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{SECT 0 {PARA 0 "" 0 "" {TEXT -1 26 "Here are the examples for " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 36 "Getting
 Control With Linear Algebra." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 41 "We do each of the examples in that paper.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 13 "with(linalg):" }}}{PARA 0 "" 0 "" {TEXT -1 10 "Example 2." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "crossprod([1,1,1],[1,-1,1]);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Dis:=t->(2*t-1)^2+2^2+(
-2*t-3)^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "diff(Dis(t),t
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Dis(-1/2);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 10 "Example 3." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "u:=[0,1
,2,3];\ny1:=[1,1,1,1];\ny2:=[1,-1,1,-1];" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 121 "eq1:=dotprod(u,y1)=alpha*dotprod(y1,y1)+beta*dotprod
(y2,y1);\neq2:=dotprod(u,y2)=alpha*dotprod(y1,y2)+beta*dotprod(y2,y2);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "solve(\{eq1,eq2\},\{alp
ha,beta\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(%);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "u-alpha*y1-beta*y2;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "alpha:='alpha';\nbeta:='beta
';" }}}{PARA 0 "" 0 "" {TEXT -1 9 "Example 4" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 38 "u:=[1,2,3];\ny1:=[1,1,1];\ny2:=[1,-1,1];" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "eq1:=dotprod(u,y1)-alpha*do
tprod(y1,y1)-beta*dotprod(y2,y1)=1;\neq2:=dotprod(u,y2)-alpha*dotprod(
y1,y2)-beta*dotprod(y2,y2)=2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 30 "solve(\{eq1,eq2\},\{alpha,beta\});" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "assign(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
34 "u-alpha*y1-beta*y2;\nnorm((u-%),2);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "alpha:='alpha';\nbeta:='beta';" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 9 "Example 5" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "u:=[1,2,3,4];\ny1:=[1,1,1,1]
;\ny2:=[1,-1,1,-1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "eq1
:=dotprod(u,y1)-alpha*dotprod(y1,y1)-beta*dotprod(y2,y1)=3;\neq2:=dotp
rod(u,y2)-alpha*dotprod(y1,y2)-beta*dotprod(y2,y2)=5;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "solve(\{eq1,eq2\},\{alpha,beta\});
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(%);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "evalm(u-alpha*y1-beta*y2);\nnorm((u
-%),2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "alpha:='alpha';
\nbeta:='beta';" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 9 "Example 6" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 26 "M:=matrix([[0,1],[-1,0]]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "exponential(M,t);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "Q0:=vector([0,1]);\nQ1:=vector([1,0]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "A:=evalm(Q1-exponential(M,Pi)&*Q0);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "simplify(sin(Pi-s),trig
);\nsimplify(cos(Pi-s),trig);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 165 "eq1:=0-A[1]=\nalpha*int(sin(s)*sin(s),s=0..Pi)+beta*int(-cos(s)
*sin(s),s=0..Pi);\neq2:=0-A[2]=\nalpha*int(sin(s)*(-cos(s)),s=0..Pi)+b
eta*int(-cos(s)*(-cos(s)),s=0..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "solve(\{eq1,eq2\},\{alpha,beta\});" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 10 "assign(%);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "v:=t->-alpha*sin(t)-beta*(-cos(t));" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 5 "v(t);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 65 "dsolve(\{diff(y(t),t,t)+y(t)=v(t),y(0)=Q0[1],D(y)(0)=
Q0[2]\},y(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "ysol:=una
pply(rhs(%),t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "simplify
(ysol(t),trig);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "ysol(0);
 simplify(D(ysol)(0));\nysol(Pi); simplify(D(ysol)(Pi));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(ysol(t),t=0..Pi);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "What fo
llows is some extra problems. This first extra problem starts at [0,0]
 with the differential equation" }}{PARA 0 "" 0 "" {TEXT -1 47 "      \+
                    y'' + 2 y' + 5 y = u." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "c:=2; k:=5;\n(c^2-4*k)/4;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "if (c^2-4*k)/4<0 then 'Expon
entiallyDampedOscillations' \n  else 'Overdamped' fi;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 37 "M:=matrix([[0,1],[-k,-c]]);\nb:=[0,1];" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "exponential(M,t);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Q0:=vector([0,0]);\nQ1:=vect
or([1,0]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "A:=evalm(Q1-e
xponential(M,Pi)&*Q0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "p
air:=evalm(exponential(M,Pi-s)&*b);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "y1:=unapply(pair[1],s);\ny2:=unapply(pair[2],s);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "eq1:=0-A[1]=\nalpha*int(y1(
s)*y1(s),s=0..Pi)+beta*int(y2(s)*y1(s),s=0..Pi);\neq2:=0-A[2]=\nalpha*
int(y1(s)*y2(s),s=0..Pi)+beta*int(y2(s)*y2(s),s=0..Pi);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "solve(\{eq1,eq2\},\{alpha,beta\});
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(%);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "v:=t->-alpha*y1(t)-beta*y2(t);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "simplify(v(t));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "dsolve(\{diff(y(t),t,t)-M[2,2]*diff
(y(t),t)-M[2,1]*y(t)=v(t),\n     y(0)=Q0[1],D(y)(0)=Q0[2]\},y(t)):" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "ysol:=unapply(rhs(%),t);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "evalf(simplify(ysol(0)));
 evalf(simplify(D(ysol)(0)));\nevalf(simplify(ysol(Pi))); evalf(simpli
fy(D(ysol)(Pi)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(y
sol(t),t=0..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 
"This second extra problem starts at [0,0] with y'' + 2 y' + 5 y = u.
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "c:=10; k:=21;\n(c^2-4*k)/4;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "if (c^2-4*k)/4<0 then 'Expon
entiallyDampedOscillations' \n  else 'Overdamped' fi;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 37 "M:=matrix([[0,1],[-k,-c]]);\nb:=[0,1];" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "exponential(M,t);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Q0:=vector([0,0]);\nQ1:=vect
or([1,0]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "A:=evalm(Q1-e
xponential(M,Pi)&*Q0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "p
air:=evalm(exponential(M,Pi-s)&*b);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "y1:=unapply(pair[1],s);\ny2:=unapply(pair[2],s);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "eq1:=0-A[1]=\nalpha*int(y1(
s)*y1(s),s=0..Pi)+beta*int(y2(s)*y1(s),s=0..Pi);\neq2:=0-A[2]=\nalpha*
int(y1(s)*y2(s),s=0..Pi)+beta*int(y2(s)*y2(s),s=0..Pi);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "solve(\{eq1,eq2\},\{alpha,beta\});
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(%);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "v:=t->-alpha*y1(t)-beta*y2(t);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "simplify(v(t));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "dsolve(\{diff(y(t),t,t)-M[2,2]*diff
(y(t),t)-M[2,1]*y(t)=v(t),\n     y(0)=Q0[1],D(y)(0)=Q0[2]\},y(t)):" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "ysol:=unapply(rhs(%),t):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "simplify(ysol(t),trig);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "simplify(ysol(0)); simplif
y(D(ysol)(0));\nsimplify(ysol(Pi)); simplify(D(ysol)(Pi));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(ysol(t),t=0..Pi);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "alpha:='alpha': beta:='beta';" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{MARK "2 0" 36 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
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