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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 23 "Least squares problems." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 19 "Original Model     " }{XPPEDIT 18 0 "y = \+
sqrt(x/(a*x+b));" "6#/%\"yG-%%sqrtG6#*&%\"xG\"\"\",&*&%\"aGF*F)F*F*%\"
bGF*!\"\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 24 "Linearized model:       " }{XPPEDIT 18 0 "x*y^2*a+
y^2*b = x;" "6#/,&*(%\"xG\"\"\"*$%\"yG\"\"#F'%\"aGF'F'*&F)F*%\"bGF'F'F
&" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 9 "The data:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "x \+
:= Vector([0.1,0.5,1,2,3,4,5]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 48 "y := Vector([0.2,0.37,0.45,0.5,0.52,0.53,0.54]):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "A := Matrix(7,2): b := Vector(7):" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "for i from 1 to 7 do\n   \+
A[i,1] := x[i]*y[i]^2:\n   A[i,2] := y[i]^2:\n   b[i] := x[i]:\nend do
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "u := LinearAlgebra[Lea
stSquares](A,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG-%'RTABLEG6%
\"*giVW\"-%'MATRIXG6#7$7#$\"3!QL*3J_xXI!#<7#$\"3YH2n*)3'R'>F0&%'Vector
G6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "a, b := u[1
], u[2]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f := t-> sqrt(t
/(a*t+b));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"tG6\"6$%)op
eratorG%&arrowGF(-%%sqrtG6#*&9$\"\"\",&*&%\"aGF1F0F1F1%\"bGF1!\"\"F(F(
F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "pts := [seq([x[j],y[j
]],j=1..7)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(f(t),t
=0..5.5);" }}{PARA 13 "" 1 "" {GLPLOT2D 356 237 237 {PLOTDATA 2 "6%-%'
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0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot1 := plot(f(t),t=0..5.5):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot2 := plot(pts,style=poin
t,symbolsize=30):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plots[
display](\{plot1,plot2\});" }}{PARA 13 "" 1 "" {GLPLOT2D 356 237 237 
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P%*=6@LFiq$\"32Bb9O0\\V_FE7$$\"3o*\\(oHmfKMFiq$\"33uU%f)**Qd_FE7$$\"3u
**\\i?9QcNFiq$\"3CO*\\WO>>F&FE7$$\"3%HLL3c2wm$Fiq$\"337n.0=B%G&FE7$$\"
3s***\\i?ljy$Fiq$\"3[m#H-^mmH&FE7$$\"3SmT&QGxR*QFiq$\"3v)owE*)[tI&FE7$
$\"3Y++v3di6SFiq$\"3VNSs;)R%=`FE7$$\"3\"G$3-3PKATFiq$\"3XpPrOgNG`FE7$$
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p#)4U&FE-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-F$6&7)7$$\"3/+++++++5FE$\"35+
++++++?FE7$$\"3++++++++]FE$\"3%**************p$FE7$$\"\"\"F)$\"35+++++
++XFE7$$\"\"#F)F__l7$$\"\"$F)$\"3;+++++++_FE7$$\"\"%F)$\"3E+++++++`FE7
$$\"\"&F)$\"3M+++++++aFEF_^l-%&STYLEG6#%&POINTG-%'SYMBOLG6$%(DEFAULTG
\"#I-%+AXESLABELSG6%Q\"t6\"Q!Fgal-%%FONTG6#Faal-%%VIEWG6$;F($\"#bFe^lF
aal" 1 2 0 1 10 0 2 9 1 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 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Example 2:     " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "original \+
model:    " }{XPPEDIT 18 0 "y = c+ln(x/a+b/x+5);" "6#/%\"yG,&%\"cG\"\"
