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3-upGhv****\\FB7$$\"3xaZFB7$$\"3CLLeR<*fT\"Fex$\"3XK2:/y17ZFB7$$\"3'******\\)H
xe9Fex$\"3I>.(ftoNm%FB7$$\"3Ymm\"H!o-*\\\"Fex$\"3dyp!p,Olh%FB7$$\"3))*
**\\7k.6a\"Fex$\"3ss-=DCBmXFB7$$\"3emmmT9C#e\"Fex$\"30YXTa\"*=;XFB7$$
\"3\"****\\i!*3`i\"Fex$\"3-/=\\Jp8jWFB7$$\"3QLLL$*zym;Fex$\"3$3$yIrNh6
WFB7$$\"3GLL$3N1#4Fex$\"3
*=!\\ewds,TFB7$$\"3/++v.Uac>Fex$\"3SBy/k%RC0%FB7$$\"\"#Fgh[m$\"3A+++++
++SFB-Fch[m6&Feh[mFhh[m$\")AR!)\\Fjh[mFfh[m-F\\i[m6#F`h\\m-%+AXESLABEL
SG6%Q\"x6\"Q%y(x)F]i\\m-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!+++++5!#5$\"++
+++@!\"*;$!+++++]!#6$\"++++]5F\\j\\m" 1 2 0 1 10 0 2 9 1 4 2 1.000000
45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 " An order 9 discrete numerical
solution with 20 equally spaced data points can be constructed as fol
lows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 155 "de := diff(y(x),x)=1/(1+x^2)-2*y(x)^2;\nic := y(0)=0
;\nsoln := desolveTA(\{de,ic\},x=0..2,stepsize=fixed,order=9):\ng := x
-> x/(1+x^2);\ncomparewithfcn(soln,g);" }}{PARA 11 "" 1 "" {XPPMATH
20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&*&\"\"\"F/,&F/F/*$)F,\"\"#F/F/
!\"\"F/*&F3F/)F)F3F/F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"y
G6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)op
eratorG%&arrowGF(*&,&*$)9$\"\"#\"\"\"F2F2F2!\"\"F0F2F(F(F(" }}{PARA 6
"" 1 "" {TEXT -1 85 " 0 0 function val: \+
0 rel err: infinity" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+
0.1 0.099009901 function val: 0.09900990099 rel
err: 1.0100e-10" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.2 0.
1923076924 function val: 0.1923076923 rel err: 5.2000e-10" }}
{PARA 6 "" 1 "" {TEXT -1 87 " 0.3 0.2752293579 functio
n val: 0.2752293578 rel err: 3.6333e-10" }}{PARA 6 "" 1 ""
{TEXT -1 87 " 0.4 0.3448275863 function val: 0.3448
275862 rel err: 2.9000e-10" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+
0.5 0.4000000001 function val: 0.4 rel err: \+
2.5000e-10" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.6 0.441176
4706 function val: 0.4411764706 rel err: 0.0000e+00" }}{PARA
6 "" 1 "" {TEXT -1 87 " 0.7 0.4697986577 function val:
0.4697986577 rel err: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1
87 " 0.8 0.4878048781 function val: 0.4878048781 \+
rel err: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 0.9 \+
0.4972375691 function val: 0.497237569 rel err: 2.0111e-1
0" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1 0.5 fu
nction val: 0.5 rel err: 0.0000e+00" }}{PARA 6 "" 1 "
" {TEXT -1 87 " 1.1 0.4977375566 function val: 0.49
77375566 rel err: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+
1.2 0.4918032787 function val: 0.4918032787 rel err
: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1.3 0.4832
713755 function val: 0.4832713755 rel err: 0.0000e+00" }}
{PARA 6 "" 1 "" {TEXT -1 87 " 1.4 0.472972973 functio
n val: 0.4729729729 rel err: 2.1143e-10" }}{PARA 6 "" 1 ""
{TEXT -1 87 " 1.5 0.4615384615 function val: 0.4615
384616 rel err: 2.1667e-10" }}{PARA 6 "" 1 "" {TEXT -1 87 " \+
1.6 0.4494382023 function val: 0.4494382022 rel err: \+
2.2250e-10" }}{PARA 6 "" 1 "" {TEXT -1 87 " 1.7 0.437017
9949 function val: 0.4370179948 rel err: 2.2882e-10" }}{PARA
6 "" 1 "" {TEXT -1 87 " 1.8 0.4245283019 function val:
0.4245283019 rel err: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1
87 " 1.9 0.4121475054 function val: 0.4121475054 \+
rel err: 0.0000e+00" }}{PARA 6 "" 1 "" {TEXT -1 87 " 2 \+
0.4 function val: 0.4 rel err: 0.0000e+0
0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 6 "" 1 "" {TEXT -1
48 " Maximum relative error: 5.2000e-10" }}{PARA 6 "" 1 "
" {TEXT -1 59 " obtained for the input value: \+
0.2" }}{PARA 6 "" 1 "" {TEXT -1 67 " excluding any cases
where the function value is zero." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 22 "A continuous solution " }{TEXT 270 2 "gn
" }{TEXT -1 114 " (numerical function) based on 20 equally spaced data
points can be constructed as follows by using the procedure " }{TEXT
0 9 "desolveTA" }{TEXT -1 18 " with the option \"" }{TEXT 270 16 "outp
ut=procedure" }{TEXT -1 3 "\". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "de := diff(y(x),x)=1/(1+x^2
)-2*y(x)^2;\nic := y(0)=0;\ngn := desolveTA(\{de,ic\},x=0..2,stepsize=
fixed,order=9,output=procedure);\ng := x -> x/(1+x^2);" }}{PARA 11 ""
1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&*&\"\"\"F/,&F/F/
*$)F,\"\"#F/F/!\"\"F/*&F3F/)F)F3F/F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#gnGf*6#
'%)x_taylorG%)realconsG61%#xkG%#ykG%#jFG%#jMG%#jSG%\"nG%\"hG%\"jG%%dat
aG%#xxG%+saveDigitsG%$sumG%%termG%%factG%#dkG6#%>Copyright~2002~by~Pet
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$*3*)3$\\q*FU$!/g(\\3HT!eFU$!/=-Jd]?aFH$\"/5'Q[Y*=6!#7$\"/s-4p!ei*F[o$
!/tFWOEnWFK$!/YDIhe$Q$FN$\"/Mn)H7H,$!\"*$\"/hmKn5v=FQ7-$\"/++++++?FU$
\"/y.Cp2B>FU$\"/8BW'Rd())FU$!/1&*)orD0\"FH$!/7=IK61RFH$\"/R,#HYa\"=F[o
$\"/GkTL^?SF[o$!/?:v]nkhFK$\"/&GiMO^g$F[o$\"/Qz(pZ$3LFdo$!/E_;f:q6FQ7-
$\"/++++++IFU$\"/7YzNH_FFU$\"/o4PzGfwFU$!/1YcBB[8FH$!/.qXIo*)>FH$\"/F2
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1D)F[o$!/J$RMvkG%FK$\"/!R&[vTv&)FK$\"/VEUq$fP*FN7-$FTFH$\"/LB++++]FU$!
%mYFU$!/q1********\\FU$\"/)f)*******\\\"FH$!/*R,++++$FH$\"/++++].G!#A$
\"/'=$)******\\%F[o$!/bT******\\JFK$\"/#e++++E\"FN$!/+++D&)RJ!#>7-$\"/
++++++6FH$\"/h\"ecvt(\\FU$!.&3oim*H%FU$!/ShB,P[OFU$\"/rH>:H17FH$!/S'[H
F[o$!/t2))\\dh>FK$\"/zC6\"R_1\"FN$!/TN
y'>J:$FN7-$FJFH$\"/i0(yK!=\\FU$!.V!yK%FH$!/\"y$=hCcFFH$!/mr^@M#*[F[o$\"/m*
oG%)H(H(HZFU$!/;4&G!p&4\"FU$
!/(fWwNG7\"FU$\"/I0R'>uS&FU$!/&Rf)p%3f\"FH$\"/HlmVhqQFH$!/%Q%HE!\\%eFH
$!/#R(*e9=i\"F[o$\"/xHQ:6LBFK$!/n@?A!=f\"FN7-$\"/++++++:FH$\"/aZ:YQ:YF
U$!/.$G&>V$=\"FU$!/!)o\\BRalFX$\"/]4[*f)**RFU$!/<`KI'[B\"FH$\"/+umb\"z
B$FH$!/tdY^LMlFH$\"/.W?$f-y$FU$\"/to[/+)4\"FK$!/htkrf0#*FK7-$\"/++++++
;FH$\"/haA?Q%\\%FU$!/B())=04B\"FU$!/Sf&f/27$FX$\"/E!\\i(4;HFU$!/)Ga\\A
HV*FU$\"/dlN'e7g#FH$!/[3n!o,3'FH$\"/o<,4BovFH$\"/D,uf)>7%F[o$!/'[!zPHl
[FK7-$\"/++++++Txlc\"FH$!/#[\")yI0@%FH$\"/-!****
oo]*FH$!/r+^_FH$\"/GFa]Z@TFU$!/bjFR6G
7FU$\"//lZx(fO#FX$\"/1+DQL85FU$!/ue(z;q&RFU$\"/s%HQ]7>\"FH$!/)))GK&e;L
FH$\"/p%yxsQG)FH$!/=^\"=gQV\"F[o$!/@bY;i+BF[o7$$FioFH$\"/i.++++SFU>8.%
'DigitsG>F\\jl-%$minG6$-%$maxG6$-%&truncG6#-%'evalhfG6#F\\jl-Fejl6#,$*
(\"\"%FE\"\"$!\"\"F\\jlFEFE,&F\\jlFE\"#5FE>8--%&evalfG6#9$>8)-%%nopsG6
#F?@$532&F?6$FEFE&F?6$Fj[mFE52Fd\\mFd[m2Fd[mFb\\m32Fd\\mFb\\m52Fd[mFd
\\m2Fb\\mFd[mY6$Q\\oindependent~variable~is~outside~of~the~interpolati
on~interval:~%1F<;-Ff[m6#Fb\\m-Ff[m6#Fd\\m>Fj[mF[\\m>8&FC>8(,&Fj[mFEFE
FE@%Fa\\mC$?(F8'-Fejl6#,&*FE\"\"#FEFh]m
FEFE*&Fh^mFEFj]mFEFE@%1&F?6$Fc^mFEFd[m>Fh]mFc^m>Fj]mFc^m@$/Fc^mFj[mC$>
Fh]m,&Fj[mFEFEF`[m>Fj]mFj[mC$?(FFc^mFd^m@%1Fd[mF]_m>Fh]m
Fc^m>Fj]mFc^m@$Fb_mC$>Fh]mFe_m>Fj]mFj[m>8$&F?6$Fh]mFE>8%&F?6$Fh]mFi^m>
8*,&Fd[mFEFd`mF`[m>8/Fh`m>81-Ff[m6#FE?(8+FEFE,&-F\\\\m6#&F?6#Fh]mFEFi^
mF`[m%%trueGC&>Faam*(FaamFEF\\amFEFeamF`[m>82&F?6$Fh]m,&FeamFEFi^mFE>8
0*&F`bmFEFaamFE>F_am,&F_amFEFebmFE>F\\jlF[jl-Ff[m6#F_amF%\"gGf*6#%\"xG6\"6$%)operatorG%&arrow
GF(*&,&*$)9$\"\"#\"\"\"F2F2F2!\"\"F0F2F(F(F(" }}}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The code for the numerica
l function " }{TEXT 270 2 "gn" }{TEXT -1 79 " can be viewed as follows
. You need to re-execute the commands to see the code." }}{PARA 0 ""
0 "" {TEXT -1 215 "It contains a data consisting of numerical values \+
for the solution and appropriate derivatives to be used in local Taylo
r series. This allows the function to interpolate between the points o
f the discrete solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "interface(verboseproc=2);\neval(gn)
;\ninterface(verboseproc=0);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 37 "The values of the numerical function " }
{TEXT 270 2 "gn" }{TEXT -1 53 " are close to those given by the analyt
ical solution " }{TEXT 270 1 "g" }{TEXT -1 2 ". " }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "xx := sqrt(2
);\nevalf(g(xx));\ngn(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG*$
\"\"##\"\"\"F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+1_/9Z!#5" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+3_/9Z!#5" }}}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "The numerical function "
}{TEXT 270 2 "gn" }{TEXT -1 51 " cannot be evaluated outside the origi
nal interval " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to \+
" }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 22 " specified when u
sing " }{TEXT 0 9 "desolveTA" }{TEXT -1 16 " to constuct it." }}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "xx
:= 1.234567891;\nevalf(g(xx));\ngn(xx);" }}{PARA 11 "" 1 "" {XPPMATH
20 "6#>%#xxG$\"+\"*ycM7!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+1!4
5*[!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+1!45*[!#5" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 ""
}}{PARA 0 "" 0 "" {TEXT -1 33 "It is better to allow the use of " }
{TEXT 265 25 "adaptive stepsize control" }{TEXT -1 18 " with the optio
n \"" }{TEXT 270 13 "adaptive=true" }{TEXT -1 6 "\" or \"" }{TEXT 270
17 "stepsize=variable" }{TEXT -1 3 "\". " }}{PARA 0 "" 0 "" {TEXT -1
65 "If the \"steps\" option is omitted then this is the default option
." }}{PARA 0 "" 0 "" {TEXT -1 18 "With this option, " }{TEXT 0 9 "deso
lveTA" }{TEXT -1 137 " controls the stepsize used in the construction \+
of the underlying discrete solution by estimating the error after each
step is performed." }}{PARA 0 "" 0 "" {TEXT -1 274 "If the error is t
oo large, the step is repeated. If the error is within the required bo
unds, the the step is accepted, and the next step is performed with a \+
step-size which is chosen on the basis of a numerical comparison of th
e error estimate with the required error bound." }}{PARA 0 "" 0 ""
{TEXT 265 5 "Notes" }{TEXT -1 2 ": " }}{PARA 15 "" 0 "" {TEXT -1 40 "I
f you want to see how this works, give " }{TEXT 0 9 "desolveTA" }
{TEXT -1 13 " the option \"" }{TEXT 270 9 "info=true" }{TEXT -1 53 "\"
. \nYou can get varying amounts of information with \"" }{TEXT 270 6 "
info=n" }{TEXT -1 27 "\", where n=0, 1, 2, 3 or 4." }}{PARA 15 "" 0 "
" {TEXT -1 50 "When plotting the graph of the numerical function " }
{TEXT 270 2 "gn" }{TEXT -1 17 ", the expression " }{TEXT 270 5 "gn(x)
" }{TEXT -1 31 " must be enclosed in quotes as " }{TEXT 0 7 "'gn(x)'"
}{TEXT -1 27 ", to delay its evaluation. " }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "de := diff(y(x),x)=1
/(1+x^2)-2*y(x)^2;\nic := y(0)=0;\ngn := desolveTA(\{de,ic\},x=0..2,in
fo=true);\nplot('gn(x)',x=0..2,thickness=2,color=brown,labels=[`x`,`y(
x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"
xGF,,&*&\"\"\"F/,&F/F/*$)F,\"\"#F/F/!\"\"F/*&F3F/)F)F3F/F4" }}{PARA
11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6&%5abs~err~estimate~->~G$\"\"!F%%2abs~err~bound~->~G$\"&
++\"!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"\"\"%$~~~G$\"+++++
5!#6F%7$F&$\"/+***4+!****!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Fincr
easing~step-size~by~a~factor~of~5G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&92\"
!#=%2abs~err~bound~->~G$\"&&)*\\!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
(%%stepG\"\"#%$~~~G$\"+++++]!#6F%7$$\"+++++gF($\"/(o5[x%yf!#:" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjust~step-sizeG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#%Freducing~step-size~and~repeating~stepG" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&I#G!#;%2abs~err~bound~->~G$
\"&-B)F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"\"$%$~~~G$\"/j(RT
(zlt!#:F%7$$\"/wRT(zlL\"!#9$\"//&>a@JJ\"F," }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#%@using~error~to~adjust~step-sizeG" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estima
te~->~G$\"&&4V!#;%2abs~err~bound~->~G$\"&JJ\"!#:" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6(%%stepG\"\"%%$~~~G$\"/d$o-))yP(!#:F%7$$\"/73W&oV2#!#9$
\"/8A!y!z))>F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adj
ust~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 ""
1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&O8$!#;%2abs~err~bound~-
>~G$\"&)))>!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"\"&%$~~~G$
\"/wVy?qAu!#:F%7$$\"/]#>vQm\"G!#9$\"/!40!og4EF," }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#%@using~error~to~adjust~step-sizeG" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estima
te~->~G$\"&1&*)!#<%2abs~err~bound~->~G$\"&'4E!#:" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6(%%stepG\"\"'%$~~~G$\"/#eN&GNO!)!#:F%7$$\"/3GPSF?O!#9$\"
/zqi*o2?$F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjust
~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%Freducing~step-size~and~repeating~stepG" }}{PARA 11
"" 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&+5\"!#:%2abs~err~bou
nd~->~G$\"&3?$F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"\"(%$~~~G
$\"/8jKGx*)*)!#:F%7$$\"/Ra?8D>X!#9$\"/qf'Q%z_PF," }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#%@using~error~to~adjust~step-sizeG" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estima
te~->~G$\"&tD\"!#:%2abs~err~bound~->~G$\"&Gv$F&" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6(%%stepG\"\")%$~~~G$\"/L1mcv-!*!#:F%7$$\"/-:()o_>a!#9$\"
/-a6q7*=%F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjust
~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&sX'!#;%2abs~err~bound~->~G
$\"&\"*=%!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"\"*%$~~~G$\"/
YT,5zQ!*!#:F%7$$\"/~G$\"&R/\"!#;%2abs~err~bound~->~G$\"&s^%!#:" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6(%%stepG\"#5%$~~~G$\"/(>L9jv!)*!#:F%7$$\"/Pi,B;/t!#9$\"/
g?V>/jZF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjust~s
tep-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#%Freducing~step-size~and~repeating~stepG" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&ee\"!#:%2abs~err~boun
d~->~G$\"&Iw%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"#6%$~~~G$
\"/EY?Bk06!#9F%7$$\"/j3AY!)4%)F($\"/)H\"*fTf#\\F(" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#%@using~error~to~adjust~step-sizeG" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estima
te~->~G$\"&()[\"!#:%2abs~err~bound~->~G$\"&f#\\F&" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6(%%stepG\"#7%$~~~G$\"/B#[3f26\"!#9F%7$$\"/'3pqj0_*F($\"/
f_<9(R*\\F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adjust
~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&ZD)!#;%2abs~err~bound~->~G
$\"&S*\\!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"#8%$~~~G$\"/)f
6#fvE6!#9F%7$$\"/o!G'>tk5!#8$\"/+7K1=!*\\F(" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#%@using~error~to~adjust~step-sizeG" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estima
te~->~G$\"&Wf%!#;%2abs~err~bound~->~G$\"&-*\\!#:" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6(%%stepG\"#9%$~~~G$\"/KS]%yS@\"!#9F%7$$\"/r%y!)Rh=\"!#8$
\"/F;nV-G\\F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adju
st~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1
"" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&+7$!#;%2abs~err~bound~->~
G$\"&!G\\!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"#:%$~~~G$\"/e
;NY+(Q\"!#9F%7$$\"/POr-%[K\"!#8$\"/LX*[0&3[F(" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#%@using~error~to~adjust~step-sizeG" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estima
te~->~G$\"&*>P!#:%2abs~err~bound~->~G$\"&&3[F&" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6(%%stepG\"#;%$~~~G$\"/A'z*>-X;!#9F%7$$\"/*f6ZU$*[\"!#8$
\"/%Q&Q!fzi%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~error~to~adj
ust~step-sizeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 ""
1 "" {XPPMATH 20 "6&%5abs~err~estimate~->~G$\"&1.\"!#:%2abs~err~bound~
->~G$\"&!GYF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"#<%$~~~G$\"/
5p;r,>:!#9F%7$$\"/!HG=W7k\"!#8$\"/R#>!>QVWF(" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#%@using~error~to~adjust~step-sizeG" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estima
te~->~G$\"&B[(!#;%2abs~err~bound~->~G$\"&MW%!#:" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6(%%stepG\"#=%$~~~G$\"/*fk%Gn)e\"!#9F%7$$\"/]Zn96+=!#8$\"
/~G$\"&(=d!#;%2abs~err~bound~->~G$
\"&^C%!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%%stepG\"#>%$~~~G$\"/1\"4
J;'3!#8$\"/pUkU'\\.%F(" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#%@using~error~to~adjust~step-sizeG" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%5abs~err~estima
te~->~G$\"&&>I!#C%2abs~err~bound~->~G$\"&].%!#:" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6(%%stepG\"#?%$~~~G$\"-RV,p-H!#8F%7$$\"/++++++?F($\"/2,++
++S!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*last~stepG" }}{PARA 11 ""
1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=the~total
~number~of~steps~isG\"#?" }}{PARA 13 "" 1 "" {GLPLOT2D 501 277 277
{PLOTDATA 2 "6'-%'CURVESG6#7S7$$\"\"!F)F(7$$\"+M3VfV!#6$\"+4;;^VF-7$$
\"+#H[D:)F-$\"+=4s)4)F-7$$\"+e0$=C\"!#5$\"+,1(HA\"F87$$\"+3RBr;F8$\"+m
U#ei\"F87$$\"+zjf)4#F8$\"+>225?F87$$\"+'4;[\\#F8$\"+gXj[BF87$$\"+j'y]!
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\"+<_M(=%F8$\"+-9niNF87$$\"+4y_qXF8$\"+d8u!y$F87$$\"+]1!>+&F8$\"+e?\"4
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\"+Ca`)[%F87$$\"+<=C#o'F8$\"+W\"=&>YF87$$\"+n#pS1(F8$\"+4u[7ZF87$$\"+j
`A3vF8$\"+X2Z,[F87$$\"+n(y8!zF8$\"+.*GW'[F87$$\"+j.tK$)F8$\"+.'oz\"\\F
87$$\"+)3zMu)F8$\"+X/Eb\\F87$$\"+#H_?<*F8$\"+b^Q\")\\F87$$\"+!G;cc*F8$
\"+:M2&*\\F87$$\"+4#G,***F8$\"+hv****\\F87$$\"+!o2J/\"!\"*$\"+s-b&*\\F
87$$\"+%Q#\\\"3\"Fas$\"+(\\&p%)\\F87$$\"+;*[H7\"Fas$\"+g:dm\\F87$$\"+q
vxl6Fas$\"+M7vT\\F87$$\"+`qn27Fas$\"+HQG7\\F87$$\"+cp@[7Fas$\"+#zs&z[F
87$$\"+3'HKH\"Fas$\"++#H\"R[F87$$\"+xanL8Fas$\"+I]l*z%F87$$\"+v+'oP\"F
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\"+.o-*\\\"Fas$\"+Fas$
\"+xds,TF87$$\"+/Uac>Fas$\"+k%RC0%F87$$\"\"#F)$\"+++++SF8-%*THICKNESSG
6#Fdz-%+AXESLABELSG6$%\"xG%%y(x)G-%'COLOURG6&%$RGBG$\")#)eqk!\")$\"))e
qk\"Fe[lFf[l-%%VIEWG6$;F(Fcz%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2
1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "The following segment of code c
ompares the value of the numerical solution " }{TEXT 270 5 "gn(x)" }
{TEXT -1 43 " with the value of the analytical solution " }{XPPEDIT
18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 " x =
sqrt(3)" "6#/%\"xG-%%sqrtG6#\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "
" {TEXT -1 165 "The numerical solution is constructed so as to give va
lues which are correct to 10 digits, but values which are correct to a
couple of extra digits can be obtained. " }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "g := x -> x/(1+x^2);
\nxx := sqrt(3); \n'gn'(xx)=gn(xx);\n'g'(xx)=evalf(g(xx));\n'gn'(xx)=e
valf[14](gn(xx));\n'g'(xx)=evalf[14](g(xx));" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,&*$)9$\"\"#
\"\"\"F2F2F2!\"\"F0F2F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG*
$\"\"$#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#gnG6#*$\"\"$
#\"\"\"\"\"#$\"+>q7IV!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#*
$\"\"$#\"\"\"\"\"#$\"+?q7IV!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#
gnG6#*$\"\"$#\"\"\"\"\"#$\"/U#*=q7IV!#9" }}{PARA 11 "" 1 "" {XPPMATH
20 "6#/-%\"gG6#*$\"\"$#\"\"\"\"\"#$\"/A#*=q7IV!#9" }}}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "We can construct a soluti
on which goes both to the left and right of the initial point" }
{XPPEDIT 18 0 " ``(0,0)" "6#-%!G6$\"\"!F&" }{TEXT -1 1 "." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "de :
= diff(y(x),x)=1/(1+x^2)-2*y(x)^2;\nic := y(0)=0;\ngn := desolveTA(\{d
e,ic\},x=-4..4);\nplot('gn(x)',x=-4..4,thickness=2,color=brown,labels=
[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%
\"yG6#%\"xGF,,&*&\"\"\"F/,&F/F/*$)F,\"\"#F/F/!\"\"F/*&F3F/)F)F3F/F4" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 13 "
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{TEXT -1 9 "Example 5" }{TEXT 285 46 " .. a differential equation for \+
which Maple's " }{TEXT 0 6 "dsolve" }{TEXT 304 36 " cannot find an ana
lytical solution " }}{PARA 0 "" 0 "" {TEXT -1 36 "Consider the differe
ntial equation " }{XPPEDIT 18 0 "dy/dx = 1/(1+x^2)-2*y^2+y;" "6#/*&%#
dyG\"\"\"%#dxG!\"\",(*&F&F&,&F&F&*$%\"xG\"\"#F&F(F&*&F.F&*$%\"yGF.F&F(
F1F&" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "x=0" "6#/%\"
xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }
{TEXT -1 28 " with the initial condition " }{XPPEDIT 18 0 "y(0)=0" "6#
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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
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olve(\{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%dif
fG6$-%\"yG6#%\"xGF,,(*&\"\"\"F/,&F/F/*$)F,\"\"#F/F/!\"\"F/*&F3F/)F)F3F
/F4F)F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}}
{PARA 0 "" 0 "" {TEXT -1 68 "\nMaple can plot a numerical solution alo
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ws=medium,color=blue,linecolor=coral);" }}{PARA 11 "" 1 "" {XPPMATH
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