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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 16 "Worked Example 9" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots
):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 37 "Define the characteristic polynomials" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "rho:=theta->theta
^2-4*theta/3+1/3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "sigma:
=theta->2/3*theta^2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 28 "and the stability polynomial" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "pi:=theta
->rho(theta)-lambda*h*sigma(theta);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 12 " Substitute " }{XPPEDIT 18 0 "z = h*lambda;" "6#/%\"zG*&%\"hG\"
\"\"%'lambdaGF'" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "piz:=theta->rho(theta)-z*sig
ma(theta);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 33 "Define the boundary of the region" }}{PARA 0 "" 0 "" 
{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "rat:=solve
(piz(theta),z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "rat1:=su
bs(theta=exp(I*phi),rat);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 17 "Plot the boundary" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "complexplot
(rat1,phi=0..2*Pi,numpoints=500);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Check whether region is inside \+
or outside closed curve" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "rs:=solve(subs(z=1,piz(theta))=0,th
eta);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 16 "Must be outside." }}}}{MARK "18" 0 }{VIEWOPTS 1 1 0 1 1 
1803 }