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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 63 "Worked Example 5 - Determ
ining The Order of a Numerical Method " }}{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 49 "Define a \+
function to represent the exact solution" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "x:=t->x(t):" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Us
e some aliases for the derivatives of x" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "alias(x1=D(x),x2=(D@@
2)(x),x3=(D@@3)(x),x4=(D@@4)(x),x5=(D@@5)(x),x6=(D@@6)(x),x7=(D@@7)(x)
,x8=(D@@8)(x)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 69 "Use the taylor command to obtain a Taylor series exp
ansion of x(tn+h)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 24 "ts:=taylor(x(t),t=tn,9):" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Replace t-tn by h
 in the expansion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 21 "tsh:=subs(t-tn=h,ts):" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 64 "Convert to a polyno
mial so that we can perform the algebra later" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "p:=unapply(
convert(tsh,polynom),h):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 25 "Repeat above for x'(tn+h)" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "dts:=t
aylor(x1(t),t=tn,8):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "dts
h:=subs(t-tn=h,dts):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "dp:
=unapply(convert(dtsh,polynom),h):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Euler's Method" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "simplify
(p(h)-x(tn)-h*x1(tn));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 16 "Trapezoidal Rule" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "simplify(p(
h)-x(tn)-h*(x1(tn)+dp(h))/2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " \+
" }}{PARA 0 "" 0 "" {TEXT -1 14 "Quade's Method" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "simplify(p(
4*h)-8*(p(3*h)-p(h))/19-x(tn)-6*h*(dp(4*h)+4*dp(3*h)+4*dp(h)+x1(tn))/1
9);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 
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