y(the)55 b(set)g(of)g(lo)5 b(cal)55 b(minimizers.)76 b(It)55 b(has)h(the)f(follo)-5 b(wing)56 b(\014elds:)900 7363 y Fi(num)p 1228 7363 V 61 w(glob)-8 b(al)p 1687 7363 V 62 w(minima)83 b Fo({)55 b(\()p Fp(unsigned)63 b(in)-5 b(t)p Fo(\))54 b(total)h(n)-5 b(um)g(b)5 b(er)57 b(of)e(global)h(minima;)900 7588 y Fi(gm)p 1122 7588 V 61 w(index)84 b Fo({)47 b(\()p Fp(unsigned)54 b(in)-5 b(t)53 b(*)p Fo(\))47 b(list)h(of)f(indices)h(of)f(generated)g (minimizers,)j(whic)-5 b(h)48 b(are)f(the)1066 7787 y(global)h(ones)g (\(elemen)-5 b(ts)48 b(0)f(to)g(\()p Fi(num)p 3363 7787 V 61 w(glob)-8 b(al)p 3822 7787 V 62 w(minima)21 b Fg(\000)g Fo(1\))47 b(of)g(the)g(list\))h(and)g(the)f(lo)5 b(cal)47 b(ones)1066 7986 y(\(the)55 b(remaining)h(elemen)-5 b(ts)56 b(of)f(the)g(list\).)900 8254 y(The)g(elemen)-5 b(ts)56 b(of)f(the)g(list)g Fi(GKLS)p 3135 8254 V 62 w(glob)p Fo(.)p Fi(gm)p 3734 8254 V 62 w(index)h Fo(are)f(indices)h(to)f(a)g(sp) 5 b(eci\014c)55 b(minimizer)h(in)900 8453 y(the)f(\014rst)h(structure)f Fi(GKLS)p 2704 8453 V 62 w(minima)g Fo(c)-5 b(haracterized)55 b(b)-5 b(y)55 b(the)g(follo)-5 b(wing)56 b(\014elds:)900 8721 y Fi(lo)-8 b(c)g(al)p 1224 8721 V 61 w(min)83 b Fo({)56 b(\()p Fp(double)62 b(**)p Fo(\))56 b(list)g(of)f(lo)5 b(cal)55 b(minimizers)i(co)5 b(ordinates;)932 8946 y Fi(f)83 b Fo({)55 b(\()p Fp(double)63 b(*)p Fo(\))55 b(list)h(of)f(lo)5 b(cal)55 b(minima)i(v)-9 b(alues;)900 9170 y Fi(rho)83 b Fo({)55 b(\()p Fp(double)63 b(*)p Fo(\))55 b(list)h(of)f(attraction)f(regions)i(radii;)900 9395 y Fi(p)-8 b(e)g(ak)83 b Fo({)55 b(\()p Fp(double)62 b(*)p Fo(\))56 b(list)f(of)h(parameters)f Fh(\015)3737 9420 y Fe(i)3848 9395 y Fo(v)-9 b(alues)55 b(from)h(\(10\);)900 9620 y Fi(w)p 1020 9620 V 61 w(rho)83 b Fo({)56 b(\()p Fp(double)62 b(*)p Fo(\))55 b(list)h(of)f(parameters)h Fh(w)3876 9645 y Fe(i)3986 9620 y Fo(v)-9 b(alues)56 b(from)f(\(23\).)900 9887 y(The)68 b(\014elds)h(of)e(these)h (structures)h(can)f(b)5 b(e)67 b(useful)i(if)f(one)g(needs)g(to)g (study)g(prop)5 b(erties)68 b(of)g(a)900 10087 y(sp)5 b(eci\014c)55 b(generated)g(test)g(function)g(more)h(deeply)-14 b(.)p eop %%Page: 4 5 4 4 bop 900 499 a Fl(4)282 b Fj(\001)316 b Fl(M.)51 b(Gaviano,)g(D.)g (E.)g(Kvasov,)f(D.)h(Lera,)g(and)h(Y)-13 b(a.)50 b(D.)h(Sergey)l(ev)900 931 y