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%%BeginProcSet: texnansi.enc
% @psencodingfile{
%   author = "Y&Y, Inc.",
%   version = "1.1",
%   date = "1 December 1996",
%   filename = "texnansi.enc",
%   email = "help@YandY.com",
%   address = "45 Walden Street // Concord, MA 01742, USA",
%   codetable = "ISO/ASCII",
%   checksum = "xx",
%   docstring = "Encoding for fonts in Adobe Type 1 format for use with TeX."
% }
%
% The idea is to have all 228 characters normally included in Type 1 text
% fonts (plus a few more) available for typesetting.  This is effectively
% the character set in Adobe Standard Encoding, ISO Latin 1, plus a few more.
%
% Character code assignments were made as follows:
%
% (1) The character layout largely matches `ASCII' in the 32 -- 126 range,
% except for `circumflex' in 94 and `tilde' in 126, to match `TeX text'
% (`asciicircumflex' and `asciitilde' appear in 158 and 142 instead).
%
% (2) The character layout matches `Windows ANSI' in almost all places,
% except for `quoteright' in 39 and `quoteleft' in 96 to match ASCII
% (`quotesingle' and `grave' appear in 129 and 18 instead).
%
% (3) The character layout matches `TeX typewriter' used by CM text fonts
% in most places (except for discordant positions such as hungarumlaut
% (instead of braceright), dotaccent (instead of underscore) etc.
%
% (4) Remaining characters are assigned arbitrarily to the `control character'
% range (0 -- 31), avoiding 0, 9, 10 and 13 in case we meet dumb software
% - similarly one should really avoid 127 and 128 if possible.
% In addition, the 8 open slots in Windows ANSI between 128 and 159 are used.
%
% (5) Y&Y Lucida Bright includes some extra ligatures and such; ff, ffi, ffl,
% and `dotlessj,' these are included 11 -- 15, and 17.
%
% (6) Hyphen appears both at 45 and 173 for compatibility with both ASCII
% and Windows ANSI.
%
% (7) It doesn't really matter where ligatures appear (both real, such as ffi,
% and pseudo such as ---) since these should not be accessed directly, only
% via ligature information in the TFM file.
%
% SAMPLE USAGE (in `psfonts.map' file for DVIPS):
% 
% lbr LucidaBright "TeXnANSIEncoding ReEncodeFont" <texnansi.enc <lbr.pfb
%
% This tells DVIPS that the font called `lbr' in TeX has PostScript 
% FontName `LucidaBright.'  It also asks DVIPS to expand the file `lbr.pfb'
% into PFA form, to include the attached `texnansi.enc' encoding vector,
% and to then actually reencode the font based on that encoding vector.
%
% Revised 1996 June 1 by adding second position for `fl' to avoid Acrobat bug.
% Revised 1996 June 1 by adding second position for `fraction' for same reason.
%
/TeXnANSIEncoding [
/.notdef /uni20AC /.notdef /.notdef % 0, 1, 2, 3
/fraction %	4
/dotaccent %	5
/hungarumlaut %	6
/ogonek	%	7
/fl	%	8
/.notdef % /fraction %	9	not used (see 4), backward compatability only
/cwm	%	10	not used, except boundary char internally maybe
/ff    %	11
/fi    %	12
/.notdef % /fl    %	13	not used (see 8), backward compatability only
/ffi   %	14
/ffl   %	15
/dotlessi %	16
/dotlessj %	17
/grave %	18
/acute %	19
/caron %	20
/breve %	21
/macron %	22
/ring  %	23
/cedilla %	24
/germandbls %	25
/ae    %	26
/oe    %	27
/oslash %	28
/AE    %	29
/OE    %	30
/Oslash %	31
/space %	32	% /suppress in TeX text
/exclam %	33
/quotedbl %	34	% /quotedblright in TeX text
/numbersign %	35
/dollar %	36
/percent %	37
/ampersand %	38
/quoteright %	39	% /quotesingle in ANSI
/parenleft %	40
/parenright %	41
/asterisk %	42
/plus  %	43
/comma %	44
/hyphen %	45
/period %	46
/slash %	47
/zero  %	48
/one   %	49
/two   %	50
/three %	51
/four  %	52
/five  %	53
/six   %	54
/seven %	55
/eight %	56
/nine  %	57
/colon %	58
/semicolon %	59
/less  %	60	% /exclamdown in Tex text
/equal %	61
/greater %	62	% /questiondown in TeX text
/question %	63
/at %	64
/A %	65
/B %	66
/C %	67
/D %	68
/E %	69
/F %	70
/G %	71
/H %	72
/I %	73
/J %	74
/K %	75
/L %	76
/M %	77
/N %	78
/O %	79
/P %	80
/Q %	81
/R %	82
/S %	83
/T %	84
/U %	85
/V %	86
/W %	87
/X %	88
/Y %	89
/Z %	90
/bracketleft %	91
/backslash %	92	% /quotedblleft in TeX text
/bracketright %	93
/circumflex %	94	% /asciicircum in ASCII
/underscore %	95	% /dotaccent in TeX text
/quoteleft %	96	% /grave accent in ANSI
/a %	97
/b %	98
/c %	99
/d %	100
/e %	101
/f %	102
/g %	103
/h %	104
/i %	105
/j %	106
/k %	107
/l %	108
/m %	109
/n %	110
/o %	111
/p %	112
/q %	113
/r %	114
/s %	115
/t %	116
/u %	117
/v %	118
/w %	119
/x %	120
/y %	121
/z %	122
/braceleft %	123	% /endash in TeX text
/bar   %	124	% /emdash in TeX test
/braceright %	125	% /hungarumlaut in TeX text
/tilde %	126	% /asciitilde in ASCII
/dieresis %	127	not used (see 168), use higher up instead
/Lslash	%	128	this position is unfortunate, but now too late to fix
/quotesingle %	129
/quotesinglbase %	130
/florin %	131
/quotedblbase %	132
/ellipsis %	133
/dagger %	134
/daggerdbl %	135
/circumflex %	136
/perthousand %	137
/Scaron %	138
/guilsinglleft %	139
/OE    %	140
/Zcaron %	141
/asciicircum %	142
/minus %	143
/lslash %	144
/quoteleft %	145
/quoteright %	146
/quotedblleft %	147
/quotedblright %	148
/bullet %	149
/endash %	150
/emdash %	151
/tilde %	152
/trademark %	153
/scaron %	154
/guilsinglright %	155
/oe    %	156
/zcaron %	157
/asciitilde %	158
/Ydieresis %	159
/nbspace %	160	% /space (no break space)
/exclamdown %	161
/cent  %	162
/sterling %	163
/currency %	164
/yen   %	165
/brokenbar %	166
/section %	167
/dieresis %	168
/copyright %	169
/ordfeminine %	170
/guillemotleft %	171
/logicalnot %	172
/sfthyphen %	173 % /hyphen (hanging hyphen)
/registered %	174
/macron %	175
/degree %	176
/plusminus %	177
/twosuperior %	178
/threesuperior %	179
/acute %	180
/mu    %	181
/paragraph %	182
/periodcentered %	183
/cedilla %	184
/onesuperior %	185
/ordmasculine %	186
/guillemotright %	187
/onequarter %	188
/onehalf %	189
/threequarters %	190
/questiondown %	191
/Agrave %	192
/Aacute %	193
/Acircumflex %	194
/Atilde %	195
/Adieresis %	196
/Aring %	197
/AE    %	198
/Ccedilla %	199
/Egrave %	200
/Eacute %	201
/Ecircumflex %	202
/Edieresis %	203
/Igrave %	204
/Iacute %	205
/Icircumflex %	206
/Idieresis %	207
/Eth   %	208
/Ntilde %	209
/Ograve %	210
/Oacute %	211
/Ocircumflex %	212
/Otilde %	213
/Odieresis %	214
/multiply %	215	% OE in T1
/Oslash %	216
/Ugrave %	217
/Uacute %	218
/Ucircumflex %	219
/Udieresis %	220
/Yacute %	221
/Thorn %	222
/germandbls %	223 % SS in T1
/agrave %	224
/aacute %	225
/acircumflex %	226
/atilde %	227
/adieresis %	228
/aring %	229
/ae    %	230
/ccedilla %	231
/egrave %	232
/eacute %	233
/ecircumflex %	234
/edieresis %	235
/igrave %	236
/iacute %	237
/icircumflex %	238
/idieresis %	239
/eth   %	240
/ntilde %	241
/ograve %	242
/oacute %	243
/ocircumflex %	244
/otilde %	245
/odieresis %	246
/divide %	247	% oe in T1
/oslash %	248
/ugrave %	249
/uacute %	250
/ucircumflex %	251
/udieresis %	252
/yacute %	253
/thorn %	254
/ydieresis %	255	% germandbls in T1
] def

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3 3 bop Black Black Black Black 1241 274 a Fu(The)33
b(Univ)m(ersit)m(y)h(of)e(Southern)h(Mississippi)p Black
Black 625 781 a Ft(P)-11 b(ARALLEL)44 b(MONTE)g(CARLO)h(COMPUT)-11
b(A)g(TION)1111 1051 y(OF)44 b(INV)-15 b(ARIANT)44 b(MEASURES)p
Black Black 2019 1505 a Fu(b)m(y)p Black Black 1644 1734
a(Zizhong)31 b(John)i(W)-8 b(ang)p Black Black 1760 2310
a(A)33 b(Dissertation)1333 2435 y(Submitted)f(to)g(the)h(Graduate)f(Sc)
m(ho)s(ol)1185 2559 y(of)g(The)i(Univ)m(ersit)m(y)f(of)f(Southern)i
(Mississippi)1175 2684 y(in)e(P)m(artial)e(F)-8 b(ul\014llmen)m(t)30
b(of)i(the)h(Requiremen)m(ts)1241 2809 y(for)f(the)h(Degree)g(of)f(Do)s
(ctor)g(of)g(Philosoph)m(y)1986 3273 y(Appro)m(v)m(ed:)2695
3394 y Fs(Jiu)38 b(Ding)p 1986 3514 1854 4 v 1986 3642
a Fu(Director)p 1986 3809 V 1986 4048 V 1986 4286 V 1986
4525 V 1986 4763 V 1986 4891 a(Dean)32 b(of)g(the)i(Graduate)e(Sc)m(ho)
s(ol)p Black Black 1803 5387 a(August)h(2000)p Black
Black eop
%%Page: 2 5
2 4 bop Black Black Black Black 1112 150 a Fn(A)l(CKNO)l(WLEDGEMENTS)
479 851 y Fu(F)-8 b(or)40 b(their)g(assistance)h(in)f(the)h(course)g
(of)f(m)m(y)g(do)s(ctoral)f(study)-8 b(,)44 b(man)m(y)c(p)s(eople)h
(deserv)m(e)300 972 y(thanks.)65 b(Sp)s(ecial)38 b(thank)h(m)m(ust)h
(go)e(to)h(m)m(y)h(advisor)f(Jiu)f(Ding)g(for)h(his)g(ongoing)e(helps)j
(and)300 1092 y(supp)s(orts)32 b(in)e(m)m(y)h(progress)h(of)e(course)j
(study)-8 b(,)32 b(researc)m(h,)h(m)m(y)e(job)g(h)m(un)m(ting,)g(and)g
(other)h(\014elds.)300 1212 y(His)26 b(con)m(tin)m(ual)g(encouragemen)m
(t,)j(advise,)g(and)e(understanding)g(are)g(all)e(greatly)h
(appreciated.)446 1333 y(I)36 b(will)c(b)s(e)k(alw)m(a)m(ys)f(grateful)
f(to)h(James)g(Ca)m(v)m(en)m(y)-8 b(,)38 b(Ras)d(P)m(andey)-8
b(,)38 b(Marcin)c(P)m(aprzyc)m(ki,)k(and)300 1453 y(Ra)m(y)43
b(Seyfarth.)75 b(They)45 b(are)e(all)d(m)m(y)j(PhD)g(committee)e(mem)m
(b)s(ers)i(and)g(m)m(y)g(course)h(profes-)300 1574 y(sors,)31
b(and)f(they)i(ga)m(v)m(e)f(me)e(v)m(ery)j(helpful)d(directions)g(and)h
(commen)m(ts)g(during)g(the)g(prosp)s(ectus)300 1694
y(preparation)42 b(and)h(the)h(writing)d(of)i(m)m(y)g(dissertation.)74
b(In)43 b(particular)e(I)i(really)f(appreciate)300 1814
y(that)35 b(Marcin)g(P)m(aprzyc)m(ki)i(ga)m(v)m(e)f(me)e(abundan)m(t)i
(ideas)f(in)g(m)m(y)g(n)m(umerical)f(exp)s(erimen)m(ts)h(with)300
1935 y(parallel)c(computers.)47 b(Besides,)35 b(he)f(help)s(ed)g(me)f
(a)g(lot)g(for)g(job)g(h)m(un)m(ting.)47 b(Ra)m(y)34
b(Seyfarth)g(also)300 2055 y(merits)c(particular)g(thanks)i(for)f(his)g
(v)-5 b(alued)31 b(advise)h(in)f(m)m(y)g(parallel)e(computing)h(w)m
(ork.)44 b(I)32 b(ad-)300 2175 y(mire)g(James)h(Ca)m(v)m(en)m(y)j(for)d
(his)g(graceful)f(teac)m(hing)h(st)m(yle)h(and)f(resp)s(ectable)h(p)s
(ersonalit)m(y)-8 b(.)45 b(Ras)300 2296 y(P)m(andey's)35
b(profound)d(kno)m(wledge)i(and)f(sc)m(holarly)f(c)m(haracter)h
(impress)f(me)g(greatly)-8 b(.)446 2416 y(I)29 b(m)m(ust)g(thank)h
(Joseph)g(Kolibal)25 b(for)k(his)f(time-consuming)f(w)m(ork)j(on)e(the)
i(uniform)d(disser-)300 2537 y(tation)33 b(\(Latex\))i(pac)m(k)-5
b(age.)50 b(Indeed)36 b(it)d(sa)m(v)m(ed)j(me)f(a)f(lot)f(of)h(time)f
(in)h(t)m(yping)g(m)m(y)h(dissertation)300 2657 y(with)d(this)g(pac)m
(k)-5 b(age.)446 2777 y(I)23 b(thank)h(Charles)f(W)-8
b(righ)m(t)22 b(for)h(his)f(e\016cien)m(t)i(managemen)m(t)e(of)h
(Wiglaf)d(parallel)g(computers.)300 2898 y(He)30 b(is)f(a)g(w)m
(arm-hearted,)h(helpful,)f(and)g(friendly)g(p)s(erson)h(and)f(he)h(ga)m
(v)m(e)g(me)f(a)g(lot)f(of)h(supp)s(orts)300 3018 y(in)j(m)m(y)g
(parallel)e(computing)i(exp)s(erimen)m(ts.)446 3139 y(I)41
b(am)e(also)h(grateful)f(to)h(W)-8 b(allace)39 b(Py)m(e,)44
b(Je\013)d(Stuart)g(and)f(man)m(y)h(other)f(professors)i(and)300
3259 y(friends)k(whose)h(en)m(th)m(usiastic)f(helps)g(to)f(me)g(during)
g(m)m(y)h(do)s(ctoral)e(study)j(will)c(alw)m(a)m(ys)k(b)s(e)300
3379 y(memorized.)446 3500 y(Finally)-8 b(,)22 b(I)h(w)m(ould)f(lik)m
(e)g(to)g(dedicate)h(this)f(dissertation)g(to)g(m)m(y)g(grandparen)m
(ts)i(who)f(brough)m(t)300 3620 y(me)46 b(up,)j(to)d(m)m(y)g(wife)g
(Huiqing)f(Y)-8 b(ang)45 b(who)i(is)e(also)g(m)m(y)h(fello)m(w)f
(classmate)h(of)f(the)i(Ph.D.)300 3740 y(program)37 b(in)h(scien)m
(ti\014c)h(computing,)g(and)g(to)f(our)h(lo)m(v)m(ely)g(daugh)m(ter)g
(Angela)f(W)-8 b(ang,)40 b(for)e(all)300 3861 y(the)24
b(happiness)h(of)e(our)h(family)d(life)h(here)j(at)f(The)g(Univ)m
(ersit)m(y)h(of)e(Southern)i(Mississippi)e(during)300
3981 y(the)33 b(past)g(y)m(ears.)p Black 2043 5764 a(ii)p
Black eop
%%Page: 3 6
3 5 bop Black Black Black Black 1181 150 a Fn(T)-13 b(ABLE)53
b(OF)h(CONTENTS)300 851 y Fm(A)m(CKNO)m(WLEDGEMENTS)72
b Fr(:)56 b(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f
(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)158 b Fu(ii)304 1096 y
Fm(LIST)38 b(OF)g(ILLUSTRA)-9 b(TIONS)69 b Fr(:)55 b(:)h(:)f(:)h(:)f(:)
h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)152
b Fu(iv)305 1341 y Fm(LIST)38 b(OF)g(T)-9 b(ABLES)74
b Fr(:)55 b(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f
(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)159
b Fu(v)305 1586 y Fm(LIST)38 b(OF)g(ABBREVIA)-9 b(TIONS)86
b Fr(:)56 b(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f
(:)h(:)f(:)h(:)f(:)h(:)172 b Fu(vi)300 1887 y Fm(1)90
b(INTR)m(ODUCTION)84 b Fr(:)55 b(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)
h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)166
b Fm(1)300 2105 y(2)90 b(FR)m(OBENIUS-PERR)m(ON)35 b(OPERA)-9
b(TORS)72 b Fr(:)56 b(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f
(:)h(:)154 b Fm(8)300 2323 y(3)90 b(ULAM'S)39 b(METHOD)88
b Fr(:)56 b(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f
(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)114 b Fm(12)300
2541 y(4)90 b(PIECEWISE)35 b(LINEAR)h(MARK)m(O)m(V)h(APPR)m(O)m(XIMA)-9
b(TION)90 b Fr(:)55 b(:)g(:)h(:)f(:)118 b Fm(20)300 2759
y(5)90 b(QUASI-MONTE)38 b(CARLO)f(ALGORITHM)67 b Fr(:)55
b(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)92
b Fm(25)300 2977 y(6)e(P)-9 b(ARALLEL)36 b(ALGORITHMS)82
b Fr(:)56 b(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g
(:)h(:)f(:)h(:)f(:)107 b Fm(29)300 3195 y(7)90 b(NUMERICAL)37
b(RESUL)-9 b(TS)81 b Fr(:)56 b(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f
(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)106 b
Fm(36)300 3315 y Fu(7.1)168 b(Numerical)30 b(Results)j(from)f(the)h
(Quasi-Mon)m(te)f(Carlo)g(Approac)m(h)681 b(36)300 3435
y(7.2)168 b(Numerical)30 b(Results)j(from)f(the)h(P)m(arallel)d
(Computation)1017 b(39)300 3556 y(7.3)168 b(An)32 b(Applied)g
(Computational)e(Problem)1592 b(45)300 3774 y Fm(8)90
b(CONCLUDING)37 b(REMARKS)85 b Fr(:)55 b(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)
h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)110 b
Fm(53)305 4060 y(APPENDIX)300 4278 y(A)61 b(BRIEF)37
b(MA)-9 b(TH)36 b(REVIEW)84 b Fr(:)56 b(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g
(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)111
b Fm(54)300 4496 y(B)66 b(A)37 b(SOUR)m(CE)g(CODE)g(IN)g(C)90
b Fr(:)55 b(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h
(:)f(:)h(:)f(:)h(:)f(:)h(:)114 b Fm(61)305 4741 y(BIBLIOGRAPHY)99
b Fr(:)55 b(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h
(:)f(:)h(:)f(:)g(:)h(:)f(:)h(:)f(:)h(:)f(:)h(:)183 b
Fu(89)p Black 2029 5764 a(iii)p Black eop
%%Page: 4 7
4 6 bop Black Black Black Black 1066 150 a Fn(LIST)54
b(OF)h(ILLUSTRA)-13 b(TIONS)446 851 y Fu(1.1)100 b(The)33
b(orbits)f(of)g Fr(S)6 b Fu(\()p Fr(x)p Fu(\))28 b(=)g(4)p
Fr(x)p Fu(\(1)22 b Fq(\000)g Fr(x)p Fu(\))33 b(with)g(\(ab)s(o)m(v)m
(e\))g Fr(x)2656 866 y Fp(0)2723 851 y Fu(=)28 b(0)p
Fr(:)p Fu(037)k(and)g(\(b)s(elo)m(w\))671 972 y Fr(x)726
987 y Fp(0)793 972 y Fu(=)c(0)p Fr(:)p Fu(038)51 b(.)f(.)g(.)g(.)g(.)f
(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)
g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p Black 132 w(1)p
Black 446 1092 a(1.2)100 b(The)37 b(frequency)i(of)d(the)h(orbits)f(of)
g Fr(S)6 b Fu(\()p Fr(x)p Fu(\))35 b(=)f(4)p Fr(x)p Fu(\(1)25
b Fq(\000)g Fr(x)p Fu(\))37 b(of)f(10000)g(iterations)671
1212 y(and)c(100)g(partitions)f(in)h([0)p Fr(;)17 b Fu(1])32
b(with)g(di\013eren)m(t)h(initial)c(states)135 b(.)50
b(.)f(.)h(.)g(.)g(.)g(.)g(.)p Black 132 w(3)p Black 446
1416 a(2.1)100 b(The)30 b(\014xed)h(densit)m(y)f Fr(f)1493
1380 y Fl(\003)1562 1416 y Fu(of)f Fr(P)43 b Fu(for)29
b Fr(S)6 b Fu(\()p Fr(x)p Fu(\))28 b(=)f(4)p Fr(x)p Fu(\(1)16
b Fq(\000)g Fr(x)p Fu(\))30 b(with)f(100)g(partitions)f(in)671
1536 y([0,1])90 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)
g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f
(.)h(.)g(.)g(.)g(.)g(.)p Black 83 w(10)p Black 446 1740
a(7.1)100 b Fr(L)737 1703 y Fp(1)809 1740 y Fu(error)32
b(with)h(c)m(hange)g(of)f Fr(n)h Fu(for)f Fr(S)2003 1755
y Fp(1)2094 1740 y Fu(.)50 b(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)
g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p Black 83 w(39)p
Black 446 1860 a(7.2)100 b(E\016ciency)34 b(of)e(UMP)-8
b(A)33 b(for)f Fr(S)1760 1875 y Fp(3)1832 1860 y Fu(on)h(Wiglaf)73
b(.)49 b(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)
g(.)p Black 83 w(42)p Black 446 1980 a(7.3)100 b(E\016ciency)34
b(of)e(CP)-8 b(A)33 b(for)f Fr(S)1668 1995 y Fp(3)1740
1980 y Fu(on)h(Wiglaf)88 b(.)50 b(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g
(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p Black 83 w(42)p
Black 446 2101 a(7.4)100 b(E\016ciency)34 b(of)e(UMP)-8
b(A)33 b(for)f Fr(S)1760 2116 y Fp(3)1832 2101 y Fu(on)h(Sw)m(eetgum)80
b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p
Black 83 w(44)p Black 446 2221 a(7.5)100 b(E\016ciency)34
b(of)e(CP)-8 b(A)33 b(for)f Fr(S)1668 2236 y Fp(3)1740
2221 y Fu(on)h(Sw)m(eetgum)95 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p Black 83 w(44)p Black
446 2342 a(7.6)100 b(E\016ciency)34 b(comparison)d(of)h(UMP)-8
b(A)34 b(for)e Fr(S)2276 2357 y Fp(3)2348 2342 y Fu(on)g(Sw)m(eetgum)h
(and)g(Wiglaf)58 b(.)50 b(.)g(.)p Black 83 w(45)p Black
446 2462 a(7.7)100 b(The)33 b(graph)g(of)f(DPLL)110 b(.)50
b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g
(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p Black 83 w(47)p
Black 446 2582 a(7.8)100 b(E\016ciency)34 b(of)e(UMP)-8
b(A)33 b(for)f(DPLL)h(on)f(Wiglaf)63 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g
(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p Black 83 w(49)p
Black 446 2703 a(7.9)100 b(E\016ciency)34 b(of)e(the)h(CP)-8
b(A)33 b(for)f(DPLL)g(on)h(Wiglaf)64 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g
(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p Black 83 w(49)p Black
446 2823 a(7.10)h(E\016ciency)34 b(of)e(UMP)-8 b(A)33
b(for)f(DPLL)h(on)f(Sw)m(eetgum)71 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g(.)f
(.)h(.)g(.)g(.)g(.)g(.)p Black 83 w(51)p Black 446 2943
a(7.11)h(E\016ciency)34 b(of)e(CP)-8 b(A)33 b(for)f(DPLL)g(on)h(Sw)m
(eetgum)86 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g
(.)p Black 83 w(51)p Black 446 3064 a(7.12)h(E\016ciency)34
b(Comparison)d(of)h(UMP)-8 b(A)34 b(for)e(DPLL)g(on)h(Sw)m(eetgum)g
(and)g(Wiglaf)p Black 131 w(52)p Black 446 3184 a(7.13)51
b(The)33 b(stationary)f(PDF)g(graph)h(of)f(DPLL)g(with)g
Fr(n)c Fu(=)g(4096)77 b(.)50 b(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p
Black 83 w(52)p Black Black 2031 5764 a(iv)p Black eop
%%Page: 5 8
5 7 bop Black Black Black Black 1411 150 a Fn(LIST)54
b(OF)h(T)-13 b(ABLES)446 851 y Fu(3.1)100 b(P)m(erformance)33
b(comparison)e(b)s(et)m(w)m(een)k(IA)e(and)f(GA)101 b(.)50
b(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p Black
83 w(19)p Black 446 1055 a(7.1)100 b Fr(L)737 1019 y
Fp(1)809 1055 y Fu(error/time)31 b(comparison)g(for)h(the)h(logistic)d
(mo)s(del)78 b(.)50 b(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p
Black 83 w(37)p Black 446 1175 a(7.2)100 b Fr(L)737 1139
y Fp(1)809 1175 y Fu(error/time)31 b(comparison)g(for)h
Fr(S)2010 1190 y Fp(2)2094 1175 y Fu(.)50 b(.)g(.)g(.)f(.)h(.)g(.)g(.)g
(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p
Black 83 w(37)p Black 446 1295 a(7.3)100 b Fr(L)737 1259
y Fp(1)809 1295 y Fu(error)32 b(comparisons)g(of)g(the)h(QMC)h(to)e
(the)h(MC)k(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p
Black 83 w(38)p Black 446 1416 a(7.4)100 b Fr(L)737 1380
y Fp(1)809 1416 y Fu(error/time)31 b(comparison)g(of)h(QMC,)i(MC,)f
(and)g(Exact)g(for)f(logistic)e(mo)s(del)p Black 106
w(40)p Black 446 1536 a(7.5)100 b Fr(L)737 1500 y Fp(1)809
1536 y Fu(error)32 b(comparison)g(of)g(QMC,)h(MC,)h(and)e(Exact)i(for)e
Fr(S)2868 1551 y Fp(2)2940 1536 y Fu(.)50 b(.)g(.)g(.)f(.)h(.)g(.)g(.)g
(.)g(.)p Black 83 w(40)p Black 446 1657 a(7.6)100 b(P)m(erformance)33
b(of)f(UMP)-8 b(A)33 b(and)g(CP)-8 b(A)33 b(for)f Fr(S)2306
1672 y Fp(3)2378 1657 y Fu(on)h(Wiglaf)65 b(.)50 b(.)g(.)g(.)g(.)f(.)h
(.)g(.)g(.)g(.)g(.)p Black 83 w(41)p Black 446 1777 a(7.7)100
b(P)m(erformance)33 b(of)f(UMP)-8 b(A)33 b(and)g(CP)-8
b(A)33 b(for)f Fr(S)2306 1792 y Fp(3)2378 1777 y Fu(on)h(Sw)m(eetgum)73
b(.)50 b(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p Black 83 w(43)p
Black 446 1897 a(7.8)100 b(P)m(erformance)33 b(of)f(UMP)-8
b(A)33 b(and)g(CP)-8 b(A)33 b(for)f(DPLL)g(on)h(Wiglaf)56
b(.)50 b(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)p Black 83 w(48)p
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Fq(\002)g(\001)e(\001)g(\001)h(\002)300 3528 y Fu([)p
Fr(a)378 3543 y Fo(d)419 3528 y Fr(;)f(b)504 3543 y Fo(d)544
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1173 3664 y Fi(Z)1273 3690 y Fo(x)1313 3699 y Fg(1)1228
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3814 y Fo(d)1014 4070 y Fu(=)1173 3934 y Fi(Z)1289 4070
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(\001)31 b Fr(;)17 b(x)968 4645 y Fo(d)1009 4630 y Fu(\))28
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4 v 1188 4699 a Fr(@)5 b(x)1299 4714 y Fp(1)1357 4699
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Fp([)p Fo(a)2561 4732 y Ff(d)2597 4720 y Fo(;x)2657 4732
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5048 y Fm(Example)26 b(2)d(The)k(bak)m(er)g(transformation)p
Fu(.)40 b(Let)23 b Fr(X)35 b Fu(=)28 b([0)p Fr(;)17 b
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1749 5478 V 1749 5535 a(2)1794 5500 y Fr(y)j Fu(+)1975
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Fr(x)f Fq(\024)g Fu(1)p Fr(;)44 b Fu(0)27 b Fq(\024)i
Fr(y)h Fq(\024)f Fu(1)p Fr(:)p Black 3591 5441 a Fu(\(2.15\))p
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11 19 bop Black 300 10 a Fk(CHAPTER)34 b(2.)76 b(FR)m(OBENIUS-PERR)m
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a(Since)33 b Fr(S)621 238 y Fl(\000)p Fp(1)715 274 y
Fu(\([0)p Fr(;)17 b(x)p Fu(])22 b Fq(\002)h Fu([0)p Fr(;)17
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251 36 4 v 1573 309 a(2)1618 274 y Fr(x)p Fu(])23 b Fq(\002)g
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b Fq(\024)h Fr(y)j(<)2625 235 y Fp(1)p 2625 251 V 2625
309 a(2)2670 274 y Fu(,)i(w)m(e)g(ha)m(v)m(e)667 574
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Black 284 w Fu(\(2.16\))p Black 300 854 a(F)-8 b(or)485
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b Fr(y)j Fq(\024)d Fu(1,)595 1128 y Fr(S)661 1087 y Fl(\000)p
Fp(1)755 1128 y Fu(\([0)p Fr(;)17 b(x)p Fu(])22 b Fq(\002)g
Fu([0)p Fr(;)17 b(y)t Fu(]\))27 b(=)g(\([0)p Fr(;)1651
1061 y Fu(1)p 1651 1105 49 4 v 1651 1197 a(2)1710 1128
y Fr(x)p Fu(])22 b Fq(\002)h Fu([0)p Fr(;)17 b Fu(1]\))k
Fq([)i Fu(\([)2333 1061 y(1)p 2333 1105 V 2333 1197 a(2)2392
1128 y Fr(;)2445 1061 y Fu(1)p 2445 1105 V 2445 1197
a(2)2526 1128 y(+)2634 1061 y(1)p 2634 1105 V 2634 1197
a(2)2693 1128 y Fr(x)p Fu(])g Fq(\002)f Fu([0)p Fr(;)17
b Fu(2)p Fr(y)25 b Fq(\000)e Fu(1]\))p Fr(;)p Black 211
w Fu(\(2.17\))p Black 300 1374 a(hence,)633 1670 y Fr(P)14
b(f)d Fu(\()p Fr(x;)17 b(y)t Fu(\))81 b(=)1308 1602 y
Fr(@)1364 1566 y Fp(2)p 1246 1647 221 4 v 1246 1738 a
Fr(@)5 b(x@)g(y)1476 1670 y Fq(f)1526 1534 y Fi(Z)1636
1533 y Ff(x)p 1636 1545 35 3 v 1638 1586 a Fg(2)1581
1759 y Fp(0)1701 1670 y Fr(ds)1815 1534 y Fi(Z)1914 1560
y Fp(1)1870 1759 y(0)1970 1670 y Fr(f)11 b Fu(\()p Fr(s;)17
b(t)p Fu(\))p Fr(dt)k Fu(+)2435 1534 y Fi(Z)2545 1533
y Fg(1)p 2545 1545 31 3 v 2545 1586 a(2)2585 1560 y Fp(+)2650
1533 y Ff(x)p 2650 1545 35 3 v 2652 1586 a Fg(2)2491
1759 y Fp(0)2716 1670 y Fr(ds)2830 1534 y Fi(Z)2929 1560
y Fp(2)p Fo(y)r Fl(\000)p Fp(1)2884 1759 y(0)3112 1670
y Fr(f)11 b Fu(\()p Fr(s;)17 b(t)p Fu(\))p Fr(dt)p Fq(g)1077
1924 y Fu(=)83 b Fr(f)11 b Fu(\()1343 1857 y(1)p 1343
1901 49 4 v 1343 1993 a(2)1423 1924 y(+)1531 1857 y(1)p
1531 1901 V 1531 1993 a(2)1590 1924 y Fr(x;)17 b Fu(2)p
Fr(y)25 b Fq(\000)e Fu(1\))p Fr(;)2107 1857 y Fu(1)p
2107 1901 V 2107 1993 a(2)2193 1924 y Fq(\024)28 b Fr(y)j
Fq(\024)d Fu(1)p Fr(:)300 2170 y Fu(In)33 b(summary)-8
b(,)32 b(w)m(e)h(ha)m(v)m(e)1010 2442 y Fr(P)14 b(f)d
Fu(\()p Fr(x;)17 b(y)t Fu(\))26 b(=)1503 2302 y Fi(\032)1619
2381 y Fr(f)11 b Fu(\()1726 2342 y Fp(1)p 1726 2358 36
4 v 1726 2416 a(2)1771 2381 y Fr(x;)17 b Fu(2)p Fr(y)t
Fu(\))p Fr(;)428 b Fu(0)28 b Fq(\024)g Fr(y)j(<)2838
2342 y Fp(1)p 2838 2358 V 2838 2416 a(2)1619 2502 y Fr(f)11
b Fu(\()1726 2462 y Fp(1)p 1726 2479 V 1726 2536 a(2)1793
2502 y Fu(+)1901 2462 y Fp(1)p 1901 2479 V 1901 2536
a(2)1946 2502 y Fr(x;)17 b Fu(2)p Fr(y)26 b Fq(\000)c
Fu(1\))p Fr(;)2474 2462 y Fp(1)p 2474 2479 V 2474 2536
a(2)2547 2502 y Fq(\024)28 b Fr(y)j Fq(\024)d Fu(1)p
Fr(:)p Black 3591 2442 a Fu(\(2.18\))p Black 300 2715
a(Since)33 b Fr(P)14 b Fu(1)27 b(=)g(1,)33 b(the)g(Leb)s(esgue)g
(measure)g Fr(m)g Fu(is)f(in)m(v)-5 b(arian)m(t)31 b(under)j
Fr(S)6 b Fu(.)446 2835 y(The)24 b(existence)g(of)f(an)g(absolutely)f
(con)m(tin)m(uous)h(\014nite)g(in)m(v)-5 b(arian)m(t)21
b(measure)i(is)g(equiv)-5 b(alen)m(t)22 b(to)300 2956
y(that)28 b(of)g(a)g(non)m(trivial)e(solution)h(to)h(the)g(\014xed)i(p)
s(oin)m(t)d(equation)h Fr(P)14 b(f)2786 2920 y Fl(\003)2853
2956 y Fu(=)27 b Fr(f)3015 2920 y Fl(\003)3083 2956 y
Fu(for)g(the)i(F)-8 b(rob)s(enius-)300 3076 y(P)m(erron)33
b(op)s(erator)f Fr(P)46 b Fu(asso)s(ciated)33 b(with)f
Fr(S)6 b Fu(.)p Black 300 3238 a Fj(The)-5 b(or)g(em)34
b(2.2.)p Black 48 w Fu([36])26 b(Let)g Fr(f)38 b Fq(2)28
b Fr(L)1505 3202 y Fp(1)1571 3238 y Fu(b)s(e)f(a)e(densit)m(y)i
(function.)41 b(If)26 b(the)h(Ces\023)-49 b(aro)27 b(a)m(v)m(erages)g
(sequence)1622 3548 y Fr(A)1695 3563 y Fo(n)1742 3548
y Fr(f)38 b Fq(\021)1948 3481 y Fu(1)p 1943 3525 59 4
v 1943 3616 a Fr(n)2034 3424 y Fo(n)p Fl(\000)p Fp(1)2028
3453 y Fi(X)2043 3663 y Fo(i)p Fp(=0)2189 3548 y Fr(P)2266
3507 y Fo(i)2293 3548 y Fr(f)p Black 1250 w Fu(\(2.19\))p
Black 300 3860 a(is)29 b(w)m(eakly)h(pre-compact,)g(then)g(it)e(con)m
(v)m(erges)k(strongly)d(to)g(some)g(in)m(v)-5 b(arian)m(t)28
b(densit)m(y)i Fr(f)3573 3824 y Fl(\003)3640 3860 y Fq(2)e
Fr(L)3800 3824 y Fp(1)300 3981 y Fu(of)k Fr(P)14 b Fu(,)32
b(i.e.,)g Fr(P)14 b(f)867 3944 y Fl(\003)934 3981 y Fu(=)27
b Fr(f)1096 3944 y Fl(\003)1168 3981 y Fu(and)1558 4201
y(lim)1534 4260 y Fo(n)p Fl(!1)1735 4201 y Fq(k)p Fr(A)1858
4216 y Fo(n)1904 4201 y Fr(f)33 b Fq(\000)23 b Fr(f)2144
4160 y Fl(\003)2183 4201 y Fq(k)28 b Fu(=)f(0)p Fr(:)p
Black 1151 w Fu(\(2.20\))p Black Black Black eop
%%Page: 12 21
12 20 bop Black Black Black Black 1714 122 a Fn(Chapter)53
b(3)p Black Black 1357 554 a(ULAM'S)g(METHOD)477 1255
y Fu(No)m(w)41 b(w)m(e)g(in)m(tro)s(duce)f(the)g(idea)f(b)s(ehind)h
(Ulam's)e(piecewise)j(constan)m(t)g(appro)m(ximations)300
1376 y(for)30 b(computing)e(the)j(\014xed)g(densit)m(y)g(of)f(the)h(F)
-8 b(rob)s(enius-P)m(erron)29 b(op)s(erator.)42 b(W)-8
b(e)31 b(\014rst)g(consider)300 1496 y(the)g(one)g(dimensional)d(case,)
k(whic)m(h)f(w)m(as)g(initially)26 b(prop)s(osed)31 b(b)m(y)h(Ulam)d
(in)g(his)i(famous)e(b)s(o)s(ok)300 1616 y([47])j(on)h(mathematical)c
(problems.)446 1737 y(Assume)44 b(that)e Fr(S)50 b Fu(:)45
b([0)p Fr(;)17 b Fu(1])44 b Fq(!)g Fu([0)p Fr(;)17 b
Fu(1])42 b(is)g(a)h(nonsingular)e(transformation)f(suc)m(h)k(that)f
(the)300 1857 y(corresp)s(onding)28 b(F)-8 b(rob)s(enius-P)m(erron)27
b(op)s(erator)g Fr(P)41 b Fu(:)28 b Fr(L)2297 1821 y
Fp(1)2337 1857 y Fu(\(0)p Fr(;)17 b Fu(1\))27 b Fq(!)g
Fr(L)2775 1821 y Fp(1)2815 1857 y Fu(\(0)p Fr(;)17 b
Fu(1\))27 b(has)h(a)f(\014xed)i(densit)m(y)300 1978 y
Fr(f)359 1941 y Fl(\003)398 1978 y Fu(.)53 b(T)-8 b(o)36
b(illustrate)e(the)i(basic)f(idea)g(b)s(ehind)h(the)g(probabilit)m(y)e
(argumen)m(t)h(of)h(Ulam's)e(for)i(the)300 2098 y(motiv)-5
b(ation)35 b(of)i(his)h(metho)s(d,)g(w)m(e)h(\014rst)f(use)h(a)f
(simple)e(partition)f(of)j(the)g(in)m(terv)-5 b(al)37
b([0)p Fr(;)17 b Fu(1])37 b(in)m(to)300 2218 y(three)c(subin)m(terv)-5
b(als,)33 b(and)g(then)g(generalize)f(it)f(to)h(a)h(partition)d(of)i
Fr(n)h Fu(subin)m(terv)-5 b(als.)446 2339 y(Supp)s(ose)40
b([0)p Fr(;)17 b Fu(1])37 b(is)h(divided)g(in)m(to)g(three)h(subin)m
(terv)-5 b(als)38 b Fr(I)2567 2354 y Fp(1)2606 2339 y
Fr(;)17 b(I)2693 2354 y Fp(2)2733 2339 y Fr(;)g(I)2820
2354 y Fp(3)2859 2339 y Fu(.)60 b(Let)39 b Fr(f)49 b
Fu(b)s(e)38 b(a)g(piecewise)300 2459 y(constan)m(t)44
b(densit)m(y)g(suc)m(h)h(that)e(the)h(p)s(ossibilit)m(y)d(of)i
Fr(I)2323 2474 y Fp(1)2406 2459 y Fu(is)f Fr(a)2565 2474
y Fp(1)2605 2459 y Fu(,)k(the)e(probabilit)m(y)d(of)i
Fr(I)3531 2474 y Fp(2)3613 2459 y Fu(is)g Fr(a)3773 2474
y Fp(2)3813 2459 y Fu(,)300 2580 y(and)37 b(the)g(probabilit)m(y)d(of)j
Fr(I)1327 2595 y Fp(3)1403 2580 y Fu(is)f Fr(a)1556 2595
y Fp(3)1596 2580 y Fu(.)55 b(Ulam's)36 b(idea)g(is)g(that)g
Fr(P)14 b(f)47 b Fu(can)37 b(b)s(e)g(appro)m(ximated)f(b)m(y)h(a)300
2700 y(piecewise)i(constan)m(t)h(densit)m(y)f(with)f(the)h
(probabilities)d(of)i Fr(I)2598 2715 y Fp(1)2638 2700
y Fr(;)17 b(I)2725 2715 y Fp(2)2764 2700 y Fr(;)39 b
Fu(and)f Fr(I)3068 2715 y Fp(3)3146 2700 y Fu(to)g(b)s(e)h
Fr(b)3451 2715 y Fp(1)3491 2700 y Fr(;)17 b(b)3576 2715
y Fp(2)3616 2700 y Fu(,)40 b(and)300 2820 y Fr(b)341
2835 y Fp(3)381 2820 y Fu(.)j(W)-8 b(e)33 b(w)m(an)m(t)h(to)e(\014nd)h
(ho)m(w)g Fr(b)1416 2835 y Fp(1)1456 2820 y Fr(;)17 b(b)1541
2835 y Fp(2)1581 2820 y Fu(,)32 b(and)h Fr(b)1871 2835
y Fp(3)1944 2820 y Fu(dep)s(end)g(on)g Fr(a)2469 2835
y Fp(1)2508 2820 y Fr(;)17 b(a)2603 2835 y Fp(2)2643
2820 y Fu(,)32 b(and)h Fr(a)2943 2835 y Fp(3)2983 2820
y Fu(.)446 2941 y(F)-8 b(rom)32 b(the)j(de\014nition)d(of)i(the)g(F)-8
b(rob)s(enius-P)m(erron)33 b(op)s(erator)g Fr(P)14 b
Fu(,)34 b(the)g(new)g(probabilit)m(y)e Fr(b)3800 2956
y Fp(1)300 3061 y Fu(of)k Fr(I)458 3076 y Fp(1)533 3061
y Fu(should)g(carry)g(the)h(old)e(probabilit)m(y)f(of)h
Fr(S)2121 3025 y Fl(\000)p Fp(1)2215 3061 y Fu(\()p Fr(I)2296
3076 y Fp(1)2336 3061 y Fu(\))g(whic)m(h)i(is)e(the)i(disjoin)m(t)e
(union)g(of)h(the)300 3181 y(three)d(parts)g(of)f Fr(S)976
3145 y Fl(\000)p Fp(1)1070 3181 y Fu(\()p Fr(I)1151 3196
y Fp(1)1191 3181 y Fu(\))g(in)g Fr(I)1418 3196 y Fp(1)1457
3181 y Fr(;)17 b(I)1544 3196 y Fp(2)1584 3181 y Fu(,)32
b(and)h Fr(I)1876 3196 y Fp(3)1915 3181 y Fu(:)1229 3377
y Fr(I)1272 3392 y Fp(1)1334 3377 y Fq(\\)22 b Fr(S)1488
3336 y Fl(\000)p Fp(1)1582 3377 y Fu(\()p Fr(I)1663 3392
y Fp(1)1703 3377 y Fu(\))p Fr(;)17 b(I)1828 3392 y Fp(2)1889
3377 y Fq(\\)23 b Fr(S)2044 3336 y Fl(\000)p Fp(1)2138
3377 y Fu(\()p Fr(I)2219 3392 y Fp(1)2258 3377 y Fu(\))p
Fr(;)17 b(I)2383 3392 y Fp(3)2444 3377 y Fq(\\)23 b Fr(S)2599
3336 y Fl(\000)p Fp(1)2693 3377 y Fu(\()p Fr(I)2774 3392
y Fp(1)2813 3377 y Fu(\))p Fr(:)300 3573 y Fu(So)32 b(w)m(e)i(can)f
(obtain)492 3792 y Fr(b)533 3807 y Fp(1)656 3792 y Fu(=)815
3657 y Fi(Z)870 3882 y Fo(I)901 3891 y Fg(1)956 3792
y Fr(P)14 b(f)d Fu(\()p Fr(x)p Fu(\))p Fr(dx)27 b Fu(=)1460
3657 y Fi(Z)1515 3882 y Fo(S)1562 3863 y Fh(\000)p Fg(1)1644
3882 y Fp(\()p Fo(I)1702 3891 y Fg(1)1737 3882 y Fp(\))1785
3792 y Fr(f)11 b Fu(\()p Fr(x)p Fu(\))p Fr(dx)656 4068
y Fu(=)815 3932 y Fi(Z)870 4158 y Fo(S)917 4139 y Fh(\000)p
Fg(1)999 4158 y Fp(\()p Fo(I)1057 4167 y Fg(1)1092 4158
y Fp(\))p Fl(\\)p Fo(I)1197 4167 y Fg(1)1252 4068 y Fr(f)g
Fu(\()p Fr(x)p Fu(\))p Fr(dx)23 b Fu(+)1669 3932 y Fi(Z)1724
4158 y Fo(S)1771 4139 y Fh(\000)p Fg(1)1853 4158 y Fp(\()p
Fo(I)1911 4167 y Fg(1)1946 4158 y Fp(\))p Fl(\\)p Fo(I)2051
4167 y Fg(2)2106 4068 y Fr(f)11 b Fu(\()p Fr(x)p Fu(\))p
Fr(dx)23 b Fu(+)2523 3932 y Fi(Z)2578 4158 y Fo(S)2625
4139 y Fh(\000)p Fg(1)2707 4158 y Fp(\()p Fo(I)2765 4167
y Fg(1)2800 4158 y Fp(\))p Fl(\\)p Fo(I)2905 4167 y Fg(3)2960
4068 y Fr(f)11 b Fu(\()p Fr(x)p Fu(\))p Fr(dx)656 4354
y Fu(=)825 4287 y Fr(m)p Fu(\()p Fr(I)991 4302 y Fp(1)1053
4287 y Fq(\\)22 b Fr(S)1207 4250 y Fl(\000)p Fp(1)1301
4287 y Fu(\()p Fr(I)1382 4302 y Fp(1)1422 4287 y Fu(\)\))p
825 4331 673 4 v 1039 4422 a Fr(m)p Fu(\()p Fr(I)1205
4437 y Fp(1)1245 4422 y Fu(\))1530 4354 y Fq(\001)f Fr(a)1630
4369 y Fp(1)1692 4354 y Fu(+)1800 4287 y Fr(m)p Fu(\()p
Fr(I)1966 4302 y Fp(2)2028 4287 y Fq(\\)i Fr(S)2183 4250
y Fl(\000)p Fp(1)2277 4287 y Fu(\()p Fr(I)2358 4302 y
Fp(1)2397 4287 y Fu(\)\))p 1800 4331 V 2015 4422 a Fr(m)p
Fu(\()p Fr(I)2181 4437 y Fp(2)2220 4422 y Fu(\))2505
4354 y Fq(\001)f Fr(a)2606 4369 y Fp(2)2668 4354 y Fu(+)2776
4287 y Fr(m)p Fu(\()p Fr(I)2942 4302 y Fp(3)3004 4287
y Fq(\\)g Fr(S)3158 4250 y Fl(\000)p Fp(1)3252 4287 y
Fu(\()p Fr(I)3333 4302 y Fp(1)3373 4287 y Fu(\)\))p 2776
4331 V 2990 4422 a Fr(m)p Fu(\()p Fr(I)3156 4437 y Fp(3)3196
4422 y Fu(\))3481 4354 y Fq(\001)f Fr(a)3581 4369 y Fp(3)3621
4354 y Fr(;)300 4600 y Fu(or)32 b(in)g(this)g(w)m(a)m(y)-8
b(,)1390 4881 y Fr(b)1431 4896 y Fp(1)1498 4881 y Fu(=)1656
4756 y Fp(3)1602 4786 y Fi(X)1617 4996 y Fo(i)p Fp(=1)1772
4813 y Fr(m)p Fu(\()p Fr(I)1938 4828 y Fo(i)1989 4813
y Fq(\\)22 b Fr(S)2143 4777 y Fl(\000)p Fp(1)2238 4813
y Fu(\()p Fr(I)2319 4828 y Fp(1)2358 4813 y Fu(\))p 1772
4858 624 4 v 1968 4949 a Fr(m)p Fu(\()p Fr(I)2134 4964
y Fo(i)2162 4949 y Fu(\))2428 4881 y Fq(\001)g Fr(a)2529
4896 y Fo(i)2557 4881 y Fr(:)p Black 1055 w Fu(\(3.1\))p
Black 300 5159 a(By)33 b(the)g(same)g(tok)m(en,)1390
5439 y Fr(b)1431 5454 y Fp(2)1498 5439 y Fu(=)1656 5315
y Fp(3)1602 5345 y Fi(X)1617 5554 y Fo(i)p Fp(=1)1772
5372 y Fr(m)p Fu(\()p Fr(I)1938 5387 y Fo(i)1989 5372
y Fq(\\)22 b Fr(S)2143 5336 y Fl(\000)p Fp(1)2238 5372
y Fu(\()p Fr(I)2319 5387 y Fp(2)2358 5372 y Fu(\))p 1772
5416 V 1968 5508 a Fr(m)p Fu(\()p Fr(I)2134 5523 y Fo(i)2162
5508 y Fu(\))2428 5439 y Fq(\001)g Fr(a)2529 5454 y Fo(i)2557
5439 y Fr(;)p Black 1055 w Fu(\(3.2\))p Black Black 2021
5764 a(12)p Black eop
%%Page: 13 22
13 21 bop Black 300 10 a Fk(CHAPTER)34 b(3.)76 b(ULAM'S)33
b(METHOD)1907 b Fu(13)p Black 1390 352 a Fr(b)1431 367
y Fp(3)1498 352 y Fu(=)1656 227 y Fp(3)1602 257 y Fi(X)1617
467 y Fo(i)p Fp(=1)1772 285 y Fr(m)p Fu(\()p Fr(I)1938
300 y Fo(i)1989 285 y Fq(\\)22 b Fr(S)2143 248 y Fl(\000)p
Fp(1)2238 285 y Fu(\()p Fr(I)2319 300 y Fp(3)2358 285
y Fu(\))p 1772 329 624 4 v 1968 420 a Fr(m)p Fu(\()p
Fr(I)2134 435 y Fo(i)2162 420 y Fu(\))2428 352 y Fq(\001)g
Fr(a)2529 367 y Fo(i)2557 352 y Fr(:)p Black 1055 w Fu(\(3.3\))p
Black 446 653 a(In)31 b(general,)f(if)e([0)p Fr(;)17
b Fu(1])30 b(is)f(divided)h(in)m(to)f Fr(n)h Fu(subin)m(terv)-5
b(als)30 b Fr(I)2537 668 y Fp(1)2577 653 y Fr(;)17 b(I)2664
668 y Fp(2)2703 653 y Fr(;)g Fq(\001)g(\001)g(\001)31
b Fr(;)17 b(I)2983 668 y Fo(n)3059 653 y Fu(and)31 b(if)d
Fr(f)41 b Fu(is)29 b(a)h(piece-)300 774 y(wise)i(constan)m(t)h(densit)m
(y)g(suc)m(h)g(that)f(the)h(probabilit)m(y)d(of)h Fr(I)2489
789 y Fo(i)2549 774 y Fu(is)h Fr(a)2698 789 y Fo(i)2726
774 y Fu(,)g(then)h(the)f(new)h(probabilit)m(y)300 894
y(of)f Fr(I)454 909 y Fo(j)523 894 y Fu(is)g Fr(b)662
909 y Fo(j)732 894 y Fu(with)1374 1186 y Fr(b)1415 1201
y Fo(j)1479 1186 y Fu(=)1633 1062 y Fo(n)1583 1091 y
Fi(X)1598 1301 y Fo(i)p Fp(=1)1753 1119 y Fr(m)p Fu(\()p
Fr(I)1919 1134 y Fo(i)1970 1119 y Fq(\\)22 b Fr(S)2124
1082 y Fl(\000)p Fp(1)2219 1119 y Fu(\()p Fr(I)2300 1134
y Fo(j)2336 1119 y Fu(\)\))p 1753 1163 659 4 v 1966 1254
a Fr(m)p Fu(\()p Fr(I)2132 1269 y Fo(i)2161 1254 y Fu(\))2444
1186 y Fq(\001)g Fr(a)2545 1201 y Fo(i)2573 1186 y Fr(:)p
Black 1039 w Fu(\(3.4\))p Black 300 1487 a(Let)35 b Fr(a)c
Fu(=)f([)p Fr(a)743 1502 y Fp(1)783 1487 y Fr(;)17 b(a)878
1502 y Fp(2)917 1487 y Fr(;)g Fq(\001)g(\001)g(\001)31
b Fr(;)17 b(a)1205 1502 y Fo(n)1252 1487 y Fu(])35 b(and)f
Fr(b)e Fu(=)e([)p Fr(b)1752 1502 y Fp(1)1792 1487 y Fr(;)17
b(b)1877 1502 y Fp(2)1917 1487 y Fr(;)g Fq(\001)g(\001)g(\001)31
b Fr(;)17 b(b)2195 1502 y Fo(n)2242 1487 y Fu(],)35 b(and)g(let)e(the)i
Fr(n)24 b Fq(\002)g Fr(n)34 b Fu(matrix)f Fr(P)3492 1502
y Fo(n)3570 1487 y Fu(=)d([)p Fr(p)3752 1502 y Fo(ij)3813
1487 y Fu(])300 1608 y(b)s(e)j(de\014ned)h(b)m(y)1514
1875 y Fr(p)1563 1890 y Fo(ij)1651 1875 y Fu(=)1764 1807
y Fr(m)p Fu(\()p Fr(I)1930 1822 y Fo(i)1981 1807 y Fq(\\)23
b Fr(S)2136 1771 y Fl(\000)p Fp(1)2230 1807 y Fu(\()p
Fr(I)2311 1822 y Fo(j)2347 1807 y Fu(\)\))p 1764 1852
V 1978 1943 a Fr(m)p Fu(\()p Fr(I)2144 1958 y Fo(i)2172
1943 y Fu(\))2433 1875 y Fr(:)p Black 1179 w Fu(\(3.5\))p
Black 300 2138 a(Then)34 b(w)m(e)f(ha)m(v)m(e)h Fr(b)28
b Fu(=)g Fr(aP)1210 2153 y Fo(n)1257 2138 y Fu(.)43 b(Since)33
b Fr(P)1645 2153 y Fo(n)1725 2138 y Fu(is)f(a)g(nonnegativ)m(e)h
(matrix)e(and)1205 2297 y Fo(n)1155 2327 y Fi(X)1165
2537 y Fo(j)t Fp(=1)1315 2422 y Fr(p)1364 2437 y Fo(ij)1452
2422 y Fu(=)1606 2297 y Fo(n)1556 2327 y Fi(X)1566 2537
y Fo(j)t Fp(=1)1726 2354 y Fr(m)p Fu(\()p Fr(I)1892 2369
y Fo(i)1943 2354 y Fq(\\)22 b Fr(S)2097 2318 y Fl(\000)p
Fp(1)2192 2354 y Fu(\()p Fr(I)2273 2369 y Fo(j)2309 2354
y Fu(\)\))p 1726 2399 V 1939 2490 a Fr(m)p Fu(\()p Fr(I)2105
2505 y Fo(i)2134 2490 y Fu(\))2423 2422 y(=)2536 2354
y Fr(m)p Fu(\()p Fr(I)2702 2369 y Fo(i)2730 2354 y Fu(\))p
2536 2399 233 4 v 2536 2490 a Fr(m)p Fu(\()p Fr(I)2702
2505 y Fo(i)2730 2490 y Fu(\))2806 2422 y(=)28 b(1)p
Fr(;)300 2730 y Fu(so)46 b Fr(P)496 2745 y Fo(n)588 2730
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b(rob)s(enius-P)m(erron)46 b(theorem)f(for)g(nonnegativ)m(e)300
2851 y(matrices,)52 b(there)d(is)f(a)g(nonnegativ)m(e)h(v)m(ector)h
Fr(c)2123 2815 y Fo(T)2233 2851 y Fu(=)k(\()p Fr(c)2443
2866 y Fp(1)2483 2851 y Fr(;)17 b(c)2569 2866 y Fp(2)2608
2851 y Fr(;)g Fq(\001)g(\001)g(\001)31 b Fr(;)17 b(c)2887
2866 y Fo(n)2934 2851 y Fu(\),)52 b(the)d(sum)g(of)f(whose)300
2971 y(comp)s(onen)m(ts)33 b(is)f(1,)h(suc)m(h)h(that)1245
3185 y(\()p Fr(c)1325 3200 y Fp(1)1364 3185 y Fr(;)17
b(c)1450 3200 y Fp(2)1489 3185 y Fr(;)g Fq(\001)g(\001)g(\001)31
b Fr(;)17 b(c)1768 3200 y Fo(n)1815 3185 y Fu(\))p Fr(P)1916
3200 y Fo(n)1990 3185 y Fu(=)28 b(\()p Fr(c)2174 3200
y Fp(1)2213 3185 y Fr(;)17 b(c)2299 3200 y Fp(2)2338
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3681 y Fr(f)1592 3696 y Fo(n)1640 3681 y Fu(\()p Fr(x)p
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Fr(f)3599 4233 y Fo(n)3688 4218 y Fu(will)300 4338 y(con)m(v)m(erge)32
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Fu(go)s(es)f(to)g(in\014nit)m(y)g([47)o(].)43 b(In)31
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4459 y(a)h(class)h(of)f(piecewise)h Fr(C)1221 4423 y
Fp(2)1293 4459 y Fu(and)g(stretc)m(hing)g(mappings)f([38)o(].)446
4579 y(Ulam's)46 b(metho)s(d)g(can)g(b)s(e)h(in)m(tro)s(duced)g(in)e
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5061 y(By)49 b(the)g(de\014nition)e(of)g(F)-8 b(rob)s(enius-P)m(erron)
48 b(op)s(erators)g(it)f(can)i(b)s(e)f(easily)f(pro)m(v)m(en)j(\(see)
300 5181 y([36]\))39 b(that)g(the)h(dual)f(of)g Fr(P)53
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%%Page: 14 23
14 22 bop Black 300 10 a Fk(CHAPTER)34 b(3.)76 b(ULAM'S)33
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b(if)d(\(3.8\))h(is)f(satis\014ed)i(for)e(some)h Fr(f)2101
2721 y Fl(\003)2140 2758 y Fu(,)i(then)f(the)g(rev)m(erse)h(pro)s(cess)
f(of)f(the)g(ab)s(o)m(v)m(e)300 2878 y(argumen)m(t)32
b(giv)m(es)h(that)1267 3022 y Fi(Z)1367 3048 y Fp(1)1322
3247 y(0)1406 3157 y Fu(\()p Fr(P)14 b(f)1580 3116 y
Fl(\003)1618 3157 y Fu(\()p Fr(x)p Fu(\))23 b Fq(\000)g
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54 b(F)-8 b(or)300 3912 y(the)35 b(sak)m(e)h(of)e(computing)f(appro)m
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%%Page: 15 24
15 23 bop Black 300 10 a Fk(CHAPTER)34 b(3.)76 b(ULAM'S)33
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575 y Fp(1)1629 775 y(0)1730 685 y Fu(\()p Fr(\036)1826
700 y Fo(i)1853 685 y Fu(\()p Fr(x)p Fu(\))23 b Fq(\000)f
Fr(U)10 b(\036)2240 700 y Fo(i)2269 685 y Fu(\()p Fr(x)p
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1032 y Fo(i)1792 1017 y Fr(\036)1850 1032 y Fo(i)1878
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1032 y Fo(i)2547 1017 y Fu(\()p Fr(x)p Fu(\))2678 847
y Fi(!)2774 1017 y Fr( )2837 1032 y Fo(j)2874 1017 y
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y Fi(Z)1420 1219 y Fp(1)1376 1418 y(0)1460 1328 y Fu(\(1)e
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b Fu(=)2431 1192 y Fi(Z)2531 1219 y Fp(1)2487 1418 y(0)2570
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Fu(=)e(0)p Fr(;)300 1602 y Fu(that)32 b(is,)h Fr(A)710
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b Fr(A)f Fu(is)f(singular.)p 2121 1577 89 4 v 2121 1627
4 50 v 2207 1627 V 2121 1630 89 4 v 300 1722 a Fm(Remark)46
b(3.1)41 b Fu(The)h(purp)s(ose)g(of)e(the)h(algorithm)d(is)i(to)g
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1842 y Fu(span)q Fq(f)p Fr( )609 1857 y Fp(1)648 1842
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1986 y Fi(Z)992 2012 y Fp(1)948 2211 y(0)1049 2121 y
Fu([)p Fr(\036)1134 2136 y Fo(n)1180 2121 y Fu(\()p Fr(x)p
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y Fu(\()p Fr(S)6 b Fu(\()p Fr(x)p Fu(\)\)])17 b Fr(f)11
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b(;)17 b(n:)p Black 510 w Fu(\(3.10\))p Black 300 2383
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Fr(f)1676 2398 y Fo(n)1763 2383 y Fu(can)f(b)s(e)f(nonnegativ)m(e)i(in)
d(general,)k(although)c(Ulam's)300 2503 y(metho)s(d)32
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Fr(\036)1754 2638 y Fo(n)1800 2623 y Fq(g)g Fu(and)g
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(er)i(Ulam's)e(sc)m(heme:)45 b(divide)300 2744 y(the)36
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3093 y Fo(i)1397 3078 y Fu(=)27 b Fr(\037)1561 3093 y
Fo(I)1592 3103 y Ff(i)1622 3078 y Fr(;)45 b( )1757 3093
y Fo(i)1813 3078 y Fu(=)27 b(1)1965 3093 y Fo(i)1993
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a(Then,)40 b(a)e(simple)e(calculation)f(giv)m(es)j(that)g(the)g(\()p
Fr(i;)17 b(j)6 b Fu(\))37 b(elemen)m(t)h(of)f(the)h(matrix)e
Fr(A)i Fu(in)f(the)300 3413 y(general)32 b(algorithm)d(is)300
3686 y Fr(a)351 3701 y Fo(ij)495 3686 y Fu(=)654 3551
y Fi(Z)753 3577 y Fp(1)709 3776 y(0)793 3686 y Fu(\()p
Fr(\037)892 3701 y Fo(I)923 3711 y Ff(i)953 3686 y Fu(\()p
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3711 y Ff(i)1328 3686 y Fu(\()p Fr(S)6 b Fu(\()p Fr(x)p
Fu(\)\)\)1)1688 3701 y Fo(j)1724 3686 y Fu(\()p Fr(x)p
Fu(\))p Fr(dx)28 b Fu(=)2093 3551 y Fi(Z)2193 3577 y
Fp(1)2148 3776 y(0)2249 3686 y Fr(\037)2310 3701 y Fo(I)2341
3711 y Ff(i)2371 3686 y Fu(\()p Fr(x)p Fu(\)1)2551 3701
y Fo(j)2587 3686 y Fu(\()p Fr(x)p Fu(\))p Fr(dx)23 b
Fq(\000)2947 3551 y Fi(Z)3046 3577 y Fp(1)3002 3776 y(0)3102
3686 y Fr(\037)3163 3703 y Fo(S)3210 3684 y Fh(\000)p
Fg(1)3293 3703 y Fp(\()p Fo(I)3351 3713 y Ff(i)3377 3703
y Fp(\))3409 3686 y Fu(\()p Fr(x)p Fu(\)1)3589 3701 y
Fo(j)3625 3686 y Fu(\()p Fr(x)p Fu(\))p Fr(dx)495 3956
y Fu(=)83 b Fr(n)729 3820 y Fi(Z)784 4046 y Fo(I)815
4056 y Ff(i)841 4046 y Fl(\\)p Fo(I)919 4056 y Ff(j)972
3956 y Fr(dx)22 b Fq(\000)h Fr(n)1275 3820 y Fi(Z)1330
4046 y Fo(S)1377 4027 y Fh(\000)p Fg(1)1459 4046 y Fp(\()p
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4056 y Ff(j)1702 3956 y Fr(dx)28 b Fu(=)f Fr(\016)1982
3971 y Fo(ij)2065 3956 y Fq(\000)2175 3888 y Fr(m)p Fu(\()p
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3903 y Fo(j)2795 3888 y Fu(\))p 2175 3933 659 4 v 2383
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4625 y Fr(P)931 4640 y Fo(n)1006 4625 y Fu(=)27 b([)p
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y Fo(j)2205 4557 y Fu(\)\))p 1622 4602 V 1836 4693 a
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4625 y Fr(;)72 b(i;)17 b(j)34 b Fu(=)27 b(1)p Fr(;)17
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Fq(\032)g Fr(R)2386 4979 y Fo(d)2471 5015 y Fu(b)s(e)d(a)f(b)s(ounded)h
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b Fr(T)2736 5271 y Fo(h)2832 5256 y Fu(=)52 b Fq(f)p
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5530 y Fo(i)1459 5515 y Fq(2)c Fr(L)1620 5478 y Fp(1)1660
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5515 y Fu(.)44 b(Let)33 b(\001)3795 5530 y Fo(h)p Black
Black eop
%%Page: 16 25
16 24 bop Black 300 10 a Fk(CHAPTER)34 b(3.)76 b(ULAM'S)33
b(METHOD)1907 b Fu(16)p Black 300 274 a(b)s(e)31 b(the)g
Fr(n)p Fu(-dimensional)d(subspace)33 b(of)d Fr(L)1804
238 y Fp(1)1844 274 y Fu(\(\012\))h(spanned)h(b)m(y)g(1)2582
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Fu(1)1795 802 y Fo(i)1850 787 y Fu(=)2004 662 y Fo(n)1954
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1307 y Fr(p)1535 1322 y Fo(ij)1624 1307 y Fu(=)1737 1240
y Fr(m)p Fu(\(\012)1930 1255 y Fo(i)1981 1240 y Fq(\\)c
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1376 a Fr(m)p Fu(\(\012)2157 1391 y Fo(i)2186 1376 y
Fu(\))2460 1307 y Fr(:)p Black 1104 w Fu(\(3.14\))p Black
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(densit)m(y)g Fr(f)2487 1568 y Fo(h)2569 1553 y Fq(2)e
Fu(\001)2753 1568 y Fo(h)2836 1553 y Fu(of)g Fr(P)3015
1568 y Fo(h)3098 1553 y Fu(to)g(appro)m(ximate)g(a)300
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m(ergence)i(of)d(Ulam's)f(metho)s(d)h(as)g Fr(h)g Fq(!)f
Fu(0,)j(w)m(e)300 1794 y(need)34 b(to)e(de\014ne)i(a)e(discretized)h
(op)s(erator)f Fr(Q)1963 1809 y Fo(h)2035 1794 y Fu(:)c
Fr(L)2156 1758 y Fp(1)2196 1794 y Fu(\(\012\))g Fq(!)f
Fr(L)2563 1758 y Fp(1)2603 1794 y Fu(\(\012\))33 b(b)m(y)1628
2066 y Fr(Q)1705 2081 y Fo(h)1750 2066 y Fr(f)38 b Fu(=)1990
1941 y Fo(n)1940 1971 y Fi(X)1955 2181 y Fo(i)p Fp(=1)2100
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1245 w Fu(\(3.15\))p Black 300 2344 a(where)1561 2563
y Fr(f)1609 2578 y Fo(i)1665 2563 y Fu(=)1884 2496 y(1)p
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y Fo(i)2000 2631 y Fu(\))2065 2427 y Fi(Z)2120 2653 y
Fp(\012)2171 2663 y Ff(i)2218 2563 y Fr(f)11 b(dm)p Black
1178 w Fu(\(3.16\))p Black 300 2816 a(is)35 b(the)h(a)m(v)m(erage)g(v)
-5 b(alue)35 b(of)g Fr(f)46 b Fu(o)m(v)m(er)36 b(\012)1668
2831 y Fo(i)1697 2816 y Fu(.)52 b(It)35 b(is)g(ob)m(vious)g(that)h
Fr(Q)2632 2831 y Fo(h)2712 2816 y Fu(is)f(a)g(Mark)m(o)m(v)i(op)s
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2951 y Fo(h)698 2936 y Fu(\))27 b(=)h Fr(n)p Fu(.)44
b(Moreo)m(v)m(er)34 b Fr(Q)1501 2951 y Fo(h)1578 2936
y Fu(is)e(a)h(Galerkin)e(pro)5 b(jection)32 b(on)m(to)g(\001)2920
2951 y Fo(h)2998 2936 y Fu(in)g(the)h(sense)h(that)1280
3132 y Fr(<)28 b(Q)1461 3147 y Fo(h)1506 3132 y Fr(f)k
Fq(\000)23 b Fr(f)5 b(;)17 b Fu(1)1832 3147 y Fo(i)1888
3132 y Fr(>)p Fu(=)27 b(0)p Fr(;)44 b(i)28 b Fu(=)g(1)p
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Black 300 3328 a(for)32 b(all)f Fr(f)38 b Fq(2)28 b Fr(L)831
3291 y Fp(1)871 3328 y Fu(\(\012\).)44 b(Also)32 b(it)f(is)h(easy)i(to)
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b(an)m(y)h Fr(f)38 b Fq(2)28 b Fr(L)3539 3291 y Fl(1)3615
3328 y Fu(\(\012\).)446 3448 y(Some)h(basic)f(prop)s(erties)h(of)f
(Ulam's)g(metho)s(d)g(are)h(summarized)f(in)g(the)h(follo)m(wing)d
(prop)s(o-)300 3568 y(sition.)p Black 300 3720 a Fj(Pr)-5
b(op)g(osition)34 b(3.1.)p Black 48 w Fu(\(i\))k(Let)g(\001)1429
3684 y Fl(0)1429 3746 y Fo(h)1512 3720 y Fu(=)f Fq(f)1675
3645 y Fi(P)1780 3671 y Fo(n)1780 3749 y(i)p Fp(=1)1915
3720 y Fr(a)1966 3735 y Fo(i)1994 3720 y Fu(1)2043 3735
y Fo(i)2109 3720 y Fu(:)h Fr(a)2225 3735 y Fo(i)2291
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b Fr(P)3478 3735 y Fo(h)3523 3720 y Fu(\(\001)3642 3684
y Fl(0)3642 3746 y Fo(h)3687 3720 y Fu(\))f Fq(\032)300
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y Fo(h)2159 3840 y Fu(.)446 3961 y(\(ii\))f(lim)744 3976
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b(some)g Fr(f)741 4488 y Fo(h)814 4473 y Fq(2)c Fu(\001)989
4437 y Fl(0)989 4499 y Fo(h)1034 4473 y Fu(.)446 4594
y(\(ii\))i(It)i(is)g(enough)g(to)f(assume)i(that)e Fr(f)39
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4927 y Fq(j)p Fr(f)1650 4942 y Fo(i)1700 4927 y Fq(\000)23
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Black 469 w Fu(\(3.19\))p Black Black Black eop
%%Page: 17 26
17 25 bop Black 300 10 a Fk(CHAPTER)34 b(3.)76 b(ULAM'S)33
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49 4 v 82 x Fu(3)2588 5422 y(\))p Fr(:)p Black 938 w
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%%Page: 24 33
24 32 bop Black 300 10 a Fk(CHAPTER)34 b(4.)76 b(PIECEWISE)35
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%%Page: 25 34
25 33 bop Black Black Black Black 1714 150 a Fn(Chapter)53
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2180 y Fu(+)22 b Fr(t)1581 2195 y Fo(w)1638 2180 y Fu(\()p
Fr(p)g Fq(\000)h Fu(1\)\))f(+)g(\()p Fr(t)2165 2195 y
Fo(s)2224 2180 y Fu(+)g Fr(nt)2415 2195 y Fo(w)2472 2180
y Fu(\))p Fr(pl)r(og)2684 2195 y Fp(2)2723 2180 y Fr(p)g
Fu(+)g Fr(pn)2999 2152 y Fp(2)3039 2180 y Fr(M)10 b(t)3178
2195 y Fo(pr)r(od)3333 2112 y Fr(:)p Black 231 w Fu(\(6.13\))p
Black 300 2385 a(The)34 b(e\016ciency)g(of)e(the)h(UMP)-8
b(A)33 b(is)677 2659 y Fr(E)g Fu(=)897 2592 y Fr(s)p
896 2636 49 4 v 896 2727 a(p)982 2659 y Fu(=)1823 2592
y Fr(n)1881 2555 y Fp(2)1921 2592 y Fu(\()p Fr(t)1994
2607 y Fo(mc)2113 2592 y Fu(+)22 b Fr(M)10 b(t)2350 2607
y Fo(pr)r(od)2495 2592 y Fu(\))p 1096 2636 2165 4 v 1096
2727 a Fr(n)1154 2699 y Fp(2)1194 2727 y Fu(\()p Fr(t)1267
2742 y Fo(mc)1386 2727 y Fu(+)22 b Fr(t)1519 2742 y Fo(w)1576
2727 y Fu(\()p Fr(p)g Fq(\000)h Fu(1\)\))e(+)h(\()p Fr(t)2102
2742 y Fo(s)2161 2727 y Fu(+)g Fr(nt)2352 2742 y Fo(w)2409
2727 y Fu(\))p Fr(pl)r(og)2621 2742 y Fp(2)2661 2727
y Fr(p)g Fu(+)g Fr(pn)2937 2699 y Fp(2)2976 2727 y Fr(M)10
b(t)3115 2742 y Fo(pr)r(od)3270 2659 y Fr(:)p Black 294
w Fu(\(6.14\))p Black 300 2932 a(If)33 b Fr(n)f Fu(is)g(k)m(ept)i
(\014xed,)g(then)f(w)m(e)h(can)f(get)f(the)h(relation)e(b)s(et)m(w)m
(een)k Fr(E)j Fu(and)33 b Fr(p)f Fu(as)h(follo)m(ws.)758
3206 y Fr(E)6 b Fu(\()p Fr(p)p Fu(\))83 b(=)1937 3139
y Fr(n)1995 3102 y Fp(2)2035 3139 y Fu(\()p Fr(t)2108
3154 y Fo(mc)2228 3139 y Fu(+)22 b Fr(M)10 b(t)2465 3154
y Fo(pr)r(od)2609 3139 y Fu(\))p 1213 3183 2159 4 v 1213
3274 a Fr(pl)r(og)1387 3289 y Fp(2)1426 3274 y Fr(p)p
Fu(\()p Fr(t)1548 3289 y Fo(s)1607 3274 y Fu(+)23 b Fr(nt)1799
3289 y Fo(w)1856 3274 y Fu(\))f(+)g Fr(pn)2121 3246 y
Fp(2)2160 3274 y Fu(\()p Fr(t)2233 3289 y Fo(w)2312 3274
y Fu(+)g Fr(M)10 b(t)2549 3289 y Fo(pr)r(od)2694 3274
y Fu(\))22 b(+)g Fr(n)2910 3246 y Fp(2)2950 3274 y Fu(\()p
Fr(t)3023 3289 y Fo(mc)3142 3274 y Fq(\000)h Fr(t)3277
3289 y Fo(w)3334 3274 y Fu(\))1044 3471 y(=)1554 3403
y Fr(K)1637 3418 y Fo(n)p 1213 3448 811 4 v 1213 3539
a Fr(pl)r(og)1387 3554 y Fp(2)1426 3539 y Fr(p)g Fu(+)f
Fr(pK)1735 3505 y Fp(1)1728 3567 y Fo(d)1796 3539 y Fu(+)g
Fr(K)1984 3505 y Fp(2)1977 3567 y Fo(d)2034 3471 y Fr(;)300
3744 y Fu(where)31 b(a)e(series)h(of)g Fr(K)7 b Fu(s)30
b(are)f(di\013eren)m(t)h(constan)m(ts.)44 b(F)-8 b(rom)28
b(the)i(form)m(ula)e(w)m(e)j(can)f(de\014nitely)f(sa)m(y)300
3864 y(that)j(for)g(the)h(UMP)-8 b(A,)34 b(when)g Fr(p)e
Fu(increases,)i Fr(E)k Fu(will)31 b(decrease.)446 3985
y(If)i Fr(p)f Fu(is)g(k)m(ept)i(\014xed,)g(w)m(e)g(can)e(get)h(the)g
(relation)e(b)s(et)m(w)m(een)j Fr(E)39 b Fu(and)33 b
Fr(n)f Fu(as)h(follo)m(ws.)706 4259 y Fr(E)6 b Fu(\()p
Fr(n)p Fu(\))84 b(=)1942 4191 y Fr(n)2000 4155 y Fp(2)2040
4191 y Fu(\()p Fr(t)2113 4206 y Fo(mc)2232 4191 y Fu(+)22
b Fr(M)10 b(t)2469 4206 y Fo(pr)r(od)2614 4191 y Fu(\))p
1171 4236 2254 4 v 1171 4327 a Fr(n)1229 4298 y Fp(2)1268
4327 y Fu(\()p Fr(t)1341 4342 y Fo(mc)1461 4327 y Fu(+)22
b Fr(M)10 b(pt)1747 4342 y Fo(pr)r(od)1914 4327 y Fu(+)22
b Fr(t)2047 4342 y Fo(w)2103 4327 y Fu(\()p Fr(p)g Fq(\000)h
Fu(1\)\))f(+)g Fr(nt)2650 4342 y Fo(w)2707 4327 y Fr(pl)r(og)2881
4342 y Fp(2)2920 4327 y Fr(p)g Fu(+)g Fr(t)3124 4342
y Fo(s)3161 4327 y Fr(pl)r(og)3335 4342 y Fp(2)3375 4327
y Fr(p)1002 4534 y Fu(=)1391 4467 y Fr(n)1449 4431 y
Fp(2)1488 4467 y Fr(E)1560 4482 y Fo(u)p 1171 4511 656
4 v 1171 4603 a Fr(n)1229 4574 y Fp(2)1290 4603 y Fu(+)g
Fr(nK)1536 4568 y Fp(1)1529 4630 y Fo(d)1598 4603 y Fu(+)g
Fr(K)1786 4568 y Fp(2)1779 4630 y Fo(d)1836 4534 y Fr(:)300
4807 y Fu(F)-8 b(rom)45 b(the)i(form)m(ula)e(w)m(e)i(kno)m(w)h(that)e
(with)g(the)h(increase)g(of)f Fr(n)p Fu(,)k(the)d(e\016ciency)h
Fr(E)53 b Fu(in)46 b(the)300 4928 y(UMP)-8 b(A)39 b(will)d(sho)m(w)k(a)
e(complicated)f(b)s(eha)m(vior.)60 b(That)39 b(means)f(that)h
Fr(E)44 b Fu(ma)m(y)38 b(decrease,)k(and)300 5048 y(also)47
b(under)h(some)g(conditions)f(it)g(will)e(increase)j(dep)s(ending)g(on)
g(the)g(parameters)g(of)g(the)300 5169 y(parallel)30
b(computers.)44 b(When)33 b Fr(n)g Fu(go)s(es)g(to)f(in\014nit)m(y)-8
b(,)32 b Fr(E)38 b Fu(will)31 b(approac)m(h)i(a)f(constan)m(t)1293
5431 y Fr(E)1365 5446 y Fo(u)1438 5431 y Fu(=)1829 5364
y Fr(t)1864 5379 y Fo(mc)1984 5364 y Fu(+)22 b Fr(M)10
b(t)2221 5379 y Fo(pr)r(od)p 1551 5408 1093 4 v 1551
5500 a Fr(t)1586 5515 y Fo(mc)1706 5500 y Fu(+)22 b Fr(M)10
b(pt)1992 5515 y Fo(pr)r(od)2159 5500 y Fu(+)22 b Fr(t)2292
5515 y Fo(w)2349 5500 y Fu(\()p Fr(p)g Fq(\000)g Fu(1\))2654
5431 y Fr(:)p Black 910 w Fu(\(6.15\))p Black Black Black
eop
%%Page: 34 43
34 42 bop Black 300 10 a Fk(CHAPTER)34 b(6.)76 b(P)-8
b(ARALLEL)33 b(ALGORITHMS)1528 b Fu(34)p Black 446 274
a(No)m(w)42 b(w)m(e)g(tak)m(e)g(a)f(lo)s(ok)f(at)h(the)h(equation)e
(\(6.15\).)69 b(When)42 b(the)g(pro)s(cessor)g(n)m(um)m(b)s(er)f
Fr(p)g Fu(is)300 395 y(1,)i Fr(E)491 410 y Fo(u)577 395
y Fu(is)e(1.)69 b(The)42 b(larger)e Fr(p)h Fu(is,)i(the)e(smaller)e
Fr(E)2142 410 y Fo(u)2228 395 y Fu(will)g(b)s(e.)69 b(If)41
b Fr(p)g Fu(go)s(es)h(to)e(in\014nit)m(y)-8 b(,)43 b
Fr(E)3602 410 y Fo(u)3688 395 y Fu(will)300 515 y(approac)m(h)d(0.)62
b(This)40 b(is)e(b)s(ecause)j(with)d(the)i(UMP)-8 b(A)40
b(the)f(parallel)e(time)g Fr(T)3116 530 y Fo(par)r(a)3304
515 y Fu(of)i(\(6.4\))f(con-)300 635 y(tains)31 b(a)g(sequen)m(tial)g
(part)h Fr(M)10 b(n)1445 599 y Fp(2)1485 635 y Fr(t)1520
650 y Fo(pr)r(od)1696 635 y Fu(whic)m(h)31 b(will)e(de\014nitely)j
(degrade)g(the)f(e\016ciency)i(when)g Fr(n)300 756 y
Fu(increases.)44 b(Numerical)28 b(exp)s(erimen)m(ts)i(in)g(the)g(next)h
(c)m(hapter)g(will)d(illustrate)g(this)h(conclusion.)446
876 y(F)-8 b(rom)31 b Fr(T)759 891 y Fo(par)r(a)941 876
y Fu(of)h(equation)g(\(6.4\),)g(w)m(e)i(obtain)d(the)i(cost)g(of)f(the)
h(UMP)-8 b(A)34 b(is)594 1096 y Fr(C)h Fu(=)27 b Fr(pT)908
1111 y Fo(par)r(a)1085 1096 y Fu(=)g Fr(n)1246 1055 y
Fp(2)1286 1096 y Fu(\()p Fr(t)1359 1111 y Fo(mc)1478
1096 y Fu(+)22 b Fr(t)1611 1111 y Fo(w)1668 1096 y Fu(\()p
Fr(p)g Fq(\000)h Fu(1\)\))f(+)g(\()p Fr(t)2195 1111 y
Fo(s)2254 1096 y Fu(+)g Fr(nt)2445 1111 y Fo(w)2502 1096
y Fu(\))p Fr(pl)r(og)2714 1111 y Fp(2)2753 1096 y Fr(p)g
Fu(+)g Fr(pn)3029 1055 y Fp(2)3069 1096 y Fr(M)10 b(t)3208
1111 y Fo(pr)r(od)3353 1096 y Fr(:)p Black 211 w Fu(\(6.16\))p
Black 446 1316 a(W)-8 b(e)40 b(no)m(w)h(analyze)f(the)g(complete)f
(parallel)e(algorithm)f(\(CP)-8 b(A\).)41 b(The)g(time)d(of)h(the)h(b)s
(est)300 1437 y(sequen)m(tial)33 b(algorithm)c(is)j(the)h(same)g(as)f
(\(6.12\),)g(that)h(is,)1124 1657 y Fr(T)1195 1592 y
Fh(0)1181 1681 y Fo(seq)1312 1657 y Fu(=)28 b Fr(n)1474
1615 y Fp(2)1514 1657 y Fr(t)1549 1672 y Fo(mc)1668 1657
y Fu(+)22 b Fr(M)10 b(t)1905 1672 y Fo(pr)r(od)2050 1657
y Fr(n)2108 1615 y Fp(2)2175 1657 y Fu(=)28 b Fr(n)2337
1615 y Fp(2)2377 1657 y Fu(\()p Fr(t)2450 1672 y Fo(mc)2569
1657 y Fu(+)22 b Fr(M)10 b(t)2806 1672 y Fo(pr)r(od)2951
1657 y Fu(\))p Fr(:)575 b Fu(\(6.17\))300 1877 y(Considering)32
b(\(6.8\))g(and)h(\(6.17\),)f(the)h(sp)s(eedup)h(of)e(the)h(CP)-8
b(A)33 b(is)f(obtained)g(as)559 2173 y Fr(s)605 2109
y Fh(0)659 2173 y Fu(=)795 2097 y Fr(T)866 2037 y Fh(0)852
2121 y Fo(seq)p 772 2150 206 4 v 772 2241 a Fr(T)843
2194 y Fh(0)829 2266 y Fo(par)r(a)1016 2173 y Fu(=)1874
2106 y Fr(pn)1981 2070 y Fp(2)2021 2106 y Fu(\()p Fr(t)2094
2121 y Fo(mc)2213 2106 y Fu(+)22 b Fr(M)10 b(t)2450 2121
y Fo(pr)r(od)2595 2106 y Fu(\))p 1129 2150 2250 4 v 1129
2241 a Fr(n)1187 2213 y Fp(2)1227 2241 y Fr(t)1262 2256
y Fo(mc)1381 2241 y Fu(+)22 b Fr(M)10 b(n)1641 2213 y
Fp(2)1682 2241 y Fr(t)1717 2256 y Fo(pr)r(od)1883 2241
y Fu(+)22 b Fr(M)10 b Fu(\()p Fr(t)2158 2256 y Fo(s)2218
2241 y Fu(+)22 b Fr(t)2351 2256 y Fo(w)2408 2241 y Fu(\))p
Fr(pl)r(og)2620 2256 y Fp(2)2659 2241 y Fr(p)g Fu(+)g
Fr(t)2863 2256 y Fo(w)2920 2241 y Fr(M)10 b Fu(\()p Fr(p)23
b Fq(\000)f Fu(1\))p Fr(n)3388 2173 y(:)p Black 176 w
Fu(\(6.18\))p Black 300 2456 a(The)34 b(e\016ciency)g(is)620
2738 y Fr(E)698 2673 y Fh(0)752 2738 y Fu(=)866 2671
y Fr(s)912 2611 y Fh(0)p 866 2715 73 4 v 878 2806 a Fr(p)976
2738 y Fu(=)1848 2671 y Fr(n)1906 2634 y Fp(2)1946 2671
y Fu(\()p Fr(t)2019 2686 y Fo(mc)2138 2671 y Fu(+)22
b Fr(M)10 b(t)2375 2686 y Fo(pr)r(od)2520 2671 y Fu(\))p
1089 2715 2228 4 v 1089 2806 a Fr(n)1147 2778 y Fp(2)1187
2806 y Fu(\()p Fr(t)1260 2821 y Fo(mc)1380 2806 y Fu(+)22
b Fr(M)10 b(t)1617 2821 y Fo(pr)r(od)1762 2806 y Fu(\))22
b(+)g Fr(M)10 b Fu(\()p Fr(t)2097 2821 y Fo(s)2156 2806
y Fu(+)22 b Fr(t)2289 2821 y Fo(w)2346 2806 y Fu(\))p
Fr(pl)r(og)2558 2821 y Fp(2)2597 2806 y Fr(p)h Fu(+)f
Fr(t)2802 2821 y Fo(w)2858 2806 y Fr(M)10 b Fu(\()p Fr(p)23
b Fq(\000)g Fu(1\))p Fr(n)3327 2738 y(:)p Black 237 w
Fu(\(6.19\))p Black 300 3015 a(If)33 b Fr(n)f Fu(is)g(k)m(ept)i
(\014xed,)g(w)m(e)g(can)e(get)h(the)g(relation)e(b)s(et)m(w)m(een)j
Fr(E)2531 2979 y Fl(0)2587 3015 y Fu(and)f Fr(p)f Fu(as)h(follo)m(ws.)
647 3289 y Fr(E)725 3248 y Fl(0)749 3289 y Fu(\()p Fr(p)p
Fu(\))82 b(=)1949 3221 y Fr(n)2007 3185 y Fp(2)2047 3221
y Fu(\()p Fr(t)2120 3236 y Fo(mc)2239 3221 y Fu(+)22
b Fr(M)10 b(t)2476 3236 y Fo(pr)r(od)2621 3221 y Fu(\))p
1125 3266 2358 4 v 1125 3357 a Fr(pl)r(og)1299 3372 y
Fp(2)1339 3357 y Fr(pM)g Fu(\()p Fr(t)1565 3372 y Fo(s)1624
3357 y Fu(+)22 b Fr(t)1757 3372 y Fo(w)1814 3357 y Fu(\))g(+)g
Fr(pt)2056 3372 y Fo(w)2113 3357 y Fr(M)10 b(n)23 b Fu(+)f
Fr(n)2454 3328 y Fp(2)2494 3357 y Fu(\()p Fr(t)2567 3372
y Fo(mc)2686 3357 y Fu(+)g Fr(M)10 b(t)2923 3372 y Fo(pr)r(od)3068
3357 y Fu(\))22 b Fq(\000)h Fr(t)3263 3372 y Fo(w)3320
3357 y Fr(M)10 b(n)956 3554 y Fu(=)1469 3486 y Fr(K)1552
3501 y Fo(d)p 1125 3531 811 4 v 1125 3622 a Fr(pl)r(og)1299
3637 y Fp(2)1339 3622 y Fr(p)22 b Fu(+)g Fr(pK)1647 3588
y Fp(1)1640 3650 y Fo(d)1708 3622 y Fu(+)g Fr(K)1896
3588 y Fp(2)1889 3650 y Fo(d)1946 3554 y Fr(:)300 3831
y Fu(F)-8 b(rom)25 b(the)h(form)m(ula)f(w)m(e)i(also)e(kno)m(w)i(that)f
(for)g(the)h(CP)-8 b(A,)27 b(when)g Fr(p)f Fu(increases,)j
Fr(E)3233 3795 y Fl(0)3282 3831 y Fu(will)24 b(decrease.)300
3951 y(If)33 b Fr(p)f Fu(is)g(k)m(ept)i(\014xed,)g(w)m(e)f(can)g(get)g
(the)g(relation)d(b)s(et)m(w)m(een)35 b Fr(E)2522 3915
y Fl(0)2578 3951 y Fu(and)e Fr(n)f Fu(as)h(follo)m(ws.)708
4225 y Fr(E)786 4184 y Fl(0)809 4225 y Fu(\()p Fr(n)p
Fu(\))83 b(=)1954 4157 y Fr(n)2012 4121 y Fp(2)2051 4157
y Fu(\()p Fr(t)2124 4172 y Fo(mc)2244 4157 y Fu(+)22
b Fr(M)10 b(t)2481 4172 y Fo(pr)r(od)2626 4157 y Fu(\))p
1195 4202 2228 4 v 1195 4293 a Fr(n)1253 4264 y Fp(2)1293
4293 y Fu(\()p Fr(t)1366 4308 y Fo(mc)1485 4293 y Fu(+)22
b Fr(M)10 b(t)1722 4308 y Fo(pr)r(od)1867 4293 y Fu(\))22
b(+)g Fr(nt)2118 4308 y Fo(w)2175 4293 y Fr(M)10 b Fu(\()p
Fr(p)23 b Fq(\000)g Fu(1\))e(+)h Fr(M)10 b Fu(\()p Fr(t)2872
4308 y Fo(s)2932 4293 y Fu(+)22 b Fr(t)3065 4308 y Fo(w)3122
4293 y Fu(\))p Fr(pl)r(og)3334 4308 y Fp(2)3373 4293
y Fr(p)1026 4501 y Fu(=)1474 4433 y Fr(n)1532 4397 y
Fp(2)p 1195 4478 656 4 v 1195 4569 a Fr(n)1253 4540 y
Fp(2)1315 4569 y Fu(+)g Fr(nK)1561 4535 y Fp(1)1554 4597
y Fo(d)1623 4569 y Fu(+)g Fr(K)1811 4535 y Fp(2)1804
4597 y Fo(d)1860 4501 y Fr(:)300 4792 y Fu(So)34 b(w)m(e)h(can)f(see)h
(that)e(with)g(the)i(increase)f(of)f(the)i(problem)d(size)i
Fr(n)p Fu(,)g(the)h(e\016ciency)g Fr(E)3532 4732 y Fh(0)3592
4792 y Fu(of)f(the)300 4912 y(CP)-8 b(A)40 b(will)d(also)h(exhibit)h
(complicated)f(prop)s(erties)h(dep)s(ending)g(on)h(the)f(parameters)h
(of)f(the)300 5033 y(parallel)30 b(computers.)44 b(When)33
b Fr(n)g Fu(go)s(es)g(to)f(in\014nit)m(y)-8 b(,)32 b
Fr(E)2299 4973 y Fh(0)2358 5033 y Fu(will)e Fj(eventual)5
b(ly)35 b(incr)-5 b(e)g(ase)66 b Fu(to)33 b(1.)446 5153
y(According)g(to)f(\(6.8\),)g(w)m(e)i(get)e(the)h(cost)g(of)f(the)h(CP)
-8 b(A)34 b(is)550 5373 y Fr(C)627 5308 y Fh(0)681 5373
y Fu(=)27 b Fr(pT)904 5308 y Fh(0)890 5398 y Fo(par)r(a)1066
5373 y Fu(=)h Fr(n)1228 5332 y Fp(2)1268 5373 y Fu(\()p
Fr(t)1341 5388 y Fo(mc)1460 5373 y Fu(+)22 b Fr(M)10
b(t)1697 5388 y Fo(pr)r(od)1842 5373 y Fu(\))22 b(+)g
Fr(M)10 b Fu(\()p Fr(t)2177 5388 y Fo(s)2237 5373 y Fu(+)22
b Fr(t)2370 5388 y Fo(w)2427 5373 y Fu(\))p Fr(pl)r(og)2639
5388 y Fp(2)2678 5373 y Fr(p)g Fu(+)g Fr(t)2882 5388
y Fo(w)2939 5373 y Fr(M)10 b(n)p Fu(\()p Fr(p)23 b Fq(\000)g
Fu(1\))p Fr(:)p Black 166 w Fu(\(6.20\))p Black Black
Black eop
%%Page: 35 44
35 43 bop Black 300 10 a Fk(CHAPTER)34 b(6.)76 b(P)-8
b(ARALLEL)33 b(ALGORITHMS)1528 b Fu(35)p Black 446 274
a(Finally)30 b(w)m(e)k(compare)e(the)h(p)s(erformance)f(of)g(the)h(t)m
(w)m(o)g(parallel)d(algorithms.)41 b(Comparing)300 395
y(\(6.16\))32 b(and)g(\(6.20\),)g(in)g(most)g(cases,)i(if)e(it)f(is)h
(true)h(that)381 615 y(\()p Fr(p)22 b Fq(\000)h Fu(1\))p
Fr(n)p Fu(\()p Fr(t)808 630 y Fo(w)864 615 y Fu(\()p
Fr(n)g Fq(\000)f Fr(M)10 b Fu(\))23 b(+)f Fr(t)1380 630
y Fo(pr)r(od)1524 615 y Fr(nM)10 b Fu(\))23 b(+)f(\()p
Fr(t)1918 630 y Fo(w)1975 615 y Fu(\()p Fr(n)h Fq(\000)f
Fu(1\))g Fq(\000)h Fu(\()p Fr(t)2475 630 y Fo(w)2554
615 y Fu(+)f Fr(t)2687 630 y Fo(s)2724 615 y Fu(\)\()p
Fr(M)32 b Fq(\000)23 b Fu(1\)\))p Fr(pl)r(og)3325 630
y Fp(2)3364 615 y Fr(p)k(>)h Fu(0)p Black -2 w(\(6.21\))p
Black 300 835 a(w)m(e)48 b(can)e(obtain)g Fr(C)58 b(>)52
b(C)1301 775 y Fh(0)1327 835 y Fu(.)85 b(This)47 b(means)g(that)f(the)h
(cost)g(of)f(the)h(Ulam)e(matrix)g(parallel)300 955 y(algorithm)22
b(\(UMP)-8 b(A\))25 b(is)g(greater)g(than)g(the)g(cost)g(of)g(the)g
(complete)f(parallel)f(algorithm)e(\(CP)-8 b(A\))300
1075 y(in)28 b(most)h(cases.)44 b(Therefore,)31 b(the)e(CP)-8
b(A)30 b(is)f(more)f(e\016cien)m(t)i(than)g(the)f(UMP)-8
b(A,)30 b(whic)m(h)g(will)d(also)300 1196 y(b)s(e)33
b(sho)m(wn)h(from)d(the)i(n)m(umerical)e(results)i(in)f(the)h(next)g(c)
m(hapter.)446 1316 y(In)38 b(summary)-8 b(,)38 b(when)g(the)g(n)m(um)m
(b)s(er)g(of)f(pro)s(cessors)i Fr(p)e Fu(increases,)j(the)d
(e\016ciency)i(of)e(b)s(oth)300 1437 y(the)h(UMP)-8 b(A)38
b(and)f(the)g(CP)-8 b(A)38 b(is)f(decreasing.)57 b(When)38
b(the)g(n)m(um)m(b)s(er)g Fr(n)f Fu(of)g(sub-in)m(terv)-5
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%%Page: 41 50
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  hpt 0 360 arc stroke } def
/Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def
/DiaW { stroke [] 0 setdash vpt add M
  hpt neg vpt neg V hpt vpt neg V
  hpt vpt V hpt neg vpt V Opaque stroke } def
/BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M
  0 vpt2 neg V hpt2 0 V 0 vpt2 V
  hpt2 neg 0 V Opaque stroke } def
/TriUW { stroke [] 0 setdash vpt 1.12 mul add M
  hpt neg vpt -1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt 1.62 mul V Opaque stroke } def
/TriDW { stroke [] 0 setdash vpt 1.12 mul sub M
  hpt neg vpt 1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt -1.62 mul V Opaque stroke } def
/PentW { stroke [] 0 setdash gsave
  translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  Opaque stroke grestore } def
/CircW { stroke [] 0 setdash 
  hpt 0 360 arc Opaque stroke } def
/BoxFill { gsave Rec 1 setgray fill grestore } def
end
%%EndProlog
gnudict begin
gsave
50 50 translate
0.050 0.050 scale
0 setgray
newpath
(Helvetica) findfont 120 scalefont setfont
1.000 UL
LTb
276 240 M
63 0 V
3057 0 R
-63 0 V
204 240 M
(0) Rshow
276 580 M
63 0 V
3057 0 R
-63 0 V
204 580 M
(1) Rshow
276 920 M
63 0 V
3057 0 R
-63 0 V
204 920 M
(2) Rshow
276 1260 M
63 0 V
3057 0 R
-63 0 V
-3129 0 R
(3) Rshow
276 1600 M
63 0 V
3057 0 R
-63 0 V
-3129 0 R
(4) Rshow
276 1940 M
63 0 V
3057 0 R
-63 0 V
-3129 0 R
(5) Rshow
276 2280 M
63 0 V
3057 0 R
-63 0 V
-3129 0 R
(6) Rshow
276 240 M
0 63 V
0 2073 R
0 -63 V
276 120 M
(0) Cshow
773 240 M
0 63 V
0 2073 R
0 -63 V
773 120 M
(1) Cshow
1269 240 M
0 63 V
0 2073 R
0 -63 V
0 -2193 R
(2) Cshow
1766 240 M
0 63 V
0 2073 R
0 -63 V
0 -2193 R
(3) Cshow
2262 240 M
0 63 V
0 2073 R
0 -63 V
0 -2193 R
(4) Cshow
2759 240 M
0 63 V
0 2073 R
0 -63 V
0 -2193 R
(5) Cshow
3255 240 M
0 63 V
0 2073 R
0 -63 V
0 -2193 R
(6) Cshow
1.000 UL
LTb
276 240 M
3120 0 V
0 2136 V
-3120 0 V
276 240 L
1.000 UL
LT0
2829 2253 M
(DPLL) Rshow
3076 2253 Pnt
276 994 Pnt
279 991 Pnt
282 989 Pnt
285 986 Pnt
288 984 Pnt
292 982 Pnt
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1911 2226 Pnt
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1961 2339 Pnt
1964 2346 Pnt
1967 2353 Pnt
1970 2360 Pnt
1973 2367 Pnt
1976 2374 Pnt
1980 246 Pnt
1983 253 Pnt
1986 260 Pnt
1989 267 Pnt
1992 274 Pnt
1995 282 Pnt
1998 289 Pnt
2001 296 Pnt
2004 303 Pnt
2008 310 Pnt
2011 318 Pnt
2014 325 Pnt
2017 332 Pnt
2020 339 Pnt
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 @endspecial Black 910 5282 a(Figure)e(7.9:)43 b(E\016ciency)34
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eop
%%Page: 51 60
51 59 bop Black 300 10 a Fk(CHAPTER)34 b(7.)76 b(NUMERICAL)34
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 @endspecial Black 849 2268 a(Figure)31 b(7.10:)43 b(E\016ciency)34
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g(the)g(Sw)m(eetgum)g(if)f(the)h(problem)e(size)i Fr(n)g
Fu(is)f(v)m(ery)i(large.)p Black Black eop
%%Page: 52 61
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b(RESUL)-8 b(TS)1678 b Fu(52)p Black Black Black Black
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 @endspecial Black 325 2268 a(Figure)32 b(7.12:)43 b(E\016ciency)34
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(algorithm)e(is)j(v)m(ery)i(con)m(v)m(enien)m(t)g(and)e(e\016cien)m(t.)
p Black Black Black 720 5048 a @beginspecial 50 @llx
50 @lly 230 @urx 176 @ury 3240 @rwi @setspecial
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%%Page: 61 70
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1928 y(metho)s(d.)40 b Fb(Sto)-5 b(chastic)35 b(A)n(nalysis)e(and)g
(Applic)-5 b(ations)p Fc(,)33 b(submitted.)p Black 300
2116 a([22])p Black 50 w(J.)d(Ding)f(and)g(Z.)g(W)-8
b(ang.)40 b(P)m(arallel)29 b(computation)g(of)h(in)m(v)-5
b(arian)m(t)28 b(measures.)39 b Fb(A)n(nnals)32 b(in)g(Op)-5
b(er)g(a-)490 2229 y(tions)34 b(R)-5 b(ese)g(ar)g(ch)p
Fc(,)32 b(to)f(app)s(ear,)f(2000.)p Black 300 2416 a([23])p
Black 50 w(J.)e(Ding)g(and)f(A.)h(Zhou.)36 b(The)27 b(pro)5
b(jection)28 b(metho)s(d)g(for)f(computing)g(m)m(ulti-dimensional)d
(abso-)490 2529 y(lutely)29 b(con)m(tin)m(uous)h(in)m(v)-5
b(arian)m(t)30 b(measures.)40 b Fb(J.)32 b(Stat.)i(Phys.)p
Fc(,)d(77\(3/4\):899{908,)37 b(1994.)p Black 300 2717
a([24])p Black 50 w(J.)g(Ding)f(and)g(A.)h(Zhou.)58 b(Piecewise)37
b(linear)e(mark)m(o)m(v)i(appro)m(ximations)f(of)h(frob)s(enius-p)s
(erron)490 2830 y(op)s(erators)23 b(asso)s(ciated)f(with)f(m)m
(ulti-dimensional)d(transformations.)27 b Fb(Nonline)-5
b(ar)26 b(A)n(nal.,)h(TMA)p Fc(,)490 2943 y(25\(4\):399{408,)36
b(1995.)p Black 300 3130 a([25])p Black 50 w(J.)29 b(Ding)f(and)g(A.)h
(Zhou.)37 b(Finite)27 b(appro)m(ximation)h(of)h(frob)s(enius-p)s(erron)
24 b(op)s(erators.)29 b(a)g(solution)490 3243 y(of)22
b(ulam's)e(conjecture)i(to)g(m)m(ulti-dimensional)17
b(transformations.)25 b Fb(Physic)-5 b(a)25 b(D)p Fc(,)c(92:61{68,)27
b(1996.)p Black 300 3431 a([26])p Black 50 w(J.)f(Ding)g(and)f(A.)i
(Zhou.)32 b(On)25 b(the)i(sp)s(ectrum)e(of)h(frob)s(enius-p)s(erron)c
(op)s(erators.)34 b Fb(J.)28 b(Math.)h(A)n(nal.)490 3544
y(Appl.)p Fc(,)i(to)g(app)s(ear.)p Black 300 3731 a([27])p
Black 50 w(P)-8 b(.)40 b(G\023)-45 b(ora)39 b(and)g(A.)g(Bo)m(y)m
(arsky)-8 b(.)68 b(Absolutely)38 b(con)m(tin)m(uous)g(in)m(v)-5
b(arian)m(t)38 b(measures)h(for)g(piecewise)490 3844
y(expanding)29 b Fa(c)966 3811 y Fp(2)1036 3844 y Fc(transformations)h
(in)f Fa(r)1842 3811 y Fo(n)1888 3844 y Fc(.)41 b Fb(Isr)-5
b(ael)34 b(J.)e(Math.)p Fc(,)f(67\(3\):272{286,)36 b(1989.)p
Black 300 4032 a([28])p Black 50 w(P)-8 b(.)30 b(G\023)-45
b(ora)29 b(and)g(A.)g(Bo)m(y)m(arsky)-8 b(.)40 b(Higher)28
b(dimensional)e(p)s(oin)m(t)i(transformations)g(and)h(asymptotic)490
4145 y(measures)h(for)g(cellular)f(automata.)42 b Fb(Comput.)34
b(Math.)f(Appl.)p Fc(,)e(19:13{31,)j(1990.)p Black 300
4332 a([29])p Black 50 w(F.)c(Hun)m(t)f(and)f(W.)i(Miller.)37
b(On)28 b(the)h(appro)m(ximation)f(of)i(in)m(v)-5 b(arian)m(t)28
b(measures.)38 b Fb(J.)31 b(Stat.)h(Phys.)p Fc(,)490
4445 y(66:535{548,)j(1992.)p Black 300 4633 a([30])p
Black 50 w(F.)k(Y.)g(Hun)m(t.)64 b(A)39 b(mon)m(te)g(carlo)f(approac)m
(h)h(to)g(the)g(appro)m(ximation)e(of)h(in)m(v)-5 b(arian)m(t)38
b(measures.)490 4746 y Fb(R)-5 b(andom)35 b(&)d(Comput)i(Dynamics)p
Fc(,)d(2\(1\):111{133,)36 b(1994.)p Black 300 4933 a([31])p
Black 50 w(F.)j(Y.)g(Hun)m(t.)64 b(Erratum:)55 b(`a)39
b(mon)m(te)h(carlo)e(approac)m(h)h(to)g(the)f(appro)m(ximation)f(of)i
(in)m(v)-5 b(arian)m(t)490 5046 y(measures'.)41 b Fb(R)-5
b(andom)35 b(&)d(Comput.)i(Dynam.)p Fc(,)d(5\(4\):361{362,)36
b(1998.)p Black 300 5234 a([32])p Black 50 w(D.)d(L.)g(Isaacson)g(and)f
(R.)g(W.)i(Madsen.)46 b Fb(Markov)35 b(Chains,)h(The)-5
b(ory)36 b(and)g(Applic)-5 b(ations)p Fc(.)49 b(John)490
5347 y(Wiley)30 b(&)g(Sons,)g(New)g(Y)-8 b(ork,)31 b(1976.)p
Black Black eop
%%Page: 91 100
91 99 bop Black 300 10 a Fk(BIBLIOGRAPHY)2663 b Fu(91)p
Black Black 300 274 a Fc([33])p Black 50 w(M.)48 b(Jablonski.)90
b(On)47 b(in)m(v)-5 b(arian)m(t)47 b(measures)g(for)g(piecewise)g
Fa(c)2718 241 y Fp(2)2758 274 y Fc(-transformations)g(of)g(the)h
Fa(n)p Fc(-)490 387 y(dimensional)28 b(cub)s(e.)40 b
Fb(A)n(nn.)31 b(Polon.)j(Math.)p Fc(,)d(XLI)s(I)s(I:185{195,)h(1983.)p
Black 300 575 a([34])p Black 50 w(G.)24 b(Keller.)k(Sto)s(c)m(hastic)c
(p)s(erturbation)e(of)h(some)h(strange)g(attractors.)32
b Fb(L)-5 b(e)g(ctur)g(e)27 b(Notes)f(in)g(Phys.)p Fc(,)490
688 y(179:192{193,)36 b(1983.)p Black 300 875 a([35])p
Black 50 w(V.)23 b(Kumar,)g(A.)g(Grama,)i(A.)d(Gupta,)j(and)d(G.)g
(Karypis.)k Fb(Intr)-5 b(o)g(duction)28 b(to)e(Par)-5
b(al)5 b(lel)27 b(Computing)p Fc(.)490 988 y(Benjamin/Cummings,)h(San)i
(F)-8 b(rancisco,)31 b(1994.)p Black 300 1176 a([36])p
Black 50 w(A.)i(Lasota)g(and)f(M.)h(Mac)m(k)m(ey)-8 b(.)49
b Fb(Chaos,)36 b(F)-7 b(r)i(actals,)37 b(and)e(Noise)p
Fc(.)46 b(Springer-V)-8 b(erlag,)32 b(New)g(Y)-8 b(ork,)490
1289 y(1994.)p Black 300 1476 a([37])p Black 50 w(A.)42
b(Lasota)g(and)f(J.)g(A.)g(Y)-8 b(ork)m(e.)75 b(On)40
b(the)i(existence)f(of)h(in)m(v)-5 b(arian)m(t)40 b(measures)h(for)g
(piecewise)490 1589 y(monotonic)31 b(transformations.)39
b Fb(T)-7 b(r)i(ans.)34 b(A)n(mer.)f(Math.)g(So)-5 b(c.)p
Fc(,)31 b(186:481{488,)k(1973.)p Black 300 1777 a([38])p
Black 50 w(T.)24 b(Y.)h(Li.)k(Finite)23 b(appro)m(ximation)g(for)h(the)
h(frob)s(enius-p)s(erron)20 b(op)s(erator,)26 b(a)e(solution)f(to)i
(ulam's)490 1890 y(conjecture.)42 b Fb(J.)32 b(Appr)-5
b(ox.)34 b(The)-5 b(ory)p Fc(,)32 b(17:177{186,)j(1976.)p
Black 300 2077 a([39])p Black 50 w(T.)45 b(Y.)g(Li.)82
b Fb(Er)-5 b(go)g(dic)47 b(the)-5 b(ory)47 b(on)f Fc([0)p
Fa(;)15 b Fc(1].)85 b(Lecture)45 b(Notes)h(at)f(the)g(Dept.)h(of)f
(Math,)k(Ky)m(oto)490 2190 y(Univ)m(ersit)m(y)-8 b(,)30
b(1988.)p Black 300 2378 a([40])p Black 50 w(T.)48 b(Y.)g(Li)f(and)g
(J.)g(A.)h(Y)-8 b(ork)m(e.)94 b(P)m(erio)s(d)47 b(three)h(implies)d(c)m
(haos.)93 b Fb(A)n(mer.)49 b(Math.)g(Monthly)p Fc(,)490
2491 y(82\(10\):985{992,)37 b(1975.)p Black 300 2679
a([41])p Black 50 w(G.)31 b(G.)g(Loren)m(tz.)42 b Fb(Appr)-5
b(oximations)35 b(of)e(F)-7 b(unctions)p Fc(.)42 b(Holt,)31
b(Rinehart)e(and)h(Winston,)g(1966.)p Black 300 2866
a([42])p Black 50 w(E.)42 b(N.)h(Lorenz.)75 b(Deterministic)41
b(nonp)s(erio)s(dic)e(\015o)m(ws.)75 b Fb(J.)43 b(A)n(tmospheric)i
(Sci.)p Fc(,)f(20:130{141,)490 2979 y(1963.)p Black 300
3167 a([43])p Black 50 w(R.)35 b(M.)h(Ma)m(y)-8 b(.)56
b(Biological)35 b(p)s(opulations)e(with)h(nono)m(v)m(erlapping)g
(generations:)50 b(stable)35 b(p)s(oin)m(ts,)490 3280
y(stable)30 b(cycles,)h(and)f(c)m(haos.)42 b Fb(Scienc)-5
b(e)p Fc(,)30 b(186:645{647,)35 b(1974.)p Black 300 3467
a([44])p Black 50 w(W.)44 b(J.)f(Morok)m(o\013.)82 b(Generating)43
b(quasi-random)f(paths)h(for)g(sto)s(c)m(hastic)i(pro)s(cesses.)79
b Fb(SIAM)490 3580 y(R)-5 b(eview)p Fc(,)31 b(40\(4\):765{788,)36
b(1998.)p Black 300 3768 a([45])p Black 50 w(R.)k(Murra)m(y)-8
b(.)67 b(Appro)m(ximation)38 b(error)h(for)g(in)m(v)-5
b(arian)m(t)38 b(densit)m(y)h(calculations.)66 b Fb(J.)41
b(Discr)-5 b(ete)40 b(&)490 3881 y(Continuous)34 b(Dyn.)e(Sys.)p
Fc(,)f(4:535{558,)j(1998.)p Black 300 4068 a([46])p Black
50 w(P)-8 b(.)41 b(P)m(ac)m(heco.)74 b Fb(Par)-5 b(al)5
b(lel)44 b(Pr)-5 b(o)g(gr)g(amming)45 b(with)e(MPI)p
Fc(.)70 b(Morgan)42 b(Kaufmann,)g(San)e(F)-8 b(rancisco,)490
4181 y(1996.)p Black 300 4369 a([47])p Black 50 w(S.)29
b(M.)h(Ulam.)39 b Fb(A)31 b(Col)5 b(le)-5 b(ction)33
b(of)f(Mathematic)-5 b(al)34 b(Pr)-5 b(oblems)p Fc(.)40
b(In)m(ter-science,)31 b(New)e(Y)-8 b(ork,)31 b(1960.)p
Black 300 4557 a([48])p Black 50 w(M.)37 b(A.)g(v)-5
b(an)36 b(Wyk)h(and)f(T.)g(S.)h(Durrani.)57 b(Classi\014cation)34
b(of)j(system)g(resp)s(onse)e(using)g(proba-)490 4669
y(bilistic)28 b(mo)s(delling.)38 b Fb(R)-5 b(ese)g(ar)g(ch)34
b(R)-5 b(ep)g(ort,)35 b(Kentr)-5 b(on,)34 b(South)f(Afric)-5
b(a)p Fc(,)31 b(1999.)p Black 300 4857 a([49])p Black
50 w(M.)h(A.)g(v)-5 b(an)31 b(Wyk)h(and)e(W.)i(H.)g(Steeb.)44
b(Chaos)31 b(in)f(electronics.)43 b(In)31 b Fb(Mathematic)-5
b(al)36 b(Mo)-5 b(del)5 b(ling:)490 4970 y(The)-5 b(ory)34
b(and)g(Applic)-5 b(ations,)34 b(No.2)p Fc(,)d(South)f(Africa,)g(1997.)
j(Klu)m(w)m(er)c(Academic)i(Publishers.)p Black 300 5158
a([50])p Black 50 w(P)-8 b(.)31 b(W)-8 b(alters.)41 b
Fb(A)n(n)32 b(Intr)-5 b(o)g(duction)35 b(to)f(Er)-5 b(go)g(dic)33
b(The)-5 b(ory)p Fc(.)42 b(Springer-V)-8 b(erlag,)30
b(New)g(Y)-8 b(ork,)32 b(1982.)p Black Black eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF