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%%Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software
%%Title: alg48v40.dvi
%%CreationDate: Sun Apr 06 17:10:40 1997
%%Pages: 32
%%PageOrder: Ascend
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%%EndComments
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D E end
%%EndProlog
%%BeginSetup
%%Feature: *Resolution 300dpi
TeXDict begin
%%PaperSize: Letter

%%EndSetup
%%Page: 1 1
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Fs(1994)i(-)e(1997)p eop
%%Page: 1 2
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1309 y(4.8)42 b(Sym)o(b)q(olic)11 b(matrix)h(manipulation)39
b(.)20 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)
f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)57
b(11)137 1376 y(4.9)42 b(Nonlinear)13 b(equations)h(and)g(Gr\177)-21
b(obner)14 b(bases)h(commands)26 b(.)20 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g
(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)57 b(12)137
1444 y(4.10)21 b(V)m(erb)q(ose)15 b(mo)q(de)e(\015ag)36
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(.)f(.)g(.)57 b(15)137 1512 y(4.11)21 b(Calculating)12
b(with)i(fractions)26 b(.)20 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g
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g(.)h(.)f(.)g(.)57 b(15)137 1579 y(4.12)21 b(Algebraic)14
b(op)q(erations)g(on)f(mo)q(dular)f(p)q(olynomials)23
b(.)e(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g
(.)g(.)h(.)f(.)g(.)57 b(15)137 1647 y(4.13)21 b(Unlimited)12
b(precision)i(in)o(teger)g(arithmetic)31 b(.)20 b(.)h(.)f(.)g(.)h(.)f
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f(.)g(.)57 b(16)137 1715 y(4.14)21 b(Adv)n(anced)14 b(algebraic)g(op)q
(erations)g(on)f(unlimited)f(precision)i(in)o(tegers)19
b(.)h(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)57
b(17)137 1783 y(4.15)21 b(Mo)q(dular)13 b(arithmetic)g(on)h(unlimited)d
(precision)k(in)o(tegers)24 b(.)c(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f
(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)57 b(18)137 1850 y(4.16)21
b(P)o(erformances)29 b(.)20 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g
(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)57 b(18)137
1918 y(4.17)21 b(Remarks)15 b(.)21 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)
g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f
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b(19)75 2027 y Fp(5)38 b(History)15 b(of)g(c)o(hanges)1301
b(20)75 2136 y(6)38 b(Con)o(tact)1524 b(21)75 2246 y(A)26
b(Simpli\014)o(cation)12 b(Rules)i(for)h(ASIM)1056 b(22)75
2355 y(B)28 b(Command)15 b(Reference)1253 b(26)965 2700
y Fo(1)p eop
%%Page: 2 3
2 2 bop 75 195 a Fq(1)67 b(Ac)n(kno)n(wledgmen)n(ts,)22
b(Cop)n(yrigh)n(t)i(&)f(Disclaimer)g(of)f(W)-6 b(arran)n(t)n(y)75
304 y Fo(All)11 b(the)h(\014les)h(of)e(the)h Fn(ALG48)f
Fo(library)g(are)h(cop)o(yrigh)o(ted)962 303 y(c)950
304 y Fm(\015)f Fo(b)o(y)h(Claude-Nicolas)f(Fiec)o(h)o(ter)i(and)e(Mik)
n(a)g(Heisk)n(anen.)75 372 y Fn(ALG48)k Fo(is)g(distributed)h(in)g(the)
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Fp(the)h(cop)o(yrigh)o(t)e(holders)g(pro)o(vide)h(the)75
421 y(program)f(\\as)h(is")g(without)e(w)o(arran)o(t)o(y)g(of)i(an)o(y)
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471 y(but)c(not)h(limited)d(to,)k(the)e(implied)e(w)o(arran)o(ties)i
(of)h(merc)o(handabil)o(i)o(t)n(y)e(and)i(\014tness)f(for)g(a)i
(particular)75 521 y(purp)q(ose.)19 b(In)14 b(no)g(ev)o(en)o(t)f(will)g
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(damages,)g(includin)o(g)75 571 y(an)o(y)23 b(general,)h(sp)q(ecial,)f
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(the)f(use)g(or)75 621 y(inabili)o(t)o(y)13 b(to)i(use)g(the)g
(program.)75 688 y Fo(This)k(v)o(ersion)h(of)i Fn(ALG48)d
Fo(is)g(a)g(GiftW)m(are)f(release.)36 b(Y)m(ou)19 b(ma)o(y)f(use)i(it)f
(as)h(long)f(as)g(y)o(ou)g(lik)o(e,)h(but)g(only)e(for)75
738 y(non-commercial)12 b(purp)q(oses)k(and)e(only)g(as)g(a)h(priv)n
(ate)f(p)q(erson.)21 b(P)o(ermission)14 b(to)g(cop)o(y)h(the)g(whole,)f
(unmo)q(di\014ed,)75 788 y Fn(ALG48)f Fo(library)h(is)h(gran)o(ted)g
(pro)o(vided)f(that)h(the)g(copies)g(are)g(not)f(made)g(or)g
(distributed)i(for)e(resale)h(\(excepting)75 838 y(nominal)c(cop)o
(ying)i(fees\))h(and)g(pro)o(vided)f(that)h(y)o(ou)f(conspicuously)h
(and)f(appropriately)g(include)h(on)f(eac)o(h)h(cop)o(y)75
888 y(this)g(cop)o(yrigh)o(t)f(notice)i(and)e(disclaimer)g(of)g(w)o
(arran)o(t)o(y)m(.)75 955 y(Sp)q(ecial)h(thanks)g(to)f(Dominique)f(Ro)q
(driguez)i(for)f(his)h(L)950 947 y Fl(A)968 955 y Fo(T)991
964 y(E)1014 955 y(X)g(v)o(ersion)g(of)f(the)i(do)q(cumen)o(tation)d
(for)i Fn(ALG48)f Fo(v2.1,)75 1005 y(on)i(whic)o(h)g(this)g(do)q(cumen)
o(t)f(is)h(based,)h(and)f(to)g(Jo)q(e)g(Horn)g(for)g(his)g(man)o(y)e
(useful)i(commen)o(ts,)e(suggestions)j(and)75 1055 y(detailed)e(bug)g
(rep)q(orts.)75 1210 y Fq(2)67 b(Ov)n(erview)75 1319
y Fn(ALG48)13 b Fo(is)h(a)f(comprehensiv)o(e)i(sym)o(b)q(olic)d(math)g
(pac)o(k)n(age)i(for)g(the)g Fn(HP48)p Fo(.)k(It)c(includes)g(commands)
e(for)h(algebraic)75 1369 y(simpli\014cation,)c(factorization,)j
(partial)f(fraction)g(expansion,)h(sym)o(b)q(olic)f(in)o(tegration,)g
(sym)o(b)q(olic)f(matrices)i(ma-)75 1418 y(nipulation,)g(and)i(for)f
(solving)g(systems)h(of)f(nonlinear)g(p)q(olynomial)e(equations.)75
1486 y Fn(ALG48)i Fo(di\013ers)i(from)d(other)i(math)f(pac)o(k)n(ages)g
(for)h(the)g Fn(HP48)f Fo(in)h(t)o(w)o(o)f(imp)q(ortan)o(t)f(asp)q
(ects:)126 1595 y(1.)20 b Fn(ALG48)e Fo(can)h(manipulate,)e(simplify)m
(,)g(and)i(factorize)g Fk(multivariate)i Fo(p)q(olynomials)16
b(and)j(functions,)g(i.e.,)179 1645 y(algebraic)13 b(expressions)j
(with)d(sev)o(eral)i(v)n(ariables.)126 1728 y(2.)20 b
Fn(ALG48)13 b Fo(only)g(do)q(es)i Fk(exact)j Fo(calculation,)13
b(using)h(unlimited)e(precision)i(in)o(tegers)h(and)f(adv)n(anced)g
(computer)179 1778 y(algebra)e(algorithms)e(\(as)i(opp)q(osed)h(to)g
(doing)e(appro)o(ximate)g(calculation)g(using)h(\015oating)f(p)q(oin)o
(t)h(n)o(um)o(b)q(ers)179 1828 y(and)f(n)o(umerical)f(algorithms\).)15
b(This)c(not)h(only)e(means)h(y)o(ou)g(will)f(not)h(get)h(wrong)f(or)g
(appro)o(ximate)f(results)179 1878 y(\(2)p Fj(:)p Fo(00001,)h(4)p
Fj(:)p Fo(9999)h(and)i(the)h(lik)o(e\),)d(but)j(also)e(that)h(all)e
(and)i(only)f(exact)i(simpli\014cations)c(are)k(p)q(erformed.)75
1987 y(Belo)o(w)d(are)g(some)f(examples)f(of)15 b Fn(ALG48)10
b Fo(op)q(erations.)18 b(The)12 b(time)e(tak)o(en)i(for)f(the)i
(commands)c(on)i(a)h Fn(HP48GX)e Fo(\(with)75 2037 y(60K)k(free\))g
(are)h(giv)o(en)e(in)h(brac)o(k)o(ets.)137 2146 y Fm(\017)21
b Fo(Simpli\014cation)11 b(of)i(m)o(ultiv)n(ariate)e(p)q(olynomials)g
(and)j(rational)e(functions:)586 2298 y(1)d Fm(\000)663
2246 y Fo(1)g Fm(\000)777 2217 y Fj(y)p 739 2236 97 2
v 739 2274 a(x)g Fo(+)h Fj(y)p 663 2289 178 2 v 663 2340
a Fo(1)f Fm(\000)776 2312 y Fj(x)p 739 2330 97 2 v 739
2368 a(x)g Fo(+)h Fj(y)p 470 2383 492 2 v 470 2434 a(y)491
2417 y Fl(2)520 2434 y Fm(\000)643 2406 y Fj(x)p 566
2424 178 2 v 566 2475 a Fo(1)f(+)679 2447 y Fj(x)p 643
2466 97 2 v 643 2504 a(y)i Fm(\000)e Fj(x)749 2434 y
Fo(\()796 2406 y Fj(xy)p 770 2424 V 770 2462 a(y)i Fm(\000)f
Fj(x)881 2434 y Fm(\000)f Fj(x)p Fo(\))1009 2392 y Fn(RSIM)d
Fo([1.8s])k Fm(\))22 b Fo(in)o(v\()p Fj(x)1367 2375 y
Fl(2)1394 2392 y Fo(+)10 b Fj(xy)h Fo(+)e Fj(y)1553 2375
y Fl(2)1573 2392 y Fo(\))965 2700 y(2)p eop
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3 3 bop 137 195 a Fm(\017)21 b Fo(including)13 b(p)q(olynomial)o(s)f
(and)h(rational)g(functions)h(with)f(non-rational)g(sub)q(expressions:)
477 269 y(cos\()p Fj(a)p Fo(\))7 b(sin\()p Fj(a)p Fo(\))j
Fm(\000)f Fo(cos)q(\()p Fj(a)p Fo(\))g Fm(\000)h Fo(sin\()p
Fj(a)p Fo(\))f(+)h(1)p 477 288 610 2 v 477 326 a(cos\()p
Fj(a)p Fo(\))d(sin\()p Fj(a)p Fo(\))j(+)f(cos)q(\()p
Fj(a)p Fo(\))g Fm(\000)h Fo(sin\()p Fj(a)p Fo(\))f Fm(\000)h
Fo(1)1133 297 y Fn(RSIM)c Fo([0.9s])k Fm(\))1401 269
y Fo(sin\()p Fj(a)p Fo(\))f Fm(\000)g Fo(1)p 1401 288
177 2 v 1401 326 a(sin\()p Fj(a)p Fo(\))g(+)g(1)137 414
y Fm(\017)21 b Fo(Complete)13 b(factorization)g(of)g(p)q(olynomials)e
(and)j(rational)e(functions:)342 509 y(75)p Fj(x)408
494 y Fl(9)435 509 y Fm(\000)d Fo(435)p Fj(x)563 494
y Fl(8)590 509 y Fo(+)h(852)p Fj(x)719 494 y Fl(7)746
509 y Fm(\000)f Fo(576)p Fj(x)874 494 y Fl(6)901 509
y Fm(\000)g Fo(663)p Fj(x)1029 494 y Fl(5)1056 509 y
Fo(+)h(3027)p Fj(x)1206 494 y Fl(4)1232 509 y Fm(\000)g
Fo(4911)p Fj(x)1382 494 y Fl(3)1408 509 y Fo(+)g(3402)p
Fj(x)1558 494 y Fl(2)1585 509 y Fm(\000)f Fo(735)p Fj(x)922
584 y Fn(FCTR)d Fo([4.9s])k Fm(\))645 658 y Fo(3)p Fj(x)f
Fm(\001)g Fo(\()p Fj(x)760 643 y Fl(2)787 658 y Fm(\000)h
Fo(3)p Fj(x)f Fo(+)g(1\))g Fm(\001)g Fo(\()p Fj(x)1031
643 y Fl(4)1059 658 y Fo(+)g Fj(x)g Fm(\000)h Fo(5\))f
Fm(\001)g Fo(\(5)p Fj(x)f Fm(\000)i Fo(7\))1390 643 y
Fl(2)137 759 y Fm(\017)21 b Fo(including)13 b(p)q(olynomial)o(s)f(and)h
(rational)g(functions)h(in)f(sev)o(eral)i(v)n(ariables:)503
836 y(3)p Fj(x)548 821 y Fl(5)567 836 y Fj(y)c Fo(+)e(9)p
Fj(x)684 821 y Fl(4)702 836 y Fj(y)723 821 y Fl(2)752
836 y Fm(\000)g Fo(3)p Fj(x)838 821 y Fl(3)857 836 y
Fj(y)878 821 y Fl(2)906 836 y Fo(+)h(21)p Fj(x)1014 821
y Fl(3)1032 836 y Fj(y)h Fm(\000)e Fo(2)p Fj(x)1149 821
y Fl(3)1176 836 y Fo(+)h Fj(x)1242 821 y Fl(2)1260 836
y Fj(y)1281 821 y Fl(2)1310 836 y Fm(\000)g Fo(6)p Fj(x)1397
821 y Fl(2)1415 836 y Fj(y)h Fo(+)e(5)p Fj(x)1532 821
y Fl(2)647 886 y Fo(+3)p Fj(xy)745 871 y Fl(3)773 886
y Fo(+)h(17)p Fj(xy)g Fm(\000)g Fo(14)p Fj(x)e Fm(\000)i
Fj(y)1090 871 y Fl(3)1118 886 y Fo(+)g(7)p Fj(y)1202
871 y Fl(2)1230 886 y Fm(\000)g Fo(5)p Fj(y)h Fo(+)e(35)912
953 y Fn(FCTR)d Fo([11.2s])k Fm(\))656 1028 y Fo(\()p
Fj(x)696 1013 y Fl(2)724 1028 y Fo(+)g(3)p Fj(xy)g Fm(\000)g
Fj(y)h Fo(+)e(7\))g Fm(\001)g Fo(\(3)p Fj(x)1083 1013
y Fl(3)1101 1028 y Fj(y)j Fm(\000)d Fo(2)p Fj(x)g Fo(+)g
Fj(y)1290 1013 y Fl(2)1319 1028 y Fo(+)g(5\))137 1128
y Fm(\017)21 b Fo(Simpli\014cation)11 b(of)i(non-rational)f
(expressions:)638 1175 y Fm(p)p 673 1175 282 2 v 34 x
Fj(x)697 1197 y Fl(3)724 1209 y Fo(+)e Fj(x)790 1197
y Fl(2)817 1209 y Fm(\000)g Fj(x)f Fm(\000)g Fo(1)p 602
1228 389 2 v 602 1236 a Fi(p)p 643 1236 190 2 v 643 1277
a Fo(12)685 1243 y Fm(p)p 720 1243 21 2 v 34 x Fo(5)g(+)g(49)g
Fm(\001)g Fo(\()p Fj(x)g Fo(+)g(1\))1037 1237 y Fn(ASIM)d
Fo([4.3s])k Fm(\))1313 1177 y(p)p 1348 1177 96 2 v 32
x Fj(x)f Fm(\000)g Fo(1)p 1304 1228 148 2 v 1304 1270
a(3)1325 1236 y Fm(p)p 1360 1236 21 2 v 34 x Fo(5)g(+)g(2)137
1365 y Fm(\017)21 b Fo(including)13 b(exp)q(onen)o(tial)g(functions:)
513 1442 y Fj(x)7 b Fo(exp\(3)g(ln)o(\()p Fj(x)p Fo(\))i(+)h(ln)o(\()p
Fj(x)866 1427 y Fl(2)885 1442 y Fo(\)\))f Fm(\000)h Fo(1)p
513 1461 477 2 v 580 1513 a(ln)621 1467 y Fi(\020)646
1478 y(p)p 687 1478 210 2 v 687 1513 a Fo(exp)q(\()p
Fj(x)791 1501 y Fl(2)818 1513 y Fm(\000)g Fo(1\))897
1467 y Fi(\021)1035 1470 y Fn(ASIM)c Fo([1.8s])k Fm(\))23
b Fo(2)p Fj(x)1343 1453 y Fl(4)1370 1470 y Fo(+)10 b(2)p
Fj(x)1457 1453 y Fl(2)1484 1470 y Fo(+)f(2)137 1618 y
Fm(\017)21 b Fo(and)13 b(trigonometric)g(functions:)517
1695 y(cos)q(\(asin\(sin\()p Fj(x)p Fo(\))c Fm(\000)g
Fo(cos)q(\()p Fj(x)p Fo(\)\)\))978 1680 y Fl(2)p 517
1713 480 2 v 549 1751 a Fo(ln\(sin\()p Fj(x)p Fo(\))e(cos)q(\()p
Fj(x)p Fo(\))g(tan\()p Fj(x)p Fo(\)\))1043 1723 y Fn(ASIM)g
Fo([8.6s])j Fm(\))1311 1695 y Fo(sin\()p Fj(x)p Fo(\))d(cos)q(\()p
Fj(x)p Fo(\))p 1311 1713 226 2 v 1337 1751 a(ln\(sin\()p
Fj(x)p Fo(\)\))137 1840 y Fm(\017)21 b Fo(P)o(artial)13
b(fraction)g(expansion)h(along)f(one)h(or)g(sev)o(eral)g(v)n(ariables:)
269 1916 y Fj(x)293 1901 y Fl(3)312 1916 y Fj(y)333 1901
y Fl(2)361 1916 y Fm(\000)c Fo(3)p Fj(x)448 1901 y Fl(3)466
1916 y Fj(y)h Fo(+)f(3)p Fj(x)584 1901 y Fl(3)611 1916
y Fm(\000)f Fj(x)676 1901 y Fl(2)695 1916 y Fj(y)716
1901 y Fl(2)744 1916 y Fo(+)h(2)p Fj(x)831 1901 y Fl(2)849
1916 y Fj(y)h Fm(\000)f Fo(3)p Fj(x)967 1901 y Fl(2)994
1916 y Fm(\000)f Fj(xy)1080 1901 y Fl(4)1109 1916 y Fo(+)g(5)p
Fj(xy)1216 1901 y Fl(3)1244 1916 y Fm(\000)h Fo(8)p Fj(xy)1352
1901 y Fl(2)1380 1916 y Fo(+)g(5)p Fj(xy)g Fm(\000)g
Fj(y)1560 1901 y Fl(3)1588 1916 y Fo(+)g(3)p Fj(y)1672
1901 y Fl(2)1700 1916 y Fm(\000)g Fo(2)p Fj(y)p 269 1935
1516 2 v 491 1973 a(x)515 1961 y Fl(2)534 1973 y Fj(y)555
1961 y Fl(2)583 1973 y Fm(\000)g Fo(3)p Fj(x)670 1961
y Fl(2)688 1973 y Fj(y)h Fo(+)f(2)p Fj(x)806 1961 y Fl(2)833
1973 y Fm(\000)f Fj(xy)919 1961 y Fl(3)948 1973 y Fo(+)g(2)p
Fj(xy)1055 1961 y Fl(2)1084 1973 y Fo(+)g Fj(xy)i Fm(\000)e
Fo(2)p Fj(x)g Fo(+)h Fj(y)1338 1961 y Fl(3)1366 1973
y Fm(\000)g Fo(3)p Fj(y)1450 1961 y Fl(2)1478 1973 y
Fo(+)g(2)p Fj(y)944 2033 y Fn(PF)c Fo([8.3s])k Fm(\))668
2108 y Fj(y)h Fo(+)f Fj(x)f Fo(+)855 2080 y Fj(x)p 820
2098 94 2 v 820 2136 a(y)i Fm(\000)f Fo(2)928 2108 y
Fm(\000)1009 2080 y Fj(x)p 974 2098 V 974 2136 a(y)h
Fm(\000)f Fo(1)1082 2108 y(+)1166 2080 y Fj(y)p 1128
2098 97 2 v 1128 2136 a(x)f Fm(\000)h Fj(y)1239 2108
y Fo(+)1322 2080 y Fj(y)p 1285 2098 96 2 v 1285 2136
a(x)f Fm(\000)h Fo(1)137 2222 y Fm(\017)21 b Fo(Rational)12
b(function)h(in)o(tegration:)492 2307 y(27)p Fj(x)558
2292 y Fl(7)586 2307 y Fm(\000)c Fo(42)p Fj(x)693 2292
y Fl(6)720 2307 y Fm(\000)h Fo(106)p Fj(x)849 2292 y
Fl(5)876 2307 y Fm(\000)f Fo(47)p Fj(x)983 2292 y Fl(4)1010
2307 y Fo(+)h(224)p Fj(x)1139 2292 y Fl(3)1165 2307 y
Fm(\000)g Fo(147)p Fj(x)1294 2292 y Fl(2)1321 2307 y
Fo(+)f(313)p Fj(x)g Fo(+)g(138)p 467 2325 1121 2 v 467
2363 a(15)p Fj(x)533 2351 y Fl(8)560 2363 y Fm(\000)g
Fo(15)p Fj(x)667 2351 y Fl(7)694 2363 y Fo(+)h(15)p Fj(x)802
2351 y Fl(6)829 2363 y Fm(\000)f Fo(60)p Fj(x)936 2351
y Fl(5)964 2363 y Fo(+)g(90)p Fj(x)1071 2351 y Fl(4)1098
2363 y Fm(\000)h Fo(105)p Fj(x)1227 2351 y Fl(3)1253
2363 y Fo(+)g(45)p Fj(x)1361 2351 y Fl(2)1388 2363 y
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1354 y(v)n(ariables.)25 b(More)16 b(precisely)m(,)h Fn(PF)f
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212 1803 y Fj(x)236 1788 y Fl(3)254 1803 y Fj(y)275 1788
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911 2310 V 911 2348 a(x)d Fm(\000)i Fo(1)1020 2320 y
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37 x Fo(2\))g Fm(\000)1186 2048 y Fo(1)p 1186 2067 V
1186 2105 a(4)1211 2040 y Fm(p)p 1246 2040 V 37 x Fo(2)g
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582 V 497 618 a Fo(5\))g Fm(\000)600 590 y Fo(1)p 590
609 42 2 v 590 647 a(10)636 582 y Fm(p)p 671 582 21 2
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339 y(in)o(tegers.)h(These)13 b(commands)d(tak)o(e)i(three)h(argumen)o
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Fo(,)g(and)g(a)f(mo)q(dulus)g Fj(N)5 b Fo(\),)12 b(except)75
389 y Fn(MODINV)17 b Fo(whic)o(h)g(tak)o(es)i(only)e(t)o(w)o(o)g
(argumen)o(ts)g(\()p Fj(A)i Fo(and)e Fj(N)5 b Fo(\).)30
b(Here)19 b(again)e(the)i(argumen)o(ts)e(can)h(b)q(e)g(giv)o(en)g(as)75
439 y(\(in)o(teger\))d(real)e(n)o(um)o(b)q(ers,)h(binary)f(in)o
(tegers,)h(or)g(strings.)75 538 y Fn(MOD+)59 b Fp({)20
b Fo(\()p Fj(A)10 b Fo(+)f Fj(B)r Fo(\))k(mo)q(d)d Fj(N)75
620 y Fn(MOD-)59 b Fp({)20 b Fo(\()p Fj(A)10 b Fm(\000)f
Fj(B)r Fo(\))k(mo)q(d)d Fj(N)75 702 y Fn(MOD*)59 b Fp({)20
b Fo(\()p Fj(A)10 b Fm(\001)f Fj(B)r Fo(\))j(mo)q(d)e
Fj(N)75 784 y Fn(MOD/)59 b Fp({)20 b Fo(\()p Fj(A)10
b Fm(\001)f Fj(C)s Fo(\))i(mo)q(d)f Fj(N)5 b Fo(,)13
b(where)i Fj(C)i Fo(is)c(the)i(in)o(v)o(erse)g(mo)q(dulo)c
Fj(N)19 b Fo(of)13 b Fj(B)75 866 y Fn(MODPOW)i Fp({)266
832 y Fi(\000)285 866 y Fj(A)316 851 y Ff(B)345 832 y
Fi(\001)376 866 y Fo(mo)q(d)10 b Fj(N)75 948 y Fn(MODINV)15
b Fp({)20 b Fo(in)o(v)o(erse)15 b(mo)q(dulo)d Fj(N)18
b Fo(of)c Fj(A)75 1047 y Fo(If)g Fj(A)g Fo(and)f Fj(N)19
b Fo(are)14 b(relativ)o(ely)f(prime)g(n)o(um)o(b)q(ers)h(\(with)g
Fj(A)d(<)h(N)5 b Fo(\),)13 b(the)i(in)o(v)o(erse)g(mo)q(dulo)c
Fj(N)19 b Fo(of)13 b Fj(A)h Fo(is)g(the)h(\(unique\))75
1097 y(n)o(um)o(b)q(er)e Fj(C)k Fo(that)d(satis\014es)798
1146 y(\()p Fj(A)9 b Fm(\001)g Fj(C)s Fo(\))i(mo)q(d)g
Fj(N)16 b Fo(=)c(1)p Fj(:)75 1220 y Fo(If)e(no)h(suc)o(h)g(in)o(v)o
(erse)h(exists,)f(i.e.,)f(if)g Fj(A)h Fo(and)g Fj(N)k
Fo(are)c(not)g(relativ)o(ely)f(prime,)g(then)h Fn(MODINV)f
Fo(returns)i Fn(#0h)p Fo(.)k(Similarly)l(,)75 1269 y
Fn(MOD/)d Fo(returns)i Fn(#0h)f Fo(if)f(its)h(second)h(and)e(third)h
(argumen)o(ts)f(are)i(not)e(relativ)o(ely)h(prime.)75
1403 y Fh(4.16)56 b(P)n(erformances)75 1498 y Fo(Algebraic)14
b(computations,)f(lik)o(e)h(the)h(simpli\014cation,)c(factorization,)i
(partial)h(fraction)g(expansion)g(or)g(in)o(tegra-)75
1547 y(tion)g(of)g(rational)f(functions,)h(are)h(complex)f(op)q
(erations)g(and)h(are)g(generally)f(time-consuming)d(for)k(non-trivial)
75 1597 y(problems.)i(Therefore,)c(ev)o(en)g(though)f
Fn(ALG48)f Fo(is)h(written)h(in)f(sysRPL)h(and)f(mac)o(hine)f
(language,)g(an)o(y)h(op)q(eration)75 1647 y(that)i(in)o(v)o(olv)o(es)f
(suc)o(h)h(op)q(erations)h(is)e(not)h(instan)o(taneous)g(on)g(the)h
Fn(HP48)p Fo(.)75 1715 y Fn(ALG48)e Fo(v)o(ersion)h(4.0)e(can)i(nev)o
(ertheless)i(p)q(erform)d(most)g(simpli\014cations)e(quite)j(quic)o
(kly)m(.)j(Section)d(2)f(giv)o(es)h(some)75 1765 y(examples)d(with)g
(their)h(timings.)j([The)c(times)g(giv)o(en)g(throughout)g(this)h(do)q
(cumen)o(t)f(w)o(ere)h(obtained)g(on)f(a)g Fn(HP48GX)75
1814 y Fo(with)k(appro)o(ximately)f(60Kb)h(of)g(free)i(memory)m(.)o(])j
(Ev)o(en)d(complex)d(simpli\014cations)f(can)j(b)q(e)h(handled)e(in)g
(a)h(rea-)75 1864 y(sonable)c(amoun)o(t)e(of)h(time.)16
b(F)m(or)11 b(instance,)i Fn(ALG48)d Fo(v)o(ersion)i(4.0)f(tak)o(es)h
(only)f(8s)h(to)g(simplify)d(the)j(relativ)o(ely)f(large)75
1914 y(three-v)n(ariable)j(rational)f(b)q(elo)o(w.)137
1991 y Fl(6)p Ff(x)173 1978 y Fa(6)189 1991 y Fg(\000)p
Fl(126)p Ff(x)285 1978 y Fa(4)299 1991 y Ff(y)317 1978
y Fa(3)333 1991 y Ff(z)q Fl(+78)p Ff(x)428 1978 y Fa(4)444
1991 y Ff(y)q(z)479 1978 y Fa(2)495 1991 y Fl(+)p Ff(x)539
1978 y Fa(4)556 1991 y Ff(y)q Fl(+)p Ff(x)618 1978 y
Fa(4)634 1991 y Ff(z)q Fl(+13)p Ff(x)729 1978 y Fa(3)745
1991 y Fg(\000)p Fl(21)p Ff(x)824 1978 y Fa(2)839 1991
y Ff(y)857 1978 y Fa(4)873 1991 y Ff(z)q Fg(\000)p Fl(21)p
Ff(x)969 1978 y Fa(2)984 1991 y Ff(y)1002 1978 y Fa(3)1018
1991 y Ff(z)1035 1978 y Fa(2)1051 1991 y Fl(+13)p Ff(x)1129
1978 y Fa(2)1145 1991 y Ff(y)1163 1978 y Fa(2)1179 1991
y Ff(z)1196 1978 y Fa(2)1212 1991 y Fl(+13)p Ff(x)1290
1978 y Fa(2)1306 1991 y Ff(y)q(z)1341 1978 y Fa(3)1357
1991 y Fg(\000)p Fl(21)p Ff(xy)1454 1978 y Fa(3)1469
1991 y Ff(z)q Fl(+13)p Ff(xy)q(z)1599 1978 y Fa(2)1614
1991 y Fl(+2)p Ff(xy)q Fl(+2)p Ff(xz)q Fl(+2)p 137 2000
1677 2 v 288 2024 a(9)p Ff(x)324 2015 y Fa(5)339 2024
y Fl(+2)p Ff(x)400 2015 y Fa(4)416 2024 y Ff(y)q(z)q
Fg(\000)p Fl(189)p Ff(x)547 2015 y Fa(3)562 2024 y Ff(y)580
2015 y Fa(3)596 2024 y Ff(z)q Fl(+117)p Ff(x)708 2015
y Fa(3)723 2024 y Ff(y)q(z)758 2015 y Fa(2)774 2024 y
Fl(+3)p Ff(x)835 2015 y Fa(3)851 2024 y Fg(\000)p Fl(42)p
Ff(x)930 2015 y Fa(2)945 2024 y Ff(y)963 2015 y Fa(4)979
2024 y Ff(z)996 2015 y Fa(2)1013 2024 y Fl(+26)p Ff(x)1091
2015 y Fa(2)1106 2024 y Ff(y)1124 2015 y Fa(2)1140 2024
y Ff(z)1157 2015 y Fa(3)1174 2024 y Fl(+18)p Ff(x)1252
2015 y Fa(2)1267 2024 y Fg(\000)p Fl(63)p Ff(xy)1364
2015 y Fa(3)1379 2024 y Ff(z)q Fl(+39)p Ff(xy)q(z)1509
2015 y Fa(2)1525 2024 y Fl(+4)p Ff(xy)q(z)q Fl(+6)745
2084 y Fn(RSIM)6 b Fo([8.1s])k Fm(\))1013 2066 y Fl(6)p
Ff(x)1049 2053 y Fa(3)1064 2066 y Fl(+)p Ff(xy)q Fl(+)p
Ff(xz)q Fl(+1)p 1013 2075 218 2 v 1027 2098 a(9)p Ff(x)1063
2090 y Fa(2)1078 2098 y Fl(+2)p Ff(xy)q(z)q Fl(+3)1235
2084 y Fj(:)75 2199 y Fn(ALG48)j Fo(is)i(also)f(v)o(ery)h(fast)f(at)h
(simplifyi)o(ng)d(p)q(olynomial)o(s)g(\(m)o(ultiplying)f(out)k(the)g
(pro)q(ducts)h(and)e(collecting)g(the)75 2248 y(terms)g(of)f(same)g
(degree\).)19 b(F)m(or)14 b(instance,)g Fn(RSIM)f Fo(tak)o(es)h(only)g
(1s)g(to)f(simplify)e(the)k(follo)o(wing)c(expression:)189
2337 y(\()p Fj(x)f Fo(+)f(1\))317 2320 y Fl(12)361 2337
y Fm(\000)h Fo(\()p Fj(x)f Fm(\000)g Fo(1\))530 2320
y Fl(12)607 2337 y Fn(RSIM)d Fo([1.1s])k Fm(\))23 b Fo(24)p
Fj(x)936 2320 y Fl(11)979 2337 y Fo(+)10 b(440)p Fj(x)1108
2320 y Fl(9)1135 2337 y Fo(+)f(1584)p Fj(x)1284 2320
y Fl(7)1311 2337 y Fo(+)g(1584)p Fj(x)1460 2320 y Fl(5)1487
2337 y Fo(+)g(440)p Fj(x)1615 2320 y Fl(3)1642 2337 y
Fo(+)h(24)p Fj(x:)75 2426 y Fo(As)18 b(a)f(comparison,)f(the)i(program)
e Fn(EXCO)g Fo(\(Expand)h(&)h(Collect)f(completely\),)g(describ)q(ed)i
(in)e Fn(HP)p Fo('s)f(Adv)n(anced)75 2476 y(User's)e(Reference)g(Man)o
(ual)e(\(p.2-20\),)g(w)o(as)h(not)f(able)h(to)f(obtain)h(the)g
(solution)f(in)g(10)h(hours.)18 b(Using)12 b(extrap)q(ola-)75
2526 y(tion)g(from)g(the)h(time)f(tak)o(en)h(b)o(y)g
Fn(EXCO)f Fo(to)h(p)q(erform)f(simpler)g(binomial)e(expansions,)j(John)
g(Stebbins)h(estimated)75 2575 y(that)g(it)f(w)o(ould)h(tak)o(e)f
Fn(EXCO)g Fo(more)g(than)h(18)g(da)o(ys)f(\(!\))19 b(to)13
b(\014nd)h(the)h(solution)e(of)g(this)h(example.)954
2700 y(18)p eop
%%Page: 19 20
19 19 bop 75 195 a Fo(The)13 b(factorization)e(of)g(p)q(olynomials)f
(is)i(comparativ)o(ely)e(slo)o(w,)h(esp)q(ecially)h(for)g(m)o(ultiv)n
(ariate)d(problems.)17 b(Simple)75 245 y(factorizations,)d(ho)o(w)o(ev)
o(er,)h(are)g(p)q(erformed)f(quite)h(fast.)21 b(F)m(or)14
b(instance,)h Fn(FCTR)f Fo(tak)o(es)h(only)f(sligh)o(tly)f(more)h(than)
75 295 y(2s)g(on)g(the)g(follo)o(wing)d(example:)415
399 y(3)p Fj(x)460 384 y Fl(2)478 399 y Fj(y)g Fo(+)e(9)p
Fj(x)595 384 y Fl(2)623 399 y Fm(\000)g Fj(xy)709 384
y Fl(3)738 399 y Fm(\000)g Fo(5)p Fj(xy)845 384 y Fl(2)873
399 y Fm(\000)h Fo(2)p Fj(xy)g Fm(\000)g Fo(18)p Fj(x)e
Fo(+)i Fj(y)1169 384 y Fl(4)1197 399 y Fm(\000)g Fj(y)1260
384 y Fl(3)1289 399 y Fo(+)f(6)p Fj(y)1372 384 y Fl(2)1401
399 y Fm(\000)g Fj(y)i Fo(+)f(5)870 474 y Fn(FCTR)c Fo([2.2s])k
Fm(\))630 548 y Fo(\(3)p Fj(x)f Fm(\000)h Fj(y)763 533
y Fl(2)791 548 y Fo(+)g Fj(y)h Fm(\000)e Fo(5\))h Fm(\001)e
Fo(\()p Fj(xy)j Fo(+)f(3)p Fj(x)f Fm(\000)g Fj(y)1201
533 y Fl(2)1230 548 y Fm(\000)g Fo(1\))p Fj(:)75 644
y Fo(Ev)o(en)k(some)g(seemingly)e(complex)h(factorizations)h(can)g(b)q
(e)h(p)q(erformed)e(quic)o(kly)m(,)g(lik)o(e)g(the)h(follo)o(wing)e
(one,)i(tak)o(en)75 693 y(from)f(Mathematica's)h(b)q(o)q(ok,)g(that)h
Fn(ALG48)e Fo(solv)o(es)j(in)e(15s,)168 780 y Fl(4096)p
Ff(x)255 768 y Fa(8)269 780 y Fg(\000)p Fl(14336)p Ff(x)399
768 y Fa(7)413 780 y Ff(y)q Fl(+43008)p Ff(x)560 768
y Fa(7)574 780 y Fl(+16768)p Ff(x)703 768 y Fa(6)717
780 y Ff(y)735 768 y Fa(2)751 780 y Fg(\000)p Fl(155904)p
Ff(x)898 768 y Fa(6)911 780 y Ff(y)q Fl(+169344)p Ff(x)1075
768 y Fa(6)1089 780 y Fg(\000)p Fl(5600)p Ff(x)1202 768
y Fa(5)1216 780 y Ff(y)1234 768 y Fa(3)1250 780 y Fl(+195552)p
Ff(x)1396 768 y Fa(5)1410 780 y Ff(y)1428 768 y Fa(2)1444
780 y Fg(\000)p Fl(635040)p Ff(x)1591 768 y Fa(5)1604
780 y Ff(y)q Fl(+296352)p Ff(x)1768 768 y Fa(5)176 830
y Fg(\000)p Fl(1919)p Ff(x)289 818 y Fa(4)303 830 y Ff(y)321
818 y Fa(4)337 830 y Fg(\000)p Fl(83244)p Ff(x)467 818
y Fa(4)481 830 y Ff(y)499 818 y Fa(3)515 830 y Fl(+849366)p
Ff(x)661 818 y Fa(4)675 830 y Ff(y)693 818 y Fa(2)709
830 y Fg(\000)p Fl(1148364)p Ff(x)873 818 y Fa(4)886
830 y Ff(y)q Fl(+194481)p Ff(x)1050 818 y Fa(4)1064 830
y Fl(+700)p Ff(x)1159 818 y Fa(3)1174 830 y Ff(y)1192
818 y Fa(5)1208 830 y Fg(\000)p Fl(9744)p Ff(x)1321 818
y Fa(3)1335 830 y Ff(y)1353 818 y Fa(4)1369 830 y Fg(\000)p
Fl(433944)p Ff(x)1516 818 y Fa(3)1529 830 y Ff(y)1547
818 y Fa(3)1563 830 y Fl(+1629936)p Ff(x)1726 818 y Fa(3)1740
830 y Ff(y)1758 818 y Fa(2)314 880 y Fg(\000)p Fl(777924)p
Ff(x)461 867 y Fa(3)475 880 y Ff(y)q Fl(+262)p Ff(x)588
867 y Fa(2)603 880 y Ff(y)621 867 y Fa(6)637 880 y Fl(+8568)p
Ff(x)749 867 y Fa(2)763 880 y Ff(y)781 867 y Fa(5)798
880 y Fl(+15876)p Ff(x)927 867 y Fa(2)941 880 y Ff(y)959
867 y Fa(4)975 880 y Fg(\000)p Fl(963144)p Ff(x)1122
867 y Fa(2)1135 880 y Ff(y)1153 867 y Fa(3)1169 880 y
Fl(+1166886)p Ff(x)1332 867 y Fa(2)1345 880 y Ff(y)1363
867 y Fa(2)1380 880 y Fl(+28)p Ff(xy)1476 867 y Fa(7)1491
880 y Fl(+1680)p Ff(xy)1621 867 y Fa(6)427 930 y Fl(+31752)p
Ff(xy)574 917 y Fa(5)588 930 y Fl(+148176)p Ff(xy)752
917 y Fa(4)766 930 y Fg(\000)p Fl(777924)p Ff(xy)931
917 y Fa(3)944 930 y Fl(+)p Ff(y)987 917 y Fa(8)1003
930 y Fl(+84)p Ff(y)1080 917 y Fa(7)1096 930 y Fl(+2646)p
Ff(y)1207 917 y Fa(6)1222 930 y Fl(+37044)p Ff(y)1350
917 y Fa(5)1364 930 y Fl(+194481)p Ff(y)1509 917 y Fa(4)645
1029 y Fn(FCTR)6 b Fo([15.2s])j Fm(\))23 b Fo(\()p Fj(x)9
b Fm(\000)h Fj(y)q Fo(\))1056 1014 y Fl(4)1075 1029 y
Fo(\(8)p Fj(x)f Fo(+)h Fj(y)h Fo(+)e(21\))1317 1014 y
Fl(4)1335 1029 y Fj(:)75 1135 y Fn(ALG48)k Fo(v)o(ersion)i(4.0)e(uses)j
(Berlek)n(amp)d(p-adic)h(factorization)g(algorithm)e(and,)i(giv)o(en)g
(enough)g(time,)f(can)h(com-)75 1185 y(pute)k(the)g(complete)e
(factorization)h(of)f(virtually)g(an)o(y)h(p)q(olynomials)d(\(up)k(to)f
(degree)h(256\),)f(ev)o(en)h(if)f(they)h(are)75 1234
y(square-free)d(and)f(con)o(tain)f(high-degree)i(factors.)j(E.g.,)75
1324 y Fj(x)99 1307 y Fl(40)136 1324 y Fo(+)r Fj(x)194
1307 y Fl(30)232 1324 y Fm(\000)r Fj(x)290 1307 y Fl(20)328
1324 y Fm(\000)r Fo(2)p Fj(x)407 1307 y Fl(15)444 1324
y Fm(\000)r Fj(x)502 1307 y Fl(10)540 1324 y Fm(\000)r
Fo(2)p Fj(x)619 1307 y Fl(5)640 1324 y Fm(\000)r Fo(1)42
b Fn(FCTR)6 b Fo([125s])k Fm(\))23 b Fo(\()p Fj(x)1049
1307 y Fl(20)1086 1324 y Fm(\000)r Fj(x)1144 1307 y Fl(15)1182
1324 y Fo(+)r Fj(x)1240 1307 y Fl(10)1277 1324 y Fm(\000)r
Fj(x)1335 1307 y Fl(5)1356 1324 y Fm(\000)r Fo(1\))r
Fm(\001)r Fo(\()p Fj(x)1483 1307 y Fl(20)1521 1324 y
Fo(+)r Fj(x)1579 1307 y Fl(15)1617 1324 y Fo(+)r Fj(x)1675
1307 y Fl(10)1713 1324 y Fo(+)r Fj(x)1771 1307 y Fl(5)1792
1324 y Fo(+)r(1\);)75 1414 y Fj(x)99 1396 y Fl(18)141
1414 y Fm(\000)7 b Fj(y)201 1396 y Fl(18)279 1414 y Fn(FCTR)f
Fo([16s])k Fm(\))23 b Fo(\()p Fj(x)7 b Fm(\000)g Fj(y)q
Fo(\)\()p Fj(x)g Fo(+)g Fj(y)q Fo(\)\()p Fj(x)816 1396
y Fl(2)843 1414 y Fm(\000)g Fj(xy)i Fo(+)e Fj(y)995 1396
y Fl(2)1014 1414 y Fo(\)\()p Fj(x)1070 1396 y Fl(2)1096
1414 y Fo(+)g Fj(xy)h Fo(+)f Fj(y)1247 1396 y Fl(2)1267
1414 y Fo(\)\()p Fj(x)1323 1396 y Fl(6)1349 1414 y Fm(\000)g
Fj(x)1412 1396 y Fl(3)1430 1414 y Fj(y)1451 1396 y Fl(3)1478
1414 y Fo(+)g Fj(y)1538 1396 y Fl(6)1557 1414 y Fo(\)\()p
Fj(x)1613 1396 y Fl(6)1639 1414 y Fo(+)g Fj(x)1702 1396
y Fl(3)1720 1414 y Fj(y)1741 1396 y Fl(3)1768 1414 y
Fo(+)g Fj(y)1828 1396 y Fl(6)1847 1414 y Fo(\))p Fj(:)75
1487 y Fo(Ev)o(en)15 b(large)f(m)o(ultiv)n(ariate)e(p)q(olynomials)g
(can)j(b)q(e)g(handled)f(in)g(a)h(reasonable)g(amoun)o(t)e(of)h(time.)k
(F)m(or)d(instance,)75 1537 y Fn(ALG48)g Fo(can)g(factor)h(the)g(n)o
(umerator)f(and)h(the)g(denominator)e(of)h(the)h(large)g(three-v)n
(ariable)g(rational)e(ab)q(o)o(v)o(e)h(in)75 1587 y(ab)q(out)f(a)f(min)
o(ute.)137 1683 y Fl(6)p Ff(x)173 1671 y Fa(6)189 1683
y Fg(\000)p Fl(126)p Ff(x)285 1671 y Fa(4)299 1683 y
Ff(y)317 1671 y Fa(3)333 1683 y Ff(z)q Fl(+78)p Ff(x)428
1671 y Fa(4)444 1683 y Ff(y)q(z)479 1671 y Fa(2)495 1683
y Fl(+)p Ff(x)539 1671 y Fa(4)556 1683 y Ff(y)q Fl(+)p
Ff(x)618 1671 y Fa(4)634 1683 y Ff(z)q Fl(+13)p Ff(x)729
1671 y Fa(3)745 1683 y Fg(\000)p Fl(21)p Ff(x)824 1671
y Fa(2)839 1683 y Ff(y)857 1671 y Fa(4)873 1683 y Ff(z)q
Fg(\000)p Fl(21)p Ff(x)969 1671 y Fa(2)984 1683 y Ff(y)1002
1671 y Fa(3)1018 1683 y Ff(z)1035 1671 y Fa(2)1051 1683
y Fl(+13)p Ff(x)1129 1671 y Fa(2)1145 1683 y Ff(y)1163
1671 y Fa(2)1179 1683 y Ff(z)1196 1671 y Fa(2)1212 1683
y Fl(+13)p Ff(x)1290 1671 y Fa(2)1306 1683 y Ff(y)q(z)1341
1671 y Fa(3)1357 1683 y Fg(\000)p Fl(21)p Ff(xy)1454
1671 y Fa(3)1469 1683 y Ff(z)q Fl(+13)p Ff(xy)q(z)1599
1671 y Fa(2)1614 1683 y Fl(+2)p Ff(xy)q Fl(+2)p Ff(xz)q
Fl(+2)419 1758 y Fn(FCTR)7 b Fo([59s])j Fm(\))23 b Fo(\()p
Fj(x)711 1743 y Fl(3)738 1758 y Fm(\000)10 b Fo(21)p
Fj(xy)867 1743 y Fl(3)885 1758 y Fj(z)i Fo(+)d(13)p Fj(xy)q(z)1065
1743 y Fl(2)1093 1758 y Fo(+)h(2\))f Fm(\001)g Fo(\(6)p
Fj(x)1263 1743 y Fl(3)1290 1758 y Fo(+)h Fj(xy)h Fo(+)e
Fj(xz)i Fo(+)f(1\);)288 1876 y Fl(9)p Ff(x)324 1864 y
Fa(5)339 1876 y Fl(+2)p Ff(x)400 1864 y Fa(4)416 1876
y Ff(y)q(z)q Fg(\000)p Fl(189)p Ff(x)547 1864 y Fa(3)562
1876 y Ff(y)580 1864 y Fa(3)596 1876 y Ff(z)q Fl(+117)p
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Fo(ev)o(en)p 1456 1681 V 493 1733 V 586 1685 a Fm(p)p
621 1685 47 2 v 33 x Fj(x)645 1706 y Ff(n)p 759 1733
2 52 v 785 1688 a Fm(p)p 819 1688 24 2 v 819 1718 a Fj(x)9
b Fm(\001)g Fj(x)897 1703 y Fl(\()p Ff(n)p Fg(\000)p
Fl(1\))p Ff(=)p Fl(2)p 1045 1733 2 52 v 1071 1718 a Fj(n)14
b Fo(o)q(dd)p 1456 1733 V 493 1783 2 50 v 572 1742 a
Fm(p)p 606 1742 76 2 v 606 1768 a Fj(x)9 b Fm(\001)g
Fj(y)p 759 1783 2 50 v 831 1738 a Fm(p)p 865 1738 24
2 v 865 1768 a Fj(x)g Fm(\001)919 1742 y(p)p 954 1742
22 2 v 26 x Fj(y)p 1045 1783 2 50 v 1456 1783 V 493 1836
2 54 v 573 1786 a Fi(p)p 614 1786 67 2 v 614 1822 a Fj(x=y)p
759 1836 2 54 v 835 1792 a Fm(p)p 870 1792 24 2 v 30
x Fj(x=)915 1796 y Fm(p)p 949 1796 22 2 v 949 1822 a
Fj(y)p 1045 1836 2 54 v 1456 1836 V 493 1886 2 50 v 581
1840 a Fm(p)p 616 1840 57 2 v 31 x(\000)p Fj(x)p 759
1886 2 50 v 852 1841 a Fm(p)p 886 1841 24 2 v 886 1871
a Fj(x)g Fm(\001)g Fj(i)p 1045 1886 2 50 v 1456 1886
V 493 1941 2 56 v 531 1890 a Fi(p)p 573 1890 149 2 v
36 x Fj(a)595 1896 y Fm(p)p 630 1896 24 2 v 30 x Fj(x)f
Fo(+)i Fj(b)p 759 1941 2 56 v 107 w(c)f Fo(+)g Fj(d)919
1896 y Fm(p)p 954 1896 24 2 v 30 x Fj(x)p 1045 1941 2
56 v 93 w Fo(for)14 b(in)o(teger)g(solutions)p 1456 1941
V 494 1943 963 2 v 850 2028 a Fp(Exp)q(onen)o(tial)p
562 2045 827 2 v 561 2095 2 50 v 587 2080 a(expression)p
827 2095 V 81 w(simpli)o(\014ed)p 1144 2095 V 80 w(condition)p
1387 2095 V 562 2096 827 2 v 561 2146 2 50 v 637 2131
a Fo(exp\(0\))p 827 2146 V 223 w(1)p 1144 2146 V 1387
2146 V 561 2196 V 637 2181 a(exp\(1\))p 827 2196 V 224
w Fj(e)p 1144 2196 V 1387 2196 V 561 2246 V 621 2231
a Fo(exp\()p Fm(\000)p Fo(1\))p 827 2246 V 164 w(in)o(v)o(\()p
Fj(e)p Fo(\))p 1144 2246 V 1387 2246 V 561 2296 V 612
2281 a(exp)q(\()p Fj(\031)c Fm(\001)f Fj(i)p Fo(\))p
827 2296 V 183 w Fm(\000)p Fo(1)p 1144 2296 V 1387 2296
V 561 2345 V 635 2331 a(exp)q(\()p Fj(x)p Fo(\))p 827
2345 V 211 w Fj(e)985 2315 y Ff(x)p 1144 2345 V 1170
2331 a Fo(if)k Fj(x)g Fo(is)h(real)p 1387 2345 V 561
2395 V 602 2380 a(exp\(ln\()p Fj(x)p Fo(\)\))p 827 2395
V 187 w Fj(x)p 1144 2395 V 1387 2395 V 561 2445 V 619
2430 a Fo(exp\()p Fm(\000)p Fj(x)p Fo(\))p 827 2445 V
113 w(in)o(v)o(\(exp)q(\()p Fj(x)p Fo(\)\))p 1144 2445
V 1387 2445 V 561 2495 V 599 2480 a(exp\()p Fj(x)9 b
Fo(+)h Fj(y)q Fo(\))p 827 2495 V 63 w(exp\()p Fj(x)p
Fo(\))g Fm(\001)e Fo(exp)q(\()p Fj(y)q Fo(\))p 1144 2495
V 1387 2495 V 561 2545 V 599 2530 a(exp\()p Fj(x)h Fm(\000)h
Fj(y)q Fo(\))p 827 2545 V 64 w(exp)q(\()p Fj(x)p Fo(\))p
Fj(=)d Fo(exp\()p Fj(y)q Fo(\))p 1144 2545 V 1387 2545
V 561 2595 V 624 2580 a(exp)q(\()p Fj(xy)q Fo(\))p 827
2595 V 152 w(exp)q(\()p Fj(y)q Fo(\))1034 2565 y Ff(x)p
1144 2595 V 1170 2580 a Fo(if)13 b(x)h(is)f(real)p 1387
2595 V 562 2596 827 2 v 954 2700 a(22)p eop
%%Page: 23 24
23 23 bop 807 188 a Fp(ALOG)15 b(F)l(unction)p 551 205
848 2 v 550 255 2 50 v 576 240 a(expression)p 817 255
V 92 w(simpli)o(\014ed)p 1154 255 V 90 w(condition)p
1398 255 V 551 257 848 2 v 550 306 2 50 v 619 291 a Fo(alog)o(\()p
Fj(n)p Fo(\))p 817 306 V 175 w(in)o(teger)p 1154 306
V 1398 306 V 550 356 V 577 341 a(alog)n(\(log\()p Fj(x)p
Fo(\)\))p 817 356 V 183 w Fj(x)p 1154 356 V 1398 356
V 550 406 V 603 391 a Fo(alog)o(\()p Fm(\000)p Fj(x)p
Fo(\))p 817 406 V 113 w(in)o(v)o(\(alog)o(\()p Fj(x)p
Fo(\)\))p 1154 406 V 1398 406 V 550 456 V 583 441 a(alog)o(\()p
Fj(x)9 b Fo(+)h Fj(y)q Fo(\))p 817 456 V 58 w(alog)n(\()p
Fj(x)p Fo(\))g Fm(\001)e Fo(alog)o(\()p Fj(y)q Fo(\))p
1154 456 V 1398 456 V 550 506 V 583 491 a(alog)o(\()p
Fj(x)h Fm(\000)h Fj(y)q Fo(\))p 817 506 V 62 w(alog)o(\()p
Fj(x)p Fo(\))p Fj(=)p Fo(alog)o(\()p Fj(y)q Fo(\))p 1154
506 V 1398 506 V 550 555 V 609 540 a(alog)o(\()p Fj(xy)q
Fo(\))p 817 555 V 152 w(alog)o(\()p Fj(y)q Fo(\))1039
525 y Ff(x)p 1154 555 V 1180 540 a Fo(if)j(x)h(is)g(real)p
1398 555 V 551 557 848 2 v 777 646 a Fp(Natural)h(Logarithm)p
702 663 547 2 v 701 713 2 50 v 727 698 a(expression)p
967 713 V 62 w(simpli\014ed)p 1247 713 V 702 714 547
2 v 701 764 2 50 v 791 749 a Fo(ln\(1\))p 967 764 V 219
w(0)p 1247 764 V 701 814 V 792 799 a(ln\()p Fj(e)p Fo(\))p
967 814 V 220 w(1)p 1247 814 V 701 864 V 775 849 a(ln\()p
Fm(\000)p Fo(1\))p 967 864 V 178 w Fj(\031)c Fm(\001)e
Fj(i)p 1247 864 V 701 914 V 794 899 a Fo(ln\()p Fj(i)p
Fo(\))p 967 914 V 178 w Fj(\031)h Fm(\001)f Fj(i=)p Fo(2)p
1247 914 V 701 963 V 742 948 a(ln\(exp\()p Fj(x)p Fo(\)\))p
967 963 V 168 w Fj(x)p 1247 963 V 701 1013 V 779 998
a Fo(ln)o(\()p Fj(xy)q Fo(\))p 967 1013 V 103 w(ln\()p
Fj(x)p Fo(\))g(+)g(ln\()p Fj(y)q Fo(\))p 1247 1013 V
701 1063 V 768 1048 a(ln\()p Fj(x=y)q Fo(\))p 967 1063
V 92 w(ln\()p Fj(x)p Fo(\))g Fm(\000)g Fo(ln\()p Fj(y)q
Fo(\))p 1247 1063 V 701 1113 V 780 1098 a(ln)o(\()p Fj(x)854
1083 y Ff(y)874 1098 y Fo(\))p 967 1113 V 147 w Fj(y)i
Fm(\001)e Fo(ln)o(\()p Fj(x)p Fo(\))p 1247 1113 V 701
1163 V 746 1148 a(ln\(in)o(v)o(\()p Fj(x)p Fo(\)\))p
967 1163 V 119 w Fm(\000)e Fo(ln\()p Fj(x)p Fo(\))p 1247
1163 V 701 1212 V 772 1197 a(ln\()823 1168 y Fm(p)p 858
1168 24 2 v 29 x Fj(x)p Fo(\))p 967 1212 2 50 v 144 w(ln\()p
Fj(x)p Fo(\))p Fj(=)p Fo(2)p 1247 1212 V 702 1214 547
2 v 778 1303 a Fp(Base)15 b(10)h(Logarithm)p 683 1320
585 2 v 682 1370 2 50 v 708 1355 a(expression)p 948 1370
V 81 w(simpli\014ed)p 1266 1370 V 683 1371 585 2 v 682
1421 2 50 v 763 1406 a Fo(log)o(\(1\))p 948 1421 V 229
w(0)p 1266 1421 V 682 1471 V 708 1456 a(log\(alog)o(\()p
Fj(x)p Fo(\)\))p 948 1471 V 172 w Fj(x)p 1266 1471 V
682 1521 V 750 1506 a Fo(log\()p Fj(xy)q Fo(\))p 948
1521 V 93 w(log\()p Fj(x)p Fo(\))9 b(+)g(log\()p Fj(y)q
Fo(\))p 1266 1521 V 682 1571 V 740 1556 a(log\()p Fj(x=y)q
Fo(\))p 948 1571 V 82 w(log\()p Fj(x)p Fo(\))g Fm(\000)g
Fo(log\()p Fj(y)q Fo(\))p 1266 1571 V 682 1620 V 751
1605 a(log\()p Fj(x)845 1590 y Ff(y)865 1605 y Fo(\))p
948 1620 V 146 w Fj(y)i Fm(\001)e Fo(log\()p Fj(x)p Fo(\))p
1266 1620 V 682 1670 V 717 1655 a(log\(in)o(v\()p Fj(x)p
Fo(\)\))p 948 1670 V 119 w Fm(\000)e Fo(log\()p Fj(x)p
Fo(\))p 1266 1670 V 682 1720 V 744 1705 a(log\()814 1675
y Fm(p)p 848 1675 24 2 v 848 1705 a Fj(x)p Fo(\))p 948
1720 2 50 v 145 w(log)o(\()p Fj(x)p Fo(\))p Fj(=)p Fo(2)p
1266 1720 V 683 1722 585 2 v 719 1811 a Fp(T)l(rigonometri)o(c)14
b(F)l(unctions)p 694 1827 562 2 v 693 1877 2 50 v 725
1862 a(expression)p 972 1877 V 70 w(simpli\014ed)p 1255
1877 V 694 1879 562 2 v 693 1929 2 50 v 728 1914 a Fo(sin\(asin\()p
Fj(x)p Fo(\)\))p 972 1929 V 164 w Fj(x)p 1255 1929 V
693 1982 2 53 v 726 1967 a Fo(sin\(acos\()p Fj(x)p Fo(\)\))p
972 1982 V 1040 1932 a Fm(p)p 1075 1932 114 2 v 35 x
Fo(1)9 b Fm(\000)g Fj(x)1170 1955 y Fl(2)p 1255 1982
2 53 v 693 2034 V 724 2019 a Fo(sin\(atan)o(\()p Fj(x)p
Fo(\)\))p 972 2034 V 75 w Fj(x=)1063 1985 y Fm(p)p 1097
1985 114 2 v 1097 2019 a Fo(1)g(+)h Fj(x)1193 2008 y
Fl(2)p 1255 2034 2 53 v 693 2084 2 50 v 764 2069 a Fo(sin\()p
Fm(\000)p Fj(x)p Fo(\))p 972 2084 V 138 w Fm(\000)d Fo(sin\()p
Fj(x)p Fo(\))p 1255 2084 V 693 2134 V 723 2119 a(cos)q(\(acos\()p
Fj(x)p Fo(\)\))p 972 2134 V 160 w Fj(x)p 1255 2134 V
693 2187 2 53 v 726 2172 a Fo(cos\(asin\()p Fj(x)p Fo(\)\))p
972 2187 V 1040 2137 a Fm(p)p 1075 2137 114 2 v 35 x
Fo(1)i Fm(\000)g Fj(x)1170 2160 y Fl(2)p 1255 2187 2
53 v 693 2240 V 721 2225 a Fo(cos)q(\(atan\()p Fj(x)p
Fo(\)\))p 972 2240 V 73 w(1)p Fj(=)1061 2190 y Fm(p)p
1096 2190 114 2 v 35 x Fo(1)g(+)g Fj(x)1191 2213 y Fl(2)p
1255 2240 2 53 v 693 2290 2 50 v 762 2275 a Fo(cos\()p
Fm(\000)p Fj(x)p Fo(\))p 972 2290 V 154 w(cos\()p Fj(x)p
Fo(\))p 1255 2290 V 693 2339 V 719 2324 a(tan\(atan\()p
Fj(x)p Fo(\)\))p 972 2339 V 155 w Fj(x)p 1255 2339 V
693 2392 2 53 v 724 2377 a Fo(tan\(asin)o(\()p Fj(x)p
Fo(\)\))p 972 2392 V 75 w Fj(x=)1063 2343 y Fm(p)p 1097
2343 114 2 v 1097 2377 a Fo(1)g Fm(\000)h Fj(x)1193 2365
y Fl(2)p 1255 2392 2 53 v 693 2445 V 721 2430 a Fo(tan\(acos)q(\()p
Fj(x)p Fo(\)\))p 972 2445 V 1018 2396 a Fm(p)p 1053 2396
114 2 v 34 x Fo(1)f Fm(\000)g Fj(x)1148 2418 y Fl(2)1166
2430 y Fj(=x)p 1255 2445 2 53 v 693 2495 2 50 v 759 2480
a Fo(tan\()p Fm(\000)p Fj(x)p Fo(\))p 972 2495 V 130
w Fm(\000)e Fo(tan\()p Fj(x)p Fo(\))p 1255 2495 V 693
2545 V 771 2530 a(asin\(0\))p 972 2545 V 208 w(0)p 1255
2545 V 693 2595 V 767 2580 a(atan)o(\(0\))p 972 2595
V 204 w(0)p 1255 2595 V 694 2596 562 2 v 954 2700 a(23)p
eop
%%Page: 24 25
24 24 bop 751 188 a Fp(Hyp)q(erb)q(olic)14 b(F)l(unctions)p
653 205 645 2 v 652 255 2 50 v 707 240 a(expression)p
977 255 V 112 w(simpli)o(\014ed)p 1296 255 V 653 257
645 2 v 652 306 2 50 v 752 291 a Fo(sinh\(0\))p 977 306
V 248 w(0)p 1296 306 V 652 356 V 687 341 a(sinh\(asinh\()p
Fj(x)p Fo(\)\))p 977 356 V 182 w Fj(x)p 1296 356 V 652
409 2 53 v 682 394 a Fo(sinh\(atanh\()p Fj(x)p Fo(\)\))p
977 409 V 93 w Fj(x=)1086 359 y Fm(p)p 1120 359 114 2
v 1120 394 a Fo(1)9 b Fm(\000)h Fj(x)1216 382 y Fl(2)p
1296 409 2 53 v 652 459 2 50 v 750 444 a Fo(cosh\(0\))p
977 459 V 246 w(1)p 1296 459 V 652 509 V 682 494 a(cosh)q(\(acosh\()p
Fj(x)p Fo(\)\))p 977 509 V 178 w Fj(x)p 1296 509 V 652
562 2 53 v 684 547 a Fo(cosh)q(\(asinh\()p Fj(x)p Fo(\)\))p
977 562 V 1063 512 a Fm(p)p 1098 512 114 2 v 35 x Fo(1)f(+)g
Fj(x)1193 535 y Fl(2)p 1296 562 2 53 v 652 614 V 680
599 a Fo(cosh)q(\(atanh\()p Fj(x)p Fo(\)\))p 977 614
V 91 w(1)p Fj(=)1084 565 y Fm(p)p 1119 565 114 2 v 34
x Fo(1)g Fm(\000)g Fj(x)1214 587 y Fl(2)p 1296 614 2
53 v 652 664 2 50 v 747 649 a Fo(tanh\(0\))p 977 664
V 244 w(0)p 1296 664 V 652 714 V 678 699 a(tanh\(atanh\()p
Fj(x)p Fo(\)\))p 977 714 V 173 w Fj(x)p 1296 714 V 652
767 2 53 v 682 752 a Fo(tanh\(asinh\()p Fj(x)p Fo(\)\))p
977 767 V 55 w Fm(\000)p Fj(i)h Fm(\001)e Fj(x=)1124
717 y Fm(p)p 1158 717 114 2 v 1158 752 a Fo(1)h(+)h Fj(x)1254
740 y Fl(2)p 1296 767 2 53 v 652 817 2 50 v 741 802 a
Fo(asinh\(0\))p 977 817 V 238 w(0)p 1296 817 V 652 866
V 737 852 a(atanh\(0\))p 977 866 V 233 w(0)p 1296 866
V 653 868 645 2 v 890 967 a Fp(Division)p 696 983 558
2 v 695 1033 2 50 v 742 1018 a(expression)p 1003 1033
V 68 w(simpli\014)o(ed)p 1253 1033 V 696 1035 558 2 v
695 1084 2 50 v 799 1069 a Fo(in)o(v\()p Fj(i)p Fo(\))p
1003 1084 V 204 w Fm(\000)p Fj(i)p 1253 1084 V 695 1134
V 751 1119 a Fo(in)o(v)o(\(in)o(v\()p Fj(x)p Fo(\)\))p
1003 1134 V 167 w Fj(x)p 1253 1134 V 695 1184 V 799 1169
a(x=)844 1139 y Fm(p)p 878 1139 24 2 v 878 1169 a Fj(x)p
1003 1184 2 50 v 1100 1139 a Fm(p)p 1134 1139 24 2 v
1134 1169 a Fj(x)p 1253 1184 2 50 v 695 1234 V 727 1219
a Fo(sin\()p Fj(x)p Fo(\))p Fj(=)d Fo(cos\()p Fj(x)p
Fo(\))p 1003 1234 V 98 w(tan\()p Fj(x)p Fo(\))p 1253
1234 V 695 1284 V 725 1269 a(sin\()p Fj(x)p Fo(\))p Fj(=)g
Fo(tan)o(\()p Fj(x)p Fo(\))p 1003 1284 V 98 w(cos\()p
Fj(x)p Fo(\))p 1253 1284 V 695 1333 V 721 1319 a(cos)q(\()p
Fj(x)p Fo(\))i Fm(\001)g Fo(tan\()p Fj(x)p Fo(\))p 1003
1333 V 96 w(sin\()p Fj(x)p Fo(\))p 1253 1333 V 695 1383
V 725 1368 a(tan\()p Fj(x)p Fo(\))p Fj(=)e Fo(sin)o(\()p
Fj(x)p Fo(\))p 1003 1383 V 74 w(1)p Fj(=)g Fo(cos\()p
Fj(x)p Fo(\))p 1253 1383 V 696 1385 558 2 v 816 1483
a Fp(Absolute)13 b(V)l(alue)p 717 1500 516 2 v 716 1550
2 50 v 742 1535 a(expression)p 982 1550 V 47 w(simpli\014ed)p
1232 1550 V 717 1551 516 2 v 716 1601 2 50 v 746 1586
a Fo(abs\(abs\()p Fj(x)p Fo(\)\))p 982 1601 V 96 w(abs\()p
Fj(x)p Fo(\))p 1232 1601 V 716 1651 V 759 1636 a(abs\(re)q(\()p
Fj(x)p Fo(\)\))p 982 1651 V 121 w(re\()p Fj(x)p Fo(\))p
1232 1651 V 716 1701 V 753 1686 a(abs\(im)n(\()p Fj(x)p
Fo(\)\))p 982 1701 V 110 w(im)n(\()p Fj(x)p Fo(\))p 1232
1701 V 716 1751 V 791 1736 a(abs)q(\()p Fj(\031)q Fo(\))p
982 1751 V 187 w Fj(\031)p 1232 1751 V 716 1801 V 794
1786 a Fo(abs\()p Fj(e)p Fo(\))p 982 1801 V 193 w Fj(e)p
1232 1801 V 716 1850 V 797 1835 a Fo(abs\()p Fj(i)p Fo(\))p
982 1850 V 195 w(1)p 1232 1850 V 716 1900 V 776 1885
a(abs\()p Fm(\000)p Fj(x)p Fo(\))p 982 1900 V 126 w(abs\()p
Fj(x)p Fo(\))p 1232 1900 V 717 1902 516 2 v 874 2000
a Fp(Real)i(P)o(art)p 702 2017 547 2 v 701 2067 2 50
v 727 2052 a(expression)p 967 2067 V 62 w(simpli\014ed)p
1247 2067 V 702 2068 547 2 v 701 2118 2 50 v 756 2103
a Fo(re)q(\(re)q(\()p Fj(x)p Fo(\)\))p 967 2118 V 149
w(re\()p Fj(x)p Fo(\))p 1247 2118 V 701 2168 V 750 2153
a(re)q(\(im)n(\()p Fj(x)p Fo(\)\))p 967 2168 V 138 w(im)n(\()p
Fj(x)p Fo(\))p 1247 2168 V 701 2218 V 736 2203 a(re)q(\(conj\()p
Fj(x)p Fo(\)\))p 967 2218 V 129 w(re\()p Fj(x)p Fo(\))p
1247 2218 V 701 2268 V 743 2253 a(re)q(\(abs\()p Fj(x)p
Fo(\)\))p 967 2268 V 124 w(abs\()p Fj(x)p Fo(\))p 1247
2268 V 701 2317 V 789 2302 a(re)q(\()p Fj(\031)q Fo(\))p
967 2317 V 215 w Fj(\031)p 1247 2317 V 701 2367 V 792
2352 a Fo(re\()p Fj(e)p Fo(\))p 967 2367 V 221 w Fj(e)p
1247 2367 V 701 2417 V 794 2402 a Fo(re)q(\()p Fj(i)p
Fo(\))p 967 2417 V 223 w(0)p 1247 2417 V 701 2467 V 773
2452 a(re)q(\()p Fm(\000)p Fj(x)p Fo(\))p 967 2467 V
151 w Fm(\000)p Fo(re)q(\()p Fj(x)p Fo(\))p 1247 2467
V 701 2517 V 753 2502 a(re)q(\()p Fj(x)9 b Fo(+)h Fj(y)q
Fo(\))p 967 2517 V 77 w(re)q(\()p Fj(x)p Fo(\))f(+)g(re)q(\()p
Fj(y)q Fo(\))p 1247 2517 V 701 2566 V 753 2551 a(re)q(\()p
Fj(x)g Fm(\000)h Fj(y)q Fo(\))p 967 2566 V 77 w(re)q(\()p
Fj(x)p Fo(\))f Fm(\000)g Fo(re)q(\()p Fj(y)q Fo(\))p
1247 2566 V 702 2568 547 2 v 954 2700 a(24)p eop
%%Page: 25 26
25 25 bop 814 188 a Fp(Imaginary)15 b(P)o(art)p 690 205
570 2 v 689 255 2 50 v 715 240 a(expression)p 956 255
V 74 w(simpli\014ed)p 1259 255 V 690 257 570 2 v 689
306 2 50 v 733 291 a Fo(im)n(\(im)o(\()p Fj(x)p Fo(\)\))p
956 306 V 184 w(0)p 1259 306 V 689 356 V 739 341 a(im)n(\(re)q(\()p
Fj(x)p Fo(\)\))p 956 356 V 190 w(0)p 1259 356 V 689 406
V 719 391 a(im)n(\(conj\()p Fj(x)p Fo(\)\))p 956 406
V 113 w Fm(\000)p Fo(im)n(\()p Fj(x)p Fo(\))p 1259 406
V 689 456 V 726 441 a(im)n(\(abs)q(\()p Fj(x)p Fo(\)\))p
956 456 V 177 w(0)p 1259 456 V 689 506 V 772 491 a(im)n(\()p
Fj(\031)q Fo(\))p 956 506 V 223 w(0)p 1259 506 V 689
555 V 775 540 a(im)n(\()p Fj(e)p Fo(\))p 956 555 V 226
w(0)p 1259 555 V 689 605 V 777 590 a(im)n(\()p Fj(i)p
Fo(\))p 956 605 V 229 w(1)p 1259 605 V 689 655 V 756
640 a(im)n(\()p Fm(\000)p Fj(x)p Fo(\))p 956 655 V 151
w Fm(\000)p Fo(im)n(\()p Fj(x)p Fo(\))p 1259 655 V 689
705 V 736 690 a(im)n(\()p Fj(x)9 b Fo(+)h Fj(y)q Fo(\))p
956 705 V 72 w(im)n(\()p Fj(x)p Fo(\))f(+)g(im)o(\()p
Fj(y)q Fo(\))p 1259 705 V 689 755 V 736 740 a(im)n(\()p
Fj(x)g Fm(\000)h Fj(y)q Fo(\))p 956 755 V 72 w(im)n(\()p
Fj(x)p Fo(\))f Fm(\000)g Fo(im)o(\()p Fj(y)q Fo(\))p
1259 755 V 690 756 570 2 v 868 855 a Fp(Conjugate)p 651
871 649 2 v 650 921 2 50 v 686 906 a(expression)p 938
921 V 114 w(simpli\014e)o(d)p 1298 921 V 651 923 649
2 v 650 973 2 50 v 676 958 a Fo(conj\(conj\()p Fj(x)p
Fo(\)\))p 938 973 V 193 w Fj(x)p 1298 973 V 650 1022
V 696 1007 a Fo(conj\(re)q(\()p Fj(x)p Fo(\)\))p 938
1022 V 180 w(re\()p Fj(x)p Fo(\))p 1298 1022 V 650 1072
V 690 1057 a(conj\(im)n(\()p Fj(x)p Fo(\)\))p 938 1072
V 169 w(im)n(\()p Fj(x)p Fo(\))p 1298 1072 V 650 1122
V 683 1107 a(conj\(abs\()p Fj(x)p Fo(\)\))p 938 1122
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w(2)h Fm(g)p 770 1507 2 50 v 269 1557 V 331 1542 a Fo('sym)o(b')49
b Fm(!)42 b Fo('-sym)o(b')p 770 1557 V 269 1606 V 353
1591 a(#z1)73 b Fm(!)e Fo(#z2)p 770 1606 V 270 1608 501
2 v 137 1671 a Fm(\017)21 b Fn(AINV)13 b Fo(-)g(Algebraic)h(in)o(v)o
(erse)h(command)p 270 1715 879 2 v 269 1765 2 50 v 362
1750 a Fp(Lev)o(el)g(1)81 b Fm(!)125 b Fp(Lev)o(el)14
b(1)p 1148 1765 V 270 1766 879 2 v 269 1816 2 50 v 295
1801 a Fm(ff)f Fo(sq-matrix)g Fm(gg)g(!)g(ff)h Fo(\(sq-matrix\))p
Fn(^)p Fo(-1)d Fm(gg)p 1148 1816 V 269 1866 V 378 1851
a Fo('sym)o(b')95 b Fm(!)86 b Fo('INV\(sym)o(b\)')121
b(\()p Fm(\003)p Fo(\))p 1148 1866 V 269 1916 V 426 1901
a(x)145 b Fm(!)156 b Fo('1/x')192 b(\()p Fm(\003)p Fo(\))p
1148 1916 V 270 1917 879 2 v 137 1980 a Fm(\017)21 b
Fn(MDET)13 b Fo(-)g(Sym)o(b)q(olic)f(matrix)g(determinan)o(t)p
270 2024 690 2 v 269 2074 2 50 v 362 2059 a Fp(Lev)o(el)j(1)81
b Fm(!)h Fp(Lev)o(el)14 b(1)p 959 2074 V 270 2076 690
2 v 269 2125 2 50 v 295 2110 a Fm(ff)f Fo(sq-matrix)g
Fm(gg)g(!)g Fo('det\(sq-matrix\)')p 959 2125 V 270 2127
690 2 v 137 2190 a Fm(\017)21 b Fn(MLU)13 b Fo(-)h(Sym)o(b)q(olic)d
(matrix)h(LU)i(decomp)q(osition)p 270 2234 958 2 v 269
2284 2 50 v 380 2269 a Fp(Lev)o(el)h(1)99 b Fm(!)f Fp(Lev)o(el)15
b(2)134 b(Lev)o(el)15 b(1)p 1227 2284 V 270 2285 958
2 v 269 2335 2 50 v 295 2320 a Fm(ff)e Fo(sq-matrix)p
527 2320 13 2 v 14 w(1)g Fm(gg)h(!)f(ff)g Fo(sq-matrix)p
916 2320 V 13 w(2)h Fm(gg)112 b Fj(n)p 1227 2335 2 50
v 270 2337 958 2 v 137 2400 a Fm(\017)21 b Fn(MTRN)13
b Fo(-)g(Sym)o(b)q(olic)f(matrix)g(transp)q(ose)p 270
2443 655 2 v 269 2493 2 50 v 354 2478 a Fp(Lev)o(el)j(1)72
b Fm(!)h Fp(Lev)o(el)14 b(1)p 923 2493 V 270 2495 655
2 v 269 2545 2 50 v 295 2530 a Fm(ff)f Fo(matrix)p 475
2530 13 2 v 13 w(1)h Fm(gg)f(!)h(ff)f Fo(matrix)p 812
2530 V 13 w(2)h Fm(gg)p 923 2545 2 50 v 269 2595 V 339
2580 a(f)f Fo(v)o(ector)i Fm(g)57 b(!)32 b(ff)13 b Fo(matrix)f
Fm(gg)p 923 2595 V 270 2596 655 2 v 954 2700 a Fo(28)p
eop
%%Page: 29 30
29 29 bop 137 195 a Fm(\017)21 b Fn(MIDN)13 b Fo(-)g(Sym)o(b)q(olic)f
(iden)o(tit)o(y)i(matrix)p 270 240 801 2 v 269 290 2
50 v 295 275 a Fp(Lev)o(el)h(1)e Fm(!)205 b Fp(Lev)o(el)14
b(1)p 1070 290 V 270 292 801 2 v 269 342 2 50 v 358 327
a Fo(n)77 b Fm(!)14 b(ff)f Fo(\()p Fj(n)c Fm(\002)h Fj(n)p
Fo(\))k(iden)o(tit)o(y-matrix)d Fm(gg)p 1070 342 V 270
343 801 2 v 137 408 a(\017)21 b Fn(Z<->S)13 b Fo(or)g
Fn(ZS)h Fo(-)g(Con)o(v)o(ersion)f(from)f(unlimited)g(precision)i(in)o
(teger)h(to)e(string)p 270 453 418 2 v 269 503 2 50 v
295 488 a Fp(Lev)o(el)i(1)e Fm(!)h Fp(Lev)o(el)g(1)p
687 503 V 270 504 418 2 v 269 554 2 50 v 343 539 a Fo(#z)62
b Fm(!)c Fo("z")p 687 554 V 269 604 V 340 589 a("z")g
Fm(!)k Fo(#z)p 687 604 V 269 654 V 360 639 a(z)80 b Fm(!)62
b Fo(#z)p 687 654 V 270 656 418 2 v 137 720 a Fm(\017)21
b Fn(GCD)13 b Fo(-)h(Greatest)h(Commo)o(n)c(Divisor)i(command)p
270 757 758 2 v 269 807 2 50 v 308 792 a Fp(Lev)o(el)h(2)106
b(Lev)o(el)15 b(1)56 b Fm(!)28 b Fp(Lev)o(el)15 b(1)p
1027 807 V 270 809 758 2 v 269 858 2 50 v 314 843 a Fo('p)q(oly)p
407 843 13 2 v 14 w(1')117 b('p)q(oly)p 662 843 V 14
w(2')62 b Fm(!)14 b Fo('p)q(oly)p 918 843 V 13 w(gcd')p
1027 858 2 50 v 269 908 V 332 893 a('p)q(oly')192 b(x)120
b Fm(!)94 b Fo(z)p 1027 908 V 269 958 V 372 943 a(x)193
b('p)q(oly')79 b Fm(!)94 b Fo(z)p 1027 958 V 269 1008
V 363 993 a(z1)216 b(z2)112 b Fm(!)83 b Fo(z3)p 1027
1008 V 269 1058 V 363 1043 a(z1)148 b(#z2/"z2")43 b Fm(!)66
b Fo(#z3)p 1027 1058 V 269 1107 V 295 1093 a(#z1/"z1")49
b(z2/#z2/"z2")13 b Fm(!)66 b Fo(#z3)p 1027 1107 V 270
1109 758 2 v 137 1174 a Fm(\017)21 b Fn(LCM)13 b Fo(-)h(Least)g(Common)
d(Multiple)i(command)p 270 1219 761 2 v 269 1269 2 50
v 308 1254 a Fp(Lev)o(el)h(2)106 b(Lev)o(el)15 b(1)56
b Fm(!)29 b Fp(Lev)o(el)15 b(1)p 1030 1269 V 270 1270
761 2 v 269 1320 2 50 v 314 1305 a Fo('p)q(oly)p 407
1305 13 2 v 14 w(1')117 b('p)q(oly)p 662 1305 V 14 w(2')62
b Fm(!)14 b Fo('p)q(oly)p 918 1305 V 13 w(lcm')p 1030
1320 2 50 v 269 1370 V 332 1355 a('p)q(oly')192 b(x)120
b Fm(!)14 b Fo('p)q(oly)p 918 1355 13 2 v 13 w(lcm')p
1030 1370 2 50 v 269 1420 V 372 1405 a(x)193 b('p)q(oly')79
b Fm(!)14 b Fo('p)q(oly)p 918 1405 13 2 v 13 w(lcm')p
1030 1420 2 50 v 269 1470 V 363 1455 a(z1)216 b(z2)112
b Fm(!)84 b Fo(z3)p 1030 1470 V 269 1519 V 363 1504 a(z1)148
b(#z2/"z2")43 b Fm(!)67 b Fo(#z3)p 1030 1519 V 269 1569
V 295 1554 a(#z1/"z1")49 b(z2/#z2/"z2")13 b Fm(!)67 b
Fo(#z3)p 1030 1569 V 270 1571 761 2 v 137 1635 a Fm(\017)21
b Fn(RAT->)13 b Fo(-)g(Rational)f(to)i(stac)o(k)g(command)p
270 1672 958 2 v 269 1722 2 50 v 384 1707 a Fp(Lev)o(el)g(1)102
b Fm(!)45 b Fp(Lev)o(el)14 b(2)132 b(Lev)o(el)14 b(1)p
1227 1722 V 270 1724 958 2 v 269 1774 2 50 v 295 1759
a Fo('rational)e(function')h Fm(!)h Fo('n)o(umerator')48
b('denominator')p 1227 1774 V 269 1823 V 447 1809 a(x)166
b Fm(!)108 b Fo(x)244 b(1.0)p 1227 1823 V 270 1825 958
2 v 137 1890 a Fm(\017)21 b Fn(ASIM)13 b Fo({)g(Algebraic)h
(simpli\014cation)d(command)p 270 1935 501 2 v 269 1985
2 50 v 316 1970 a Fp(Lev)o(el)j(1)35 b Fm(!)f Fp(Lev)o(el)14
b(1)p 770 1985 V 270 1986 501 2 v 269 2036 2 50 v 314
2021 a Fo('sym)o(b)p 424 2021 13 2 v 13 w(1')32 b Fm(!)f
Fo('sym)o(b)p 683 2021 V 14 w(2')p 770 2036 2 50 v 269
2086 V 295 2071 a Fm(f)14 b Fo(s-list)p 418 2071 13 2
v 14 w(1)g Fm(g)g(!)f(f)g Fo(s-list)p 677 2071 V 15 w(2)h
Fm(g)p 770 2086 2 50 v 269 2136 V 342 2121 a Fo(\()p
Fj(x;)7 b(y)q Fo(\))62 b Fm(!)583 2106 y Fg(0)595 2121
y Fj(x)9 b Fo(+)g Fj(y)q(i)704 2106 y Fg(0)p 770 2136
V 269 2186 V 380 2171 a Fj(z)101 b Fm(!)d Fj(z)p 770
2186 V 270 2187 501 2 v 137 2252 a Fm(\017)21 b Fn(MOD+)13
b Fo({)g(Mo)q(dular)h(addition)p 270 2289 1026 2 v 269
2339 2 50 v 338 2324 a Fp(Lev)o(el)g(3)136 b(Lev)o(el)14
b(2)112 b(Lev)o(el)14 b(1)33 b Fm(!)13 b Fp(Lev)o(el)i(1)p
1295 2339 V 270 2340 1026 2 v 269 2390 2 50 v 295 2375
a Fo(z1/#z1/"z1")49 b(z2/#z2/"z2")g(n/#n/"n")13 b Fm(!)51
b Fo(#z3)p 1295 2390 V 270 2392 1026 2 v 137 2456 a Fm(\017)21
b Fn(MOD-)13 b Fo({)g(Mo)q(dular)h(substraction)p 270
2493 V 269 2543 2 50 v 338 2528 a Fp(Lev)o(el)g(3)136
b(Lev)o(el)14 b(2)112 b(Lev)o(el)14 b(1)33 b Fm(!)13
b Fp(Lev)o(el)i(1)p 1295 2543 V 270 2545 1026 2 v 269
2595 2 50 v 295 2580 a Fo(z1/#z1/"z1")49 b(z2/#z2/"z2")g(n/#n/"n")13
b Fm(!)51 b Fo(#z3)p 1295 2595 V 270 2596 1026 2 v 954
2700 a(29)p eop
%%Page: 30 31
30 30 bop 137 195 a Fm(\017)21 b Fn(MOD*)13 b Fo({)g(Mo)q(dular)h(m)o
(ultiplicatio)o(n)p 270 241 1026 2 v 269 290 2 50 v 338
276 a Fp(Lev)o(el)g(3)136 b(Lev)o(el)14 b(2)112 b(Lev)o(el)14
b(1)33 b Fm(!)13 b Fp(Lev)o(el)i(1)p 1295 290 V 270 292
1026 2 v 269 342 2 50 v 295 327 a Fo(z1/#z1/"z1")49 b(z2/#z2/"z2")g
(n/#n/"n")13 b Fm(!)51 b Fo(#z3)p 1295 342 V 270 344
1026 2 v 137 408 a Fm(\017)21 b Fn(MOD/)13 b Fo({)g(Mo)q(dular)h
(division)p 270 449 V 269 499 2 50 v 338 484 a Fp(Lev)o(el)g(3)136
b(Lev)o(el)14 b(2)112 b(Lev)o(el)14 b(1)33 b Fm(!)13
b Fp(Lev)o(el)i(1)p 1295 499 V 270 501 1026 2 v 269 550
2 50 v 295 535 a Fo(z1/#z1/"z1")49 b(z2/#z2/"z2")g(n/#n/"n")13
b Fm(!)51 b Fo(#z3)p 1295 550 V 270 552 1026 2 v 137
617 a Fm(\017)21 b Fn(MODPOW)12 b Fo({)i(Mo)q(dular)f(exp)q(onen)o
(tiation)p 270 662 V 269 712 2 50 v 338 697 a Fp(Lev)o(el)h(3)136
b(Lev)o(el)14 b(2)112 b(Lev)o(el)14 b(1)33 b Fm(!)13
b Fp(Lev)o(el)i(1)p 1295 712 V 270 714 1026 2 v 269 764
2 50 v 295 749 a Fo(z1/#z1/"z1")49 b(z2/#z2/"z2")g(n/#n/"n")13
b Fm(!)51 b Fo(#z3)p 1295 764 V 270 765 1026 2 v 137
830 a Fm(\017)21 b Fn(MODINV)12 b Fo({)i(In)o(v)o(erse)h(mo)q(dulo)d
Fj(N)p 270 867 741 2 v 269 917 2 50 v 338 902 a Fp(Lev)o(el)i(2)112
b(Lev)o(el)14 b(1)33 b Fm(!)13 b Fp(Lev)o(el)i(1)p 1009
917 V 270 919 741 2 v 269 969 2 50 v 295 954 a Fo(z1/#z1/"z1")49
b(n/#n/"n")13 b Fm(!)51 b Fo(#z2)p 1009 969 V 270 970
741 2 v 137 1034 a Fm(\017)21 b Fn(PRIM?)13 b Fo({)g(Prime)g(testing)p
270 1080 442 2 v 269 1130 2 50 v 307 1115 a Fp(Lev)o(el)h(1)26
b Fm(!)13 b Fp(Lev)o(el)i(1)p 711 1130 V 270 1131 442
2 v 269 1181 2 50 v 295 1166 a Fo(z/#z/"z")f Fm(!)57
b Fo(0/1)p 711 1181 V 270 1183 442 2 v 137 1247 a Fm(\017)21
b Fn(PRIM+)13 b Fo({)g(Next)i(prime)p 270 1293 V 269
1342 2 50 v 307 1327 a Fp(Lev)o(el)f(1)26 b Fm(!)13 b
Fp(Lev)o(el)i(1)p 711 1342 V 270 1344 442 2 v 269 1394
2 50 v 295 1379 a Fo(z/#z/"z")f Fm(!)59 b Fo(#n)p 711
1394 V 270 1396 442 2 v 137 1460 a Fm(\017)21 b Fn(PRIM-)13
b Fo({)g(Previous)i(prime)p 270 1505 V 269 1555 2 50
v 307 1540 a Fp(Lev)o(el)f(1)26 b Fm(!)13 b Fp(Lev)o(el)i(1)p
711 1555 V 270 1557 442 2 v 269 1607 2 50 v 295 1592
a Fo(z/#z/"z")f Fm(!)59 b Fo(#n)p 711 1607 V 270 1608
442 2 v 137 1673 a Fm(\017)21 b Fn(RORD)13 b Fo({)g(Reorder)i(p)q
(olynomial)p 270 1718 696 2 v 269 1768 2 50 v 295 1753
a Fp(Lev)o(el)g(2)89 b(Lev)o(el)14 b(1)53 b Fm(!)14 b
Fp(Lev)o(el)g(1)p 965 1768 V 270 1770 696 2 v 269 1820
2 50 v 301 1805 a Fo('p)q(oly)p 394 1805 13 2 v 14 w(1')147
b('x')104 b Fm(!)20 b Fo('p)q(oly)p 891 1805 V 14 w(2')p
965 1820 2 50 v 269 1869 V 359 1855 a Fj(z)208 b Fo('x')104
b Fm(!)78 b Fj(z)p 965 1869 V 269 1919 V 301 1904 a Fo('p)q(oly)p
394 1904 13 2 v 14 w(1')55 b Fm(f)14 b Fo('x')f('y')f('z')i
Fm(g)f(!)20 b Fo('p)q(oly)p 891 1904 V 14 w(2')p 965
1919 2 50 v 269 1969 V 359 1954 a Fj(z)116 b Fm(f)14
b Fo('x')f('y')f('z')i Fm(g)f(!)78 b Fj(z)p 965 1969
V 270 1971 696 2 v 137 2035 a Fm(\017)21 b Fn(PF)13 b
Fo({)h(P)o(artial)f(fraction)g(expansion)p 270 2081 732
2 v 269 2131 2 50 v 297 2116 a Fp(Lev)o(el)i(2)105 b(Lev)o(el)14
b(1)67 b Fm(!)16 b Fp(Lev)o(el)e(1)p 1001 2131 V 270
2132 732 2 v 269 2182 2 50 v 295 2167 a Fo('sym)o(b)p
405 2167 13 2 v 13 w(1')49 b Fm(f)28 b Fo('x')13 b('y')f('z')28
b Fm(g)13 b(!)h Fo('sym)o(b)p 934 2167 V 13 w(2')p 1001
2182 2 50 v 269 2232 V 361 2217 a Fj(z)118 b Fm(f)28
b Fo('x')13 b('y')f('z')28 b Fm(g)13 b(!)80 b Fj(z)p
1001 2232 V 269 2282 V 550 2267 a Fo('sym)o(b)p 660 2267
13 2 v 13 w(1')64 b Fm(!)14 b Fo('sym)o(b)p 934 2267
V 13 w(2')p 1001 2282 2 50 v 269 2332 V 616 2317 a Fj(z)133
b Fm(!)80 b Fj(z)p 1001 2332 V 270 2333 732 2 v 137 2398
a Fm(\017)21 b Fn(RINT)13 b Fo({)g(Rational)f(function)i(in)o
(tegration)p 270 2443 626 2 v 269 2493 2 50 v 297 2478
a Fp(Lev)o(el)h(2)51 b(Lev)o(el)15 b(1)f Fm(!)h Fp(Lev)o(el)g(1)p
895 2493 V 270 2495 626 2 v 269 2545 2 50 v 295 2530
a Fo('sym)o(b)p 405 2530 13 2 v 13 w(1')102 b('x')65
b Fm(!)13 b Fo('sym)o(b)p 827 2530 V 13 w(2')p 895 2545
2 50 v 269 2595 V 361 2580 a Fj(z)171 b Fo('x')65 b Fm(!)48
b Fo('z*x')p 895 2595 V 270 2596 626 2 v 954 2700 a(30)p
eop
%%Page: 31 32
31 31 bop 137 195 a Fm(\017)21 b Fn(GBASIS)12 b Fo({)i(Gr\177)-21
b(obner)14 b(basis)g(of)f(a)h(system)g(of)f(p)q(olynomial)d(equations)p
270 242 912 2 v 269 292 2 50 v 346 277 a Fp(Lev)o(el)15
b(2)140 b(Lev)o(el)14 b(1)53 b Fm(!)71 b Fp(Lev)o(el)14
b(1)p 1181 292 V 270 294 912 2 v 269 346 2 52 v 295 331
a Fm(f)g Fo('eq1')f('eq2')g Fm(g)49 b(f)14 b Fo('x')f('y')f('z')i
Fm(g)f(!)h(f)f Fo('eq1')g('eq2')g Fm(g)1145 310 y Fg(0)p
1181 346 V 270 347 912 2 v 137 413 a Fm(\017)21 b Fn(GSOLVE)12
b Fo({)i(Solutions)f(of)g(a)h(system)g(of)f(p)q(olynomial)d(equations)p
270 461 1448 2 v 269 510 2 50 v 346 495 a Fp(Lev)o(el)15
b(2)140 b(Lev)o(el)14 b(1)53 b Fm(!)40 b Fp(Lev)o(el)14
b Fj(n)9 b Fo(+)h(1)26 b(.)6 b(.)h(.)66 b Fp(Lev)o(el)15
b(2)110 b(Lev)o(el)14 b(1)p 1716 510 V 270 512 1448 2
v 269 562 2 50 v 295 547 a Fm(f)g Fo('eq1')f('eq2')g
Fm(g)49 b(f)14 b Fo('x')f('y')f('z')i Fm(g)f(!)h(f)f
Fo('eq1')g('eq2')g Fm(g)1145 559 y Ff(n)1168 547 y Fo(.)6
b(.)h(.)f Fm(f)14 b Fo('eq1')f('eq2')g Fm(g)1474 559
y Fl(1)1606 547 y Fo(n)p 1716 562 V 270 564 1448 2 v
137 630 a Fm(\017)21 b Fn(GSIMP)13 b Fo({)g(Reduction)h(of)f(an)h
(expression)h(giv)o(en)e(a)h(system)g(of)f(side)h(relations)p
270 677 998 2 v 269 727 2 50 v 295 712 a Fp(Lev)o(el)h(3)100
b(Lev)o(el)15 b(2)140 b(Lev)o(el)14 b(1)54 b Fm(!)13
b Fp(Lev)o(el)h(1)p 1266 727 V 270 728 998 2 v 269 778
2 50 v 300 763 a Fo('sym)o(b1')53 b Fm(f)14 b Fo('eq1')f('eq2')g
Fm(g)49 b(f)14 b Fo('x')f('y')g('z')g Fm(g)h(!)k Fo('sym)o(b2')p
1266 778 V 270 780 998 2 v 954 2700 a(31)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF