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{SECT 0 {EXCHG {PARA 200 "" 0 "" {TEXT 202 36 "Andrew Sommese -- Septe
mber 29, 2004" }{TEXT 202 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 
1 203 32 "DQ:= (f,x,h) -> (f(x+h)-f(x))/h;" }{MPLTEXT 1 203 0 "" }}
{PARA 202 "" 1 "" {XPPMATH 20 "6#>I#DQG6\"f*6%I\"fGF%I\"xGF%I\"hGF%F%6
$I)operatorGF%I&arrowGF%F%*&,&-9$6#,&9%\"\"\"9&F5F5-F16#F4!\"\"F5F6F9F
%F%F%" }{TEXT 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 
19 "f:= x -> exp(-x*x);" }{MPLTEXT 1 203 0 "" }}{PARA 202 "" 1 "" 
{XPPMATH 20 "6#>I\"fG6\"f*6#I\"xGF%F%6$I)operatorGF%I&arrowGF%F%-I$exp
GF%6#,$*&9$\"\"\"F1F2!\"\"F%F%F%" }{TEXT 204 0 "" }}}{EXCHG {PARA 201 
"> " 0 "" {MPLTEXT 1 203 45 "trueDeriv := evalf(subs(x=0.3,diff(f(x),x
)));" }{MPLTEXT 1 203 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I*trueD
erivG6\"$!+7re$[&!#5" }{TEXT 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" 
{MPLTEXT 1 203 15 "x:=array(1..4);" }{MPLTEXT 1 203 0 "" }}{PARA 202 "
" 1 "" {XPPMATH 20 "6#>I\"xG6\"-I&arrayGI*protectedGF(6$;\"\"\"\"\"%7
\"" }{TEXT 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 16 "y
:= array(1..4);" }{MPLTEXT 1 203 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 
"6#>I\"yG6\"-I&arrayGI*protectedGF(6$;\"\"\"\"\"%7\"" }{TEXT 204 0 "" 
}}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 43 "for j from 1 to 4 do \+
x[j] := 1/2^(j+1); od;" }{MPLTEXT 1 203 0 "" }}}{EXCHG {PARA 201 "> " 
0 "" {MPLTEXT 1 203 48 "for j from 1 to 4 do y[j] := DQ(f,0.3,x[j]); o
d;" }{MPLTEXT 1 203 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 
22 "approx:=interp(x,y,0);" }{MPLTEXT 1 203 0 "" }}{PARA 202 "" 1 "" 
{XPPMATH 20 "6#>I'approxG6\"$!+t_U$[&!#5" }{TEXT 204 0 "" }}}{EXCHG 
{PARA 201 "> " 0 "" {MPLTEXT 1 203 21 "for j from 1 to 4 do " }
{MPLTEXT 1 203 0 "" }}{PARA 203 "> " 0 "" {MPLTEXT 1 0 18 "trueDeriv-y
[j];od;" }{MPLTEXT 1 0 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#$\"+o2#
\\^\"!#5" }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#$\"*k7/^)
!#5" }{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#$\"*;icZ%!#5" 
}{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#$\"*'ps!H#!#5" }
{TEXT 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 17 "trueDe
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!'R=;!#5" }{TEXT 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 
203 21 "for k from 1 to 12 do" }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0
 "" {MPLTEXT 1 203 15 "x:=array(1..k):" }{MPLTEXT 1 203 0 "" }}{PARA 
201 "> " 0 "" {MPLTEXT 1 203 16 "y:= array(1..k):" }{MPLTEXT 1 203 0 "
" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 40 "for j from 1 to k do x[j] :
= 1/2^(j+1); " }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 
203 28 " y[j] := DQ(f,0.3,x[j]); od:" }{MPLTEXT 1 203 0 "" }}{PARA 
203 "> " 0 "" {MPLTEXT 1 0 22 "approx:=interp(x,y,0):" }{MPLTEXT 1 0 
0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 26 "print(k,trueDeriv-approx
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" }}{PARA 202 "" 1 "" {XPPMATH 20 "6$\"\"\"$\"+o2#\\^\"!#5" }{TEXT 
204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6$\"\"#$\"*g<;(=!#5" }{TEXT 
204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6$\"\"$$!(H!*f$!#5" }{TEXT 
204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6$\"\"%$!'R=;!#5" }{TEXT 
204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6$\"\"&$\"$E$!#5" }{TEXT 
204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6$\"\"'$\"$;$!#5" }{TEXT 
204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6$\"\"($!$y%!#5" }{TEXT 204 
0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6$\"\")$\"$)\\!#5" }{TEXT 204 0 
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{TEXT 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 45 "trueDe
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202 "" 1 "" {XPPMATH 20 "6#>I*trueDerivG6\"$!?+z77%[?\"pti6re$[&!#I" }
{TEXT 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 21 "for k \+
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1 203 15 "x:=array(1..k):" }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" 
{MPLTEXT 1 203 16 "y:= array(1..k):" }{MPLTEXT 1 203 0 "" }}{PARA 201 
"> " 0 "" {MPLTEXT 1 203 40 "for j from 1 to k do x[j] := 1/2^(j+1); "
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= DQ(f,0.3,x[j]); od:" }{MPLTEXT 1 203 0 "" }}{PARA 203 "> " 0 "" 
{MPLTEXT 1 0 22 "approx:=interp(x,y,0):" }{MPLTEXT 1 0 0 "" }}{PARA 
201 "> " 0 "" {MPLTEXT 1 203 26 "print(k,trueDeriv-approx);" }
{MPLTEXT 1 203 0 "" }{MPLTEXT 1 203 4 "\nod:" }{MPLTEXT 1 203 0 "" }}
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{XPPMATH 20 "6$\"\"&$\"8y$*=@)y--(*yON!#I" }{TEXT 204 0 "" }}{PARA 
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204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6$\"\"*$!+DvW,X!#I" }{TEXT 
204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6$\"#5$\")\"R<$))!#I" }
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{TEXT 204 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6$\"#7$\"%q\")!#I" }
{TEXT 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 12 "Digits
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itsG6\"\"#g" }{TEXT 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 
203 45 "trueDeriv := evalf(subs(x=0.3,diff(f(x),x)));" }{MPLTEXT 1 
203 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I*trueDerivG6\"$!gn`\")*f
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201 "> " 0 "" {MPLTEXT 1 203 21 "for k from 1 to 12 do" }{MPLTEXT 1 
203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 15 "x:=array(1..k):" }
{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 16 "y:= array
(1..k):" }{MPLTEXT 1 203 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 40 
"for j from 1 to k do x[j] := 1/2^(j+1); " }{MPLTEXT 1 203 0 "" }}
{PARA 201 "> " 0 "" {MPLTEXT 1 203 28 " y[j] := DQ(f,0.3,x[j]); od:" }
{MPLTEXT 1 203 0 "" }}{PARA 203 "> " 0 "" {MPLTEXT 1 0 22 "approx:=int
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26 "print(k,trueDeriv-approx);" }{MPLTEXT 1 203 0 "" }{MPLTEXT 1 203 
4 "\nod:" }{MPLTEXT 1 203 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6$\"\"
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