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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart;with(linalg)
:" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm an
d trace have been redefined and unprotected\n" }}}{PARA 0 "" 0 "" 
{TEXT -1 139 "We use Maple to show the numerical calculations in the e
xample in 7.1 in Bretscher's text.  This can be modified to compute va
lues for #52." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "A:=matrix(2
,2,[0.86,0.08,-0.12,1.14]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-
%'matrixG6#7$7$$\"#')!\"#$\"\")F,7$$!#7F,$\"$9\"F," }}}{PARA 0 "" 0 "
" {TEXT -1 24 "Enter an initial vector." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "x0:=vector([100,300]);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#x0G-%'vectorG6#7$\"$+\"\"$+$" }}}{PARA 0 "" 0 "" {TEXT -1 61 
"Compute powers of A times x_0 using the recursive property.  " }
{TEXT 256 69 "Notice that % refers to whatever object or value was las
t calculated!" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "for i to 10
 do evalm(A&*%); od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$
$\"&+5\"!\"#$\"&+I$F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7
$$\"(++@\"!\"%$\"(++j$F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG
6#7$$\"*++5L\"!\"'$\"*++I*RF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'ve
ctorG6#7$$\"+++5k9!\"($\"+++I#R%F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%'vectorG6#7$$\"+++^5;!\"($\"+++`J[F)" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'vectorG6#7$$\"++5cr<!\"($\"++Io9`F)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'vectorG6#7$$\"++rr[>!\"($\"++8:YeF)" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'vectorG6#7$$\"+5))eV@!\"($\"+IkwIkF)" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$$\"+\"pZzN#!\"($\"+tI%Q2(F)" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$$\"+gCu$f#!\"($\"+!QF7y
(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "%[2]/%[1];" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#$\"+++++I!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 
138 "If you change the 10 in the FOR loop above to other values (and r
e-enter x0), this ratio will remain 3 because x0 is an eigenvector for
 A." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "x0:=vector([200,100]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G-%'vectorG6#7$\"$+#\"$+\"" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
31 "for i to 10 do evalm(A&*%); od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#-%'vectorG6#7$$\"&+!=!\"#$\"%+!*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#-%'vectorG6#7$$\"(++i\"!\"%$\"'++\")F)" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'vectorG6#7$$\"*++!e9!\"'$\")++!H(F)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'vectorG6#7$$\"+++?78!\"($\"++++hl!\")" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$$\"+++)4=\"!\"($\"+++!\\!f!\")" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$$\"++?)G1\"!\"($\"+++T9`
!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$$\"++!Qfc*!\")$
\"++!pHy%F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$$\"++UM4'
)!\")$\"++@n/VF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$$\"+
!y4%[x!\")$\"+!*[?uQF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#
7$$\"+-)oN(p!\")$\"+,Wy'[$F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 10 "%[2]/%[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++]!#5" }}}
{PARA 0 "" 0 "" {TEXT -1 140 "If you change the 10 in the FOR loop abo
ve to other values (and re-enter x0), this ratio will remain 0.5 becau
se x0 is an eigenvector for A." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "x0:=vector([1000,1000]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G-%'vectorG6#7$\"%+5F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "for i to 10 do evalm(A&*%); \+
od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$$\"&+S*!\"#$\"'+?
5F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$$\"(++!*)!\"%$\")
++]5F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$$\"*++S\\)!\"'
$\"+++?!4\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$$\"++++
x\")!\"($\"+++!49\"!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6
#7$$\"+++%\\%z!\"($\"++?]-7!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'
vectorG6#7$$\"+++l%z(!\"($\"++I^v7!\"'" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'vectorG6#7$$\"++%4Qs(!\"($\"+?!\\0O\"!\"'" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'vectorG6#7$$\"++`\"4t(!\"($\"+q,Me9!\"'" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$$\"+%Hf_\"y!\"($\"+5otp:!\"'" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$$\"+TC\"p(z!\"($\"+^or
&p\"!\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "%[2]/%[1];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+]4yD@!\"*" }}}{PARA 0 "" 0 "" 
{TEXT -1 113 "If you change the 10 in the FOR loop above to other valu
es (and re-enter x0), this ratio will vary because x0 is " }{TEXT 257 
3 "not" }{TEXT -1 22 " an eigenvector for A." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
151 "To calculate the coyote population after i years when x0=[1000,10
00] and keep track of the year, use the closed formula for c(t) in the
 FOR loop below." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "for i to
 10 do 200*(1.1)^i+800*(0.9)^i,i; od;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6$$\"%+%*!\"\"\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"&+!*)!\"#
\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"'+%\\)!\"$\"\"$" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$$\"(+q<)!\"%\"\"%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$$\")+%\\%z!\"&\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$
$\"*+]Yz(!\"'\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"++%4Qs(!\"(\"
\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"++`\"4t(!\"(\"\")" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$$\"+%Hf_\"y!\"(\"\"*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$$\"+TC\"p(z!\"(\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 155 "To calculate the roadrunner population \+
after i years when x0=[1000,1000] and keep track of the year, use the \+
closed formula for r(t) in the FOR loop below." }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 45 "for i to 10 do 600*(1.1)^i+400*(0.9)^i,i; od;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"&+-\"!\"\"\"\"\"" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6$$\"'+]5!\"#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$
$\"(+-4\"!\"$\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\")+!49\"!\"%\"
\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"*+-D?\"!\"&\"\"&" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$$\"++I^v7!\"'\"\"'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$$\"+?!\\0O\"!\"'\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6$$\"+q,Me9!\"'\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+6otp:!\"'
\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+_or&p\"!\"'\"#5" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
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