\"-%#lnG6#,(*&%\"xGF'%\"aG!\"\"F'*&%\"bGF'F-F/F'\"\"&F'F'" }{TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "li
nearized model:    " }{XPPEDIT 18 0 "alpha*x+beta*1/x-gamma*exp(y) = -
5;" "6#/,(*&%&alphaG\"\"\"%\"xGF'F'*(%%betaGF'F'F'F(!\"\"F'*&%&gammaGF
'-%$expG6#%\"yGF'F+,$\"\"&F+" }{TEXT -1 7 "   with" }}{PARA 0 "" 0 "" 
{TEXT -1 6 "      " }{XPPEDIT 18 0 "alpha = 1/a;" "6#/%&alphaG*&\"\"\"
F&%\"aG!\"\"" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "beta = b;" "6#/%%bet
aG%\"bG" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "gamma = exp(-c);" "6#/%&g
ammaG-%$expG6#,$%\"cG!\"\"" }{TEXT -1 14 "    implying  " }{XPPEDIT 
18 0 "a = 1/alpha;" "6#/%\"aG*&\"\"\"F&%&alphaG!\"\"" }{TEXT -1 4 ",  \+
 " }{XPPEDIT 18 0 "b = beta;" "6#/%\"bG%%betaG" }{TEXT -1 4 ",   " }
{XPPEDIT 18 0 "c = -ln(gamma);" "6#/%\"cG,$-%#lnG6#%&gammaG!\"\"" }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "The data:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "x := Vector([1,1.5,2,3.5,4,4.5,5]):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "y := Vector([1.14,1.05,
1.015,1.03,1.05,1.07,1.09]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 33 "A := Matrix(7,3): b := Vector(7):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 108 "for i from 1 to 7 do\n   A[i,1] := x[i]:\n   A[i,2] \+
:= 1/x[i]:\n   A[i,3] := -exp(y[i]):\n   b[i] := -5:\nend do:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "A, b;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$-%'RTABLEG6%\"*OkZW\"-%'MATRIXG6#7)7%\"\"\"F,$!+l$on7$!
\"*7%$\"#:!\"\"$\"+nmmmm!#5$!+=6ldGF/7%\"\"##F,F:$!+(Rj$fFF/7%$\"#NF3$
\"+dG9dGF6$!+Ne1,GF/7%\"\"%#F,FFF77%$\"#XF3$\"+AAAAAF6$!++&z`\"HF/7%\"
\"&#F,FP$!+tSFuHF/%'MatrixG-F$6%\"*gS[W\"-F(6#7)7#!\"&FenFenFenFenFenF
en&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "
u := LinearAlgebra[LeastSquares](A,b);  # alpha, beta, gamma" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"uG-%'RTABLEG6%\"*!=%[W\"-%'MATRIXG6#7%7#
$\"3tD`X[BhP[!#=7#$\"335^RO<!f\"H!#<7#$\"3_bX*z8[jo#F4&%'VectorG6#%'co
lumnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "a, b, c := 1/u[1],
 u[2], -ln(u[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6%%\"aG%\"bG%\"c
G6%$\"+ba8n?!\"*$\"335^RO<!f\"H!#<$!+=q#=))*!#5" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 28 "g := t -> c + ln(t/a+b/t+5);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"gGf*6#%\"tG6\"6$%)operatorG%&arrowGF(,&%\"cG\"\"
\"-%#lnG6#,(*&9$F.%\"aG!\"\"F.*&%\"bGF.F4F6F.\"\"&F.F.F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(g(t),t=0.5..5.5);" }}
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F-7$$\"3*QLLeRwX5)F*$\"3o'[COd?z?\"F-7$$\"3EMLL$eI8k)F*$\"3CwE\"y30d=
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-$\"3ZGDB5CFM5F-7$$\"3+++v=S2LPF-$\"32Z6&[kb#Q5F-7$$\"3Jmmm\"p)=MQF-$
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{PARA 0 "" 0 "" {TEXT -1 26 "#2.  For a given model    " }{XPPEDIT 18 
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{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "#3.  Extr
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