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{SECT 0 {EXCHG {PARA 3 "" 0 "" {TEXT -1 53 "Solutions to the SIAM 100-
Dollar, 100-Digit Challenge" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 207 "Team: Gaston Gonnet, Informatik, ETH, Zurich, \+
and Robert Israel, Mathematics, UBC.\nMost of these solutions are the \+
ones that actually produced the results we sent in, but I (R.I.) have \+
changed a few of them." }}{PARA 0 "" 0 "" {TEXT -1 32 "All these solut
ions use Maple.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 99 "Each problem had an answer that was a single real number,
 of which 10 correct digits were required." }}{PARA 0 "" 0 "" {TEXT 
-1 28 "The complete results are at " }{URLLINK 17 "http://www.comlab.o
x.ac.uk/oucl/work/nick.trefethen/hundred.html" 4 "http://www.comlab.ox
.ac.uk/oucl/work/nick.trefethen/hundred.html" "" }{TEXT -1 55 ".\nOur \+
team was one of 20 first prize winners.  See our " }{URLLINK 17 "prize
" 4 "http://www.math.ubc.ca/~israel/challenge/Prize.jpg" "" }{TEXT -1 
1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 10 "P
roblem 1:" }}{PARA 0 "" 0 "" {TEXT -1 8 "What is " }{XPPEDIT 18 0 "lim
it(int(x^(-1)*cos(x^(-1)*log(x)), x=epsilon..1), epsilon = 0)" "6#-%&l
imitG6$-%$intG6$*&)%\"xG,$\"\"\"!\"\"F--%$cosG6#*&)F+,$F-F.F--%$logG6#
F+F-F-/F+;%(epsilonGF-/F:\"\"!" }{TEXT -1 2 " ?" }}}{SECT 1 {PARA 5 "
" 0 "" {TEXT -1 9 "Solution:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Us
e a change of variables " }{XPPEDIT 18 0 "y = - x^(-1)*log(x)" "6#/%\"
yG,$*&)%\"xG,$\"\"\"!\"\"F*-%$logG6#F(F*F+" }{TEXT -1 60 " to make the
 cosine linear in the variable of integration.  " }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "ti:= time(): Digits:= 20:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 47 "J:= Int(x^(-1)*cos(x^(-1)*ln(x)),x=epsilon..1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG6$*&%\"xG!\"\"-%$cosG6#*&
F)F*-%#lnG6#F)\"\"\"F2/F);%(epsilonGF2" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "alias(W=LambertW): xs:= solve(y=-x^(-1)*ln(x),x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xsG*&%\"yG!\"\"-%\"WG6#F&\"\"\"" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "simplify(PDEtools[dchange](
x=xs,J));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$**%\"yG!\"\"-%$c
osG6#F'\"\"\"-%\"WGF+F,,&F,F,F-F,F(/F';\"\"!,$*&%(epsilonGF(-%#lnG6#F5
F,F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Of course, " }{XPPEDIT 
18 0 "limit(ln(1/epsilon)/epsilon, epsilon=0, right) = infinity" "6#/-
%&limitG6%*&-%#lnG6#*&\"\"\"F,%(epsilonG!\"\"F,F-F./F-\"\"!%&rightG%)i
nfinityG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 
"integrand:= cos(y)*LambertW(y)/(1+LambertW(y))/y;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%*integrandG**%\"yG!\"\"-%$cosG6#F&\"\"\"-%\"WGF*F+,
&F+F+F,F+F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "answer:= Int
(integrand,y=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'answe
rG-%$IntG6$**%\"yG!\"\"-%$cosG6#F)\"\"\"-%\"WGF-F.,&F.F.F/F.F*/F);\"\"
!%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 148 "This is still no
t a very pleasant oscillatory integral, although the singularity at 0 \+
is removable.  We break it up into 3 pieces.  For y from 0 to " }
{XPPEDIT 18 0 "199*Pi" "6#*&\"$*>\"\"\"%#PiGF%" }{TEXT -1 277 ", use o
rdinary numerical integration, (but on the original form of the integr
al, to avoid the apparent singularity at y=0 as well as not having to \+
evaluate W numerically; and using the _Dexp method, as otherwise Maple
 will have trouble because it's close to the singularity at " }
{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 4 ").  " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "Int(1/x * cos(ln(x)/x) , x=subs( y=
199*Pi, xs ) ... 1 ); \nr1:= evalf(Int(op(%),_Dexp)); " }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG!\"\"-%$cosG6#*&F'F(-%#lnG6#F'\"\"
\"F0/F';,$*&%#PiGF(-%\"WG6#,$F5\"$*>F0#F0F:F0" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#r1G$\"5q:U$*f:\\pLK!#?" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 18 "For each interval " }{XPPEDIT 18 0 "(2*n-1)*Pi < y" "6#2*
&,&*&\"\"#\"\"\"%\"nGF(F(F(!\"\"F(%#PiGF(%\"yG" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "`` < (2*n+1)*Pi;" "6#2%!G*&,&*&\"\"#\"\"\"%\"nGF)F)F)F)
F)%#PiGF)" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "100 <= n;" "6#1\"$+\"%
\"nG" }{TEXT -1 26 ",  write the integral as  " }{XPPEDIT 18 0 "int(co
s(t)*W(t+2*n*Pi)/(t+2*n*Pi)/(1+W(t+2*n*Pi)),t = -Pi .. Pi);" "6#-%$int
G6$**-%$cosG6#%\"tG\"\"\"-%\"WG6#,&F*F+*(\"\"#F+%\"nGF+%#PiGF+F+F+,&F*
F+*(F1F+F2F+F3F+F+!\"\",&F+F+-F-6#,&F*F+*(F1F+F2F+F3F+F+F+F6/F*;,$F3F6
F3" }{TEXT -1 114 ".  Expand this in an asymptotic series in n and int
egrate term-by-term.   The terms for even n vanish by symmetry." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "map(factor,asympt(cos(t)*W(t
+2*n*Pi)/(t+2*n*Pi)/(1+W(t+2*n*Pi)), n, 8));" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#,2*,-%$cosG6#%\"tG\"\"\"-%\"WG6#,$*&%\"nGF)%#PiGF)\"\"#
F)F0!\"\",&F*F)F)F)F2F/F2#F)F1*&#F)\"\"%F)*0F%F)F(F)F*F1,&F*F)F1F)F)F0
!\"#F3!\"$F/F:F)F2*2#F)\"#;F)F%F)F(F1F*\"\"$,(*$)F*F1F)F1*&\"\")F)F*F)
F)\"\"*F)F)F0F;F3!\"&F/F;F)*&#F)\"#'*F)*0F%F)F(F?F*F7,**$)F*F?F)\"\"'*
&\"#OF)FBF)F)*&\"#zF)F*F)F)\"#kF)F)F0!\"%F3!\"(F/FTF)F2*2#F)\"$o(F)F%F
)F(F7F*\"\"&,,*$)F*F7F)\"#C*&\"$#>F)FMF)F)*&\"$A'F)FBF)F)*&\"$u*F)F*F)
F)\"$D'F)F)F0FFF3!\"*F/FFF)*&#F)\"%!o(F)*0F*FNF(FYF%F),.*$)F*FYF)\"$?
\"*&\"%+7F)FfnF)F)*&\"%E^F)FMF)F)*&\"&e<\"F)FBF)F)*&\"&VX\"F)F*F)F)\"%
wxF)F)F0!\"'F3!#6F/FapF)F2*2#F)\"&g@*F)F*\"\"(F(FNF%F),0Feo\"%S')*&\"$
?(F))F*FNF)F)*&\"&cd%F)FfnF)F)*&\"'GeDF)F*F)F)\"'\\w6F)*&\"']#[#F)FBF)
F)*&\"'7v8F)FMF)F)F)F0FUF3!#8F/FUF)-%\"OG6#*&F)F)*$)F/FDF)F2F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "J2:= map(factor,map(int, eva
l(%,O=0), t=-Pi..Pi));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#J2G,(*,%#
PiG!\"#-%\"WG6#,$*&%\"nG\"\"\"F'F/\"\"#\"\"$,(*$)F)F0F/F0*&\"\")F/F)F/
F/\"\"*F/F/,&F)F/F/F/!\"&F.!\"$#!\"\"\"\"%*&#F/\"#'*F/*.F)\"\"&,,*$)F)
F=F/\"#C*&\"$#>F/)F)F1F/F/*&\"$A'F/F4F/F/*&\"$u*F/F)F/F/\"$D'F/F/,&*$)
F'F0F/F/\"\"'F<F/F'!\"%F8!\"*F.F9F/F<*&#F/\"%!o(F/*.F)\"\"(,0*$)F)FBF/
\"%S')*&\"$?(F/)F)FRF/F/*&\"&cd%F/FEF/F/*&\"'GeDF/F)F/F/\"'\\w6F/*&\"'
]#[#F/F4F/F/*&\"'7v8F/FIF/F/F/,(FP!#?\"$?\"F/*$)F'F=F/F/F/F'!\"'F8!#8F
.!\"(F/F<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "L:= convert(%,
list): evalf(eval(L,n=100));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7%$!5?
JC/l7-$G+%!#F$!5'4FCaWtr'Rw!#K$!5+UMh'\\\\`=Z\"!#O" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 14 "Apparently by " }{XPPEDIT 18 0 "n=100" "6#/%\"nG
\"$+\"" }{TEXT -1 18 " we only need the " }{XPPEDIT 18 0 "n^(-3)" "6#)
%\"nG,$\"\"$!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "n^(-5)" "6#)%\"
nG,$\"\"&!\"\"" }{TEXT -1 48 " terms.  We'll use these for n from 100 \+
to 1000." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "J3:= op(1,J2)+o
p(2,J2);\nr2:= add(evalf(J3),n=100..1000);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%#J3G,&*,%#PiG!\"#-%\"WG6#,$*&%\"nG\"\"\"F'F/\"\"#\"\"
$,(*$)F)F0F/F0*&\"\")F/F)F/F/\"\"*F/F/,&F)F/F/F/!\"&F.!\"$#!\"\"\"\"%*
&#F/\"#'*F/*.F)\"\"&,,*$)F)F=F/\"#C*&\"$#>F/)F)F1F/F/*&\"$A'F/F4F/F/*&
\"$u*F/F)F/F/\"$D'F/F/,&*$)F'F0F/F/\"\"'F<F/F'!\"%F8!\"*F.F9F/F<" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r2G$!5$4GeT<Er\"Q?!#D" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 87 "For n > 1000, use the asymptotics of the \+
Euler-Maclaurin form for the summation of the " }{XPPEDIT 18 0 "n^(-3)
" "6#)%\"nG,$\"\"$!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "n^(-5)" "
6#)%\"nG,$\"\"&!\"\"" }{TEXT -1 7 " terms." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 43 "J4:= map(factor,asympt(eulermac(J3,n),n)); " }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#J4G,,*,-%\"WG6#,$*&%\"nG\"\"\"%#PiG
F-\"\"#F/,&F'F-F/F-F-F.!\"#,&F'F-F-F-!\"$F,F1#F-\"\"%*.#F-\"\")F-F.F1F
'\"\"$,(*$)F'F/F-F/*&F8F-F'F-F-\"\"*F-F-F2!\"&F,F3F-*0#F-\"#KF-F'F5,**
$)F'F9F-\"\"'*&\"#OF-F<F-F-*&\"#zF-F'F-F-\"#kF-F-,&*$)F.F/F-F-F/!\"\"F
-F.!\"%F2!\"(F,FPF-*0#F-\"$#>F-F'\"\"&,,*$)F'F5F-\"#C*&FTF-FEF-F-*&\"$
A'F-F<F-F-*&\"$u*F-F'F-F-\"$D'F-F-,&FMF-FFFOF-F.FPF2!\"*F,F?F--%\"OG6#
*&F-F-*$)F,FFF-FOF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "r3:=
 -evalf(eval(J4,\{O=0, n=1001\}));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%#r3G$!5;c:SwX$*3r@!#F" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "
r:= r1+r2+r3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG$\"5[;!zxnJuOB$
!#?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Actual answer, as given by
 Trefethen to 40 digits:" }}{PARA 0 "" 0 "" {TEXT -1 61 "             \+
0.3233674316 7777876139 9370087952 1704466510..." }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 20 "(time()-ti)*seconds;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$%(secondsG$\"&z*[!\"$" }}}}{EXCHG {PARA 4 "" 0 "" 
{TEXT -1 10 "Problem 2:" }}{PARA 0 "" 0 "" {TEXT -1 34 "A photon movin
g at speed 1 in the " }{XPPEDIT 18 0 "x-y" "6#,&%\"xG\"\"\"%\"yG!\"\"
" }{TEXT -1 17 " plane starts at " }{XPPEDIT 18 0 "t=0" "6#/%\"tG\"\"!
" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "(x,y) = (0.5,0.1)" "6#/6$%\"xG%\"
yG6$-%&FloatG6$\"\"&!\"\"-F)6$\"\"\"F," }{TEXT -1 55 " heading due eas
t.  Around every integer lattice point " }{XPPEDIT 18 0 "``(i,j);" "6#
-%!G6$%\"iG%\"jG" }{TEXT -1 43 " in the plane, a circular mirror of ra
dius " }{XPPEDIT 18 0 "1/3" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 61 " ha
s been erected.  How far from the origin is the photon at " }{XPPEDIT 
18 0 "t=10" "6#/%\"tG\"#5" }{TEXT -1 1 "?" }}}{SECT 1 {PARA 5 "" 0 "" 
{TEXT -1 9 "Solution:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 181 "You have
 to be careful to use enough digits here, because there are 13 reflect
ions, and each reflection from a curved surface amplifies the error.  \+
We needed 11 additional digits.  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
27 "The state of the photon is " }{XPPEDIT 18 0 "[x,y,dx,dy,s]" "6#7'%
\"xG%\"yG%#dxG%#dyG%\"sG" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "[x,y]
" "6#7$%\"xG%\"yG" }{TEXT -1 18 " is the position, " }{XPPEDIT 18 0 "[
dx,dy]" "6#7$%#dxG%#dyG" }{TEXT -1 15 " the velocity, " }{XPPEDIT 18 
0 "s" "6#%\"sG" }{TEXT -1 41 " the distance travelled.  We always have
 " }{XPPEDIT 18 0 "dx^2 + dy^2 = 1" "6#/,&*$%#dxG\"\"#\"\"\"*$%#dyGF'F
(F(" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 53 "Starting  in thi
s state, you hit a mirror centred at " }{XPPEDIT 18 0 "[mx, my]" "6#7$
%#mxG%#myG" }{TEXT -1 4 " if " }{XPPEDIT 18 0 "abs((mx-x)*dy-(my-y)*dx
) < 1/3;" "6#2-%$absG6#,&*&,&%#mxG\"\"\"%\"xG!\"\"F+%#dyGF+F+*&,&%#myG
F+%\"yGF-F+%#dxGF+F-*&F+F+\"\"$F-" }}{PARA 0 "" 0 "" {TEXT -1 90 "(ass
uming that mirror is in the forward direction, and you don't hit anyth
ing else first)." }}{PARA 0 "" 0 "" {TEXT -1 11 "If so, let " }
{XPPEDIT 18 0 "p = [px, py]" "6#/%\"pG7$%#pxG%#pyG" }{TEXT -1 38 " be \+
the point where it hits.  This is " }{XPPEDIT 18 0 "[x + t*dx, y + t*d
y]" "6#7$,&%\"xG\"\"\"*&%\"tGF&%#dxGF&F&,&%\"yGF&*&F(F&%#dyGF&F&" }
{TEXT -1 7 " where " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "(x-mx)^2 + (y-my
)^2 + 2*(dx*(x-mx) + dy*(y-my))*t + t^2 = 1/9" "6#/,**$,&%\"xG\"\"\"%#
mxG!\"\"\"\"#F(*$,&%\"yGF(%#myGF*F+F(*(F+F(,&*&%#dxGF(,&F'F(F)F*F(F(*&
%#dyGF(,&F.F(F/F*F(F(F(%\"tGF(F(*$F8F+F(*&F(F(\"\"*F*" }{TEXT -1 26 ".
  Then the next state is " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "[px, py, d
x - h*(mx - px), dy - h*(my-py), s + sqrt((px-x)^2+(py-y)^2)] " "6#7'%
#pxG%#pyG,&%#dxG\"\"\"*&%\"hGF(,&%#mxGF(F$!\"\"F(F-,&%#dyGF(*&F*F(,&%#
myGF(F%F-F(F-,&%\"sGF(-%%sqrtG6#,&*$,&F$F(%\"xGF-\"\"#F(*$,&F%F(%\"yGF
-F<F(F(" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "h = 18*
((mx-px)*dx + (my-py)*dy)" "6#/%\"hG*&\"#=\"\"\",&*&,&%#mxGF'%#pxG!\"
\"F'%#dxGF'F'*&,&%#myGF'%#pyGF-F'%#dyGF'F'F'" }{TEXT -1 1 "." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 230 "plots[display]([plottools[c
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\nplots[arrow]([1,0],[1,2],shape=arrow,colour=blue),\nplots[arrow]([1,
0],[2,0],shape=arrow,colour=red)\n],axes=none,scaling=constrained);" }
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$\"\"#F*$!\"#F*F'7%7$$\"0++++++5\"F_s$!/++++++XF_sF'7$$\"0++++++I\"F_s
$!/++++++NF_s-%&STYLEG6#%,PATCHNOGRIDG-%'COLOURG6&%$RGBGF+F+$\"*++++\"
!\")-F$6&7$F'7$FdzFdz7%7$$\"0++++++q\"F_s$\"0++++++l\"F_sFa\\l7$$\"0++
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45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Global variables: " }{MPLTEXT 0 
21 15 "x, y, dx, dy, s" }{TEXT -1 40 " for current state; \nif going m
ainly in " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 12 " direction, " }
{MPLTEXT 0 21 6 "iy = 0" }{TEXT -1 5 " and " }{MPLTEXT 0 21 2 "ix" }
{TEXT -1 21 " is integer to go in " }{XPPEDIT 18 0 "x" "6#%\"xG" }
{TEXT -1 26 " direction to next mirror;" }}{PARA 0 "" 0 "" {TEXT -1 
19 "if going mainly in " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 12 " d
irection, " }{MPLTEXT 0 21 6 "ix = 0" }{TEXT -1 5 " and " }{MPLTEXT 0 
21 2 "iy" }{TEXT -1 21 " is integer to go in " }{XPPEDIT 18 0 "y" "6#%
\"yG" }{TEXT -1 26 " direction to next mirror." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 1164 "move:= proc()\n# return true if you hit the \+
mirror or end of path\n# and update state in this case,\n# false if yo
u don't (leaving state unchanged)\n  local mx, my, t, px, py,h,ds;\n  \+
global x,y,dx,dy,s,ix,iy;\n  if iy = 0 then # going in x direction\n  \+
  mx:= round(x+ix-.001);\n    my:= round(y+dy/dx*ix);\n    ds:= sqrt(1
+(dy/dx)^2)*abs(mx-x);\n  else # going in y direction\n    my:= round(
y+iy);\n    mx:= round(x+dx/dy*iy);\n    ds:= sqrt(1+(dx/dy)^2)*abs(my
-y);\n  fi;\n  if abs((mx-x)*dy - (my-y)*dx) < 1/3 then\n    t:= min(f
solve((x+t*dx-mx)^2 + (y+t*dy-my)^2=1/9,t));\n    px:= x+t*dx; py:= y+
t*dy;\n    ds:= sqrt((px-x)^2+(py-y)^2);\n    if s + ds >= 10 then\n  \+
    x:= x + (10-s)*dx;\n      y:= y + (10-s)*dy;\n      s:= 10;\n     \+
 return true;\n    else\n      h:= 18*((mx-px)*dx+(my-py)*dy);\n      \+
s:= s + ds;\n      dx:= dx - h*(mx-px);\n      dy:= dy - h*(my-py);\n#
 dx^2 + dy^2 should stay constant, but make sure\n      h:= sqrt(dx^2+
dy^2);\n      dx:= dx/h; dy:= dy/h;\n      x:= px; y:= py;\n      retu
rn true;\n    fi\n  # no hit, but maybe we'd go too far\n  elif s + ds
 >= 10 then\n    x:= x + (10-s)*dx;\n    y:= y + (10-s)*dy;\n    s:= 1
0;\n    return true;\n  fi;\nfalse   \nend;" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%%moveGf*6\"6)%#mxG%#myG%\"tG%#pxG%#pyG%\"hG%#dsGF&F&C
%@%/%#iyG\"\"!C%>8$-%&roundG6#,(%\"xG\"\"\"%#ixGF<$F<!\"$!\"\">8%-F86#
,&%\"yGF<*(%#dyGF<%#dxGF@F=F<F<>8**&-%%sqrtG6#,&F<F<*&FH\"\"#FI!\"#F<F
<-%$absG6#,&F6F<F;F@F<C%>FB-F86#,&FFF<F2F<>F6-F86#,&F;F<*(FIF<FHF@F2F<
F<>FK*&-FN6#,&F<F<*&FIFRFHFSF<F<-FU6#,&FBF<FFF@F<@&2-FU6#,&*&FWF<FHF<F
<*&FdoF<FIF<F@#F<\"\"$C'>8&-%$minG6#-%'fsolveG6$/,&*$),(F;F<*&F`pF<FIF
<F<F6F@FRF<F<*$),(FFF<*&F`pF<FHF<F<FBF@FRF<F<#F<\"\"*F`p>8',&F;F<F\\qF
<>8(,&FFF<F`qF<>FK-FN6#,&*$),&FdqF<F;F@FRF<F<*$),&FgqF<FFF@FRF<F<@%1\"
#5,&%\"sGF<FKF<C&>F;,&F;F<*&,&FerF<FgrF@F<FIF<F<>FF,&FFF<*&F\\sF<FHF<F
<>FgrFerO%%trueGC,>8),&*&,&F6F<FdqF@F<FIF<\"#=*(FisF<,&FBF<FgqF@F<FHF<
F<>FgrFfr>FI,&FIF<*&FesF<FhsF<F@>FH,&FHF<*&FesF<F[tF<F@>Fes-FN6#,&*$)F
IFRF<F<*$)FHFRF<F<>FI*&FIF<FesF@>FH*&FHF<FesF@>F;Fdq>FFFgqOFbsFdrC&>F;
Fjr>FFF^s>FgrFerOFbs%&falseGF&6)F;FFFIFHFgrF=F2F&" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 19 "Initial conditions." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 92 "x,y,dx,dy,s:= 0.5, 0.1, 1, 0, 0; path:= [x,y]; ix:= 1
; iy:= 0;\n_Envsignum0:= 0; Digits:= 50;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>6'%\"xG%\"yG%#dxG%#dyG%\"sG6'$\"\"&!\"\"$\"\"\"F-F/\"\"!F0" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%pathG7$$\"\"&!\"\"$\"\"\"F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ixG\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#iyG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,_Envs
ignum0G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#]" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 288 "while s < 10 do\n  if move(
) then # reassess directions\n    if abs(dx) >= abs(dy) then\n      ix
:= signum(dx); iy:= 0\n    else\n      ix:= 0; iy:= signum(dy);\n    f
i;\n    path:= path,[x,y];\n  else # go to next mirror in same directi
on\n    ix:= ix+signum(ix);\n    iy:= iy+signum(iy);\n  fi\nod:" }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "The final state:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "x,y,dx,dy,s;" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6'$!Swb+!e*=X?<Rm=<Pdx5$='4')p#HO(!#]$!S*e'=fx
8'oY)4n42wV>v8djjpU'p'F%$\"SpX!)QP$=#R]oQ,)='\\2=P+x`rLd$*F%$\"S&QK'=(
=5i_Xj1rSOoi`9nvjsq_$F%\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Th
e answer:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "ans:=sqrt(x^2+
y^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ansG$\"S*)*HG@8H5i@nU4=J!*
3;aLW>HE&**!#]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "with(plo
ts): display(\{plot([path]),seq(seq(plottools[circle]([i,j],1/3),i=-1.
.1),j=-1..2)\},scaling=constrained,axes=frame);" }}{PARA 7 "" 1 "" 
{TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}
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$F[zFfgm7$FfyFcgm7$FayFa[n7$F\\yF[gm7$FgxFhfm-%(SCALINGG6#%,CONSTRAINE
DG-%+AXESLABELSG6$Q!6\"F\\io-%*AXESSTYLEG6#%&FRAMEG-%%VIEWG6$%(DEFAULT
GFeio" 1 2 0 1 10 0 2 9 1 3 1 1.000000 45.000000 45.000000 0 0 "Curve \+
1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve \+
8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" }}}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 47 "Just for comparison, here it is with 40 d
igits." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 405 "x,y,dx,dy,s:= 0.
5, 0.1, 1, 0, 0: path:= [x,y]: ix:= 1: iy:= 0:\n_Envsignum0:= 0: Digit
s := 40:\nwhile s < 10 do\n  if move() then # reassess directions\n   \+
 if abs(dx) >= abs(dy) then\n      ix:= signum(dx); iy:= 0\n    else\n
      ix:= 0; iy:= signum(dy);\n    fi;\n    path:= path,[x,y];\n  els
e # go to next mirror in same direction\n    ix:= ix+signum(ix);\n    \+
iy:= iy+signum(iy);\n  fi\nod:\nans40:= sqrt(x^2+y^2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%&ans40G$\"IkL4A0B%4=J!*3;aLW>HE&**!#S" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "ans40 - ans;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$!,Fp6%pO!#S" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
69 "And here's what we would have got with a starting point different \+
by " }{XPPEDIT 18 0 "10^(-10)" "6#)\"#5,$F$!\"\"" }{TEXT -1 20 " in th
e y direction." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 403 "path0:= \+
path:\nx,y,dx,dy,s:= 0.5, 0.1+1e-10, 1, 0, 0: path:= [x,y]: ix:= 1: iy
:= 0:\n_Envsignum0:= 0: Digits := 40:\nwhile s < 10 do\n  if move() th
en # reassess directions\n    if abs(dx) >= abs(dy) then\n      ix:= s
ignum(dx); iy:= 0\n    else\n      ix:= 0; iy:= signum(dy);\n    fi;\n
    path:= path,[x,y];\n  else # go to next mirror in same direction\n
    ix:= ix+signum(ix);\n    iy:= iy+signum(iy);\n  fi\nod:\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "display(\{plot([path],colou
r=blue),plot([path0],colour=red),seq(seq(plottools[circle]([i,j],1/3),
i=-1..1),j=-1..2)\},scaling=constrained,axes=frame);" }}{PARA 13 "" 1 
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0 0 "Curve 1" }}}}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 10 "Problem 3:" }}
{PARA 0 "" 0 "" {TEXT -1 20 "The infinite matrix " }{XPPEDIT 18 0 "A" 
"6#%\"AG" }{TEXT -1 14 " with entries " }{XPPEDIT 18 0 "a[1,1] = 1" "6
#/&%\"aG6$\"\"\"F'F'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[1,2] = 1/2" "
6#/&%\"aG6$\"\"\"\"\"#*&F'F'F(!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "
a[2,1]=1/3" "6#/&%\"aG6$\"\"#\"\"\"*&F(F(\"\"$!\"\"" }{TEXT -1 2 ", " 
}{XPPEDIT 18 0 "a[1,3] = 1/4" "6#/&%\"aG6$\"\"\"\"\"$*&F'F'\"\"%!\"\"
" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[2,2]=1/5" "6#/&%\"aG6$\"\"#F'*&\"
\"\"F)\"\"&!\"\"" }{TEXT -1 1 "," }{XPPEDIT 18 0 "a[3,1]=1/6" "6#/&%\"
aG6$\"\"$\"\"\"*&F(F(\"\"'!\"\"" }{TEXT -1 33 ", etc., is a bounded op
erator on " }{XPPEDIT 18 0 "l^2" "6#*$%\"lG\"\"#" }{TEXT -1 13 ".  Wha
t is ||" }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 4 "||?." }}}{SECT 1 
{PARA 5 "" 0 "" {TEXT -1 9 "Solution:" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 57 "The norm is the square root of the largest eigenvalue of " }
{XPPEDIT 18 0 "A^T*A;" "6#*&)%\"AG%\"TG\"\"\"F%F'" }{TEXT -1 46 ", whi
ch can be computed by a simple iteration:" }}{PARA 0 "" 0 "" {TEXT -1 
30 "Start with a \"typical\" vector " }{XPPEDIT 18 0 "x(0)" "6#-%\"xG6
#\"\"!" }{TEXT -1 17 ", normalized so  " }{XPPEDIT 18 0 "x(0)[1] = 1" 
"6#/&-%\"xG6#\"\"!6#\"\"\"F*" }{TEXT -1 14 ", and repeat \n" }
{XPPEDIT 18 0 "y := A^T*A*x(n)" "6#>%\"yG*()%\"AG%\"TG\"\"\"F'F)-%\"xG
6#%\"nGF)" }{TEXT -1 1 "\n" }{XPPEDIT 18 0 "x(n+1) := y/y[1];" "6#>-%
\"xG6#,&%\"nG\"\"\"F)F)*&%\"yGF)&F+6#F)!\"\"" }}{PARA 0 "" 0 "" {TEXT 
-1 5 "Then " }{XPPEDIT 18 0 "y[1]" "6#&%\"yG6#\"\"\"" }{TEXT -1 585 " \+
converges to the largest eigenvalue (the second eigenvalue is much sma
ller, so it converges quite fast).  But we can only work with a finite
 number of  components, so we truncate to an m by m matrix.  I did it \+
in Maple for up to a 2500 by 2500 matrix (obtaining 1.274224152 517112
9126 which happens to have 10 correct digits, but I wasn't confident o
f the 10th), and Gaston used a C program to get the result for a 32768
 by 32768 matrix (of course we don't actually construct the matrix, bu
t work directly with vectors).  His result was 1.274224152 82109, whic
h has 13 correct digits." }}{PARA 0 "" 0 "" {TEXT -1 22 "Here's the Ma
ple code." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 389 "mulAA:= proc(
V::Array,W::Array,n)\n# multiply the n-component vector V by A^T*A\n# \+
W is another vector of the same size used for\n# intermediate calculat
ions\n          local i,j;\n            for i from 1 to n do\n        \+
    W[i]:= add(V[j]/((i+j-1)*(i+j-2)/2+i),j=1..n)\n          od;\n    \+
      for j from 1 to n do\n            V[j]:= add(W[i]/((i+j-1)*(i+j-
2)/2+i),i=1..n)\n          od;\n end:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 532 "F:= proc(n)\n# get the approximate eigenvalue for an
 n x n # truncation\n    local V,W,VP,i,eps;\n    if not type(n,intege
r) then return 'procname(n)' fi;\n    eps:= 1.0*10^(-Digits+4);\n    V
:= Array(1..n,datatype=float[8]);\n    W:= Array(1..n,datatype=float[8
]);\n    V[1]:= 1.0;\n    for i from 2 to n do V[i]:= 0 od;\n    do\n \+
     VP:= copy(V);\n      mulAA(V,W,n);\n      if max(seq(abs(V[i]-V[1
]*VP[i]),i=1..n)) < eps\n        then return sqrt(V[1]) fi;\n      for
 i from 2 to n do V[i]:= V[i]/V[1] od;\n      V[1]:= 1.0;\n    od;\n  \+
  end:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "Digits:= 15: ti
:= time(): F(20); (time()-ti)*seconds;\nti:= time():\nF(200);\n(time()
-ti)*seconds;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0(4aS)RQF\"!#9" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%(secondsG$\"%%y\"!\"$" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$\"0s_8gBUF\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,$%(secondsG$\"'!Gu\"!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
62 "Some groups used various extrapolation techniques.  We didn't." }}
}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 10 "Problem 4:" }}{PARA 0 "" 0 "" 
{TEXT -1 42 "What is the global minimum of the function" }}{PARA 0 "" 
0 "" {XPPEDIT 18 0 "exp(sin(50*x))+sin(60*exp(y))+sin(70*sin(x))+sin(s
in(80*y))-sin(10*(x+y))+(x^2+y^2)/4;" "6#,.-%$expG6#-%$sinG6#*&\"#]\"
\"\"%\"xGF,F,-F(6#*&\"#gF,-F%6#%\"yGF,F,-F(6#*&\"#qF,-F(6#F-F,F,-F(6#-
F(6#*&\"#!)F,F4F,F,-F(6#*&\"#5F,,&F-F,F4F,F,!\"\"*&,&*$F-\"\"#F,*$F4FJ
F,F,\"\"%FFF," }{TEXT -1 1 "?" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "
Solution:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Look at the function \+
for reasonably small values of x and y." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 122 "restart;\nF := (x,y) -> exp(sin(50*x)) + sin(60*exp(
y)) + sin(70*sin(x)) +\n\011sin(sin(80*y)) - sin(10*(x+y)) + 1/4*(x^2+
y^2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6$%\"xG%\"yG6\"6$%)op
eratorG%&arrowGF),0-%$expG6#-%$sinG6#,$9$\"#]\"\"\"-F26#,$-F/6#9%\"#gF
7-F26#,$-F26#F5\"#qF7-F26#-F26#,$F=\"#!)F7-F26#,&F5\"#5*&FNF7F=F7F7!\"
\"*&#F7\"\"%F7)F5\"\"#F7F7*&FRF7)F=FUF7F7F)F)F)" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 87 "plot3d(F(x,y),x=-2..2,y=-2..2,axes=box,grid=[1
00,100],style=patchcontour,shading=ZHUE);" }}{PARA 13 "" 1 "" 
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221CD40037CA9BC05BB1E-%'COLOURG6#%%ZHUEG-%*AXESSTYLEG6#%$BOXG-%&STYLEG
6#%-PATCHCONTOURG-%+AXESLABELSG6%%\"xG%\"yGQ!F/" 1 4 0 1 10 0 2 4 1 2 
2 1.000000 75.000000 113.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 161 "It looks like a nightmare for someone trying to find \+
the global minimum: there are huge numbers of local minima.  Note howe
ver that there's an easy lower bound: " }{XPPEDIT 18 0 "exp(-1)-4+(x^2
+y^2)/4 <= F(x,y);" "6#1,(-%$expG6#,$\"\"\"!\"\"F)\"\"%F**&,&*$%\"xG\"
\"#F)*$%\"yGF0F)F)F+F*F)-%\"FG6$F/F2" }{TEXT -1 114 ".  From the graph
 it looks like there are points where F(x,y) is below -2.  In fact, th
e minimum value plotted is " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
6 "P:= %:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "G:= op([1,3],P
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG-%'RTABLEG6+\"))Q:'G&%&flo
atG6#\"\")%&ArrayG%,rectangularG%(C_orderG7\"\"\"#;\"\"\"\"$+\"F2" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "CurrMin:=min(seq(seq(G[i,j],
i=1..100),j=1..100));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(CurrMinG$!
3]!Qw1r.&>G!#<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "For any point d
oing better than that, " }{XPPEDIT 18 0 "exp(-1)-4+(x^2+y^2)/4 < CurrM
in;" "6#2,(-%$expG6#,$\"\"\"!\"\"F)\"\"%F**&,&*$%\"xG\"\"#F)*$%\"yGF0F
)F)F+F*F)%(CurrMinG" }{TEXT -1 4 " so " }{XPPEDIT 18 0 "x^2 + y^2 < 4*
(CurrMin + 4 - exp(-1))" "6#2,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(*&\"\"%F(,
(%(CurrMinGF(F,F(-%$expG6#,$F(!\"\"F3F(" }{TEXT -1 1 " " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "rmax:=sqrt(4*(CurrMin+4-exp(-1.0)))
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rmaxG$\"+l_!H!=!\"*" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 144 "So we c
ertainly don't have to search farther away from the origin than that. \+
 Now suppose we search in a rectangular grid such that every point " }
{XPPEDIT 18 0 "[x,y]" "6#7$%\"xG%\"yG" }{TEXT -1 23 " in the disk of r
adius " }{XPPEDIT 18 0 "rmax" "6#%%rmaxG" }{TEXT -1 11 " is within " }
{XPPEDIT 18 0 "dx/2" "6#*&%#dxG\"\"\"\"\"#!\"\"" }{TEXT -1 8 " in the \+
" }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 15 " direction and " }
{XPPEDIT 18 0 "dy/2" "6#*&%#dyG\"\"\"\"\"#!\"\"" }{TEXT -1 8 " in the \+
" }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 27 " direction of a grid poin
t " }{XPPEDIT 18 0 "[x[i],y[j]]" "6#7$&%\"xG6#%\"iG&%\"yG6#%\"jG" }
{TEXT -1 6 ".  If " }{XPPEDIT 18 0 "[x[0],y[0]]" "6#7$&%\"xG6#\"\"!&%
\"yG6#F'" }{TEXT -1 41 " is the actual minimum (so in particular " }
{XPPEDIT 18 0 "F(x[0],y[0]) <= CurrMin" "6#1-%\"FG6$&%\"xG6#\"\"!&%\"y
G6#F*%(CurrMinG" }{TEXT -1 24 "), how much bigger than " }{XPPEDIT 18 
0 "CurrMin" "6#%(CurrMinG" }{TEXT -1 5 " can " }{XPPEDIT 18 0 "F(x[i],
y[j])" "6#-%\"FG6$&%\"xG6#%\"iG&%\"yG6#%\"jG" }{TEXT -1 77 " be?  Well
, from Taylor's Theorem (remembering that the first derivatives of " }
{XPPEDIT 18 0 "F(x,y)" "6#-%\"FG6$%\"xG%\"yG" }{TEXT -1 24 " at the mi
nimum are 0), " }{XPPEDIT 18 0 "F(x[i],y[j]) = F(x[0],y[0]) + F[x,x](x
,y)*(x[i]-x[0])^2/2 + F[x,y](x,y)*(x[i]-x[0])*(y[j]-y[0]) + F[y,y](x,y
)*(y[j]-y[0])^2/2" "6#/-%\"FG6$&%\"xG6#%\"iG&%\"yG6#%\"jG,*-F%6$&F(6#
\"\"!&F,6#F4\"\"\"*(-&F%6$F(F(6$F(F,F7*$,&&F(6#F*F7&F(6#F4!\"\"\"\"#F7
FDFCF7*(-&F%6$F(F,6$F(F,F7,&&F(6#F*F7&F(6#F4FCF7,&&F,6#F.F7&F,6#F4FCF7
F7*(-&F%6$F,F,6$F(F,F7*$,&&F,6#F.F7&F,6#F4FCFDF7FDFCF7" }{TEXT -1 16 "
 for some point " }{XPPEDIT 18 0 "(x,y)" "6$%\"xG%\"yG" }{TEXT -1 26 "
 on the line segment from " }{XPPEDIT 18 0 "[x[i],y[j]]" "6#7$&%\"xG6#
%\"iG&%\"yG6#%\"jG" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "[x[0],y[0]]" "6
#7$&%\"xG6#\"\"!&%\"yG6#F'" }{TEXT -1 20 ".  In particular if " }
{XPPEDIT 18 0 "F[x,x] <= a" "6#1&%\"FG6$%\"xGF'%\"aG" }{TEXT -1 2 ", \+
" }{XPPEDIT 18 0 "abs(F[x,y]) <= b" "6#1-%$absG6#&%\"FG6$%\"xG%\"yG%\"
bG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "F[y,y] <= c" "6#1&%\"FG6$%\"yG
F'%\"cG" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "F(x[i],y[j]) <= Cur
rMin + a*dx^2/8 + b*dx*dy/4 + c*dy^2/8" "6#1-%\"FG6$&%\"xG6#%\"iG&%\"y
G6#%\"jG,*%(CurrMinG\"\"\"*(%\"aGF1*$%#dxG\"\"#F1\"\")!\"\"F1**%\"bGF1
F5F1%#dyGF1\"\"%F8F1*(%\"cGF1*$F;F6F1F7F8F1" }{TEXT -1 4 ".   " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "a:= op([1,2],evalf(evalr(sub
s(x=INTERVAL(-rmax..rmax),y=INTERVAL(-rmax..rmax),diff(F(x,y),x$2)))))
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG$\"+94>m=!\"&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "b:= op([1,2],evalf(evalr(subs(x=IN
TERVAL(-rmax..rmax),y=INTERVAL(-rmax..rmax),abs(diff(F(x,y),x,y))))));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG$\"$+\"\"\"!" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "c:= op([1,2],evalf(evalr(subs(x=INT
ERVAL(-rmax..rmax),y=INTERVAL(-rmax..rmax),diff(F(x,y),y,y)))));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG$\"+-QrZ9!\"%" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 36 "With dx = .006 and dy = .002, I get " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "dz:=eval(a*dx^2/8 + b*dx*dy/
4 + c*dy^2/8,\{dx=.006,dy=.002\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%#dzG$\"+7Gkm:!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 471 "ti:=
 time():\ndx:= .006: dy:= .002:\ncandidates:= NULL;\nimax:= ceil(rmax/
dx);\nfor i from -imax to imax do\n  xi:= i*dx;\n  if abs(xi) > rmax t
hen jmax:= 1 \n  else jmax:= ceil(sqrt(rmax^2-xi^2)/dy)\n  fi;\n  for \+
j from -jmax to jmax do\n    yj:= j*dy;\n    Fij:= evalhf(F(xi,yj));\n
    if Fij < CurrMin + dz then\n      candidates:= candidates,[xi,yj,F
ij];\n      if Fij < CurrMin then \n        CurrMin:= Fij;\n        be
stcand:= [xi,yj];\n      fi\n    fi\n  od\nod:\n(time()-ti)*seconds;" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%+candidatesG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%imaxG\"$,$" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%(secondsG$\"'Jw?!\"$" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "nops([candidates]);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#\"#v" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "
CurrMin;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!3kV4\\>_`/L!#<" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "We've come down quite a bit from \+
the previous CurrMin.  Now we can search closer to each of the 75 cand
idates." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "dx:= .0006; dy:=
 .0002; dz:= a*dx^2/8 + b*dx*dy/4 + c*dy^2/8;candidates2:= NULL:" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dxG$\"\"'!\"%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#dyG$\"\"#!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#
dzG$\"+7Gkm:!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 411 "ti:= ti
me():\nfor cand from 1 to 75 do\n  for i from -5 to 4 do\n    xi:= can
didates[cand][1]+(i+1/2)*dx;\n    for j from -5 to 4 do\n      yj:= ca
ndidates[cand][2]+(j+1/2)*dy;\n      Fij:= evalhf(F(xi,yj));\n    if F
ij < CurrMin + dz then\n      candidates2:= candidates2,[xi,yj,Fij];\n
      if Fij < CurrMin then \n        CurrMin:= Fij;\n        bestcand
:= [xi,yj];\n      fi\n    fi\n  od\nod\nod:\n(time()-ti)*seconds;\n  \+
  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%(secondsG$\"%@P!\"$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "nops([candidates2]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "candidates2;" }}{PARA 12 "" 1 "" {XPPMATH 20 "627%$!+
+++]D!#6$\"++++0@!#5$!3`)zFn$\\@.L!#<7%F$$\"++++2@F)$!3OV-W:2E.LF,7%$!
++++!\\#F&$\"++++*4#F)$!3NmC\"fpjNI$F,7%F3$\"++++,@F)$!33v\"Gvs*z/LF,7
%F3$\"++++.@F)$!3&o6-E:KcI$F,7%F3F'$!3&ob+8(R11LF,7%F3F.$!3\")[.0G#)41
LF,7%F3$\"++++4@F)$!3_\"[gM+QdI$F,7%$!++++ICF&F:$!3!fce%>T`0LF,7%FOF?$
!30y#*HH]N1LF,7%FOF'$!3gs*R&R`x1LF,7%FOF.$!3DONl%4)z1LF,7%FOFJ$!3'e'\\
HvjU1LF,7%$!++++qBF&F'$!3)*R`%*p5L0LF,7%FjnF.$!3vfJ;WBM0LF,7%FO$\"++++
6@F)$!3Pf(*Q!HjcI$F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 237 "pl
ots[display](\{\n plots[implicitplot](diff(F(x,y),x),x=-.0255-dx/2 .. \+
-.0237+dx/2, y=0.2099-dy/2 .. 0.2111+dy/2, colour=red), plots[implicit
plot](diff(F(x,y),y),x=-.0255-dx/2 .. -.0237+dx/2, y=0.2099-dy/2 .. 0.
2111+dy/2, colour=blue) \});" }}{PARA 13 "" 1 "" {GLPLOT2D 354 266 
266 {PLOTDATA 2 "6%-%'CURVESG6U7$7$$!3ts%e5h1!RC!#>$\"3')***********z4
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7$$!3eY'3x$>!3W#F*$\"3'***********>4@F-Ff[l7$F\\\\l7$$!3`!=S9>Z3W#F*$
\"3V5S$Gv#[4@F-7$7$$!3UOK?^:*3W#F*$\"3(**********f(4@F-Fb\\l7$Fh\\l7$$
!3(oJ$oTw$4W#F*$\"3A$>)4H![+6#F-7$7$$!3MHY%*\\6)4W#F*$\"3(**********>.
6#F-F^]l7$Fd]l7$$!3#zyMn2G5W#F*$\"3*f>wWI816#F-7$7$$!3]bf!Rtq5W#F*$\"3
)**********z36#F-Fj]l7$F`^l7$$!3w_nc'\\=6W#F*$\"3%RRm*y&y66#F-7$7$$!3!
*[.1..;TCF*$\"3)**********R96#F-Ff^l7$F\\_l7$$!3,w8:,*37W#F*$\"3Njrc_Q
u6@F-7$7$$!3IY4Qd)\\7W#F*$\"3*************>6#F-Fb_l-%'COLOURG6&%$RGBG$
\"*++++\"!\")$\"\"!Fe`lFd`l-F$6U7$7$$!3O++++++!e#F*$\"3>lnr6*fi5#F-7$$
!31*\\]/!ybxDF*$\"3W7'f$QvD1@F-7$7$$!3[+++++SqDF*$\"3.P/U$e]i5#F-F^al7
$7$$!37+++++SqDF*Fgal7$$!3%)y\\`*=&znDF*$\"3&Q(Gs_![i5#F-7$7$$!3!*****
******zgDF*$\"3+$yLwDTi5#F-F]bl7$7$$!3C+++++!3c#F*Ffbl7$$!3Y(\\3RiK!eD
F*$\"3h&*zsp&Qi5#F-7$7$$!3O+++++?^DF*$\"3lo^OM>B1@F-F\\cl7$Fbcl7$$!3S#
yBP5q#[DF*$\"3D()QQ*3Hi5#F-7$7$$!3\"3+++++;a#F*$\"3gbHi8EA1@F-Fhcl7$F^
dl7$$!3M:N8Hw]QDF*$\"3*\\X*p6'>i5#F-7$7$$!3B++++++KDF*$\"3%Q]:aH8i5#F-
Fddl7$Fjdl7$$!3aK.H+_uGDF*$\"3!Gg$oO,@1@F-7$7$$!3.,++++SADF*$\"3or6vzR
?1@F-F`el7$Ffel7$$!36DoM<G)*=DF*$\"3MJ_Mk1?1@F-7$7$$!3!3+++++G^#F*$\"3
]:$Qmm%>1@F-F\\fl7$Fbfl7$$!31QbX![?#4DF*$\"3iRKp%>\">1@F-7$7$$!3E,++++
?.DF*$\"3!*)G&3c`=1@F-Fhfl7$7$$!3\"4+++++K]#F*Fagl7$$!3_p*o(*=e%*\\#F*
$\"3,BltF<=1@F-7$7$$!3o+++++g$\\#F*$\"3^V/5[g<1@F-Fggl7$7$$!3/,++++g$
\\#F*$\"3zV/5[g<1@F-7$$!3)=dRa%fp*[#F*$\"3ouR[jA<1@F-7$7$$!3],+++++%[#
F*$\"3*)H@pUn;1@F-Fhhl7$7$$!39,+++++%[#F*Fail7$$!3O_(>wuL*zCF*$\"3d&[W
>!G;1@F-7$7$$!3E,++++SuCF*$\"3(\\po)Ru:1@F-Fgil7$F]jl7$$!3Kp=Y'fr,Z#F*
$\"3>Up7VL:1@F-7$7$$!3Q,++++![Y#F*$\"38&[Q'R\"[h5#F-Fcjl7$Fijl7$$!3OP#
=@\\4/Y#F*$\"3[I-/()Q91@F-7$7$$!3[,++++?bCF*$\"39V)4?%)Qh5#F-F_[m7$Fe[
m7$$!3[@6uMuk]CF*$\"3%>B$pLW81@F-7$7$$!3g,++++gXCF*$\"3m56*paHh5#F-F[
\\m7$Fa\\m7$$!3HUF[Ca)3W#F*$\"3VE[4$)\\71@F-7$7$$!3r,+++++OCF*$\"38F1f
a-71@F-Fg\\m7$F]]m7$$!38r_\\hM7JCF*$\"3s!*QDNb61@F-7$7$$!3#=+++++kU#F*
$\"3@Hn\"['461@F-Fc]m7$Fi]m7$$!3`L3$fah8U#F*$\"3G)Hz,41h5#F-7$7$$!3$>+
++++oT#F*$\"3#>vxwn,h5#F-F_^m7$Fe^m7$$!382:%zn*f6CF*$\"3!4#*zyk'41@F-7
$7$$!30-++++?2CF*$\"3dG?=$R#41@F-F[_m7$Fa_m7$$!33A$zw&y$=S#F*$\"3/FYO3
s31@F-7$7$$!3;-++++g(R#F*$\"3<*)yL6J31@F-Fg_m7$F]`m7$$!3%4E'H&3w?R#F*$
\"3$=GU;xxg5#F-7$7$$!3G-+++++)Q#F*$\"3*Gm`@$Q21@F-Fc`m7$Fi`m7$$!3ydU%4
O9BQ#F*$\"3<[<sP$og5#F-7$7$$!3Q-++++SyBF*$\"3evwjbX11@F-F_am7$Feam7$$!
33+_x%o_DP#F*$\"3x')=h1*eg5#F-7$7$$!3]-++++!)oBF*$\"3p^#)z\"Gbg5#F-F[b
m7$7$$!3%G+++++)oBF*Fdbm7$$!3WD4%p0\"ziBF*$\"3(Qb@$y%\\g5#F-7$7$$!3i-+
+++?fBF*$\"3;8Pk5g/1@F-Fjbm7$F`cm7$$!3eCKfx%HIN#F*$\"3h/'fG0Sg5#F-7$7$
$!3s-++++g\\BF*$\"3YzB=Un.1@F-Ffcm7$F\\dm7$$!3(*QQ)o%zEVBF*$\"3B!*[BI1
.1@F-7$7$$!3&G++++++M#F*$\"3$*oDUwu-1@F-Fbdm-F^`l6&F``lFd`lFd`lFa`l-%+
AXESLABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 119 "Digits:= 30; fsolve(\{diff(F(x,y),x), di
ff(F(x,y),y)\},\{x,y\},x=-.0255-dx/2 .. -.0237+dx/2,\ny=0.2099-dy/2 ..
 0.2111+dy/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#I" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#<$/%\"yG$\"?b05\"f\"fqdNbrU71@!#I/%\"x
G$!?(H$3Lh.><vVpzISC!#J" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "
eval(F(x,y),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!?*3x8h2!GP_ZZ'ooI
$!#H" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "All 30 digits here are co
rrect." }}}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 10 "Problem 5:" }}{PARA 0 
"" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(z) = 1/GAMMA(z)" "6#/-%\"f
G6#%\"zG*&\"\"\"F)-%&GAMMAG6#F'!\"\"" }{TEXT -1 8 ", where " }
{XPPEDIT 18 0 "GAMMA(z)" "6#-%&GAMMAG6#%\"zG" }{TEXT -1 32 " is the ga
mma function, and let " }{XPPEDIT 18 0 "p(z)" "6#-%\"pG6#%\"zG" }
{TEXT -1 48 " be the cubic polynomial that best approximates " }
{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%\"zG" }{TEXT -1 39 " on the unit disk
 in the supremum norm " }{XPPEDIT 18 0 "abs(abs(`.`))[infinity]" "6#&-
%$absG6#-F%6#%\".G6#%)infinityG" }{TEXT -1 11 ".  What is " }{XPPEDIT 
18 0 "abs(abs(f - p))[infinity] " "6#&-%$absG6#-F%6#,&%\"fG\"\"\"%\"pG
!\"\"6#%)infinityG" }{TEXT -1 1 "?" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 
-1 9 "Solution:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 140 "Since the func
tions are analytic, the supremum will occur on the unit circle.  So wr
ite f(z) and p(z) as functions of t where z = exp(I*t). " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "restart;\nF:= t -> 1/GAMMA(exp(I*t)
);\nF(Pi):= 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#%\"tG6\"6$
%)operatorG%&arrowGF(*&\"\"\"F--%&GAMMAG6#-%$expG6#*&9$F-^#F-F-!\"\"F(
F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"FG6#%#PiG\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "P:= (a,t) -> \na[0] + a[1]*e
xp(I*t)+\na[2]*exp(2*I*t)+a[3]*exp(3*I*t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"PGf*6$%\"aG%\"tG6\"6$%)operatorG%&arrowGF),*&9$6#\"
\"!\"\"\"*&&F/6#F2F2-%$expG6#*&9%F2^#F2F2F2F2*&&F/6#\"\"#F2-F76#*&^#F?
F2F:F2F2F2*&&F/6#\"\"$F2-F76#*&^#FGF2F:F2F2F2F)F)F)" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "f(z)" "6#-%\"fG6#%
\"zG" }{TEXT -1 18 " has the symmetry " }{XPPEDIT 18 0 "f(conjugate(z)
) = conjugate(f(z))" "6#/-%\"fG6#-%*conjugateG6#%\"zG-F(6#-F%6#F*" }
{TEXT -1 122 ".  It is easy to show that the best approximation must h
ave that symmetry too, which means we can assume the coefficients " }
{XPPEDIT 18 0 "a[0]" "6#&%\"aG6#\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 
18 0 "a[3]" "6#&%\"aG6#\"\"$" }{TEXT -1 21 " are real.  And then " }
{XPPEDIT 18 0 "abs(F(t)-P(a,t)) = abs(F(-t)-P(a,-t));" "6#/-%$absG6#,&
-%\"FG6#%\"tG\"\"\"-%\"PG6$%\"aGF+!\"\"-F%6#,&-F)6#,$F+F1F,-F.6$F0,$F+
F1F1" }{TEXT -1 25 ", so it suffices to take " }{XPPEDIT 18 0 "t" "6#%
\"tG" }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[0,Pi]" "6#7$\"
\"!%#PiG" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 68 "It's convenient to have the conjugates of these f
unctions available." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Fbar
:= t -> 1/GAMMA(exp(-I*t));Fbar(Pi):= 0;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%%FbarGf*6#%\"tG6\"6$%)operatorG%&arrowGF(*&\"\"\"F--%&GAMMAG6#
-%$expG6#*&^#!\"\"F-9$F-F6F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-
%%FbarG6#%#PiG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Pbar
:= (a,t) -> P(a,-t); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%PbarGf*6$%
\"aG%\"tG6\"6$%)operatorG%&arrowGF)-%\"PG6$9$,$9%!\"\"F)F)F)" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "We have to minimize with respect t
o " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 29 " the maximum with respe
ct to " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 5 " of  " }{XPPEDIT 18 
0 "abs(F(t)-P(a,t))^2" "6#*$-%$absG6#,&-%\"FG6#%\"tG\"\"\"-%\"PG6$%\"a
GF+!\"\"\"\"#" }{TEXT -1 36 " (this is a quadratic polynomial in " }
{XPPEDIT 18 0 "a[0]" "6#&%\"aG6#\"\"!" }{TEXT -1 5 ",...," }{XPPEDIT 
18 0 "a[3]" "6#&%\"aG6#\"\"$" }{TEXT -1 42 ", so it's a bit simpler to
 deal with than " }{XPPEDIT 18 0 "abs(F(t)-P(a,t))" "6#-%$absG6#,&-%\"
FG6#%\"tG\"\"\"-%\"PG6$%\"aGF*!\"\"" }{TEXT -1 92 ").  I can change th
is from a min-max question to a constrained minimization by writing it
 as" }}{PARA 0 "" 0 "" {TEXT -1 9 "minimize " }{XPPEDIT 18 0 "v" "6#%
\"vG" }{TEXT -1 18 " (with respect to " }{XPPEDIT 18 0 "a" "6#%\"aG" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "v" "6#%\"vG" }{TEXT -1 13 ") subjec
t to " }{XPPEDIT 18 0 "abs(F(t)-P(a,t))^2 - v <= 0" "6#1,&*$-%$absG6#,
&-%\"FG6#%\"tG\"\"\"-%\"PG6$%\"aGF-!\"\"\"\"#F.%\"vGF3\"\"!" }{TEXT 
-1 10 " for each " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 1 " " }{XPPEDIT 18 0 
"[0, Pi]" "6#7$\"\"!%#PiG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Now write the Karush-Kuhn-Tucker c
onditions:" }}{PARA 0 "" 0 "" {TEXT -1 18 "The Lagrangian is " }
{XPPEDIT 18 0 "H(v,a,lambda) = v+sum((abs(F(t)-P(a,t))^2-v)*lambda[t],
t);" "6#/-%\"HG6%%\"vG%\"aG%'lambdaG,&F'\"\"\"-%$sumG6$*&,&*$-%$absG6#
,&-%\"FG6#%\"tGF+-%\"PG6$F(F9!\"\"\"\"#F+F'F=F+&F)6#F9F+F9F+" }{TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Formally the sum is over all " 
}{XPPEDIT 18 0 "t*epsilon*[0,Pi]" "6#*(%\"tG\"\"\"%(epsilonGF%7$\"\"!%
#PiGF%" }{TEXT -1 81 ".  In general this might run into technical prob
lems, but they won't occur here: " }{XPPEDIT 18 0 "lambda[t];" "6#&%'l
ambdaG6#%\"tG" }{TEXT -1 40 " will be nonzero for only finitely many \+
" }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 40 ".  The Karush-Kuhn-Tucker
 conditions say" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{XPPEDIT 18 0 "diff(H(v,a,lambda),v) = ``" "6#/-%%diffG6$-%\"HG6%%\"vG
%\"aG%'lambdaGF*%!G" }{XPPEDIT 18 0 "1-sum(lambda[t],t) = 0;" "6#/,&\"
\"\"F%-%$sumG6$&%'lambdaG6#%\"tGF,!\"\"\"\"!" }{TEXT -1 1 " " }}{PARA 
0 "" 0 "" {XPPEDIT 18 0 "diff(H(v,a,lambda),a[j]) = ``" "6#/-%%diffG6$
-%\"HG6%%\"vG%\"aG%'lambdaG&F+6#%\"jG%!G" }{XPPEDIT 18 0 "sum(2*Re((P(
a,-t)-F(-t))*exp(j*I*t))*lambda[t],t) = 0;" "6#/-%$sumG6$*(\"\"#\"\"\"
-%#ReG6#*&,&-%\"PG6$%\"aG,$%\"tG!\"\"F)-%\"FG6#,$F4F5F5F)-%$expG6#*(%
\"jGF)%\"IGF)F4F)F)F)&%'lambdaG6#F4F)F4\"\"!" }{TEXT -1 5 " for " }
{XPPEDIT 18 0 "j = 0,1,2,3" "6&/%\"jG\"\"!\"\"\"\"\"#\"\"$" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "diff(H(v,a,lamb
da),lambda[t]) = ``;" "6#/-%%diffG6$-%\"HG6%%\"vG%\"aG%'lambdaG&F,6#%
\"tG%!G" }{TEXT -1 1 " " }{XPPEDIT 18 0 "abs(F(t)-P(a,t))^2-v <= 0;" "
6#1,&*$-%$absG6#,&-%\"FG6#%\"tG\"\"\"-%\"PG6$%\"aGF-!\"\"\"\"#F.%\"vGF
3\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "lambda[t] >= 0" "6#1\"\"!&%'l
ambdaG6#%\"tG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "lambda[t]*(abs(F(t
)-P(a,t))^2-v) = 0" "6#/*&&%'lambdaG6#%\"tG\"\"\",&*$-%$absG6#,&-%\"FG
6#F(F)-%\"PG6$%\"aGF(!\"\"\"\"#F)%\"vGF7F)\"\"!" }{TEXT -1 5 " for " }
{XPPEDIT 18 0 "t*epsilon*[0,Pi]" "6#*(%\"tG\"\"\"%(epsilonGF%7$\"\"!%#
PiGF%" }}{PARA 0 "" 0 "" {TEXT -1 38 "Note that this last condition im
plies " }{XPPEDIT 18 0 "lambda[t] <> 0" "6#0&%'lambdaG6#%\"tG\"\"!" }
{TEXT -1 24 " for only finitely many " }{XPPEDIT 18 0 "t" "6#%\"tG" }
{TEXT -1 8 ", since " }{XPPEDIT 18 0 "abs(F(t)-P(a,t))^2 = v;" "6#/*$-
%$absG6#,&-%\"FG6#%\"tG\"\"\"-%\"PG6$%\"aGF,!\"\"\"\"#%\"vG" }{TEXT 
-1 24 " for only finitely many " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 215 "Also, since the objective is \+
linear and the constraints are convex, we don't have to worry about lo
cal minima.  The Karush-Kuhn-Tucker conditions should have a unique so
lution, which will give us the global minimum." }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 329 "Finding the solution is not so easy, however.  I sp
ent some time exploring the parameter space, looking for approximate s
olutions.\nFor a starting point, we can use a Taylor polynomial for f(
z), corresponding to a partial sum of the Fourier series for F(t).  Th
is will give us the best approximation in the L^2 norm on the circle.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "ptayl:= taylor(1/GAMMA(
z),z=0,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ptaylG++%\"zG\"\"\"F'
%&gammaG\"\"#,&*$)%#PiGF)F'#!\"\"\"#7*&#F'F)F')F(F)F'F'\"\"$-%\"OG6#F'
\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "a0:= table([seq(j=
evalf(coeff(ptayl,z,j)),j=0..3)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%#a0G-%&TABLEG6#7&/\"\"!$F*F*/\"\"\"$F-F*/\"\"#$\"+\\m:sd!#5/\"\"$$!+
<2yelF3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot(abs(P(a0,t)
-F(t))^2,t=0..Pi);" }}{PARA 13 "" 1 "" {GLPLOT2D 353 265 265 
{PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"\"!F)$\"+GUx(='!#77$$\"+LzxZo!#6$\"+
%Gqm@'F,7$$\"+!H*f!G\"!#5$\"+V8<*G'F,7$$\"+xGm]>F6$\"+4_]CkF,7$$\"+#3o
^i#F6$\"+(>7.i'F,7$$\"+![nkH$F6$\"+I#Hv(oF,7$$\"+bz%)=RF6$\"+03%\\<(F,
7$$\"+()oGjXF6$\"+&R/oa(F,7$$\"+`lwH_F6$\"+=^/0!)F,7$$\"+4)3T*eF6$\"+o
&QEa)F,7$$\"+ykYxlF6$\"+FvO(=*F,7$$\"+&Go$zrF6$\"+3w5R)*F,7$$\"+gr'p&y
F6$\"+!f3v1\"F07$$\"+?$[t`)F6$\"+$zyK;\"F07$$\"+S;-$>*F6$\"+zGrn7F07$$
\"+HeV)y*F6$\"+tyht8F07$$\"+*3W'\\5!\"*$\"+\"*[89:F07$$\"+/9i46Fgp$\"+
hM>Y;F07$$\"+\"G*Qz6Fgp$\"+G(3a\"=F07$$\"+tc9T7Fgp$\"+j&\\%z>F07$$\"+B
A!*38Fgp$\"+m4yu@F07$$\"+#[AMP\"Fgp$\"+)yb_P#F07$$\"+.EuS9Fgp$\"+!yr%)
f#F07$$\"+!\\jD]\"Fgp$\"+)=*o9GF07$$\"+ocCp:Fgp$\"+\"*\\&y0$F07$$\"+5(
4&Q;Fgp$\"+(zK$=LF07$$\"+iU!))p\"Fgp$\"+s;h[NF07$$\"+IS#Rw\"Fgp$\"+tyS
(z$F07$$\"+B\"*>J=Fgp$\"+Q%p-0%F07$$\"+wY,(*=Fgp$\"+!3S!*G%F07$$\"+3Yp
g>Fgp$\"+%)=y2XF07$$\"+:.SJ?Fgp$\"+ml#>t%F07$$\"+QD$\\4#Fgp$\"+HZ)H\"
\\F07$$\"+Ynwi@Fgp$\"+Om3#3&F07$$\"+@YBCAFgp$\"+D![?@&F07$$\"+W_V\"H#F
gp$\"+Q%)*zK&F07$$\"+$zlYN#Fgp$\"+3%4CT&F07$$\"+O*f2U#Fgp$\"+:4JwaF07$
$\"+U!z`[#Fgp$\"+l)[o^&F07$$\"+d#HIb#Fgp$\"+Sg5RbF07$$\"+bW==EFgp$\"+b
H]WbF07$$\"+b]\"[o#Fgp$\"+3whPbF07$$\"+?R*3v#Fgp$\"+ysIAbF07$$\"+MNh6G
Fgp$\"+cN-/bF07$$\"+k]?\")GFgp$\"+9ij\"[&F07$$\"+f&[M%HFgp$\"+'[LIY&F0
7$$\"+1J\")4IFgp$\"+6g.ZaF07$$\"+XCLtIFgp$\"+)3dqV&F07$$\"+]EfTJFgp$\"
+/!pKV&F0-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"t6\"Q!Fe[l
-%%VIEWG6$;F($\"+aEfTJFgp%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
48 "Here's a fast way to approximate the maximum of " }{XPPEDIT 18 0 "
abs(F(t) - P(a,t))" "6#-%$absG6#,&-%\"FG6#%\"tG\"\"\"-%\"PG6$%\"aGF*!
\"\"" }{TEXT -1 88 ".  Instead of all t, I'll just look at 90 of them.
\nIt's more convenient to use the list " }{XPPEDIT 18 0 "[a[0],a[1],a[
2],a[3]]" "6#7&&%\"aG6#\"\"!&F%6#\"\"\"&F%6#\"\"#&F%6#\"\"$" }{TEXT 
-1 23 " rather than the table " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT 
-1 28 " (whose indices start at 0)." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 202 "eits:= evalf([seq(exp(I*i*Pi/90),i=0..90)]):\nfvals:
= evalf([seq(F(i*Pi/90),i=0..90)]):\nappnorm:= proc(L) local i;\n  max
(seq(\nabs(fvals[i]-L[1]-eits[i]*(L[2]+eits[i]*(L[3]+eits[i]*L[4]))),i
=1..91)) end:\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 " Now we genera
te random steps from the current L, accepting the step if it improves \+
the norm." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 498 "ti:= time():
\nbestL:= [a0[0],a0[1],a0[2],a0[3]]:\nbestnorm:= appnorm(bestL);\nfor \+
iter from 1 to 5000 do\n  L:= bestL + .01*exp(-.001*iter)*[stats[rando
m,normald](4)];\n  newnorm:= appnorm(L);\n  if newnorm < bestnorm then
\n    bestL:= L; bestnorm:= newnorm;\n    print (\"iteration\",iter,be
stnorm,L);\n  else\n    L:= 2*bestL-L;\n    newnorm:= appnorm(L);\n   \+
 if newnorm < bestnorm then\n      bestL:= L; bestnorm:= newnorm;\n   \+
   print (\"iteration\",iter,bestnorm,L);\n    fi\n  fi\nod:\n(time()-
ti)*seconds;   \n  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)bestnormG$\"
+&exYN#!#5" }}{PARA 12 "" 1 "" {XPPMATH 20 "6&Q*iteration6\"\"\"#$\"+K
]]JB!#57&$!+nD&>5%!#7$\"+!3CR*)*F($\"+.L_LdF($!+gvg%G'F(" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6&Q*iteration6\"\"\"'$\"+Gbi!G#!#57&$!+HH'34&!#7$
\"+U!on()*F($\"+G<%))z&F($!+#)RJpiF(" }}{PARA 12 "" 1 "" {XPPMATH 20 "
6&Q*iteration6\"\"\"($\"+o/QhA!#57&$!+<Pwn@!#7$\"+1A4w)*F($\"+13+;eF($
!+XwN5jF(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6&Q*iteration6\"\"\"*$\"+=Z
(*>A!#57&$\"+2^XT>!#7$\"+E\"R++\"!\"*$\"+\\#Q&QfF($!+%)p\"zD'F(" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6&Q*iteration6\"\"#L$\"+p*Gz@#!#57&$!+qP
l?E!#7$\"+q^1g**F($\"+p6H\"*eF($!+j[R$G'F(" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6&Q*iteration6\"\"$f\"$\"+t@t)>#!#57&$\"*tG(zi!#7$\"+<V$=
+\"!\"*$\"+>kV-gF($!+py;#>'F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6&Q*ite
ration6\"\"$2#$\"+iyNx@!#57&$\"+p^_-?!#7$\"+3r\"z+\"!\"*$\"+&y'=wgF($!
+2pJghF(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6&Q*iteration6\"\"$D$$\"+Q(G
X<#!#57&$\"+(pG*>I!#7$\"+^$eZ,\"!\"*$\"+TR^jhF($!+x6k2hF(" }}{PARA 12 
"" 1 "" {XPPMATH 20 "6&Q*iteration6\"\"$r$$\"++&[n:#!#57&$\"+T'ySS&!#7
$\"+0ND<5!\"*$\"+KiELiF($!+r(QI.'F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6
&Q*iteration6\"\"$2%$\"+)RT]:#!#57&$\"+Kz=;R!#7$\"+s,)p,\"!\"*$\"+)G4I
@'F($!+o<zlgF(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6&Q*iteration6\"\"$)f$
\"+y=%Q:#!#57&$\"+M$z&oI!#7$\"+Ixu:5!\"*$\"+SLc=iF($!+IvubgF(" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6&Q*iteration6\"\"%V5$\"+z`%G:#!#57&$\"+
c[*ey$!#7$\"+J$))e,\"!\"*$\"+rXc8iF($!+k^IggF(" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6&Q*iteration6\"\"%u5$\"+2r>Z@!#57&$\"+/nU%H&!#7$\"+`VF=5
!\"*$\"+?E*pB'F($!+^P+SgF(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6&Q*iterat
ion6\"\"%\"3\"$\"+8$Ho9#!#57&$\"+9JF;Y!#7$\"+$>Hq,\"!\"*$\"+')ec=iF($!
+L*G@0'F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6&Q*iteration6\"\"%q<$\"+E>
OY@!#57&$\"+[w**3W!#7$\"+SIc<5!\"*$\"+%=r5B'F($!+p6$o/'F(" }}{PARA 12 
"" 1 "" {XPPMATH 20 "6&Q*iteration6\"\"%!H#$\"+=H$e9#!#57&$\"+NZHrW!#7
$\"+@_/=5!\"*$\"+FxiLiF($!+I8-WgF(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6&
Q*iteration6\"\"%yC$\"+<q#\\9#!#57&$\"+%\\9:q%!#7$\"+9-2=5!\"*$\"+=T)[
B'F($!+.EsTgF(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6&Q*iteration6\"\"%tE$
\"+s]]W@!#57&$\"+)z.*z]!#7$\"+X<>=5!\"*$\"+q7YNiF($!+CL[RgF(" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6&Q*iteration6\"\"%JH$\"+s@QW@!#57&$\"+(3.m<&!
#7$\"+!e_\">5!\"*$\"+nntWiF($!+>CCQgF(" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6&Q*iteration6\"\"%'3$$\"+S7/W@!#57&$\"+j;o+^!#7$\"+&=='=5!\"*$\"+
#yS'RiF($!+Mw&*QgF(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6&Q*iteration6\"
\"%)H$$\"+bn!R9#!#57&$\"+]wU_\\!#7$\"+!oM$=5!\"*$\"+y1*pB'F($!+l)p,/'F
(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6&Q*iteration6\"\"%LQ$\"+3C$Q9#!#57
&$\"+UYH0]!#7$\"+-'f#=5!\"*$\"+TLTOiF($!+^`%*RgF(" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6&Q*iteration6\"\"%lX$\"+v:yV@!#57&$\"+!\\V<'\\!#7$\"+smS
=5!\"*$\"+[AvPiF($!+5'y//'F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6&Q*iter
ation6\"\"%aY$\"+#HCP9#!#57&$\"+qr*=&\\!#7$\"+;?S=5!\"*$\"+x6#zB'F($!+
&H0+/'F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6&Q*iteration6\"\"%SZ$\"+xmi
V@!#57&$\"+SF`()\\!#7$\"+1i[=5!\"*$\"+Se*)QiF($!+[WdRgF(" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6&Q*iteration6\"\"%,\\$\"+sWgV@!#57&$\"+n6fi\\!#7$
\"+%*4Y=5!\"*$\"+z4kQiF($!+@V%*RgF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#,$%(secondsG$\"()\\49!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "The
 last solution turns out to have 4 correct digits of the actual answer
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "a1:= table([seq(j=best
L[j+1],j=0..3)]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#a1G-%&TABLEG6#
7&/\"\"!$\"+n6fi\\!#7/\"\"\"$\"+%*4Y=5!\"*/\"\"#$\"+z4kQi!#5/\"\"$$!+@
V%*RgF7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "In our approximate sol
ution, " }{XPPEDIT 18 0 "abs(F(t) - P(a,t))" "6#-%$absG6#,&-%\"FG6#%\"
tG\"\"\"-%\"PG6$%\"aGF*!\"\"" }{TEXT -1 33 " is close to constant from
 about " }{XPPEDIT 18 0 "t=1.1" "6#/%\"tG-%&FloatG6$\"#6!\"\"" }{TEXT 
-1 4 " to " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 1 "." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "plot(abs(F(t)-P(a1,t)),t=0..Pi);" }
}{PARA 13 "" 1 "" {GLPLOT2D 353 265 265 {PLOTDATA 2 "6%-%'CURVESG6$7Y7
$$\"\"!F)$\"++^KHV!#67$$\"+:Csf&)!#7$\"+(*=&\\L%F,7$$\"+$[W>r\"F,$\"+7
hy^VF,7$$\"+Cn\"zc#F,$\"+H.pzVF,7$$\"+m*))QU$F,$\"+W:W=WF,7$$\"+\\M$e8
&F,$\"+ka@FXF,7$$\"+LzxZoF,$\"+0'o\\n%F,7$$\"+;a)o#)*F,$\"+GVZ8]F,7$$
\"+!H*f!G\"!#5$\"+n\"4nV&F,7$$\"+xGm]>FT$\"+3*[%)f'F,7$$\"+#3o^i#FT$\"
+9jpIzF,7$$\"+![nkH$FT$\"+8.X>$*F,7$$\"+bz%)=RFT$\"+qlFh5FT7$$\"+()oGj
XFT$\"+\"p2E>\"FT7$$\"+`lwH_FT$\"+**\\KB8FT7$$\"+4)3T*eFT$\"+)RemW\"FT
7$$\"+ykYxlFT$\"+oJzk:FT7$$\"+&Go$zrFT$\"+\"ps0m\"FT7$$\"+gr'p&yFT$\"+
$R1$e<FT7$$\"+?$[t`)FT$\"+i%f]%=FT7$$\"+S;-$>*FT$\"+Fg\\<>FT7$$\"+HeV)
y*FT$\"+nont>FT7$$\"+*3W'\\5!\"*$\"+fOtG?FT7$$\"+/9i46Fer$\"+Dztl?FT7$
$\"+\"G*Qz6Fer$\"+,(z$)4#FT7$$\"+tc9T7Fer$\"+FD!)=@FT7$$\"+BA!*38Fer$
\"+BLGL@FT7$$\"+#[AMP\"Fer$\"+<VsS@FT7$$\"+.EuS9Fer$\"+#>zM9#FT7$$\"+!
\\jD]\"Fer$\"+/q#H9#FT7$$\"+ocCp:Fer$\"+KI\\S@FT7$$\"+5(4&Q;Fer$\"+^!f
t8#FT7$$\"+iU!))p\"Fer$\"+K$G]8#FT7$$\"+IS#Rw\"Fer$\"+q'>N8#FT7$$\"+B
\"*>J=Fer$\"+()\\JL@FT7$$\"+wY,(*=Fer$\"+KoNM@FT7$$\"+3Ypg>Fer$\"+'Q0i
8#FT7$$\"+:.SJ?Fer$\"+xZoQ@FT7$$\"+QD$\\4#Fer$\"+bT!39#FT7$$\"+Ynwi@Fe
r$\"+g;`U@FT7$$\"+@YBCAFer$\"+/'3M9#FT7$$\"+W_V\"H#Fer$\"+M/dV@FT7$$\"
+$zlYN#Fer$\"+\\P2V@FT7$$\"+O*f2U#Fer$\"+sg6U@FT7$$\"+U!z`[#Fer$\"+mo-
T@FT7$$\"+d#HIb#Fer$\"+ze,S@FT7$$\"+bW==EFer$\"+9?QR@FT7$$\"+b]\"[o#Fe
r$\"+\\$)>R@FT7$$\"+?R*3v#Fer$\"+W-\\R@FT7$$\"+MNh6GFer$\"+^J5S@FT7$$
\"+k]?\")GFer$\"+'\\c59#FT7$$\"+f&[M%HFer$\"+c\"o>9#FT7$$\"+1J\")4IFer
$\"+s<#G9#FT7$$\"+XCLtIFer$\"+1HQV@FT7$$\"+]EfTJFer$\"+q7gV@FT-%'COLOU
RG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"t6\"Q!Fc]l-%%VIEWG6$;F($\"+
aEfTJFer%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "On clo
ser inspection, it achieves its maximum at three points in this interv
al, one of which is " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 1 "." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot(abs(F(t)-P(a1,t)),t=1.
.Pi,0.21 .. 0.21437 );" }}{PARA 13 "" 1 "" {GLPLOT2D 353 265 265 
{PLOTDATA 2 "6%-%'CURVESG6$7hn7$$\"+,+++5!\"*$\"+@<X\"*>!#57$$\"+e,n65
F*$\"+)yq2+#F-7$$\"+9.MB5F*$\"+hpu4?F-7$$\"+q/,N5F*$\"+SDQ=?F-7$$\"+E1
oY5F*$\"+&4!oE?F-7$$\"+2*))p1\"F*$\"+O@KS?F-7$$\"+)=(H(3\"F*$\"+\"QmH0
#F-7$$\"+uf856F*$\"+.'=g1#F-7$$\"+hZ(H8\"F*$\"+^a'y2#F-7$$\"+Q\\'f:\"F
*$\"+K'4')3#F-7$$\"+9^&*y6F*$\"+X\"4#)4#F-7$$\"+Ag$=?\"F*$\"+)eqm5#F-7
$$\"+IprC7F*$\"+B)*39@F-7$$\"+h/$fC\"F*$\"+O=3?@F-7$$\"+#*R9n7F*$\"+^^
ED@F-7$$\"+cZ268F*$\"+!QEO8#F-7$$\"+;!3lN\"F*$\"+80HR@F-7$$\"+%z^\"z8F
*$\"+P]8T@F-7$$\"+rbz,9F*$\"+_TTU@F-7$$\"+![(3D9F*$\"+)43K9#F-7$$\"+)Q
z$[9F*$\"+yCaV@F-7$$\"+9\\*)o9F*$\"+D;^V@F-7$$\"+R/T*[\"F*$\"+xYAV@F-7
$$\"+%o,c`\"F*$\"+QL(=9#F-7$$\"+&f#)>e\"F*$\"+ZW#*R@F-7$$\"+(>zmi\"F*$
\"+FQ)y8#F-7$$\"+%*zEn;F*$\"+W1;O@F-7$$\"+**>`:<F*$\"+&\\DX8#F-7$$\"+U
zTc<F*$\"+T%HO8#F-7$$\"+8!yR!=F*$\"+-1BL@F-7$$\"+Nn2Y=F*$\"+?BXL@F-7$$
\"+1dE#*=F*$\"+*)pCM@F-7$$\"+G&[i$>F*$\"+z^UN@F-7$$\"+*)*R@)>F*$\"+<r$
p8#F-7$$\"+wEGC?F*$\"+Z@VQ@F-7$$\"+e#R(p?F*$\"+!z.+9#F-7$$\"+qe&p6#F*$
\"+T8WT@F-7$$\"+q!e!e@F*$\"+GeVU@F-7$$\"+t&\\C?#F*$\"+a(yJ9#F-7$$\"+Q.
J[AF*$\"+M#fN9#F-7$$\"+]h<$H#F*$\"+cVcV@F-7$$\"+IheOBF*$\"+4(oK9#F-7$$
\"+6by%Q#F*$\"+jgnU@F-7$$\"+*z%4GCF*$\"+`Y*>9#F-7$$\"+4nLuCF*$\"+::@T@
F-7$$\"+^(Qi^#F*$\"+LN`S@F-7$$\"+_([?c#F*$\"+qW!*R@F-7$$\"+#R_^g#F*$\"
+8]ZR@F-7$$\"+;\"3-l#F*$\"+))>BR@F-7$$\"+@$eUp#F*$\"+4A@R@F-7$$\"+/[PS
FF*$\"+-ZTR@F-7$$\"+e/z%y#F*$\"+E')zR@F-7$$\"+z=@IGF*$\"+a$R.9#F-7$$\"
+)=d_(GF*$\"+d%p49#F-7$$\"+\\\"\\m\"HF*$\"+Z(z:9#F-7$$\"+2!*3kHF*$\"+/
XDU@F-7$$\"+#y>l+$F*$\"+(y%yU@F-7$$\"+P)f<0$F*$\"+*QFK9#F-7$$\"+V.1&4$
F*$\"+K*)\\V@F-7$$\"+]EfTJF*$\"+q7gV@F--%'COLOURG6&%$RGBG$\"#5!\"\"$\"
\"!Fg^lFf^l-%+AXESLABELSG6$Q\"t6\"Q!F\\_l-%%VIEWG6$;$\"\"\"Fg^l$\"+aEf
TJF*;$\"#@!\"#$\"&P9#!\"&" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 
"peak1:= fsolve(diff((F(t)-P(a1,t))*(Fbar(t)-Pbar(a1,t)),t), t= 1.4 ..
 1.5); peak1:= simplify(%,zero);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
&peak1G^$$\"+e-Dc9!\"*$\"\"!F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&p
eak1G$\"+e-Dc9!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "peak
2:= simplify(fsolve(diff((F(t)-P(a1,t))*(Fbar(t)-Pbar(a1,t)),t), t= 2.
2 .. 2.3),zero); v1:= evalf(abs(F(Pi)-P(a1,Pi))^2);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%&peak2G$\"+1S9rA!\"*" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#v1G$\"+0k-&f%!#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "So w
e should expect " }{XPPEDIT 18 0 "lambda[t] <> 0;" "6#0&%'lambdaG6#%\"
tG\"\"!" }{TEXT -1 21 " for three values of " }{XPPEDIT 18 0 "t" "6#%
\"tG" }{TEXT -1 6 ", say " }{XPPEDIT 18 0 "t[1]" "6#&%\"tG6#\"\"\"" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "t[2]" "6#&%\"tG6#\"\"#" }{TEXT -1 5 " \+
and " }{XPPEDIT 18 0 "t[3] = Pi;" "6#/&%\"tG6#\"\"$%#PiG" }{TEXT -1 
79 ".  We write the KKT equations with this assumption; note that in o
rder to have " }{XPPEDIT 18 0 "abs(F(t)-P(a,t))^2 <= v" "6#1*$-%$absG6
#,&-%\"FG6#%\"tG\"\"\"-%\"PG6$%\"aGF,!\"\"\"\"#%\"vG" }{TEXT -1 9 " fo
r all " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 5 " and " }{XPPEDIT 18 
0 "``=v" "6#/%!G%\"vG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "t=t[i]" "6#
/%\"tG&F$6#%\"iG" }{TEXT -1 9 " we need " }{XPPEDIT 18 0 "diff(abs(F(t
)-P(a,t))^2,t) = 0;" "6#/-%%diffG6$*$-%$absG6#,&-%\"FG6#%\"tG\"\"\"-%
\"PG6$%\"aGF/!\"\"\"\"#F/\"\"!" }{TEXT -1 5 "  at " }{XPPEDIT 18 0 "t=
t[i]" "6#/%\"tG&F$6#%\"iG" }{TEXT -1 24 ".  This is automatic at " }
{XPPEDIT 18 0 "t=Pi" "6#/%\"tG%#PiG" }{TEXT -1 1 "." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 313 "t[3]:= Pi:\nkkt:= [ 1 - (lambda[1]+lambd
a[2]+lambda[3]),\nseq(\nadd(((Pbar(a,t[i])-Fbar(t[i]))*exp(j*I*t[i])+(
P(a,t[i])-F(t[i]))*exp(-j*I*t[i]))*lambda[i],i=1..3),j=0..3),\nseq(\n(
F(t[i])-P(a,t[i]))*(Fbar(t[i])-Pbar(a,t[i])) - v, i=1..3),\nseq(eval(d
iff((F(t)-P(a,t))*(Fbar(t)-Pbar(a,t)),t), t=t[i]),i=1..2)];  \n       \+
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$kktG7,,*\"\"\"F'&%'lambdaG6#F'!
\"\"&F)6#\"\"#F+&F)6#\"\"$F+,(*&,4&%\"aG6#\"\"!F.*&&F6F*F'-%$expG6#*&^
#F+F'&%\"tGF*F'F'F'*&&F6F-F'-F<6#*&^#!\"#F'F@F'F'F'*&&F6F0F'-F<6#*&^#!
\"$F'F@F'F'F'*&F'F'-%&GAMMAG6#F;F+F+*&F:F'-F<6#*&F@F'^#F'F'F'F'*&FCF'-
F<6#*&^#F.F'F@F'F'F'*&FJF'-F<6#*&^#F1F'F@F'F'F'*&F'F'-FR6#FUF+F+F'F(F'
F'*&,4F5F.*&F:F'-F<6#*&F?F'&FAF-F'F'F'*&FCF'-F<6#*&FGF'FfoF'F'F'*&FJF'
-F<6#*&FNF'FfoF'F'F'*&F'F'-FR6#FcoF+F+*&F:F'-F<6#*&FfoF'FXF'F'F'*&FCF'
-F<6#*&FgnF'FfoF'F'F'*&FJF'-F<6#*&F\\oF'FfoF'F'F'*&F'F'-FR6#FcpF+F+F'F
,F'F'*&,*F5F.*&F.F'F:F'F+*&F.F'FCF'F'*&F.F'FJF'F+F'F/F'F',(*&,&*&,,F5F
'F9F'FBF'FIF'FPF+F'FUF'F'*&,,F5F'FTF'FYF'FhnF'F]oF+F'F;F'F'F'F(F'F'*&,
&*&,,F5F'FboF'FgoF'F[pF'F_pF+F'FcpF'F'*&,,F5F'FbpF'FfpF'FjpF'F^qF+F'Fc
oF'F'F'F,F'F'*&,*F5FH*&F.F'F:F'F'*&F.F'FCF'F+*&F.F'FJF'F'F'F/F'F',(*&,
&*&FjqF'FZF'F'*&F\\rF'FDF'F'F'F(F'F'*&,&*&F`rF'FgpF'F'*&FbrF'FhoF'F'F'
F,F'F'FaqF',(*&,&*&FjqF'FinF'F'*&F\\rF'FKF'F'F'F(F'F'*&,&*&F`rF'F[qF'F
'*&FbrF'F\\pF'F'F'F,F'F'FcrF',&*&,,F]oF'F5F+FTF+FYF+FhnF+F',,FPF'F5F+F
9F+FBF+FIF+F'F'%\"vGF+,&*&,,F^qF'F5F+FbpF+FfpF+FjpF+F',,F_pF'F5F+FboF+
FgoF+F[pF+F'F'F^tF+,&*$),*F5F+F:F'FCF+FJF'F.F'F'F^tF+,&*&,***F?F'F^oF+
-%$PsiGF_oF'FUF'F'*(F?F'F:F'FUF'F'*(FGF'FCF'FZF'F'*(FNF'FJF'FinF'F'F'F
]tF'F'*&F\\tF',***FQF+-F\\uFSF'F;F'FXF'F'*(F:F'F;F'FXF'F'*(FgnF'FCF'FD
F'F'*(F\\oF'FJF'FKF'F'F'F',&*&,***F?F'F_qF+-F\\uF`qF'FcpF'F'*(F?F'F:F'
FcpF'F'*(FGF'FCF'FgpF'F'*(FNF'FJF'F[qF'F'F'FbtF'F'*&FatF',***F`pF+-F\\
uFapF'FcoF'FXF'F'*(F:F'FcoF'FXF'F'*(FgnF'FCF'FhoF'F'*(F\\oF'FJF'F\\pF'
F'F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Here's what we have for
 these equations at our approximate solution." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 55 "evalf(eval(kkt,\{a=a1,t[1]=peak1, t[2]=peak2, v \+
= v1\}));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7,,*$\"\"\"\"\"!F&*&$F&F'
F&&%'lambdaG6#F&F&!\"\"*&$F&F'F&&F+6#\"\"#F&F-*&$F&F'F&&F+6#\"\"$F&F-,
(*&^$$!+xe;EI!#5$F'F'F&F*F&F&*&^$$!*!)>EE\"!\"*F>F&F0F&F&*&$\"*a-sG%FC
F&F5F&F&,(*&^$$!+1lriLF=F>F&F*F&F&*&^$$\"+S'Hk%RF=F>F&F0F&F&*&$FFFCF&F
5F&F-,(*&^$$\"+3vZdAF=F>F&F*F&F&*&^$$!+5$)>CQF=F>F&F0F&F&*&FEF&F5F&F&,
(*&^$$\"+[mvyQF=F>F&F*F&F&*&^$$\"+QIOG)*!#6F>F&F0F&F&*&$FFFCF&F5F&F-^$
$!'u.;F_oF>^$$\"&Gb\"F_oF>F>^$$\"+37dbI!#?F>^$$\"+#>w,$=!#>F>" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "simplify(fnormal(%),zero);" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#7,,*$\"\"\"\"\"!F&*&$F&F'F&&%'lambda
G6#F&F&!\"\"*&$F&F'F&&F+6#\"\"#F&F-*&$F&F'F&&F+6#\"\"$F&F-,(F*$!+xe;EI
!#5*&$\"*!)>EE\"!\"*F&F0F&F-*&$\"*a-sG%F?F&F5F&F&,(F*$!+1lriLF;*&$\"+S
'Hk%RF;F&F0F&F&*&$FBF?F&F5F&F-,(F*$\"+3vZdAF;*&$\"+5$)>CQF;F&F0F&F-*&F
AF&F5F&F&,(F*$\"+[mvyQF;*&$\"+QIOG)*!#6F&F0F&F&*&$FBF?F&F5F&F-$!+++u.;
!#:$\"+++!Gb\"!#;$F'F'F[oF[o" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "W
e need values for the Lagrange multipliers " }{XPPEDIT 18 0 "lambda[1]
, lambda[2], lambda[3]" "6%&%'lambdaG6#\"\"\"&F$6#\"\"#&F$6#\"\"$" }
{TEXT -1 65 ", from an overdetermined system with\n5 equations involvi
ng these." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "remove(type,%,
constant);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7',*$\"\"\"\"\"!F&*&$F&F
'F&&%'lambdaG6#F&F&!\"\"*&$F&F'F&&F+6#\"\"#F&F-*&$F&F'F&&F+6#\"\"$F&F-
,(F*$!+xe;EI!#5*&$\"*!)>EE\"!\"*F&F0F&F-*&$\"*a-sG%F?F&F5F&F&,(F*$!+1l
riLF;*&$\"+S'Hk%RF;F&F0F&F&*&$FBF?F&F5F&F-,(F*$\"+3vZdAF;*&$\"+5$)>CQF
;F&F0F&F-*&FAF&F5F&F&,(F*$\"+[mvyQF;*&$\"+QIOG)*!#6F&F0F&F&*&$FBF?F&F5
F&F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "lambdas:=linalg[lea
stsqrs](convert(%,set),\{lambda[1],lambda[2],lambda[3]\});\nsubs(lambd
as,%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(lambdasG<%/&%'lambdaG6#
\"\"\"$\"+y[9B@!#5/&F(6#\"\"#$\"+]Imp[F-/&F(6#\"\"$$\"+jI`0IF-" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7'$\"(4!f;!#5$\")T2=JF&$!)r!32)F&$!)Uw
U%*F&$\")'\\\"f8F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "Now using t
his approximate solution as a starting point, we solve the KKT equatio
ns using fsolve. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "approx
sol:= lambdas union \{seq(a[i]=a1[i],i=0..3),t[1]=peak1, t[2]=peak2, v
 = v1\};" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%*approxsolG<,/&%'lambdaG
6#\"\"\"$\"+y[9B@!#5/&F(6#\"\"#$\"+]Imp[F-/&F(6#\"\"$$\"+jI`0IF-/&%\"t
GF0$\"+1S9rA!\"*/%\"vG$\"+0k-&f%!#6/&F<F)$\"+e-Dc9F?/&%\"aG6#\"\"!$\"+
n6fi\\!#7/&FKF)$\"+%*4Y=5F?/&FKF0$\"+z4kQiF-/&FKF6$!+@V%*RgF-" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Digits:= 30: \nfsolve(conver
t(kkt,set), approxsol);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<,/&%'lambd
aG6#\"\"\"^$$\"?sz2E9U:7z&)**)3P6#!#I$!?r-fYS%\\:&=U\\<](z\"!#w/&F&6#
\"\"#^$$\"?SA7b%)>!*4f&)pi9c[F,$\"?G!G\\(\\[<@@Ou]Od6F//&F&6#\"\"$^$$
\"?)y*z=,Q%z<'GI[9IIF,$\"?ICi;2fu.tf]nO,k!#x/&%\"aGF'^$$\"?=eP;ZjCaQ)H
&=w>5!#H$\"?%\\cs8-*3e'pe(HsLcFB/&FEF2^$$\"?8`Mv>')3BSLk\">@D'F,$\"?gu
qy3(y.6e\"G\"*4[GF//&FE6#\"\"!^$$\"?XguIZHZaa7L2&>a&!#K$\"?-2GZ[kO^BC$
))>gM\"F//&FEF;^$$!?)e0vwtC7'y2/AVLgF,$\"?(3it=n/_)z,!*o3f=F//&%\"tGF2
^$$\"?(pm&=GQ4(*Gh\\uPiAFI$\"?cYU\"Gi.j\"HwQFd!G)FB/%\"vG^$$\"?YzvjsJj
OZpy#fRf%!#J$\"?\"4kC&GUhjS&*yCe%f(FB/&F`oF'^$$\"?;zfR^V'H,;3<*>.9FI$!
?$*4Rhji7pYR*o9;9\"F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "so
l:=simplify(fnormal(%),zero);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$so
lG<,/&%\"aG6#\"\"#$\"?8`Mv>')3BSLk\">@D'!#I/&%\"tGF)$\"?(pm&=GQ4(*Gh\\
uPiA!#H/&F06#\"\"\"$\"?;zfR^V'H,;3<*>.9F3/%\"vG$\"?YzvjsJjOZpy#fRf%!#J
/&%'lambdaGF6$\"?sz2E9U:7z&)**)3P6#F-/&FAF)$\"?SA7b%)>!*4f&)pi9c[F-/&F
A6#\"\"$$\"?)y*z=,Q%z<'GI[9IIF-/&F(FJ$!?)e0vwtC7'y2/AVLgF-/&F(6#\"\"!$
\"?XguIZHZaa7L2&>a&!#K/&F(F6$\"?=eP;ZjCaQ)H&=w>5F3" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 18 "And the answer is:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "ans:= subs(sol,sqrt(v));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$ansG$\"?&=+k_h%R'f/fM_L9#!#I" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 38 "This is actually correct to 30 digits." }}}}{EXCHG 
{PARA 4 "" 0 "" {TEXT -1 10 "Problem 6:" }}{PARA 0 "" 0 "" {TEXT -1 
173 "A flea starts at (0,0) on the infinite 2D integer lattice and exe
cutes a biased random walk: At each step it hops north or south with p
robability 1/4, east with probability " }{XPPEDIT 18 0 "1/4 + epsilon
" "6#,&*&\"\"\"F%\"\"%!\"\"F%%(epsilonGF%" }{TEXT -1 28 ", and west wi
th probability " }{XPPEDIT 18 0 "1/4 - epsilon" "6#,&*&\"\"\"F%\"\"%!
\"\"F%%(epsilonGF'" }{TEXT -1 98 ".  The probability that the flea ret
urns to (0,0) sometime during its wanderings is 1/2.  What is " }
{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 1 "?" }}}{SECT 1 
{PARA 5 "" 0 "" {TEXT -1 9 "Solution:" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 3 "If " }{XPPEDIT 18 0 "u(x,y)" "6#-%\"uG6$%\"xG%\"yG" }{TEXT -1 
47 " is the probability that the flea, starting at " }{XPPEDIT 18 0 "`
`(x,y);" "6#-%!G6$%\"xG%\"yG" }{TEXT -1 27 ", ever reaches (0,0), then
 " }{XPPEDIT 18 0 "u(0,0) = 1" "6#/-%\"uG6$\"\"!F'\"\"\"" }{TEXT -1 
16 " and otherwise \n" }{XPPEDIT 18 0 "u(x,y) = u(x,y+1)/4+u(x,y-1)/4+
(1/4+epsilon)*u(x-1,y)+(1/4-epsilon)*u(x+1,y);" "6#/-%\"uG6$%\"xG%\"yG
,**&-F%6$F',&F(\"\"\"F.F.F.\"\"%!\"\"F.*&-F%6$F',&F(F.F.F0F.F/F0F.*&,&
*&F.F.F/F0F.%(epsilonGF.F.-F%6$,&F'F.F.F0F(F.F.*&,&*&F.F.F/F0F.F8F0F.-
F%6$,&F'F.F.F.F(F.F." }{TEXT -1 44 ".  The probability that the flea r
eturns is " }{XPPEDIT 18 0 "v(epsilon) = u(0,1)/4+u(0,-1)/4+(1/4+epsil
on)*u(-1,0)+(1/4-epsilon)*u(1,0);" "6#/-%\"vG6#%(epsilonG,**&-%\"uG6$
\"\"!\"\"\"F.\"\"%!\"\"F.*&-F+6$F-,$F.F0F.F/F0F.*&,&*&F.F.F/F0F.F'F.F.
-F+6$,$F.F0F-F.F.*&,&*&F.F.F/F0F.F'F0F.-F+6$F.F-F.F." }{TEXT -1 1 "." 
}}{PARA 0 "" 0 "" {TEXT -1 19 "We can't calculate " }{XPPEDIT 18 0 "v
" "6#%\"vG" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "u(x,y)" "6#-%\"uG6$%\"x
G%\"yG" }{TEXT -1 113 " directly, but we can approximate it by taking \+
solving the equations in a finite region with boundary conditions " }
{XPPEDIT 18 0 "u(x,y)=0" "6#/-%\"uG6$%\"xG%\"yG\"\"!" }{TEXT -1 159 " \+
on the boundary of the region.  This corresponds to looking at the sit
uation where the flea is killed if it hits the boundary of the region.
  We want to find " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 
33 " (in this approximation) so that " }{XPPEDIT 18 0 "v(epsilon) = 1/
2" "6#/-%\"vG6#%(epsilonG*&\"\"\"F)\"\"#!\"\"" }{TEXT -1 77 " .  This \+
can be done by Newton's method: for this we need the derivatives of " 
}{XPPEDIT 18 0 "v" "6#%\"vG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "u(x,y
)" "6#-%\"uG6$%\"xG%\"yG" }{TEXT -1 17 " with respect to " }{XPPEDIT 
18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 30 ", which satisfy the equati
ons " }}{PARA 0 "" 0 "" {TEXT -1 54 "obtained by differentiating the e
quations for u and v:" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "`u'`(0,0) = 0
" "6#/-%#u'G6$\"\"!F'F'" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "`u'`(x,y) = \+
`u'`(x,y+1)/4 + `u'`(x,y-1)/4 + (1/4+epsilon)*`u'`(x-1,y) + u(x-1,y) +
 (1/4-epsilon)*`u'`(x+1,y) - u(x+1,y)" "6#/-%#u'G6$%\"xG%\"yG,.*&-F%6$
F',&F(\"\"\"F.F.F.\"\"%!\"\"F.*&-F%6$F',&F(F.F.F0F.F/F0F.*&,&*&F.F.F/F
0F.%(epsilonGF.F.-F%6$,&F'F.F.F0F(F.F.-%\"uG6$,&F'F.F.F0F(F.*&,&*&F.F.
F/F0F.F8F0F.-F%6$,&F'F.F.F.F(F.F.-F=6$,&F'F.F.F.F(F0" }{TEXT -1 10 " o
therwise" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "`v'`(epsilon) = `u'`(0,1)/4
+`u'`(0,-1)/4+(1/4+epsilon)*`u'`(-1,0)+u(-1,0)+(1/4-epsilon)*`u'`(1,0)
-u(1,0);" "6#/-%#v'G6#%(epsilonG,.*&-%#u'G6$\"\"!\"\"\"F.\"\"%!\"\"F.*
&-F+6$F-,$F.F0F.F/F0F.*&,&*&F.F.F/F0F.F'F.F.-F+6$,$F.F0F-F.F.-%\"uG6$,
$F.F0F-F.*&,&*&F.F.F/F0F.F'F0F.-F+6$F.F-F.F.-F<6$F.F-F0" }{TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 37 "I'll solve the system by iteration: \+
\n" }{XPPEDIT 18 0 "u[n+1](x,y) = u[n](x,y+1)/4+u[n](x,y-1)/4+(1/4+eps
ilon)*u[n](x-1,y)+(1/4-epsilon)*u[n](x+1,y);" "6#/-&%\"uG6#,&%\"nG\"\"
\"F*F*6$%\"xG%\"yG,**&-&F&6#F)6$F,,&F-F*F*F*F*\"\"%!\"\"F**&-&F&6#F)6$
F,,&F-F*F*F6F*F5F6F**&,&*&F*F*F5F6F*%(epsilonGF*F*-&F&6#F)6$,&F,F*F*F6
F-F*F**&,&*&F*F*F5F6F*F@F6F*-&F&6#F)6$,&F,F*F*F*F-F*F*" }{TEXT -1 1 " \+
" }}{PARA 0 "" 0 "" {TEXT -1 36 "I'll take advantage of the symmetry \+
" }{XPPEDIT 18 0 "u(x,y) = u(x,-y)" "6#/-%\"uG6$%\"xG%\"yG-F%6$F',$F(!
\"\"" }{TEXT -1 44 ", and also the fact that (if you colour the " }}
{PARA 0 "" 0 "" {TEXT -1 88 "lattice as a checkerboard) the flea alter
nates between black and white sites, so we only" }}{PARA 0 "" 0 "" 
{TEXT -1 69 "update, say, the black sites for odd n and the white ones
 for even n." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "getflea:= p
roc(eps,M)\n" }{TEXT -1 62 "Procedure to do one Newton iteration for (
2M+1) by (2M+1) grid" }{MPLTEXT 1 0 47 "\nlocal A,dA,i,j,iter,pp,pm,R,
dR,Rp,ip,ip0,iip;\n" }{TEXT -1 19 "u(x,y) = A[x+M,y+1]" }{MPLTEXT 1 0 
27 "\nA:= hfarray(1..2*M,1..M);\n" }{TEXT -1 23 "u'(x,y) = dA[x+M,y+1]
  " }{MPLTEXT 1 0 28 "\ndA:= hfarray(1..2*M,1..M);\n" }{TEXT -1 47 "us
e R to check convergence by changes in u(1,0)" }{MPLTEXT 1 0 8 "\nR:= \+
0;\n" }{TEXT -1 38 "Probabilities for east and west hops. " }{MPLTEXT 
1 0 29 "\npp:= 1/4+eps; pm:= 1/4-eps;\n" }{TEXT -1 21 "Initialize the \+
arrays" }{MPLTEXT 1 0 91 "\nfor i from 1 to 2*M do for j from 1 to M d
o A[i,j]:= 0;\n  dA[i,j]:= 0;\nod od:\nA[M,1]:= 1;\n" }{TEXT -1 71 "ip
 =0 or 1 depending whether this iteration is for black or white sites
" }{MPLTEXT 1 0 42 " \nif (-1)^M=1 then ip:= 0 else ip:= 1 fi;\n" }
{TEXT -1 42 "ip = ip0 on iterations that update u(1,0)." }{MPLTEXT 1 
0 32 "\nip0:= 1-ip;\nfor iter from 1 do " }{TEXT -1 18 "for each itera
tion" }{MPLTEXT 1 0 14 "\n  ip:= 1-ip;\n" }{TEXT -1 25 "     update si
tes on y=0\n" }{MPLTEXT 1 0 196 "  for i from 2+ip to 2*M-1 by 2 do if
 i <> M then\n      A[i,1]:= 0.5*A[i,2]+pp*A[i+1,1]+pm*A[i-1,1];\n    \+
  dA[i,1]:= 0.5*dA[i,2]+A[i+1,1]+pp*dA[i+1,1]-A[i-1,1]+pm*dA[i-1,1]   \+
fi od:\n  iip:= ip;\n  " }{TEXT -1 23 "update sites not on y=0" }
{MPLTEXT 1 0 320 "\n  for i from 2 to 2*M-1 do\n   iip:= 1-iip;\n   fo
r j from 2+iip to M-1 by 2 do\n     A[i,j]:= 0.25*(A[i,j-1]+A[i,j+1])+
pp*A[i+1,j]+pm*A[i-1,j];\n     dA[i,j]:= 0.25*(dA[i,j-1]+dA[i,j+1])+pp
*dA[i+1,j]         +A[i+1,j]+pm*dA[i-1,j]-A[i-1,j];\n     od od; \n  i
f ip=ip0 then\n    Rp:= A[M+1,1];\n    if abs(Rp-R) <= 1e-15 then \n" 
}{TEXT -1 65 "if no significant change in u(1,0), calculate v(eps) and
 next eps" }{MPLTEXT 1 0 201 "\n      R:= pp*A[M+1,1]+pm*A[M-1,1]+0.5*
A[M,2];\n      dR:= pp*dA[M+1,1]+A[M+1,1]+pm*dA[M-1,1]-A[M-1,1]+0.5*dA
[M,2];\n      print(R,dR);\n      return eps - (R-0.5)/dR;\n    else\n
     R:= Rp;\nfi fi;\nod\nend;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(getfleaGf*6$%$epsG%\"MG6/%\"AG%#d
AG%\"iG%\"jG%%iterG%#ppG%#pmG%\"RG%#dRG%#RpG%#ipG%$ip0G%$iipG6\"F7C,>8
$-%(hfarrayG6$;\"\"\",$9%\"\"#;F?FA>8%F;>8+\"\"!>8),&#F?\"\"%F?9$F?>8*
,&FLF?FN!\"\"?(8&F?F?F@%%trueG?(8'F?F?FAFUC$>&F:6$FTFWFH>&FEFenFH>&F:6
$FAF?F?@%/)FRFAF?>8.FH>F_oF?>8/,&F?F?F_oFR?(8(F?F?F7FUC'>F_oFco?(FT,&F
BF?F_oF?FB,&FAFBF?FRFU@$0FTFAC$>&F:6$FTF?,(&F:6$FTFB$\"\"&FR*&FJF?&F:6
$,&FTF?F?F?F?F?F?*&FPF?&F:6$,&FTF?F?FRF?F?F?>&FEF`p,,&FEFcpFdpFgpF?*&F
JF?&FEFhpF?F?F[qFR*&FPF?&FEF\\qF?F?>80F_o?(FTFBF?FjoFUC$>Fgq,&F?F?FgqF
R?(FW,&FBF?FgqF?FB,&FAF?F?FRFUC$>FZ,*&F:6$FT,&FWF?F?FR$\"#D!\"#*&FerF?
&F:6$FT,&FWF?F?F?F?F?*&FJF?&F:6$FipFWF?F?*&FPF?&F:6$F]qFWF?F?>Fgn,.&FE
FcrFer*&FerF?&FEFjrF?F?*&FJF?&FEF^sF?F?F]sF?*&FPF?&FEFasF?F?F`sFR@$/F_
oFboC$>8-&F:6$,&FAF?F?F?F?@%1-%$absG6#,&F_tF?FGFR$F?!#:C&>FG,(*&FJF?F`
tF?F?*&FPF?&F:6$F^rF?F?F?*&FdpF?&F:6$FAFBF?F?>8,,,*&FJF?&FEFatF?F?F`tF
?*&FPF?&FEFauF?F?F`uFR*&FdpF?&FEFduF?F?-%&printG6$FGFfuO,&FNF?*&,&FGF?
$FepFRFRF?FfuFRFR>FGF_tF7F7F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 206 "Digits:= 15; \nti:= time(): lasteps:= 0; eps:= 0.1: N:= 5:\nfor
 iter from 1 while abs(lasteps - eps) > 1e-14 do\n  N:= floor(3/2*N);
\n  lasteps:= eps:\n  eps:= evalhf(getflea(eps,N));\n  (time()-ti)*sec
onds;\nod;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#:" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(lastepsG\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"NG\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(laste
psG$\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"3*RwFL'z4+T!#=$!
3!fR&yps!f)>!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$epsG$\"3miMNL'f&
oa!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%(secondsG$\"$E'!\"$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"#5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(lastepsG$\"3miMNL'f&oa!#>" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$$\"3-2y#zF!>M^!#=$!37Q9fL*ypH#!#<" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$epsG$\"3kIqgzHw_g!#>" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,$%(secondsG$\"%>M!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG
\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(lastepsG$\"3kIqgzHw_g!#>" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"3V3\\>cJUK]!#=$!35(=DLg.V^#!#<" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$epsG$\"3#4#\\'pz<<='!#>" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,$%(secondsG$\"&FE\"!\"$" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"NG\"#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(last
epsG$\"3#4#\\'pz<<='!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"3_\\eiN8
Q-]!#=$!3u!*>-J$*)>_#!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$epsG$\"
3G4lg(3g6>'!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%(secondsG$\"&)eO!
\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"#L" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(lastepsG$\"3G4lg(3g6>'!#>" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$$\"3zfW@]\"f++&!#=$!3RW]BxQY@D!#<" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$epsG$\"3)*HuSuYR\">'!#>" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$%(secondsG$\"&Wk*!\"$" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"NG\"#\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(lastepsG$\"3)*
HuSuYR\">'!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"3'RK^[>+++&!#=$!37
%*fKY1V@D!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$epsG$\"3;\\#)=ZaR\"
>'!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%(secondsG$\"']NB!\"$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"#t" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(lastepsG$\"3;\\#)=ZaR\">'!#>" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$$\"3W%y_+++++&!#=$!3!p(=0A0V@D!#<" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$epsG$\"37#f(RZaR\">'!#>" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$%(secondsG$\"'!3V&!\"$" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"NG\"$4\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(lastepsG$\"37
#f(RZaR\">'!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"3W*)************
\\!#=$!32dm,A0V@D!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$epsG$\"3]]v
RZaR\">'!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%(secondsG$\"(&=V7!\"
$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "The latest " }{XPPEDIT 18 0 
"epsilon" "6#%(epsilonG" }{TEXT -1 10 " value of " }{XPPEDIT 18 0 ".61
9139544739755050e-1;" "6#-%&FloatG6$\"3]]vRZaR\">'!#>" }{TEXT -1 323 "
 should have more than 10 correct digits (actually it has 12, not coun
ting the initial 0).  \n\nBut this method might be very slow if we wan
ted, say, 30 digits.  Here's a different method that I thought of late
r, which works directly with the infinite lattice.  First of all, acco
rding to probability theory, the probability " }{XPPEDIT 18 0 "p" "6#%
\"pG" }{TEXT -1 45 " of return is related to the expected number " }
{XPPEDIT 18 0 "mu" "6#%#muG" }{TEXT -1 61 " of times the flea is at (0
,0) [including the first time] by " }{XPPEDIT 18 0 "mu = sum((k+1)*p^k
*(1-p),k=0..infinity)" "6#/%#muG-%$sumG6$*(,&%\"kG\"\"\"F+F+F+)%\"pGF*
F+,&F+F+F-!\"\"F+/F*;\"\"!%)infinityG" }{XPPEDIT 18 0 "``=1/(1-p)" "6#
/%!G*&\"\"\"F&,&F&F&%\"pG!\"\"F)" }{TEXT -1 25 ".  So we are looking f
or " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 9 " to make " }
{XPPEDIT 18 0 "mu = 2" "6#/%#muG\"\"#" }{TEXT -1 238 ".  We can write \+
this as the sum of the probabilities that the flea is at (0,0) at each
 time.  Noting that this means making the same number of hops north as
 south and the same number east as west, we can write this as a conver
gent series\n" }{XPPEDIT 18 0 "mu = sum(sum((2*j+2*k)!/(j!^2)/(k!^2)*(
1/16-epsilon^2)^j*(1/4)^(2*k),k = 0 .. infinity),j = 0 .. infinity);" 
"6#/%#muG-%$sumG6$-F&6$*,-%*factorialG6#,&*&\"\"#\"\"\"%\"jGF1F1*&F0F1
%\"kGF1F1F1*$-F,6#F2F0!\"\"*$-F,6#F4F0F8),&*&F1F1\"#;F8F1*$%(epsilonGF
0F8F2F1)*&F1F1\"\"%F8*&F0F1F4F1F1/F4;\"\"!%)infinityG/F2;FHFI" }
{XPPEDIT 18 0 "``=sum(c[j]*q^j,j=0..infinity)" "6#/%!G-%$sumG6$*&&%\"c
G6#%\"jG\"\"\")%\"qGF,F-/F,;\"\"!%)infinityG" }{TEXT -1 7 " where " }
{XPPEDIT 18 0 "q = 1/16 - epsilon^2" "6#/%\"qG,&*&\"\"\"F'\"#;!\"\"F'*
$%(epsilonG\"\"#F)" }{TEXT -1 49 ".  All we need to do is compute the \+
coefficients " }{XPPEDIT 18 0 "c[j]" "6#&%\"cG6#%\"jG" }{TEXT -1 31 ",
 sum the series and solve for " }{XPPEDIT 18 0 "mu = 2" "6#/%#muG\"\"#
" }{TEXT -1 87 ".   The series doesn't converge very quickly: the radi
us of convergence is 1/16 (since " }{XPPEDIT 18 0 "mu = infinity" "6#/
%#muG%)infinityG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "epsilon = 0" "6#
/%(epsilonG\"\"!" }{TEXT -1 7 ", i.e. " }{XPPEDIT 18 0 "q=1/16" "6#/%
\"qG*&\"\"\"F&\"#;!\"\"" }{TEXT -1 117 "), and we will be looking at v
alues of q rather close to 1/16.  So we'll need lots of terms to get a
n accurate value." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 159 "Maple calculates c[j] as an expression involving a hyper
geometric function.  Maple 8 can convert this to an expression involvi
ng associated Legendre functions." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 80 "cj:= sum((2*j+2*k)!/(j!)^2/(k!)^2*(1/4)^(2*k),k=0..in
finity) assuming j::posint;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cjG*
(-%*factorialG6#,$%\"jG\"\"#\"\"\"-F'6#F*!\"#-%*hypergeomG6%7$,&F*F,F,
F,,&F*F,#F,F+F,7#F,#F,\"\"%F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 37 "convert(%,`2F1`); # Maple 8 required " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*&#\"\"#\"\"$\"\"\"*2-%*factorialG6#,$*&F&F(%\"jGF(F(
F(-F+6#F/!\"#-%&GAMMAG6#,&*&F&F(F/F(F(F(F(!\"\"%#PiG#F8F&-%*LegendreQG
6%F:,&*&F&F(F/F(F(#F(F&F(F&F()F&F>F()F',&F/F(#F'\"\"%F8F8-%$expG6#*(^#
F@F(,&*&FEF(F/F(F(F(F(F(F9F(F8F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 21 "We want to calculate " }{XPPEDIT 18 0 "c[j]" "6#&%\"cG6#%\"jG" 
}{TEXT -1 55 " numerically, which Maple can do quite nicely for each \+
" }{XPPEDIT 18 0 "j" "6#%\"jG" }{TEXT -1 98 " using the hypergeom expr
ession.  If we want, say, about 30 digits, we'll want enough terms unt
il " }{XPPEDIT 18 0 "c[j]*q^j < 10^(-31)" "6#2*&&%\"cG6#%\"jG\"\"\")%
\"qGF(F))\"#5,$\"#J!\"\"" }{TEXT -1 9 " for the " }{XPPEDIT 18 0 "q" "
6#%\"qG" }{TEXT -1 58 " we'll be using.  I'll do the calculations with
 40 digits." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "Digits:= 40
: ti:= time():\nq0:= 1/16 - .06^2:\nfor j from 0 do c[j]:= evalf(cj);
\nif c[j]*q0^j < 1e-31 then break fi;\nod:\n(time()-ti)*seconds;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%(secondsG$\"'Ckc!\"$" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "j;" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#\"%n5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "c[j]*q0^j;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"I/A:%)*>U3()oSr$>?o'e*4J'*!#r" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Newt:= proc(eps) local q,j,v
,vp; \n" }{TEXT -1 47 "This does one iteration of Newton's method for \+
" }{XPPEDIT 18 0 "sum(c[j]*q^j,j = 1 .. 1067) = 2;" "6#/-%$sumG6$*&&%
\"cG6#%\"jG\"\"\")%\"qGF+F,/F+;F,\"%n5\"\"#" }{MPLTEXT 1 0 115 "\n  q:
= evalf(1/16 - eps^2);\n  v:= add(c[j]*[1,j/q]*q^j,j=0..1067);\n  q:= \+
q - (v[1]-2)/v[2];\n  sqrt(1/16-q);\n  end;\n" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%%NewtGf*6#%$epsG6&%\"qG%\"jG%\"vG%#vpG6\"F-C&>8$-%&ev
alfG6#,&#\"\"\"\"#;F6*$)9$\"\"#F6!\"\">8&-%$addG6$*(&%\"cG6#8%F67$F6*&
FFF6F0F<F6)F0FFF6/FF;\"\"!\"%n5>F0,&F0F6*&,&&F>6#F6F6F;F<F6&F>6#F;F<F<
-%%sqrtG6#,&F5F6F0F<F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
177 "epsilon:= 0.06; ti:= time():\ndo\n  oldepsilon:= epsilon:\n  epsi
lon:= Newt(epsilon);\n  print(epsilon);\n  if abs(oldepsilon-epsilon) \+
< 1e-30 then break fi;\nod:\n(time()-ti)*seconds;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(epsilonG$\"\"'!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#$\"I)[W(p%yBn$Q'4G\\Bftxka='!#T" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
\"Im\"=i<w)en#*G4#p9hYp*Q\">'!#T" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
\"I&>\"f.P`m)>'zbtzORZaR\">'!#T" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"
IhtU\\En]o^<[G%4*RZaR\">'!#T" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"I5G
J45KZ;_<[G%4*RZaR\">'!#T" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"IEGJ45K
Z;_<[G%4*RZaR\">'!#T" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%(secondsG$
\"%!Q&!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "This has 31 correct
 digits." }}}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 10 "Problem 7:" }}{PARA 
0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 
173 " be the 20,000 x 20,000 matrix whose entries are zero everywhere \+
except for the primes 2, 3, 5, 7, ..., 224737 along the main diagonal \+
and the number 1 in all the positions " }{XPPEDIT 18 0 "a[i,j]" "6#&%
\"aG6$%\"iG%\"jG" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "abs(i-j) = 1" "
6#/-%$absG6#,&%\"iG\"\"\"%\"jG!\"\"F)" }{TEXT -1 46 ",2,4,8,...,16384.
  What is the (1,1) entry of " }{XPPEDIT 18 0 "A^(-1)" "6#)%\"AG,$\"\"
\"!\"\"" }{TEXT -1 1 "?" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Soluti
on:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "This is equivalent to compu
ting " }{XPPEDIT 18 0 "x[1]" "6#&%\"xG6#\"\"\"" }{TEXT -1 15 " after s
olving " }{XPPEDIT 18 0 "A*x=b" "6#/*&%\"AG\"\"\"%\"xGF&%\"bG" }{TEXT 
-1 8 ", where " }{XPPEDIT 18 0 "b = [1,0,0,`...`]" "6#/%\"bG7&\"\"\"\"
\"!F'%$...G" }{TEXT -1 69 ". The system is diagonally dominant, and we
 use an iterative method: " }{XPPEDIT 18 0 "x(n+1) = D^(-1)*(b - (A-D)
*x(n))" "6#/-%\"xG6#,&%\"nG\"\"\"F)F)*&)%\"DG,$F)!\"\"F),&%\"bGF)*&,&%
\"AGF)F,F.F)-F%6#F(F)F.F)" }{TEXT -1 29 ", where D is the diagonal of \+
" }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 119 " (but without actually c
onstructing the matrices). Gonnet used a  Darwin program, equivalent t
o the following in Maple." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
185 "b:= Array(1..20000,datatype=float[8]): b[1]:= 1: \nx:= Array(1..2
0000,datatype=float[8]):\nprimes:= Array(1..20000,datatype=float[8]):
\nfor i from 1 to 20000 do primes[i]:= ithprime(i) od:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 507 "solvproc:= proc(N)\nlocal iter, to
terr, i, rhs, ij, newxi;\nglobal x;\nfor i from 1 to N do x[i]:= 0 od:
\nfor iter to 40 do\n    toterr := 0;\n    for i to N do\n\011     rhs
 := b[i];\n     \011ij := 1;\n\011     while ij < N do\n\011       if \+
i-ij > 0 then rhs := rhs - x[i-ij] fi;\n\011       if i+ij <= N then r
hs := rhs - x[i+ij] fi;\n\011       ij := 2*ij\n\011     od;\n\011    \+
 newxi := rhs/primes[i];\n\011     toterr := toterr + abs(newxi-x[i]);
\n\011     x[i] := newxi\n\011   od;\n    if toterr=0 then break fi;\n
    lprint(toterr);\nod:\nx[1]; \nend;" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%)solvprocGf*6#%\"NG6(%%iterG%'toterrG%\"iG%$rhsG%#ijG%&newxiG6
\"F/C%?(8&\"\"\"F39$%%trueG>&%\"xG6#F2\"\"!?(8$F3F3\"#SF5C&>8%F:?(F2F3
F3F4F5C(>8'&%\"bGF9>8(F3?(F/F3F3F/2FHF4C%@$2F:,&F2F3FH!\"\">FD,&FDF3&F
86#FNFO@$1,&F2F3FHF3F4>FD,&FDF3&F86#FVFO>FH,$FH\"\"#>8)*&FDF3&%'primes
GF9FO>F@,&F@F3-%$absG6#,&FinF3F7FOF3>F7Fin@$/F@F:[-%'lprintG6#F@&F86#F
3F/6#F8F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "ti:= time(): \+
\nevalhf(solvproc(20000));\n(time()-ti)*seconds ;" }}{PARA 6 "" 1 "" 
{TEXT -1 19 ".887865995381123986" }}{PARA 6 "" 1 "" {TEXT -1 19 ".2786
58134431981340" }}{PARA 6 "" 1 "" {TEXT -1 22 ".873116873307076269e-1
" }}{PARA 6 "" 1 "" {TEXT -1 22 ".280984403047528780e-1" }}{PARA 6 "" 
1 "" {TEXT -1 22 ".896168412303717000e-2" }}{PARA 6 "" 1 "" {TEXT -1 
22 ".282597754444424655e-2" }}{PARA 6 "" 1 "" {TEXT -1 22 ".8857087172
67995350e-3" }}{PARA 6 "" 1 "" {TEXT -1 22 ".276700748083163562e-3" }}
{PARA 6 "" 1 "" {TEXT -1 22 ".862812417589902238e-4" }}{PARA 6 "" 1 "
" {TEXT -1 22 ".268719686136664988e-4" }}{PARA 6 "" 1 "" {TEXT -1 22 "
.836254516891092076e-5" }}{PARA 6 "" 1 "" {TEXT -1 22 ".26011174730884
4605e-5" }}{PARA 6 "" 1 "" {TEXT -1 22 ".808810146402529806e-6" }}
{PARA 6 "" 1 "" {TEXT -1 22 ".251448343535394878e-6" }}{PARA 6 "" 1 "
" {TEXT -1 22 ".781623535215165102e-7" }}{PARA 6 "" 1 "" {TEXT -1 22 "
.242947694034407002e-7" }}{PARA 6 "" 1 "" {TEXT -1 22 ".75510388082236
6756e-8" }}{PARA 6 "" 1 "" {TEXT -1 22 ".234685997925699352e-8" }}
{PARA 6 "" 1 "" {TEXT -1 22 ".729389779125172184e-9" }}{PARA 6 "" 1 "
" {TEXT -1 22 ".226686206265103620e-9" }}{PARA 6 "" 1 "" {TEXT -1 23 "
.704513858008347722e-10" }}{PARA 6 "" 1 "" {TEXT -1 23 ".2189541217695
08914e-10" }}{PARA 6 "" 1 "" {TEXT -1 23 ".680502535422857584e-11" }}
{PARA 6 "" 1 "" {TEXT -1 23 ".211472595988469877e-11" }}{PARA 6 "" 1 "
" {TEXT -1 23 ".656998092145587618e-12" }}{PARA 6 "" 1 "" {TEXT -1 23 
".204068598900091197e-12" }}{PARA 6 "" 1 "" {TEXT -1 23 ".635341005751
694184e-13" }}{PARA 6 "" 1 "" {TEXT -1 23 ".199326919187803528e-13" }}
{PARA 6 "" 1 "" {TEXT -1 23 ".616146405998221652e-14" }}{PARA 6 "" 1 "
" {TEXT -1 23 ".163608756217164570e-14" }}{PARA 6 "" 1 "" {TEXT -1 23 
".409211670512742128e-15" }}{PARA 6 "" 1 "" {TEXT -1 23 ".708330759345
821984e-16" }}{PARA 6 "" 1 "" {TEXT -1 23 ".934890384403199346e-19" }}
{PARA 6 "" 1 "" {TEXT -1 23 ".545716338525474910e-22" }}{PARA 6 "" 1 "
" {TEXT -1 23 ".355549011630070350e-29" }}{PARA 6 "" 1 "" {TEXT -1 23 
".173686262298320443e-34" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3I2SoiM
y]s!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%(secondsG$\"'D4=!\"$" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "This answer is correct to 14 digit
s." }}}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 10 "Problem 8:" }}{PARA 0 "" 
0 "" {TEXT -1 15 "A square plate " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"
\"!\"\"F%" }{TEXT -1 3 " x " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"
\"F%" }{TEXT -1 19 " is at temperature " }{XPPEDIT 18 0 "u=0" "6#/%\"u
G\"\"!" }{TEXT -1 11 ".  At time " }{XPPEDIT 18 0 "t=0" "6#/%\"tG\"\"!
" }{TEXT -1 33 " the temperature is increased to " }{XPPEDIT 18 0 "u=5
" "6#/%\"uG\"\"&" }{TEXT -1 49 " along one of the four sides while bei
ng held at " }{XPPEDIT 18 0 "u=0" "6#/%\"uG\"\"!" }{TEXT -1 78 " along
 the other three sides, and heat then flows into the plate according t
o " }{XPPEDIT 18 0 "u[t]=Delta*u" "6#/&%\"uG6#%\"tG*&%&DeltaG\"\"\"F%F
*" }{TEXT -1 35 ".  When does the temperature reach " }{XPPEDIT 18 0 "
u=1" "6#/%\"uG\"\"\"" }{TEXT -1 28 " at the center of the plate?" }}}
{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Solution:" }}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 170 "This is a standard heat-equation problem with homogene
ous boundary conditions.  It will be slightly more convenient to take \+
the plate as [0,2] x [0,2], and the side with " }{XPPEDIT 18 0 "u=5" "
6#/%\"uG\"\"&" }{TEXT -1 18 " will be taken as " }{XPPEDIT 18 0 "x=2" 
"6#/%\"xG\"\"#" }{TEXT -1 67 ".  The solution is obtained as the sum o
f a steady-state solution v" }{XPPEDIT 18 0 "(x,y) = sum(a[j]*sinh(j*P
i*x/2)*sin(j*Pi*y/2),j=1..infinity)" "6#/6$%\"xG%\"yG-%$sumG6$*(&%\"aG
6#%\"jG\"\"\"-%%sinhG6#**F.F/%#PiGF/F%F/\"\"#!\"\"F/-%$sinG6#**F.F/F4F
/F&F/F5F6F//F.;F/%)infinityG" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "a[j
] = int(5/sinh(j*Pi)*sin(j*Pi*y/2),y=0..2)" "6#/&%\"aG6#%\"jG-%$intG6$
*(\"\"&\"\"\"-%%sinhG6#*&F'F-%#PiGF-!\"\"-%$sinG6#**F'F-F2F-%\"yGF-\"
\"#F3F-/F8;\"\"!F9" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "v(2,y) = 5
" "6#/-%\"vG6$\"\"#%\"yG\"\"&" }{TEXT -1 71 ", and a solution of the e
quation with homogeneous boundary conditions, " }{XPPEDIT 18 0 "w(x,y,
t) = sum(sum(b[i,j]*sin(i*Pi*x/2)*sin(j*Pi*y/2)*exp(-Pi^2*(i^2+j^2)/4*
t),i=1..infinity),j=1..infinity)" "6#/-%\"wG6%%\"xG%\"yG%\"tG-%$sumG6$
-F+6$**&%\"bG6$%\"iG%\"jG\"\"\"-%$sinG6#**F3F5%#PiGF5F'F5\"\"#!\"\"F5-
F76#**F4F5F:F5F(F5F;F<F5-%$expG6#,$**F:F;,&*$F3F;F5*$F4F;F5F5\"\"%F<F)
F5F<F5/F3;F5%)infinityG/F4;F5FK" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "
b[i,j] = - a[j]*int(sin(i*Pi*x/2)*sinh(j*Pi*x/2),x=0..2)" "6#/&%\"bG6$
%\"iG%\"jG,$*&&%\"aG6#F(\"\"\"-%$intG6$*&-%$sinG6#**F'F.%#PiGF.%\"xGF.
\"\"#!\"\"F.-%%sinhG6#**F(F.F7F.F8F.F9F:F./F8;\"\"!F9F.F:" }{TEXT -1 
9 " so that " }{XPPEDIT 18 0 "w(x,y,0) = -v(x,y)" "6#/-%\"wG6%%\"xG%\"
yG\"\"!,$-%\"vG6$F'F(!\"\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 67 "aj:= int(5/sinh(j*Pi)*sin(j*Pi*y/2),y = 0 .. 2) ass
uming j::posint;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ajG,$**,&)!\"\"
%\"jG\"\"\"F+F)F+-%%sinhG6#*&F*F+%#PiGF+F)F*F)F0F)!#5" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "bij:= -aj*int(sin(i*Pi*x/2)*sinh(j*
Pi*x/2),x = 0 .. 2) assuming i::posint,j::posint;" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%$bijG,$*4,&)!\"\"%\"jG\"\"\"F+F)F+-%%sinhG6#*&F*F+%#P
iGF+F)F*F)F0!\"#,2*(F*F+))F)%\"iG\"\"#F+^#F+F+F+*&-%$expG6#,$F/F7F+F6F
+F+*&F6F+F4F+F)**F*F+F:F+F4F+F8F+F+*(F6F+F:F+F4F+F+*&^#F)F+F*F+F+F6F)*
(FBF+F:F+F*F+F+F+,&*&F*F+F8F+F+F6F+F),&FEF+F6F)F)-F;6#,$F/F)F+F5F)\"\"
&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "bij:= simplify(evalc(c
onvert(%,exp))) assuming i::posint,j::posint;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$bijG,$*,,&)!\"\",&%\"iG\"\"\"%\"jGF,F,)F)F+F)F,F+F,,
&*$)F+\"\"#F,F,*$)F-F2F,F,F)%#PiG!\"#F-F)!#?" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 84 "We want to evaluate this at x=1, y=1.  By symmetry it i
s obvious that v(1,1) = 5/4. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "A
s for " }{XPPEDIT 18 0 "w(x,y,t)" "6#-%\"wG6%%\"xG%\"yG%\"tG" }{TEXT 
-1 13 ", the factor " }{XPPEDIT 18 0 "exp(-Pi^2*(i^2+j^2)/4*t) < 10^(-
31);" "6#2-%$expG6#,$**%#PiG\"\"#,&*$%\"iGF*\"\"\"*$%\"jGF*F.F.\"\"%!
\"\"%\"tGF.F2)\"#5,$\"#JF2" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "t > 1/
4" "6#2*&\"\"\"F%\"\"%!\"\"%\"tG" }{TEXT -1 116 " (some preliminary in
vestigation assured us that the answer we want should be somewhere bet
ween about 1/4 and 1) if " }{XPPEDIT 18 0 "i^2 + j^2 > 116" "6#2\"$;\"
,&*$%\"iG\"\"#\"\"\"*$%\"jGF(F)" }{TEXT -1 28 ".  So this approximatio
n to " }{XPPEDIT 18 0 "u(1,1,t)" "6#-%\"uG6%\"\"\"F&%\"tG" }{TEXT -1 
54 " should be accurate to approximately 30 decimals when " }{XPPEDIT 
18 0 "t > 1/4" "6#2*&\"\"\"F%\"\"%!\"\"%\"tG" }{TEXT -1 2 " :" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "U:= 5/4 + add(add(bij*sin(i
*Pi*1/2)*sin(j*Pi*1/2)*exp(-Pi^2*(i^2+j^2)/4*t),j=1..floor(sqrt(116-i^
2))),i=1..10);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"UG,<#\"\"&\"\"%
\"\"\"*(\"#?F)%#PiG!\"#-%$expG6#,$*&)F,\"\"#F)%\"tGF)#!\"\"F4F)F7*(#\"
#S\"\"$F)F,F--F/6#,$F2#!\"&F4F)F)*(\"\")F)F,F--F/6#,$F2#!#8F4F)F7*(#\"
$s\"\"#NF)F,F--F/6#,$F2#!#DF4F)F)*&#F:\"\"*F)*&F,F--F/6#,$F2#!#TF4F)F)
F7*&#F+FSF)*&F,F--F/6#,$F2#!\"*F4F)F)F7*(#FBF;F)F,F--F/6#,$F2#!#<F4F)F
)*&#F:\"#@F)*&F,F--F/6#,$F2#!#HF4F)F)F7*(#F:\"#FF)F,F--F/6#,$F2#!#XF4F
)F)*(#FB\"\"(F)F,F--F/6#,$F2#!#PF4F)F)*&#FBFSF)*&F,F--F/6#,$F2#!#`F4F)
F)F7*&#F+\"#\\F)*&F,F--F/6#,$F2#!#\\F4F)F)F7" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 13 "Now to solve " }{XPPEDIT 18 0 "u(1,1,t) = 1" "6#/-%\"uG
6%\"\"\"F'%\"tGF'" }{TEXT -1 2 ": " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "Digits:= 40: fsolve(U=1,t=1/4 .. 1);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"IyC^kD$foOV(zj$)oLqQ6SU!#S" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 36 "This actually has 39 correct digits." }}}}{EXCHG 
{PARA 4 "" 0 "" {TEXT -1 10 "Problem 9:" }}{PARA 0 "" 0 "" {TEXT -1 
13 "The integral " }{XPPEDIT 18 0 "I(alpha) = int((2+sin(10*alpha))*x^
alpha*sin(alpha/(2-x)),x = 0 .. 2);" "6#/-%\"IG6#%&alphaG-%$intG6$*(,&
\"\"#\"\"\"-%$sinG6#*&\"#5F.F'F.F.F.)%\"xGF'F.-F06#*&F'F.,&F-F.F5!\"\"
F:F./F5;\"\"!F-" }{TEXT -1 26 " depends on the parameter " }{XPPEDIT 
18 0 "alpha" "6#%&alphaG" }{TEXT -1 21 ".  What is the value " }
{XPPEDIT 18 0 "alpha*epsilon* [0,5]" "6#*(%&alphaG\"\"\"%(epsilonGF%7$
\"\"!\"\"&F%" }{TEXT -1 10 " at which " }{XPPEDIT 18 0 "I(alpha)" "6#-
%\"IG6#%&alphaG" }{TEXT -1 22 " achieves its maximum?" }}}{SECT 1 
{PARA 5 "" 0 "" {TEXT -1 9 "Solution:" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 142 "This is rather different from Gaston's solution.  The numerica
l integration works in Maple 8, but not Maple 7.\nWe want to be able t
o evaluate " }{XPPEDIT 18 0 "`I'`(alpha)" "6#-%#I'G6#%&alphaG" }{TEXT 
-1 5 " and " }{XPPEDIT 18 0 "`I''`(alpha)" "6#-%$I''G6#%&alphaG" }
{TEXT -1 80 " so that we can use Newton's method to find the location \+
of the maximum.  Write " }{XPPEDIT 18 0 "J(alpha) = int(x^alpha*sin(al
pha/(2-x)),x=0..2)" "6#/-%\"JG6#%&alphaG-%$intG6$*&)%\"xGF'\"\"\"-%$si
nG6#*&F'F.,&\"\"#F.F-!\"\"F5F./F-;\"\"!F4" }{TEXT -1 4 " so " }
{XPPEDIT 18 0 "I(alpha) = (2+sin(10*alpha))*J(alpha)" "6#/-%\"IG6#%&al
phaG*&,&\"\"#\"\"\"-%$sinG6#*&\"#5F+F'F+F+F+-%\"JG6#F'F+" }{TEXT -1 2 
", " }{XPPEDIT 18 0 "`I'`(alpha) = 10*cos(10*alpha)*J(alpha) + (2+sin(
10*alpha))*`J'`(alpha)" "6#/-%#I'G6#%&alphaG,&*(\"#5\"\"\"-%$cosG6#*&F
*F+F'F+F+-%\"JG6#F'F+F+*&,&\"\"#F+-%$sinG6#*&F*F+F'F+F+F+-%#J'G6#F'F+F
+" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "`I''`(alpha) = -100*sin(10*alp
ha)*J(alpha) + 20*cos(10*alpha)*`J'`(alpha) + (2+sin(10*alpha))*`J''`(
alpha)" "6#/-%$I''G6#%&alphaG,(*(\"$+\"\"\"\"-%$sinG6#*&\"#5F+F'F+F+-%
\"JG6#F'F+!\"\"*(\"#?F+-%$cosG6#*&F0F+F'F+F+-%#J'G6#F'F+F+*&,&\"\"#F+-
F-6#*&F0F+F'F+F+F+-%$J''G6#F'F+F+" }{TEXT -1 44 ".\n  \nWe break the i
nterval up into pieces.  " }}{PARA 0 "" 0 "" {TEXT -1 16 "On the inter
val " }{XPPEDIT 18 0 "0 .. 1/3" "6#;\"\"!*&\"\"\"F&\"\"$!\"\"" }{TEXT 
-1 30 " we use a series in powers of " }{XPPEDIT 18 0 "x" "6#%\"xG" }
{TEXT -1 6 ".  On " }{XPPEDIT 18 0 "1/3 .. 2 " "6#;*&\"\"\"F%\"\"$!\"
\"\"\"#" }{TEXT -1 27 " we use the transformation " }{XPPEDIT 18 0 "y \+
= alpha/(2-x)" "6#/%\"yG*&%&alphaG\"\"\",&\"\"#F'%\"xG!\"\"F+" }{TEXT 
-1 28 " , obtaining an integral on " }{XPPEDIT 18 0 "3*alpha/5 .. infi
nity" "6#;*(\"\"$\"\"\"%&alphaGF&\"\"&!\"\"%)infinityG" }{TEXT -1 19 "
.  On the interval " }{XPPEDIT 18 0 "3*alpha/5 .. 2*Pi " "6#;*(\"\"$\"
\"\"%&alphaGF&\"\"&!\"\"*&\"\"#F&%#PiGF&" }{TEXT -1 50 " we use Maple'
s own numerical integration, and on " }{XPPEDIT 18 0 "2*Pi .. infinity
" "6#;*&\"\"#\"\"\"%#PiGF&%)infinityG" }{TEXT -1 62 " we expand the in
tegrand in negative powers of y times sin(y)." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 21 "restart: ti:= time():" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 23 "First for the interval " }{XPPEDIT 18 0 "0 .. 1/3" "6#;
\"\"!*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 72 ": J1 is the integral, dJ1 its
 derivative and d2J1 its second derivative." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 320 "f:= x^alpha*sin(alpha/(2-x));\nS:= convert(series
(sin(alpha/(2-x)),x,40),polynom):\nintp:= proc(t) local c,xp,n; \n  if
 has(t,x) then xp,c := selectremove(has,t,x); \n    if xp = x then n:=
 1 else n:= op(2,xp) fi;\n  else n:= 0; c:= t \n  fi;\n  collect(expan
d(c),[cos,sin])*3^(-n-alpha-1)/(n+alpha+1);\nend:\nJ1:= map(intp,S):  \+
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*&)%\"xG%&alphaG\"\"\"-%$sin
G6#*&F(F),&\"\"#F)F'!\"\"F0F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 46 "dJ1:= diff(J1,alpha):\nd2J1:= diff(J1,alpha$2):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "length(J1),length(dJ1),length(d2J1)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"&5_$\"'R!R\"\"'zCM" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 52 "How big is the last term in dJ1 for a typ
ical alpha?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "evalf(eval(s
elect(has,dJ1,3^(-40-alpha)),alpha=0.8));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+CyW*)>!#T" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 " \+
Next the transformation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
229 "xy := expand(solve(alpha/(2-x)=y,x));\ng:= subs(x=xy,f)*diff(xy,y
);\ng1:= eval(g,y=3*alpha/5);\nga:= normal(diff(g,alpha));\nga1:= norm
al(eval(ga,y=3*alpha/5));\ngy:= normal(eval(diff(g,y),y=3*alpha/5));\n
gaa:= normal(diff(g,alpha$2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x
yG,&*&%\"yG!\"\"%&alphaG\"\"\"F(\"\"#F*" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"gG**),&*&%\"yG!\"\"%&alphaG\"\"\"F*\"\"#F,F+F,-%$sinG6#F)F,F
)!\"#F+F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g1G,$*()#\"\"\"\"\"$%&
alphaGF)-%$sinG6#,$F+#F*\"\"&F)F+!\"\"#\"#D\"\"*" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%#gaG*,)*&,&%&alphaG!\"\"*&\"\"#\"\"\"%\"yGF-F-F-F.F*F
)F-F.!\"#-%$sinG6#F.F-,,*&-%#lnG6#F'F-)F)F,F-F***F,F-F)F-F5F-F.F-F-*$F
8F-F*F)F**&F,F-F.F-F-F-F(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ga1G
,$**)#\"\"\"\"\"$%&alphaGF)-%$sinG6#,$F+#F*\"\"&F)F+!\"#,(*&-%#lnG6#F*
F)F+F)F)*&F1F)F+F)F)F)!\"\"F)#!#D\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#gyG,$*()#\"\"\"\"\"$%&alphaGF),(*&-%$sinG6#,$F+#F*\"\"&F)F+F)
\"#D*(F*F)-%$cosGF0F)F+F)F)*&\"#5F)F.F)!\"\"F)F+!\"##F4\"#F" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%$gaaG*,)*&,&%&alphaG!\"\"*&\"\"#\"\"\"%\"y
GF-F-F-F.F*F)F-F.!\"#-%$sinG6#F.F-,8*&)-%#lnG6#F'F,F-)F)\"\"$F-F-**\"
\"%F-F5F-)F)F,F-F.F-F**(F,F-F6F-F9F-F-**F<F-F)F-F5F-)F.F,F-F-**F<F-F=F
-F6F-F.F-F**$F9F-F-*&F:F-F=F-F-*(\"\")F-F)F-F.F-F**(F,F-F6F-F=F-F-**FE
F-F)F-F6F-F.F-F**(FEF-F6F-F@F-F-F-F(F/" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 11 "So we want " }{XPPEDIT 18 0 "int(g,y=3*alpha/5 .. infinit
y)" "6#-%$intG6$%\"gG/%\"yG;*(\"\"$\"\"\"%&alphaGF,\"\"&!\"\"%)infinit
yG" }{TEXT -1 3 ".  " }{XPPEDIT 18 0 "J2(alpha)" "6#-%#J2G6#%&alphaG" 
}{TEXT -1 22 " is the integral from " }{XPPEDIT 18 0 "3*alpha/5" "6#*(
\"\"$\"\"\"%&alphaGF%\"\"&!\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "2*
Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "dJ2(alph
a)" "6#-%$dJ2G6#%&alphaG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "d2J2(alp
ha)" "6#-%%d2J2G6#%&alphaG" }{TEXT -1 68 " its derivatives.  Maple som
etimes seems to have trouble evaluating " }{XPPEDIT 18 0 "d2J2(alpha)
" "6#-%%d2J2G6#%&alphaG" }{TEXT -1 94 " to high precision, but we don'
t need as much accuracy there, so the relative error tolerance " }
{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 31 " can be increased
 in this case." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "dH:=diff(
int(h(a,y),y=3*a/5 .. 2*Pi),a);\nd2H:=diff(int(h(a,y),y=3*a/5 .. 2*Pi)
,a$2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dHG,&-%$intG6$-%%diffG6$-
%\"hG6$%\"aG%\"yGF//F0;,$F/#\"\"$\"\"&,$%#PiG\"\"#\"\"\"*&#F5F6F:-F-6$
F/F3F:!\"\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$d2HG,*-%$intG6$-%%di
ffG6$-%\"hG6$%\"aG%\"yG-%\"$G6$F/\"\"#/F0;,$F/#\"\"$\"\"&,$%#PiGF4\"\"
\"*&#F9F:F=-F*6$-F-6$F/F7F/F=!\"\"*&#F9F:F=--&%\"DG6#F=6#F-FCF=FD*&#\"
\"*\"#DF=--&FJ6#F4FLFCF=FD" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
55 "J2:= a -> evalf(Int(subs(alpha=a,g),y=3*a/5 .. 2*Pi));\n" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#J2Gf*6#%\"aG6\"6$%)operatorG%&arrowGF(-%&
evalfG6#-%$IntG6$-%%subsG6$/%&alphaG9$%\"gG/%\"yG;,$F7#\"\"$\"\"&,$%#P
iG\"\"#F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 193 "dJ2:= a -
> evalf(Int(subs(alpha=a,ga),y=3*a/5 .. 2*Pi) - 3/5*subs(alpha=a,g1));
\nd2J2:= a -> evalf(Int(subs(alpha=a,gaa),y=3*a/5 .. 2*Pi,epsilon=10^(
-Digits/2)) - subs(alpha=a,6/5*ga1+9/25*gy));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$dJ2Gf*6#%\"aG6\"6$%)operatorG%&arrowGF(-%&evalfG6#,&
-%$IntG6$-%%subsG6$/%&alphaG9$%#gaG/%\"yG;,$F8#\"\"$\"\"&,$%#PiG\"\"#
\"\"\"*&#F?F@FD-F46$F6%#g1GFD!\"\"F(F(F(" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%%d2J2Gf*6#%\"aG6\"6$%)operatorG%&arrowGF(-%&evalfG6#,&-%$IntG6
%-%%subsG6$/%&alphaG9$%$gaaG/%\"yG;,$F8#\"\"$\"\"&,$%#PiG\"\"#/%(epsil
onG)\"#5,$%'DigitsG#!\"\"FC\"\"\"-F46$F6,&%$ga1G#\"\"'F@*&#\"\"*\"#DFL
%#gyGFLFLFKF(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "J2(0.8
),dJ2(0.8),d2J2(0.8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"+iI&)z%*!#
5$!*o%oKZF%$!+>yqn=!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 18 " For the inte
rval " }{XPPEDIT 18 0 "2*Pi .. infinity" "6#;*&\"\"#\"\"\"%#PiGF&%)inf
inityG" }{TEXT -1 27 ", we will use a series for " }{XPPEDIT 18 0 "(2-
alpha/y)^alpha;" "6#),&\"\"#\"\"\"*&%&alphaGF&%\"yG!\"\"F*F(" }{TEXT 
-1 23 " in negative powers of " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT 
-1 32 ", relying on Maple to integrate " }{XPPEDIT 18 0 "int(y^(-n)*si
n(y),y = 2*Pi .. infinity);" "6#-%$intG6$*&)%\"yG,$%\"nG!\"\"\"\"\"-%$
sinG6#F(F,/F(;*&\"\"#F,%#PiGF,%)infinityG" }{TEXT -1 11 ".  We have " 
}{XPPEDIT 18 0 "g = sum(2^(2+alpha-n)*(-alpha)^(n)*pochhammer(alpha-n+
3,n-3)/(n-2)!/y^n, n=2..infinity)" "6#/%\"gG-%$sumG6$*,)\"\"#,(F*\"\"
\"%&alphaGF,%\"nG!\"\"F,),$F-F/F.F,-%+pochhammerG6$,(F-F,F.F/\"\"$F,,&
F.F,F6F/F,-%*factorialG6#,&F.F,F*F/F/)%\"yGF.F//F.;F*%)infinityG" }
{TEXT -1 7 " where " }{XPPEDIT 18 0 "pochhammer(t,m) = t*(t+1)*`...`*(
t+m-1)" "6#/-%+pochhammerG6$%\"tG%\"mG**F'\"\"\",&F'F*F*F*F*%$...GF*,(
F'F*F(F*F*!\"\"F*" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 112 "J3:=add(2^(2+alpha-n)*(-alpha)^n*pochhammer(alpha-n+
3,n-3)/(n-2)!*int(sin(y)/y^n,y=2*Pi..infinity),n = 2 .. 30):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "dJ3:= diff(J3,alpha): d2J3:=
 diff(J3,alpha$2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "evalf
(eval([J3,dJ3,d2J3],alpha=0.8));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%
$\"+)o-l*H!#6$\"+!*H(4Y&F&$\"+MC+(z%F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 177 "t30:= add(2^(2+alpha-n)*(-alpha)^n*pochhammer(alpha-
n+3,n-3)/(n-2)!*int(sin(y)/y^n,y=2*Pi..infinity),n = 30..30); evalf(ev
al([t30,diff(t30,alpha),diff(t30,alpha$2)],alpha=0.8));" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#>%$t40G,$*,)\"\"#,&!#Q\"\"\"%&alphaGF+F+F,\"#S-%
+pochhammerG6$,&!#PF+F,F+\"#PF+,J\"C+DJXe.\"z,x;B8\\BY?$F+*&\"%g7F+)%#
PiG\"#IF+!\"\"*&\"2]irkvh<')*F+)F9\"#9F+F;*&\"+]d-x7F+)F9\"#AF+F;*&\"&
!oAF+)F9\"#GF+F+*&\",+5vPo)F+)F9\"#?F+F+*&\"<voz'Q^l!pDW)HD7F+)F9\"\"'
F+F;*(\"#kF+-%#CiG6#,$F9F(F+)F9\"#QF+F+*&\"0+Dg4fez(F+)F9\"#;F+F+*&\"5
+vV2ZjUEz9F+)F9\"#7F+F+*&\"@+DcE!>faSu$z'R\\B'*F+)F9F(F+F;*&\".+0rFYU(
F+)F9\"#=F+F;*&\"#CF+)F9\"#MF+F;*&\"$?\"F+)F9\"#KF+F+*&\"'+PiF+)F9\"#E
F+F;*&\"7]7Gb6R#y3hf#F+)F9\"#5F+F;*&FenF+)F9\"#OF+F+*&\">+]P9hcHL#Q))y
yMKF+)F9\"\"%F+F+*&\")+VKCF+)F9FboF+F+*&\"9]P4AXSDG35q_F+)F9\"\")F+F+F
+F9F*#F+\"hp++++++++C5SE>D\"p!e![!\\6]9RRG0'Q)*R``6F?+lP*40KNuy#o" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7%$!+SLM*>)!#i$!+s?VFM!#g$!+p+&*f8!#e
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "Jp:= a -> evalf(eval(J1
+J3,alpha=a))+J2(a):\nIp:= a -> evalf(2+sin(10*a))*Jp(a):\n Jp(0.8);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+HnJ65!\"*" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 41 "Here it is plotted (with low precision), " }
{XPPEDIT 18 0 "J(alpha)" "6#-%\"JG6#%&alphaG" }{TEXT -1 12 " in red an
d " }{XPPEDIT 18 0 "I(alpha)" "6#-%\"IG6#%&alphaG" }{TEXT -1 32 " in g
reen.  The maximum is near " }{XPPEDIT 18 0 "y = 0.78" "6#/%\"yG-%&Flo
atG6$\"#y!\"#" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 19 "plot([Jp,Ip],0..5);" }}{PARA 13 "" 1 "" {GLPLOT2D 430 234 234 
{PLOTDATA 2 "6&-%'CURVESG6$7jn7$\"\"!$F(F(7$$\"+sUkCF!#6$\"+h24)H\"!#5
7$$\"+X&)G\\aF-$\"+.+-IAF07$$\"+<G$R<)F-$\"+_$oT-$F07$$\"+4x&)*3\"F0$
\"+5MNFPF07$$\"+#R(*Rc\"F0$\"+BEk$z%F07$$\"+uq8Q?F0$\"+tSY1dF07$$\"+Nn
NrDF0$\"+lYQ(e'F07$$\"+(RwX5$F0$\"+%*y\"*QtF07$$\"+%eI8k$F0$\"+&zv?)zF
07$$\"+sZ3yTF0$\"+Q](R_)F07$$\"+]4\\Y_F0$\"+QWIP$*F07$$\"+'fl<u&F0$\"+
*y,qg*F07$$\"+U-/PiF0$\"+MC6:)*F07$$\"+^%o)\\nF0$\"+.G&*p**F07$$\"+fmp
isF0$\"+$o4n+\"!\"*7$$\"+$\\!)y_(F0$\"+\\Zf45Fap7$$\"+EV1$z(F0$\"+\"o(
365Fap7$$\"+f\"[#e!)F0$\"+*oO7,\"Fap7$$\"+#*>VB$)F0$\"+`()355Fap7$$\"+
j()4_))F0$\"+?:6/5Fap7$$\"+Mbw!Q*F0$\"+&yP^$**F07$$\"+#HkX#**F0$\"+1%)
\\\"y*F07$$\"+0j$o/\"Fap$\"+$*y9'e*F07$$\"+_>jU6Fap$\"+S'46:*F07$$\"+j
^Z]7Fap$\"+ht8U&)F07$$\"+)=h(e8Fap$\"+pWtFyF07$$\"+Q[6j9Fap$\"+WG-kqF0
7$$\"+\\z(yb\"Fap$\"+`HvBjF07$$\"+b/cq;Fap$\"+R]$fS&F07$$\"+<t,m<Fap$
\"+I*f@h%F07$$\"+Tj0x=Fap$\"+^qf(o$F07$$\"+#pW`(>Fap$\"+Vly\")GF07$$\"
+\"f#=$3#Fap$\"+181D?F07$$\"+t(pe=#Fap$\"+vpMZ7F07$$\"+uI,$H#Fap$\"*&H
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a'z#F07$$\"+z#=o'HFap$!+Ht'fu#F07$$\"+o=\"*zHFap$!+c@GQFF07$$\"+ca+$*H
Fap$!++(Hex#F07$$\"+W!*41IFap$!+:o/gGF07$Fex$!+CDU\"*HF07$$\"+7v')pIFa
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7$$\"+)o_$eKFap$!+#zx=<*F07$$\"+c8j$G$Fap$!+pE_E%*F07$$\"+!pqiH$Fap$!+
&>mhZ*F07$$\"+C+\"*3LFap$!+B!pEZ*F07$$\"+e$\\:K$Fap$!+?j7;%*F07$Fdy$!+
`W_2$*F07$$\"+S%p\")Q$Fap$!+X\"QsJ)F07$Fiy$!+LL?InF07$$\"+pZ1\"\\$Fap$
!+l%[p;&F07$F^z$!+Gt'o'QF07$$\"+HwrmNFap$!+F#=qP$F07$$\"+2fX$f$Fap$!+(
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\\F07$$\"+cI;+QFap$!+t/.4iF07$F][l$!+z*[J5(F07$$\"+mYhlQFap$!+ZGJNsF07
$$\"+H-ZyQFap$!+3/YFtF07$$\"+\"zD8*QFap$!+@#yyP(F07$$\"+a8=/RFap$!+(e<
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07$$\"+E67:TFap$!+jN5BLF07$F\\\\l$!+hk3+BF07$$\"+e9[$>%Fap$!+=u)Q%>F07
$$\"+JH**>UFap$!+4]08<F07$$\"+o'[KB%Fap$!+ko(Rk\"F07$$\"+0W]YUFap$!+gn
?/;F07$$\"+T,wfUFap$!+_^'=f\"F07$Fa\\l$!+noe/;F07$$\"+p(*fDVFap$!+]=%G
%=F07$Ff\\l$!+?c@;AF07$$\"+EKM-WFap$!+(oDAP#F07$$\"+$z-lU%Fap$!+*4KH\\
#F07$$\"+fBm]WFap$!+&ew[c#F07$F[]l$!+**eJzDF07$$\"+<:^-XFap$!+([w0_#F0
7$$\"+26?IXFap$!+lF*QQ#F07$$\"+)p!*yb%Fap$!++P5z@F07$F`]l$!+R\\$>#>F07
$Fe]l$!+J,**f()F-7$$\"+8^XPZFap$!+u_LfVF-7$Fj]l$!+zxB7:F-7$$\"+FM\"3%[
Fap$\"+JK0>g!#87$F_^l$\"+Bc)[E\"F-7$Fd^l$\"+r/N\\iF--Fh^l6&Fj^lF)F[_lF
)-%+AXESLABELSG6$Q!6\"F^^o-%%VIEWG6$;F)$Fd^lF(%(DEFAULTG" 1 2 0 1 10 
0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Here is a Newton iteration procedu
re.  It takes an initial " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT 
-1 29 " value, prints the values of " }{XPPEDIT 18 0 "I(alpha)" "6#-%
\"IG6#%&alphaG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "`I'`(alpha)" "6#-%#I'
G6#%&alphaG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "`I''`(alpha)" "6#-%$I
''G6#%&alphaG" }{TEXT -1 52 ", and returns the new alpha for the next \+
iteration: " }{XPPEDIT 18 0 "alpha - `I'`(alpha)/`I''`(alpha)" "6#,&%&
alphaG\"\"\"*&-%#I'G6#F$F%-%$I''G6#F$!\"\"F-" }{TEXT -1 2 ". " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 331 "Newt:= proc(a) \n local J,d
J,d2J,Ia,dI,d2I;\n J:= evalf(eval(J1+J3,alpha=a)+J2(a));\n dJ:= evalf(
eval(dJ1+dJ3,alpha=a)+dJ2(a));\n d2J:= evalf(eval(d2J1+d2J3,alpha=a)+d
2J2(a));\n Ia:= (2+sin(10*a))*J;\n dI:=10*cos(10*a)*J+(2+sin(10*a))*dJ
;\n d2I:= -100*sin(10*a)*J+20*cos(10*a)*dJ+(2+sin(10*a))*d2J;\n print(
[Ia,dI,d2I]);\n a-dI/d2I;\n end; \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%NewtGf*6#%\"aG6(%\"JG%#dJG%$
d2JG%#IaG%#dIG%$d2IG6\"F/C*>8$-%&evalfG6#,&-%%evalG6$,&%#J1G\"\"\"%#J3
GF</%&alphaG9$F<-%#J2G6#F@F<>8%-F46#,&-F86$,&%$dJ1GF<%$dJ3GF<F>F<-%$dJ
2GFCF<>8&-F46#,&-F86$,&%%d2J1GF<%%d2J3GF<F>F<-%%d2J2GFCF<>8'*&,&\"\"#F
<-%$sinG6#,$F@\"#5F<F<F2F<>8(,&*&-%$cosGF]oF<F2F<F_o*&FinF<FEF<F<>8),(
*&F[oF<F2F<!$+\"*(\"#?F<FdoF<FEF<F<*&FinF<FQF<F<-%&printG6#7%FgnFaoFho
,&F@F<*&FaoF<Fho!\"\"FepF/F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 15 "a1:=Newt(0.8);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%$\"+/!)=
BI!\"*$!+uSS'\\\"F&$!+4X(e0\"!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%#a1G$\"+F#y#ey!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "Digit
s:= 36: for i from 2 to 5 do a||i:= Newt(a||(i-1)) od;\n(time()-ti)*se
conds;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7%$\"E#45(p`cgKIuyp6))>tLI!#
N$\"EU'\\j$p<vtj4:0LP3eI6!#P$!EO-qz,_&)fIz(*pelU3o5!#L" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#a2G$\"EPxvpdT:pg*Rfs(RnLfy!#O" }}{PARA 12 "" 1 
"" {XPPMATH 20 "6#7%$\"E(>2Fy(oQ.^i$\\&['eKP.$!#N$\"?j#Qh@N%Ga3iuA7&)R
!#P$!E'yJ'zK$**=`hkD(*f_w!o5!#L" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#
a3G$\"EU0hp>>\"4gXXr.NuO$fy!#O" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7%$
\"E]DG^$osy`KO\\&['eKP.$!#N$\"4)H&Q\\opyN]&!#P$!E5&p23jao6;bvnf_w!o5!#
L" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a4G$\"EmaaFj)RClXXr.NuO$fy!#O
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7%$\"E_DG^$osy`KO\\&['eKP.$!#N$\"$
\"Q!#P$!E,Z@S8sW;h^bx'f_w!o5!#L" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#
a5G$\"EmaaFj)RClXXr.NuO$fy!#O" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&$
\"(N9v#!\"$\"\"\"%(secondsGF(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
27 "This has 33 correct digits." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 11 "Problem 10:" }}{PARA 
0 "" 0 "" {TEXT -1 243 "A particle at the center of a 10 x 1 rectangle
 undergoes Brownian motion (i.e. 2D random walk with infinitesimal ste
p lengths) till it hits the boundary.  What is the probability that it
 hits at one of the ends rather than at one of the sides?" }}}{SECT 1 
{PARA 5 "" 0 "" {TEXT -1 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 252 
"This is another standard heat equation problem, this time just for th
e steady state.  The probability is the temperature at the centre in t
he steady-state solution of the heat equation on the rectangle, where \+
the temperature is fixed at 1 on the ends (" }{XPPEDIT 18 0 "x=0" "6#/
%\"xG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x=10" "6#/%\"xG\"#5" }
{TEXT -1 22 ") and 0 on the sides (" }{XPPEDIT 18 0 "y=0" "6#/%\"yG\"
\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=1" "6#/%\"yG\"\"\"" }{TEXT 
-1 39 ").  The solution with temperature 1 at " }{XPPEDIT 18 0 "x=10" 
"6#/%\"xG\"#5" }{TEXT -1 38 " and 0 on the rest of the boundary is " }
{XPPEDIT 18 0 "u[1](x,y) = sum(a[j]*sinh(j*Pi*x)*sin(j*Pi*y),j=1..infi
nity)" "6#/-&%\"uG6#\"\"\"6$%\"xG%\"yG-%$sumG6$*(&%\"aG6#%\"jGF(-%%sin
hG6#*(F3F(%#PiGF(F*F(F(-%$sinG6#*(F3F(F8F(F+F(F(/F3;F(%)infinityG" }
{TEXT -1 7 " where " }{XPPEDIT 18 0 "a[j]*sinh(10*j*Pi)=2*int(sin(j*Pi
*y),y=0..1)" "6#/*&&%\"aG6#%\"jG\"\"\"-%%sinhG6#*(\"#5F)F(F)%#PiGF)F)*
&\"\"#F)-%$intG6$-%$sinG6#*(F(F)F/F)%\"yGF)/F9;\"\"!F)F)" }{TEXT -1 2 
": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "restart;\naj:= 2/sinh
(10*Pi*j)*int(sin(j*Pi*y),y=0..1) assuming j::integer;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#ajG,$*,\"\"#\"\"\"-%%sinhG6#,$*(\"#5F(%#PiGF(%
\"jGF(F(!\"\",&)F1F0F(F(F1F(F0F1F/F1F1" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "u1:= Sum(aj*sinh(j*Pi*x)*sin(j*Pi*y),j=1..infinity);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u1G-%$SumG6$,$*0\"\"#\"\"\"-%%s
inhG6#,$*(\"#5F+%#PiGF+%\"jGF+F+!\"\",&)F4F3F+F+F4F+F3F4F2F4-F-6#*(F3F
+F2F+%\"xGF+F+-%$sinG6#*(F3F+F2F+%\"yGF+F+F4/F3;F+%)infinityG" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "By symmetry, the solution with tem
perature 1 at " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 38 " an
d 0 on the rest of the boundary is " }{XPPEDIT 18 0 "u[2](x,y) = u[1](
10-x,y)" "6#/-&%\"uG6#\"\"#6$%\"xG%\"yG-&F&6#\"\"\"6$,&\"#5F/F*!\"\"F+
" }{TEXT -1 69 ".  By superposition, the solution with temperature 1 a
t both ends is " }{XPPEDIT 18 0 "u[1](x,y) + u[2](x,y)" "6#,&-&%\"uG6#
\"\"\"6$%\"xG%\"yGF(-&F&6#\"\"#6$F*F+F(" }{TEXT -1 26 ".  At the centr
e, we have " }{XPPEDIT 18 0 "u[1](5,1/2) + u[2](5,1/2) = 2 u[1](5,1/2)
" "6#/,&-&%\"uG6#\"\"\"6$\"\"&*&F)F)\"\"#!\"\"F)-&F'6#F-6$F+*&F)F)F-F.
F)*&F-F)-&F'6#F)6$F+*&F)F)F-F.F)" }{TEXT -1 73 ".  The series converge
s rapidly.  Here it is calculated with Digits = 40." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 49 "Digits:= 40; ans:= 2*evalf(eval(u1,\{x=5,
y=1/2\}));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#S" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%$ansG$\"I%z+\"R[?'=LrS.hA^#zzePQ!#Y" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 27 "This has 39 correct digits." }}}}}{MARK "19" 0 }
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }