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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 16 "Review of Exam 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 53 "1. Linear Least Squares problem (Practice p. 157, #7
)" }}{PARA 0 "" 0 "" {TEXT -1 36 "(a) pencil work for setting up  Ax=b
" }}{PARA 0 "" 0 "" {TEXT -1 41 "(b) solve Ax=b for least squares solu
tion" }}{PARA 0 "" 0 "" {TEXT -1 45 "(c) construct the function we are
 looking for" }}{PARA 0 "" 0 "" {TEXT -1 47 "(d) put the function and \+
data in the same graph" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "2. Linearizable Leas
t Squares Problem." }}{PARA 0 "" 0 "" {TEXT -1 40 "(a) pencil work:  l
inearize the problem " }}{PARA 0 "" 0 "" {TEXT -1 31 "(b) pencil work:
  set up Ax = b" }}{PARA 0 "" 0 "" {TEXT -1 31 "(c, d, e) same as (b,c
,d) above" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
21 "Example:   Model:    " }{XPPEDIT 18 0 "y = sqrt(t/(a+b*exp(t)));" 
"6#/%\"yG-%%sqrtG6#*&%\"tG\"\"\",&%\"aGF**&%\"bGF*-%$expG6#F)F*F*!\"\"
" }{TEXT -1 9 "     and " }}{PARA 0 "" 0 "" {TEXT -1 5 "data:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "     t:  \+
   0.5    1.0   1.5   2.0   2.5   3.0   3.5" }}{PARA 0 "" 0 "" {TEXT 
-1 52 "     y:    .37    .46   .48   .46   .42   .37   .316" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "(pencil done)" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 33 "A := Matrix(7,2): b := Vector(7):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 44 "t := Vector[row]([0.5,1.0,1.5,2,2.5,3,3.5]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"tG-%'RTABLEG6%\"*!yPX9-%'VECTORG6#
7)$\"\"&!\"\"$\"#5F/$\"#:F/\"\"#$\"#DF/\"\"$$\"#NF/&%'VectorG6#%$rowG
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "y := Vector[row]([.37,.
46 ,.48,.46,.42,.37,.316]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG-
%'RTABLEG6%\"*Sz`W\"-%'VECTORG6#7)$\"#P!\"#$\"#YF/$\"#[F/F0$\"#UF/F-$
\"$;$!\"$&%'VectorG6#%$rowG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
140 "for k from 1 to 7 do\n   A[k,1] := 1:          # first column\n  \+
 A[k,2] := exp(t[k]):  # second column\n   b[k] := t[k]/y[k]^2:  # RHS
\nend do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "A, b;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\"*SXbW\"-%'MATRIXG6#7)7$\"\"\"$
\"+r7s[;!\"*7$F,$\"+G=G=FF/7$F,$\"+q!*o\"[%F/7$F,-%$expG6#\"\"#7$F,$\"
+'R\\#=7!\")7$F,-F86#\"\"$7$F,$\"+'>X:J$F>%'MatrixG-F$6%\"*+>bW\"-F(6#
7)7#$\"+]4I_OF/7#$\"+@z*es%F/7#$\"+nmT5lF/7#$\"+Tez^%*F/7#$\"+gNB<9F>7
#$\"+q0Q\">#F>7#$\"+os/0NF>&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 38 "z := LinearAlgebra[LeastSquares](A,b);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG-%'RTABLEG6%\"*%)>&[9-%'MATRIXG6
#7$7#$\"+`,\\I?!\"*7#$\"+fL=c**!#5&%'VectorG6#%'columnG" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 27 "Now construct the function " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "g := s -> sqrt(s/(z[1]+z[2]*exp(s))
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"sG6\"6$%)operatorG%
&arrowGF(-%%sqrtG6#*&9$\"\"\",&&%\"zG6#F1F1*&&F46#\"\"#F1-%$expG6#F0F1
F1!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "plot(g,0..
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\"3RLLLe'40j\"FE$\"3_2Ug%GIkD#FE7$$\"3/++](Q&3d?FE$\"3Z\"3Kaw$\\9DFE7$
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1.000000 46.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 36 "points := [seq([t[j],y[j]],j=1..7)]:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(points,style=point,symbolsize=
30);" }}{PARA 13 "" 1 "" {GLPLOT2D 378 198 198 {PLOTDATA 2 "6'-%'CURVE
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+++DF6$\"3%)*************>%F*7$$\"\"$F0F+7$$\"3++++++++NF6$\"3-++++++g
JF*-%'COLOURG6&%$RGBG$\"#5!\"\"$F0F0FP-%&STYLEG6#%&POINTG-%'SYMBOLG6$%
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 " 0 "" {MPLTEXT 1 0 22 "plot1 := plot(g,0..4):" }}}{EXCHG {PARA 0 "> \+
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)>6HZKFE7$$\"3FLL$e#pa-NFcr$\"3KrsP^sffJFE7$$\"3!*******Rv&)zNFcr$\"3%
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3#fI2i:'yYFFE7$$\"\"%F)$\"3gp(\\lFoLm#FE-%'COLOURG6&%$RGBG$\"#5!\"\"F(
F(-F$6&7)7$$\"3++++++++]FE$\"3%**************p$FE7$$\"\"\"F)$\"3=+++++
++YFE7$$\"3++++++++:Fcr$\"3#)*************z%FE7$$\"\"#F)F\\_l7$$\"3+++
+++++DFcr$\"3%)*************>%FE7$$\"\"$F)Fg^l7$$\"3++++++++NFcr$\"3-+
+++++gJFEFj]l-%&STYLEG6#%&POINTG-%'SYMBOLG6$%(DEFAULTG\"#I-%+AXESLABEL
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4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Example: \+
 Page 186, #2" }}{PARA 0 "" 0 "" {TEXT -1 47 "(Example for solving 1x1
 \"system\" by R-K method" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "read(\"e:/340/rkmxm.txt\"):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "f := Vector(1):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f[1] := (t,x) -> 1 + (t-x)^2;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"fG6#\"\"\"f*6$%\"tG%\"xG6\"6$%)op
eratorG%&arrowGF,,&F'F'*$),&9$F'9%!\"\"\"\"#F'F'F,F,F," }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "a, b := 2, 3;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>6$%\"aG%\"bG6$\"\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "x0 := Vector([1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%#x0G-%'RTABLEG6%\"*?cxX\"-%'MATRIXG6#7#7#\"\"\"&%'VectorG6#%'column
G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "n := 10:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "t,x := rkmxm(f,a,b,x0,n);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%L~End~of~Runge-Kutta~method~for~mxm~IVP-ODE
~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%\"tG%\"xG6$-%'RTABLEG6%\"*+d
xX\"-%'MATRIXG6#7,7#$\"\"#\"\"!7#$\"+6666@!\"*7#$\"+AAAAAF77#$\"+LLLLB
F77#$\"+WWWWCF77#$\"+cbbbDF77#$\"+nmmmEF77#$\"+yxxxFF77#$\"+*)))))))GF
77#$\"+++++IF7&%'VectorG6#%'columnG%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "x[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%
\"*?exX\"-%'MATRIXG6#7,7#$\"\"\"\"\"!7#$\"+j1667!\"*7#$\"+%yRSS\"F27#$
\"+mEL$e\"F27#$\"+%4O@v\"F27#$\"+)y(p7>F27#$\"+&3mm1#F27#$\"+UsF:AF27#
$\"+KmZfBF27#$\"+\\&****\\#F2&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 36 "pts := [seq([t[j],x[1][j]],j=1..n)]:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "plot(pts);" }}{PARA 13 "" 1 
"" {GLPLOT2D 378 219 219 {PLOTDATA 2 "6%-%'CURVESG6$7,7$$\"\"#\"\"!$\"
\"\"F*7$$\"3/+++6666@!#<$\"3#******Hm56@\"F07$$\"33+++AAAAAF0$\"32+++%
yRSS\"F07$$\"37+++LLLLBF0$\"3++++mEL$e\"F07$$\"3;+++WWWWCF0$\"3#******
R4O@v\"F07$$\"3%)*****fbbbb#F0$\"3,+++)y(p7>F07$$\"3))*****pmmmm#F0$\"
3/+++&3mm1#F07$$\"3#******zxxxx#F0$\"3)******>Cx_@#F07$$\"3'*******)))
)))))GF0$\"3!)*****>jw%fBF07$$\"\"$F*$\"3)*******[&****\\#F0-%'COLOURG
6&%$RGBG$\"#5!\"\"$F*F*F\\o-%+AXESLABELSG6$Q!6\"F`o-%%VIEWG6$%(DEFAULT
GFeo" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1
" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
16 "Solve by dsolve:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "dsolve(diff(x(t),t) = 1 + (t-x(t))^2,x(t));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG*&,(\"\"\"!\"\"*$)F'\"\"#F*F**
&%$_C1GF*F'F*F*F*,&F'F*F0F*F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 32 "   The general solution is:     " }
{XPPEDIT 18 0 "x(t) = (-1+t^2+C*t)/(t+C);" "6#/-%\"xG6#%\"tG*&,(\"\"\"
!\"\"*$)F'\"\"#F*F**&%\"CGF*F'F*F*F*,&F'F*F0F*F+" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 7 "       " }{XPPEDIT 18 0 "x(2) = (3+2*C)/(2
+C);" "6#/-%\"xG6#\"\"#*&,&\"\"$\"\"\"*&F'F+%\"CGF+F+F+,&F'F+F-F+!\"\"
" }{TEXT -1 4 " = 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 29 "solve( (3+2*c)/(2+c) = 1, c);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 42 "g := t-> (-1+t^2-t)/(t-1);  # the solution" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"gGf*6#%\"tG6\"6$%)operatorG%&arrowGF(*&,(\"\"\"!
\"\"*$)9$\"\"#F.F.F2F/F.,&F2F.F.F/F/F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "How to verify the solutio
n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "A := D(g)(t);  # LHS" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"AG,&*&,&*&\"\"#\"\"\"%\"tGF*F*F*!\"\"F*,&F+F*F*F,F,F**&,(F*F,*$
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