Fr(3.3)166 b(Evaluation)56 b(of)f(a)g(generated)f(test)h (function)f(o)-5 b(r)56 b(its)g(pa)-5 b(rtial)56 b(derivatives)900 1197 y Fo(While)45 b(there)f(exists)g(a)h(structure)g Fi(GKLS)p 3528 1197 51 7 v 61 w(minima)g Fo(of)f(lo)5 b(cal)44 b(minima,)k(the)d(user)g(can)g(ev)-9 b(aluate)900 1397 y(a)64 b(test)g(function)g(\(or)g(partial)g(deriv)-9 b(ativ)k(es)64 b(of)g(D-)g(and)h(D2-t)-5 b(yp)5 b(e)63 b(functions\))h(that)g(is)h(deter-)900 1596 y(mined)56 b(b)-5 b(y)55 b(its)g(n)-5 b(um)g(b)5 b(er)56 b(\(a)e(parameter)h(to)f (the)h(subroutine)h Fi(GKLS)p 5232 1596 V 61 w(ar)-8 b(g)p 5516 1596 V 61 w(gener)g(ate)p Fo(\(\)\))55 b(within)900 1795 y(the)68 b(c)-5 b(hosen)70 b(test)e(class.)115 b(If)68 b(the)h(user)g(wishes)h(to)e(ev)-9 b(aluate)68 b(another)g(function)h (within)g(the)900 1994 y(same)48 b(class)g(he)g(should)g(deallo)5 b(cate)47 b(dynamic)h(arra)-5 b(ys)47 b(\(see)g(the)g(next)g (subsection\))h(and)g(recall)900 2194 y(the)f(generator)g Fi(GKLS)p 2372 2194 V 62 w(ar)-8 b(g)p 2657 2194 V 61 w(gener)g(ate)p Fo(\(\))48 b(\(passing)h(it)e(the)h(corresp)5 b(onding)48 b(function)g(n)-5 b(um)g(b)5 b(er\))900 2393 y(without)46 b(resetting)g(the)g(input)g(class)i(parameters)e(\(see)g (subsection)h(3.1\).)71 b(If)46 b(the)g(user)h(wishes)900 2592 y(to)55 b(c)-5 b(hange)55 b(the)g(test)g(class)h(prop)5 b(erties)55 b(he)g(should)i(reset)e(also)h(the)e(input)i(class)g (parameters.)1066 2792 y(Ev)-9 b(aluation)55 b(of)g(an)h(ND-t)-5 b(yp)5 b(e)54 b(function)i(is)g(done)f(b)-5 b(y)56 b(calling)g (subroutine)1066 2992 y Fp(double)f Fi(GKLS)p 2139 2992 V 61 w(ND)p 2448 2992 V 62 w(func)h Fo(\()p Fi(x)p Fo(\).)900 3191 y(Ev)-9 b(aluation)55 b(of)g(a)h(D-t)-5 b(yp)5 b(e)54 b(function)i(is)g(done)f(b)-5 b(y)56 b(calling)g(subroutine)1066 3391 y Fp(double)f Fi(GKLS)p 2139 3391 V 61 w(D)p 2325 3391 V 62 w(func)h Fo(\()p Fi(x)p Fo(\).)900 3590 y(Ev)-9 b(aluation)55 b(of)g(a)h(D2-t)-5 b(yp)5 b(e)54 b(function)i(is)g(done)f (b)-5 b(y)56 b(calling)g(subroutine)1066 3790 y Fp(double)f Fi(GKLS)p 2139 3790 V 61 w(D2)p 2410 3790 V 62 w(func)h Fo(\()p Fi(x)p Fo(\).)900 3990 y(All)f(these)g(subroutines)i(ha)-5 b(v)g(e)56 b(only)f(one)h(input)g(parameter)906 4309 y Fi(x)83 b Fo({)55 b(\()p Fp(double)63 b(*)p Fo(\))55 b(a)h(p)5 b(oin)-5 b(t)55 b Fh(x)46 b Fg(2)g Fd(R)3083 4249 y Fe(N)3273 4309 y Fo(where)55 b(the)g(function)h(m)-5 b(ust)56 b(b)5 b(e)55 b(ev)-9 b(aluated.)900 4622 y(All)47 b(the)g(subroutines)j(return)d(a)h(test)f(function)g(v)-9 b(alue)47 b(corresp)5 b(onding)49 b(to)e(the)g(p)5 b(oin)-5 b(t)48 b Fh(x)p Fo(.)71 b(They)900 4821 y(return)f(the)f(v)-9 b(alue)70 b(GKLS)p 2632 4821 50 7 v 60 w(MAX)p 3094 4821 V 59 w(V)-18 b(ALUE)68 b(\(de\014ned)i(in)g Fp(gkls.h)p Fo(\))f(in)i(t)-5 b(w)g(o)70 b(cases:)104 b(\(a\))69 b(v)-5 b(ec-)900 5021 y(tor)58 b Fh(x)g Fo(do)5 b(es)58 b(not)g(b)5 b(elong)58 b(to)g(the)f(admissible)j(region)f(\012)f(and)g (\(b\))g(the)f(user)i(tries)f(to)g(call)g(the)900 5220 y(subroutines)f(without)e(generating)g(a)h(test)e(function.)1066 5420 y(The)60 b(follo)-5 b(wing)60 b(subroutines)i(are)e(pro)-5 b(vided)60 b(for)g(calculating)g(the)f(partial)h(deriv)-9 b(ativ)k(es)60 b(of)900 5619 y(the)55 b(test)g(functions)h(\(see)f(App) 5 b(endix)55 b(in)g(the)g Fi(p)-8 b(ap)g(er)p Fo(\).)1066 5819 y(Ev)f(aluation)58 b(of)g(the)f(\014rst)i(order)f(partial)g(deriv) -9 b(ativ)k(e)57 b(of)h(the)f(D-t)-5 b(yp)5 b(e)58 b(test)f(functions)i (with)900 6018 y(resp)5 b(ect)51 b(to)h(the)f(v)-9 b(ariable)52 b Fh(x)2662 6043 y Fe(j)2783 6018 y Fo(\(see)g(\(A.1\){\(A.2\))e(in)i (App)5 b(endix\))51 b(is)h(done)h(b)-5 b(y)52 b(calling)g(subrou-)900 6217 y(tine)1066 6417 y Fp(double)j Fi(GKLS)p 2139 6417 51 7 v 61 w(D)p 2325 6417 V 62 w(deriv)h Fo(\()p Fi(j)p Fo(,)f Fi(x)p Fo(\).)900 6617 y(Ev)-9 b(aluation)65 b(of)f(the)h (\014rst)g(order)g(partial)g(deriv)-9 b(ativ)k(e)64 b(of)h(the)g(D2-t) -5 b(yp)5 b(e)64 b(test)g(functions)i(with)900 6816 y(resp)5 b(ect)51 b(to)h(the)f(v)-9 b(ariable)52 b Fh(x)2662 6841 y Fe(j)2783 6816 y Fo(\(see)g(\(A.3\){\(A.4\))e(in)i(App)5 b(endix\))51 b(is)h(done)h(b)-5 b(y)52 b(calling)g(subrou-)900 7015 y(tine)1066 7215 y Fp(double)j Fi(GKLS)p 2139 7215 V 61 w(D2)p 2410 7215 V 62 w(deriv1)h Fo(\()p Fi(j)p Fo(,)e Fi(x)p Fo(\).)900 7414 y(Ev)-9 b(aluation)49 b(of)f(the)h (second)g(order)g(partial)g(deriv)-9 b(ativ)k(e)48 b(of)g(the)h(D2-t)-5 b(yp)5 b(e)48 b(test)g(functions)i(with)900 7614 y(resp)5 b(ect)62 b(to)h(the)f(v)-9 b(ariables)63 b Fh(x)2771 7639 y Fe(j)2904 7614 y Fo(and)g Fh(x)3329 7639 y Fe(k)3473 7614 y Fo(\(see)g(in)g(App)5 b(endix)62 b(the)h(form)-5 b(ulae)63 b(\(A.5\){\(A.6\))e(for)900 7813 y(the)55 b(case)g Fh(j)h Fg(6)p Fo(=)47 b Fh(k)61 b Fo(and)56 b(\(A.7\){\(A.8\))d(for)i (the)g(case)g Fh(j)h Fo(=)46 b Fh(k)5 b Fo(\))56 b(is)g(done)g(b)-5 b(y)55 b(calling)h(subroutine)1066 8013 y Fp(double)f Fi(GKLS)p 2139 8013 V 61 w(D2)p 2410 8013 V 62 w(deriv2)h Fo(\()p Fi(j)p Fo(,)e Fi(k)p Fo(,)i Fi(x)p Fo(\).)900 8212 y(Input)g(parameters)f(for)h(these)f(three)g(subroutines)h(are:) 900 8532 y Fi(j,)j(k)84 b Fo({)50 b(\()p Fp(unsigned)57 b(in)-5 b(t)p Fo(\))49 b(indices)j(of)e(the)g(v)-9 b(ariables)51 b(\(that)e(m)-5 b(ust)51 b(b)5 b(e)51 b(in)f(the)g(range)h(from)g(1)f (to)1066 8731 y Fi(GKLS)p 1529 8731 V 62 w(dim)p Fo(\))k(with)h(resp)5 b(ect)55 b(to)g(whic)-5 b(h)56 b(the)f(partial)h(deriv)-9 b(ativ)k(e)54 b(is)i(ev)-9 b(aluated;)906 8977 y Fi(x)83 b Fo({)55 b(\()p Fp(double)63 b(*)p Fo(\))55 b(a)h(p)5 b(oin)-5 b(t)55 b Fh(x)46 b Fg(2)g Fd(R)3083 8916 y Fe(N)3273 8977 y Fo(where)55 b(the)g(deriv)-9 b(ativ)k(e)55 b(m)-5 b(ust)56 b(b)5 b(e)55 b(ev)-9 b(aluated.)900 9290 y(All)46 b(subroutines)i(return)e(the)g(v)-9 b(alue)46 b(of)g(a)g(sp)5 b(eci\014c)47 b(partial)f(deriv)-9 b(ativ)k(e)46 b(corresp)5 b(onding)47 b(to)f(the)900 9489 y(p)5 b(oin)-5 b(t)70 b Fh(x)g Fo(and)h(to)e(the)h(giv)-5 b(en)70 b(direction.)118 b(They)70 b(return)g(the)g(v)-9 b(alue)69 b(GKLS)p 5794 9489 50 7 v 61 w(MAX)p 6257 9489 V 59 w(V)-18 b(ALUE)900 9688 y(\(de\014ned)71 b(in)h Fp(gkls.h)p Fo(\))e(in)i(three)e(cases:) 106 b(\(a\))70 b(index)i(\()p Fh(j)80 b Fo(or)71 b Fh(k)5 b Fo(\))72 b(of)e(a)i(v)-9 b(ariable)70 b(is)i(out)f(of)g(the)900 9887 y(range)61 b([1,)p Fi(GKLS)p 1996 9887 51 7 v 62 w(dim)p Fo(];)i(\(b\))d(v)-5 b(ector)60 b Fh(x)g Fo(do)5 b(es)61 b(not)g(b)5 b(elong)60 b(to)h(the)f(admissible)i(region)f (\012;)j(\(c\))900 10087 y(the)55 b(user)h(tries)f(to)g(call)h(the)f (subroutines)h(without)g(generating)f(a)g(test)g(function.)p eop %%Page: 5 6 5 5 bop 2253 499 a Fl(GKLS-Generato)l(r)53 b(of)e(Classes)h(of)f(T)-13 b(est)52 b(F)l(unctions:)70 b Fk(User)53 b(Manual)317 b Fj(\001)283 b Fl(5)1066 931 y Fo(Subroutines)70 b(for)e(calculating)g (the)g(gradien)-5 b(ts)69 b(of)f(the)g(D-)h(and)g(D2-t)-5 b(yp)5 b(e)67 b(test)h(functions)900 1130 y(and)76 b(for)e(calculating) i(the)e(Hessian)i(matrix)f(of)g(the)f(D2-t)-5 b(yp)5 b(e)75 b(test)f(functions)i(at)e(a)h(giv)-5 b(en)900 1329 y(feasible)56 b(p)5 b(oin)-5 b(t)55 b(are)g(also)h(pro)-5 b(vided.)75 b(These)56 b(are)1066 1529 y Fp(in)-5 b(t)54 b Fi(GKLS)p 1812 1529 51 7 v 62 w(D)p 1999 1529 V 62 w(gr)-8 b(adient)55 b Fo(\()p Fi(x)p Fo(,)h Fi(g)p Fo(\),)1066 1927 y Fp(in)-5 b(t)54 b Fi(GKLS)p 1812 1927 V 62 w(D2)p 2084 1927 V 62 w(gr)-8 b(adient)55 b Fo(\()p Fi(x)p Fo(,)g Fi(g)p Fo(\),)1066 2326 y Fp(in)-5 b(t)54 b Fi(GKLS)p 1812 2326 V 62 w(D2)p 2084 2326 V 62 w(hessian)h Fo(\()p Fi(x)p Fo(,)g Fi(h)p Fo(\).)900 2525 y(Here)906 2824 y Fi(x)83 b Fo({)71 b(\()p Fp(double)80 b(*)p Fo(\))71 b(a)g(p)5 b(oin)-5 b(t)71 b Fh(x)h Fg(2)g Fd(R)3215 2764 y Fe(N)3420 2824 y Fo(where)f(the)g(gradien)-5 b(t)71 b(or)g(Hessian)g(matrix)g(m)-5 b(ust)72 b(b)5 b(e)1066 3023 y(ev)-9 b(aluated;)907 3256 y Fi(g)83 b Fo({)55 b(\()p Fp(double)63 b(*)p Fo(\))55 b(a)h(p)5 b(oin)-5 b(ter)55 b(to)g(the)g(gradien)-5 b(t)56 b(v)-5 b(ector)54 b(calculated)h(at)g Fi(x)p Fo(;)900 3488 y Fi(h)83 b Fo({)55 b(\()p Fp(double)63 b(**)p Fo(\))56 b(a)f(p)5 b(oin)-5 b(ter)55 b(to)g(the)g(Hessian)h(matrix)f(calculated)h(at)f Fi(x)p Fo(.)900 3787 y(Note)c(that)g(b)5 b(efore)51 b(calling)h(these)g (subroutines)h(the)f(user)g(m)-5 b(ust)53 b(allo)5 b(cate)51 b(dynamic)h(memory)900 3986 y(for)64 b(the)g(gradien)-5 b(t)64 b(v)-5 b(ector)63 b Fi(g)i Fo(or)f(the)g(Hessian)h(matrix)e Fi(h)h Fo(and)h(pass)g(the)f(p)5 b(oin)-5 b(ters)65 b Fi(g)f Fo(or)g Fi(h)g Fo(as)900 4185 y(parameters)56 b(of)f(the)g(subroutines.)1066 4385 y(These)46 b(subroutines)h(call)e (the)g(subroutines)i(describ)5 b(ed)45 b(ab)5 b(o)-5 b(v)g(e)46 b(for)f(calculating)g(the)g(partial)900 4584 y(deriv)-9 b(ativ)k(es)79 b(and)i(return)e(an)h(error)g(co)5 b(de)79 b(\()p Fp(GKLS)p 4322 4584 58 7 v 68 w(DERIV)p 5020 4584 V 70 w(EV)-21 b(AL)p 5597 4584 V 70 w(ERR)-5 b(OR)81 b Fo(in)f(the)900 4783 y(case)53 b(of)h(an)f(error)h(during)g (ev)-9 b(aluation)53 b(of)g(a)g(particular)h(comp)5 b(onen)-5 b(t)54 b(of)f(the)g(gradien)-5 b(t)54 b(or)f(the)900 4982 y(Hessian)j(matrix,)f(or)h Fp(GKLS)p 2828 4982 V 68 w(OK)g Fo(if)f(there)g(are)g(no)g(errors\).)900 5315 y Fr(3.4)166 b(Memo)-5 b(ry)54 b(deallo)5 b(cating)900 5580 y Fo(When)41 b(the)f(user)h(concludes)h(his)f(w)-5 b(ork)41 b(with)g(a)g(test)f(function)g(he)h(should)h(deallo)5 b(cate)40 b(dynamic)900 5779 y(arra)-5 b(ys)56 b(allo)5 b(cated)55 b(b)-5 b(y)55 b(the)g(generator.)74 b(This)56 b(is)g(done)f(b)-5 b(y)56 b(calling)g(subroutine)1066 5979 y Fp(v)-5 b(oid)54 b Fi(GKLS)p 1934 5979 51 7 v 62 w(fr)-8 b(e)g(e)55 b Fo(\()p Fp(v)-5 b(oid)p Fo(\);)900 6178 y(with)55 b(no)h(parameters.)1066 6377 y(When)c(the)g(user)h (abandons)g(the)f(test)g(class)h(he)f(should)i(deallo)5 b(cate)51 b(dynamic)h(b)5 b(oundaries)900 6576 y(v)-5 b(ectors)55 b Fi(GKLS)p 1927 6576 V 62 w(domain)p 2524 6576 V 61 w(left)h Fo(and)g Fi(GKLS)p 3641 6576 V 61 w(domain)p 4237 6576 V 61 w(right)g Fo(b)-5 b(y)55 b(calling)h (subroutine)1066 6776 y Fp(v)-5 b(oid)54 b Fi(GKLS)p 1934 6776 V 62 w(domain)p 2531 6776 V 61 w(fr)-8 b(e)g(e)55 b Fo(\()p Fp(v)-5 b(oid)p Fo(\);)900 6975 y(again)56 b(with)f(no)h(parameters.)1066 7174 y(It)43 b(should)i(b)5 b(e)43 b(\014nally)h(highligh)-5 b(ted)44 b(that)f(if)g(the)g(user,)k (after)42 b(deallo)5 b(cating)43 b(memory)-14 b(,)47 b(wishes)900 7373 y(to)82 b(return)h(to)g(the)f(same)i(class,)91 b(generation)82 b(of)h(the)g(class)g(with)g(the)g(same)g(parameters)900 7573 y(pro)5 b(duces)56 b(the)f(same)h(100)f(test)g(functions.)1066 7772 y(An)50 b(example)h(of)g(the)f(generation)g(and)h(use)h(of)e(some) h(of)g(the)f(test)g(classes)i(can)f(b)5 b(e)50 b(found)h(in)900 7971 y(the)k(\014le)g Fp(example.c)p Fo(.)900 8303 y Fc(REFERENCES)920 8536 y Fb(Ga)-12 b(viano,)56 b(M.)p Fa(,)e Fb(Kv)-12 b(aso)m(v,)55 b(D.)f(E.)p Fa(,)f Fb(Lera,)g(D.)p Fa(,)h Fb(and)g(Ser)m(geyev,)h(Y.)e(D.)h Fa(2003.)74 b(Gener)-7 b(ation)53 b(of)g(classes)h(of)1066 8702 y(test)e(functions) e(with)h(known)f(lo)-7 b(c)g(al)51 b(minima.)65 b Fm(A)l(CM)48 b(T)-12 b(rans.)47 b(Math.)g(Soft.)k(V,)g Fa(N,)g(PP{PP.)920 8884 y Fb(Knuth,)56 b(D.)f Fa(1997.)78 b Fm(The)51 b(Art)f(of)h (Computer)f(Programming,)e(V)-12 b(ol.)50 b(2:)h(Semin)l(umerical)d (Algorithms)p Fa(,)j(thir)-7 b(d)1066 9051 y(e)g(d.)66 b(A)-7 b(ddison-Wesley,)50 b(R)-7 b(e)g(ading,)49 b(Massachusetts.)920 9233 y Fb(Knuth,)j(D.)f Fa(2002.)67 b(Home)50 b(p)-7 b(age)49 b(at:)i(http://sunburn.stanfor)-7 b(d.e)g(du/)16 b Fm(~)d Fa(knuth/.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF