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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 17 "Worked Example 10" }}
{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 22 "rho:=theta->theta
^2-1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "sigma:=theta->2*th
eta;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 29 "Find the stability polynomial" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "pi:=theta->rho(theta)-lam
bda*h*sigma(theta);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "piz:
=theta->rho(theta)-z*sigma(theta);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Solve for z" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "rat:=solve(
piz(theta),z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 11 "Substitute " }{XPPEDIT 18 0 "theta = e^(i*phi);" "6#
/%&thetaG)%\"eG*&%\"iG\"\"\"%$phiGF)" }{TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "rat1:=sub
s(theta=exp(I*phi),rat);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 19 "Simplify the result" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "simplify(ra
t1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 40 "Thus empty region of absolute stability." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}}{MARK "1 0 0" 8 }{VIEWOPTS 1 1 0 1 1 1803 }