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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 257 49 "Derivative Estimates by
 Undetermined Coefficients" }}{PARA 0 "" 0 "" {TEXT 256 20 "Mth 351 Ap
ril 6 2001" }}{PARA 0 "" 0 "" {TEXT 258 16 "Bent E. Petersen" }}{PARA 
0 "" 0 "" {TEXT 259 45 "Filename: 351s2001_derivs_by_undeter_coef.mws
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 327 "In this worksheet we use Maple to obtain
s a number of derivative estimators. These estimators are useful for \+
\"differentiating\" data obtained experimentally as a table of numbers
. Symmetric estimators usually perform best, but near the beginning an
d near the end of the table it will be necessary to use nonsymmetric e
stimators." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 152 "Numeric differentiation is not a very stable process. High ord
er estimates (and high order derivatives) tend to suffer from loss of \+
significance errors." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 53 "We begin by looking at some symmetric 3-point f
omulae" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "g:=h->A1*f(a+h)+A2*f(a)+A3*f(a-h);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"gGR6#%\"hG6\"6$%)operatorG%&arrowGF(,(*&%#A1G\"\"
\"-%\"fG6#,&%\"aGF/9$F/F/F/*&%#A2GF/-F16#F4F/F/*&%#A3GF/-F16#,&F4F/F5!
\"\"F/F/F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 105 "Since we have 3 coefficients we can specify the ter
ms in the Taylor expansion of  g  up through order 2. " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ex:=ta
ylor(g(h),h=0,3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#exG++%\"hG,(*&
%#A1G\"\"\"-%\"fG6#%\"aGF*F**&%#A3GF*F+\"\"\"F**&%#A2GF*F+F1F*\"\"!,&*
&F0F1--%\"DG6#F,F-F*!\"\"*&F)F1F7F1F*\"\"\",&*&F0F1---%#@@G6$F9\"\"#F:
F-F*#F*FE*&F)F1F@F1FF\"\"#-%\"OG6#F*\"\"$" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Let's pick out the coeffi
cients of powers of  h" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "for k from 0 to 2 do A[k]:=coeff(ex
,h,k); od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"AG6#\"\"!,(*&%#A1G
\"\"\"-%\"fG6#%\"aGF+F+*&%#A3GF+F,\"\"\"F+*&%#A2GF+F,F2F+" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>&%\"AG6#\"\"\",&*&%#A3GF'--%\"DG6#%\"fG6#%\"aG
F'!\"\"*&%#A1GF'F+\"\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"AG6
#\"\"#,&*&%#A3G\"\"\"---%#@@G6$%\"DGF'6#%\"fG6#%\"aGF+#F+F'*&%#A1GF+F,
\"\"\"F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 75 "It is convenient to factor out the derivatives of  f  in \+
these coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 66 "for k from 0 to 2 do B[k]:=simplify(subs(f=
(x->x^k/k!),A[k])); od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"BG6#\"
\"!,(%#A1G\"\"\"%#A3GF*%#A2GF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%
\"BG6#\"\"\",&%#A3G!\"\"%#A1GF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%
\"BG6#\"\"#,&%#A3G#\"\"\"F'%#A1GF*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "To separate out the first deri
vative of  f  we set  B[0]=0 and B[1]=1. We have one more condition av
ailable so we set B[2]=0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "der1:=solve(\{B[0]=0,B[1]=1,B[2]=0
\},\{A1,A2,A3\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%der1G<%/%#A2G
\"\"!/%#A1G#\"\"\"\"\"#/%#A3G#!\"\"F-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "subs(der1,g(h)); taylor(%,h=0,7);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,&-%\"fG6#,&%\"aG\"\"\"%\"hGF)#F)\"\"#-F%6#,&F(F)F*!
\"\"#F0F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#++%\"hG--%\"DG6#%\"fG6#%
\"aG\"\"\",$---%#@@G6$F'\"\"$F(F*#\"\"\"\"\"'\"\"$,$---F16$F'\"\"&F(F*
#F5\"$?\"\"\"&-%\"OG6#F5\"\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 203 "We divide by  h  to separate out D(f
)(a). We see that we will have an error term of order 2. So expanding \+
just through order 2 we obtain the usual form of the central 3-point e
stimator of the derivative." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "subs(der1,g(h))/h: %=taylor(
simplify(%),h=0,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&-%\"fG6#,&
%\"aG\"\"\"%\"hGF+#F+\"\"#-F'6#,&F*F+F,!\"\"#F2F.\"\"\"F,!\"\"+'F,--%
\"DG6#F'6#F*\"\"!-%\"OG6#F+\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
1 " " }}{PARA 0 "" 0 "" {TEXT -1 68 "We can now define a function whic
h approximates the first derivative" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "D1B1F1:=unapply(subs(der1,
g(h))/h,f,a,h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'D1B1F1GR6%%\"fG%
\"aG%\"hG6\"6$%)operatorG%&arrowGF**&,&-9$6#,&9%\"\"\"9&F5#F5\"\"#-F16
#,&F4F5F6!\"\"#F<F8\"\"\"F6!\"\"F*F*F*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "The notation D1B1F1 mean
s first derivative estimate, back 1 step, forward 1 step (so central 3
 point)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
80 "To get an estimate for the second derivative we proceed in much th
e same manner:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 47 "der2:=solve(\{B[0]=0,B[1]=0,B[2]=1\},\{A1,A2,A
3\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%der2G<%/%#A1G\"\"\"/%#A3GF
(/%#A2G!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "subs(der2,g(
h)); taylor(%,h=0,7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%\"fG6#,&%
\"aG\"\"\"%\"hGF)F)-F%6#F(!\"#-F%6#,&F(F)F*!\"\"F)" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#++%\"hG---%#@@G6$%\"DG\"\"#6#%\"fG6#%\"aG\"\"#,$---F(
6$F*\"\"%F,F.#\"\"\"\"#7\"\"%,$---F(6$F*\"\"'F,F.#F8\"$g$\"\"'-%\"OG6#
F8\"\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 52 "To separate out the second derivative we divide by  " }
{XPPEDIT 18 0 "h^2;" "6#*$%\"hG\"\"#" }{TEXT -1 130 " .  We see that w
e will have an error term of order 2. So to get the usual formula we e
xpand the Taylor series (after dividing by " }{XPPEDIT 18 0 "h^2;" "6#
*$%\"hG\"\"#" }{TEXT -1 23 ") just through order 2." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "subs(der2
,g(h))/h^2; %=taylor(simplify(%),h=0,2);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#*&,(-%\"fG6#,&%\"aG\"\"\"%\"hGF*F*-F&6#F)!\"#-F&6#,&F)F*F+!\"\"F
*\"\"\"*$)F+\"\"#F3!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,(-%\"f
G6#,&%\"aG\"\"\"%\"hGF+F+-F'6#F*!\"#-F'6#,&F*F+F,!\"\"F+\"\"\"*$)F,\"
\"#F4!\"\"+'F,---%#@@G6$%\"DG\"\"#6#F'F.\"\"!-%\"OG6#F+\"\"#" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "He
re's our formula for the second derivative" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "D2B1F1:=unapply(s
ubs(der2,g(h))/h^2,f,a,h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'D2B1F
1GR6%%\"fG%\"aG%\"hG6\"6$%)operatorG%&arrowGF**&,(-9$6#,&9%\"\"\"9&F5F
5-F16#F4!\"#-F16#,&F4F5F6!\"\"F5\"\"\"*$)F6\"\"#F>!\"\"F*F*F*" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "Le
t's try now some to find some one-sided 3 point formulae. Note I am re
defining  g ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 36 "g:=h->A1*f(a)+A2*f(a+h)+A3*f(a+2*h);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"hG6\"6$%)operatorG%&arrowGF(,(*&
%#A1G\"\"\"-%\"fG6#%\"aGF/F/*&%#A2GF/-F16#,&F3F/9$F/F/F/*&%#A3GF/-F16#
,&F3F/F9\"\"#F/F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "
ex:=taylor(g(h),h=0,3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#exG++%\"
hG,(*&%#A1G\"\"\"-%\"fG6#%\"aGF*F**&%#A3GF*F+\"\"\"F**&%#A2GF*F+F1F*\"
\"!,&*&F3F1--%\"DG6#F,F-F*F**&F0F1F7F1\"\"#\"\"\",&*&F3F1---%#@@G6$F9F
<F:F-F*#F*F<*&F0F1F@F1F<\"\"#-%\"OG6#F*\"\"$" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 45 "for k from 0 to 2 do A[k]:=coeff(ex,h,k); od;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"AG6#\"\"!,(*&%#A1G\"\"\"-%\"fG6#
%\"aGF+F+*&%#A3GF+F,\"\"\"F+*&%#A2GF+F,F2F+" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%\"AG6#\"\"\",&*&%#A2GF'--%\"DG6#%\"fG6#%\"aGF'F'*&%#
A3GF'F+\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"AG6#\"\"#,&
*&%#A2G\"\"\"---%#@@G6$%\"DGF'6#%\"fG6#%\"aGF+#F+F'*&%#A3GF+F,\"\"\"F'
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "for k from 0 to 2 do B[
k]:=simplify(subs(f=(x->x^k/k!),A[k])); od;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%\"BG6#\"\"!,(%#A1G\"\"\"%#A3GF*%#A2GF*" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>&%\"BG6#\"\"\",&%#A2GF'%#A3G\"\"#" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>&%\"BG6#\"\"#,&%#A2G#\"\"\"F'%#A3GF'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "der1:=solve(\{B[0]=0,B[1]=1,
B[2]=0\},\{A1,A2,A3\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%der1G<%/
%#A2G\"\"#/%#A3G#!\"\"F(/%#A1G#!\"$F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "subs(der1,g(h)); taylor(%,h=0,7);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,(-%\"fG6#%\"aG#!\"$\"\"#-F%6#,&F'\"\"\"%\"hGF.F*-F%6
#,&F'F.F/F*#!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+/%\"hG--%\"DG6
#%\"fG6#%\"aG\"\"\",$---%#@@G6$F'\"\"$F(F*#!\"\"F3\"\"$,$---F16$F'\"\"
%F(F*#F5F<\"\"%,$---F16$F'\"\"&F(F*#!\"(\"#g\"\"&,$---F16$F'\"\"'F(F*#
F5\"#C\"\"'-%\"OG6#\"\"\"\"\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 47 "subs(der1,g(h))/h: %=taylor(simplify(%),h=0,2);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/*&,(-%\"fG6#%\"aG#!\"$\"\"#-F'6#,&F)\"\"\"%\"hG
F0F,-F'6#,&F)F0F1F,#!\"\"F,\"\"\"F1!\"\"+'F1--%\"DG6#F'F(\"\"!-%\"OG6#
F0\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "D1F2:=unapply(su
bs(der1,g(h))/h,f,a,h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%D1F2GR6%
%\"fG%\"aG%\"hG6\"6$%)operatorG%&arrowGF**&,(-9$6#9%#!\"$\"\"#-F16#,&F
3\"\"\"9&F:F6-F16#,&F3F:F;F6#!\"\"F6\"\"\"F;!\"\"F*F*F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "der2:=solve(\{B[0]=0,B[1]=0,B[2]=1
\},\{A1,A2,A3\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%der2G<%/%#A1G
\"\"\"/%#A3GF(/%#A2G!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 
"subs(der2,g(h)); taylor(%,h=0,7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
,(-%\"fG6#%\"aG\"\"\"-F%6#,&F'F(%\"hGF(!\"#-F%6#,&F'F(F,\"\"#F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#+/%\"hG---%#@@G6$%\"DG\"\"#6#%\"fG6#%
\"aG\"\"#---F(6$F*\"\"$F,F.\"\"$,$---F(6$F*\"\"%F,F.#\"\"(\"#7\"\"%,$-
--F(6$F*\"\"&F,F.#\"\"\"F<\"\"&,$---F(6$F*\"\"'F,F.#\"#J\"$g$\"\"'-%\"
OG6#FH\"\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "subs(der2,g(
h))/h^2; %=taylor(simplify(%),h=0,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#*&,(-%\"fG6#%\"aG\"\"\"-F&6#,&F(F)%\"hGF)!\"#-F&6#,&F(F)F-\"\"#F)\"
\"\"*$)F-\"\"#F3!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,(-%\"fG6#
%\"aG\"\"\"-F'6#,&F)F*%\"hGF*!\"#-F'6#,&F)F*F.\"\"#F*\"\"\"*$)F.\"\"#F
4!\"\"+'F.---%#@@G6$%\"DGF36#F'F(\"\"!-%\"OG6#F*\"\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "D2F2:=unapply(subs(der2,g(h))/h^2,f
,a,h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%D2F2GR6%%\"fG%\"aG%\"hG6
\"6$%)operatorG%&arrowGF**&,(-9$6#9%\"\"\"-F16#,&F3F49&F4!\"#-F16#,&F3
F4F8\"\"#F4\"\"\"*$)F8\"\"#F>!\"\"F*F*F*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "We have obtained 4 estima
tors:" }}{PARA 0 "" 0 "" {TEXT -1 52 "D1B1F1 = first derivative, 3 poi
nt, central, order 2" }}{PARA 0 "" 0 "" {TEXT -1 53 "D2B1F1 = second d
erivative, 3 point, central, order 2" }}{PARA 0 "" 0 "" {TEXT -1 59 "D
1F2  = first derivative, 3 point, 2 steps forward, order 2" }}{PARA 0 
"" 0 "" {TEXT -1 59 "D2F2 = second derivative, 3 point, 2 steps forwar
d, order 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 104 "The first derivative of the exponential at 0 is 1. Here are ou
r estimates (with step size 0.1 and 0.01):" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "D1B1F1(exp,0,0.1)
; D1B1F1(exp,0,0.01);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++vm,5!\"*
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+gm,+5!\"*" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 34 "D1F2(exp,0,0.1); D1F2(exp,0,0.01);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#$\"++d/k**!#5" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"++Sm****!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 117 "The second derivative of the exponential at 0 \+
is 1 (of course). Here are our estimates (with step size 0.1 and 0.01)
:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "D2B1F1(exp,0,0.1); D2B1F1(exp,0,0.01);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$\"++O$3+\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#$\"++q++5!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "D2F2(ex
p,0,0.1); D2F2(exp,0,0.01);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++A4
16!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++155!\"*" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "Note the \+
numerical results support our claims concerning the order." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 204 "To obtain higher
 order derivatives or higher order methods for low order derivatives w
e have to increase the number of points at which we evaluate  f.  Here
's a central 5 point first derivative estimator:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "g:=h->A1*f(
a-2*h)+A2*f(a-h)+A3*f(a)+A4*f(a+h)+A5*f(a+2*h);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"gGR6#%\"hG6\"6$%)operatorG%&arrowGF(,,*&%#A1G\"\"\"
-%\"fG6#,&%\"aGF/9$!\"#F/F/*&%#A2GF/-F16#,&F4F/F5!\"\"F/F/*&%#A3GF/-F1
6#F4F/F/*&%#A4GF/-F16#,&F4F/F5F/F/F/*&%#A5GF/-F16#,&F4F/F5\"\"#F/F/F(F
(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ex:=taylor(g(h),h=0,
5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "for k from 0 to 4 do
 A[k]:=coeff(ex,h,k); od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
66 "for k from 0 to 4 do B[k]:=simplify(subs(f=(x->x^k/k!),A[k])); od;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"BG6#\"\"!,,%#A1G\"\"\"%#A4GF*
%#A2GF*%#A5GF*%#A3GF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"BG6#\"\"
\",*%#A1G!\"#%#A4GF'%#A2G!\"\"%#A5G\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>&%\"BG6#\"\"#,*%#A5GF'%#A1GF'%#A4G#\"\"\"F'%#A2GF," }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>&%\"BG6#\"\"$,*%#A2G#!\"\"\"\"'%#A5G#\"\"%F'%#
A1G#!\"%F'%#A4G#\"\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"BG6#
\"\"%,*%#A4G#\"\"\"\"#C%#A2GF*%#A5G#\"\"#\"\"$%#A1GF/" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "der1:=solve(\{B[0]=0,B[1]=1,B[2]=0,
B[3]=0,B[4]=0\},\{A1,A2,A3,A4,A5\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%%der1G<'/%#A3G\"\"!/%#A5G#!\"\"\"#7/%#A1G#\"\"\"F-/%#A2G#!\"#\"\"$
/%#A4G#\"\"#F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "subs(der1
,g(h)); taylor(%,h=0,7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*-%\"fG6#
,&%\"aG\"\"\"%\"hG!\"##F)\"#7-F%6#,&F(F)F*!\"\"#F+\"\"$-F%6#,&F(F)F*F)
#\"\"#F3-F%6#,&F(F)F*F8#F1F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+)%\"h
G--%\"DG6#%\"fG6#%\"aG\"\"\",$---%#@@G6$F'\"\"&F(F*#!\"\"\"#I\"\"&-%\"
OG6#\"\"\"\"\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "subs(der
1,g(h))/h: %=taylor(simplify(%),h=0,4);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/*&,*-%\"fG6#,&%\"aG\"\"\"%\"hG!\"##F+\"#7-F'6#,&F*F+F,!\"\"#F-
\"\"$-F'6#,&F*F+F,F+#\"\"#F5-F'6#,&F*F+F,F:#F3F/\"\"\"F,!\"\"+'F,--%\"
DG6#F'6#F*\"\"!-%\"OG6#F+\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 41 "D1B2F2:=unapply(subs(der1,g(h))/h,f,a,h);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'D1B2F2GR6%%\"fG%\"aG%\"hG6\"6$%)operatorG%&arrowGF**
&,*-9$6#,&9%\"\"\"9&!\"##F5\"#7-F16#,&F4F5F6!\"\"#F7\"\"$-F16#,&F4F5F6
F5#\"\"#F?-F16#,&F4F5F6FD#F=F9\"\"\"F6!\"\"F*F*F*" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Note D1B2F2 is a 4
 th order method. Let's check it on the exponential:" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Digits:=3
0: D1B2F2(exp,0,0.1); D1B2F2(exp,0,0.01); Digits:=10:" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"?!Gv862;.(4'pim*****!#I" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"?q\"z,b*Q)pimm*********!#I" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "How about a central thi
rd derivative estimator?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "der3:=solve(\{B[0]=0,B[1]=0,B[2]=0,
B[3]=1,B[4]=0\},\{A1,A2,A3,A4,A5\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%%der3G<'/%#A3G\"\"!/%#A2G\"\"\"/%#A4G!\"\"/%#A1G#F.\"\"#/%#A5G#F+F
2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "subs(der3,g(h)); taylo
r(%,h=0,7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*-%\"fG6#,&%\"aG\"\"\"
%\"hG!\"##!\"\"\"\"#-F%6#,&F(F)F*F-F)-F%6#,&F(F)F*F)F--F%6#,&F(F)F*F.#
F)F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+)%\"hG---%#@@G6$%\"DG\"\"$6#%
\"fG6#%\"aG\"\"$,$---F(6$F*\"\"&F,F.#\"\"\"\"\"%\"\"&-%\"OG6#F8\"\"(" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "subs(der3,g(h))/h^3: %=ta
ylor(value(%),h=0,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,*-%\"fG6#
,&%\"aG\"\"\"%\"hG!\"##!\"\"\"\"#-F'6#,&F*F+F,F/F+-F'6#,&F*F+F,F+F/-F'
6#,&F*F+F,F0#F+F0\"\"\"*$)F,\"\"$F;!\"\"+'F,---%#@@G6$%\"DG\"\"$6#F'6#
F*\"\"!-%\"OG6#F+\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "D
3B2F2:=unapply(subs(der3,g(h))/h^3,f,a,h);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'D3B2F2GR6%%\"fG%\"aG%\"hG6\"6$%)operatorG%&arrowGF**
&,*-9$6#,&9%\"\"\"9&!\"##!\"\"\"\"#-F16#,&F4F5F6F9F5-F16#,&F4F5F6F5F9-
F16#,&F4F5F6F:#F5F:\"\"\"*$)F6\"\"$FE!\"\"F*F*F*" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Note D3B2F2 is a 2
 nd order method. Let's check it on the exponential." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Digits:=3
0: D3B2F2(exp,0,0.1); D3B2F2(exp,0,0.01); Digits:=10:" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"?+mfZ4\"yf$fS,D]-5!#H" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"?++qSGaS,+D+]-+5!#H" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Here's a 5 point central \+
4 th derivative estimator:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "der4:=solve(\{B[0]=0,B[1]=0,B[2]=0,
B[3]=0,B[4]=1\},\{A1,A2,A3,A4,A5\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%%der4G<'/%#A1G\"\"\"/%#A2G!\"%/%#A4GF+/%#A5GF(/%#A3G\"\"'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "subs(der4,g(h)); taylor(%,h=
0,7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,-%\"fG6#,&%\"aG\"\"\"%\"hG!
\"#F)-F%6#,&F(F)F*!\"\"!\"%-F%6#F(\"\"'-F%6#,&F(F)F*F)F0-F%6#,&F(F)F*
\"\"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+)%\"hG---%#@@G6$%\"DG\"\"%
6#%\"fG6#%\"aG\"\"%,$---F(6$F*\"\"'F,F.#\"\"\"F6\"\"'-%\"OG6#F8\"\"(" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "subs(der4,g(h))/h^4: %=ta
ylor(simplify(%),h=0,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,,-%\"f
G6#,&%\"aG\"\"\"%\"hG!\"#F+-F'6#,&F*F+F,!\"\"!\"%-F'6#F*\"\"'-F'6#,&F*
F+F,F+F2-F'6#,&F*F+F,\"\"#F+\"\"\"*$)F,\"\"%F=!\"\"+'F,---%#@@G6$%\"DG
\"\"%6#F'F4\"\"!-%\"OG6#F+\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 43 "D4B2F2:=unapply(subs(der4,g(h))/h^4,f,a,h);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%'D4B2F2GR6%%\"fG%\"aG%\"hG6\"6$%)operatorG%&arrowGF
**&,,-9$6#,&9%\"\"\"9&!\"#F5-F16#,&F4F5F6!\"\"!\"%-F16#F4\"\"'-F16#,&F
4F5F6F5F<-F16#,&F4F5F6\"\"#F5\"\"\"*$)F6\"\"%FG!\"\"F*F*F*" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Note D4B2
F2 is a 2 nd order method. Let's check it on the exponential." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 62 "Digits:=30: D4B2F2(exp,0,0.1); D4B2F2(exp,0,0.01); Digits:=10:" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?++_f*zro+Hs\"zm,5!#H" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#$\"?++++Q)Gsm\"zmm,+5!#H" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 127 "With 5 points we
 can for example go back 1 step and forward 3 steps. let's construct s
uch an estimator for the 1 st derivative." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "g:=h->A1*f(a-h)+A2*f
(a)+A3*f(a+h)+A4*f(a+2*h)+A5*f(a+3*h);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"gGR6#%\"hG6\"6$%)operatorG%&arrowGF(,,*&%#A1G\"\"\"-%\"fG6#,
&%\"aGF/9$!\"\"F/F/*&%#A2GF/-F16#F4F/F/*&%#A3GF/-F16#,&F4F/F5F/F/F/*&%
#A4GF/-F16#,&F4F/F5\"\"#F/F/*&%#A5GF/-F16#,&F4F/F5\"\"$F/F/F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ex:=taylor(g(h),h=0,5):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "for k from 0 to 4 do A[k]:=c
oeff(ex,h,k); od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "for k \+
from 0 to 4 do B[k]:=simplify(subs(f=(x->x^k/k!),A[k])); od;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>&%\"BG6#\"\"!,,%#A1G\"\"\"%#A4GF*%#A2GF*%#A
5GF*%#A3GF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"BG6#\"\"\",*%#A3GF
'%#A1G!\"\"%#A4G\"\"#%#A5G\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%
\"BG6#\"\"#,*%#A3G#\"\"\"F'%#A1GF*%#A4GF'%#A5G#\"\"*F'" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>&%\"BG6#\"\"$,*%#A4G#\"\"%F'%#A3G#\"\"\"\"\"'%#A1
G#!\"\"F/%#A5G#\"\"*\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"BG6#
\"\"%,*%#A1G#\"\"\"\"#C%#A4G#\"\"#\"\"$%#A3GF*%#A5G#\"#F\"\")" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "der1:=solve(\{B[0]=0,B[1]=1,
B[2]=0,B[3]=0,B[4]=0\},\{A1,A2,A3,A4,A5\});" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%der1G<'/%#A5G#\"\"\"\"#7/%#A3G#\"\"$\"\"#/%#A2G#!\"&
\"\"'/%#A1G#!\"\"\"\"%/%#A4G#F8F/" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "subs(der1,g(h)); taylor(%,h=0,7);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,,-%\"fG6#,&%\"aG\"\"\"%\"hG!\"\"#F+\"\"%-F%6#F(#!\"&
\"\"'-F%6#,&F(F)F*F)#\"\"$\"\"#-F%6#,&F(F)F*F8#F+F8-F%6#,&F(F)F*F7#F)
\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#++%\"hG--%\"DG6#%\"fG6#%\"aG\"
\"\",$---%#@@G6$F'\"\"&F(F*#\"\"\"\"#?\"\"&,$---F16$F'\"\"'F(F*#F5\"#C
\"\"'-%\"OG6#F5\"\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "sub
s(der1,g(h))/h: %=taylor(value(%),h=0,4);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&,,-%\"fG6#,&%\"aG\"\"\"%\"hG!\"\"#F-\"\"%-F'6#F*#!\"
&\"\"'-F'6#,&F*F+F,F+#\"\"$\"\"#-F'6#,&F*F+F,F:#F-F:-F'6#,&F*F+F,F9#F+
\"#7\"\"\"F,!\"\"+'F,--%\"DG6#F'F1\"\"!-%\"OG6#F+\"\"%" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "D1B1F3:=unapply(subs(der1,g(h))/h,f
,a,h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'D1B1F3GR6%%\"fG%\"aG%\"hG
6\"6$%)operatorG%&arrowGF**&,,-9$6#,&9%\"\"\"9&!\"\"#F7\"\"%-F16#F4#!
\"&\"\"'-F16#,&F4F5F6F5#\"\"$\"\"#-F16#,&F4F5F6FD#F7FD-F16#,&F4F5F6FC#
F5\"#7\"\"\"F6!\"\"F*F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 63 "D1B1F3 is 4 th order method. Let's check \+
it on the exponential:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "Digits:=30: D1B1F3(exp,0,0.1); D1B1
F3(exp,0,0.01); D1B1F3(exp,0,0.001); Digits:=10:" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"?(*p$)3Y'*=-tbTa++5!#H" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"?#*HgIS2e!>/0++++\"!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"
?]2m[!pT+0++++++\"!#H" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 38 "There are many variations! Experiment!" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 150 "If it i
s desired to know more about the details of the error we can always ca
lculate a Taylor polynomial. For example for the error in D1B1F3 we ha
ve:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "taylor( D(f)(a)-D1B1F3(f,a,h),h=0,8);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#++%\"hG,$---%#@@G6$%\"DG\"\"&6#%\"fG6#%\"aG#!\"\"
\"#?\"\"%,$---F)6$F+\"\"'F-F/#F2\"#C\"\"&,$---F)6$F+\"\"(F-F/#F2\"#U\"
\"'-%\"OG6#\"\"\"\"\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 124 "Thus if for example the 5 th derivative \+
of  f  at  a  happens to be  0  then  D1B1F3(f,a,h) would actually be \+
of 5 th order!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{MARK "83 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }