{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " TXT CMD" -1 256 "MS Sans Serif" 0 0 128 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "Bookmark" 20 257 "" 0 0 0 128 0 1 1 0 0 0 0 1 0 0 0 }{CSTYLE "word" -1 258 "" 0 0 128 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "bookmark" -1 259 "" 0 0 0 128 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE " " -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 0 0 128 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "M aple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "newpage" -1 256 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 1 0 -1 0 }{PSTYLE "vfill" -1 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Definition" -1 258 1 {CSTYLE "" -1 -1 "" 0 0 0 64 128 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Theorem" -1 259 1 {CSTYLE "" -1 -1 "" 0 0 219 36 36 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 } {PSTYLE "Problem" -1 260 1 {CSTYLE "" -1 -1 "" 0 0 0 0 255 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 3 4 0 0 0 0 -1 0 }{PSTYLE "dblnorm" -1 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "code" 2 262 1 {CSTYLE "" -1 -1 "Comic Sans MS" 0 0 128 0 128 1 0 1 0 0 0 0 3 0 0 }0 0 0 -1 -1 -1 3 12 0 0 0 0 -1 0 } {PSTYLE "asis" 0 263 1 {CSTYLE "" -1 -1 "Arial Narrow" 1 12 128 64 0 1 0 0 0 0 0 1 3 0 0 }0 0 0 -1 -1 -1 3 6 0 0 0 0 0 0 }{PSTYLE "subprobl em" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "diagram" -1 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "dblnorm.mws" -1 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 2 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Item" 0 267 1 {CSTYLE "" -1 -1 "Lucida Sans" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 6 -1 3 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 268 "" 0 "" {TEXT 260 24 "Review topics for exam \+ 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 18 "Sol ving equations." }{TEXT -1 331 " Know the bisection method. H ow does the intermediate value theorem play a role? How fast is it? Know Newton's Method. When does it converge? What is quadratic c onvergence? Know the secant method. How can you use these method s to invert a function? Which methods generalize to solving systems \+ of equations?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "1. Set up solving the equation " }{XPPEDIT 18 0 "x* sin(x) = .25 " "/*&%\"xG\"\"\"-%$sinG6#F$F%$\"#D!\"#" }{TEXT -1 96 " \+ using bisection, Newton's method, and the secant method. Carry out one iteration of each. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "f(x) = x*sin(x) - .25" "/-% \"fG6#%\"xG,&*&F&\"\"\"-%$sinG6#F&F)F)$\"#D!\"#!\"\"" }{TEXT -1 3 ". \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f := t -> t*sin(t)-.25; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"tG6\"6$%)operatorG%&ar rowGF(,&*&9$\"\"\"-%$sinG6#F.F/F/$!#D!\"#F/F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 10 "Bisection:" }{TEXT -1 23 " find a and b so that " }{XPPEDIT 18 0 "f(a)*f(b) < 0" "2*&-%\"fG6#%\"aG\"\"\"-F%6#%\"bGF(\"\" !" }{TEXT -1 68 ". Intermediate value theorem guarantees a solution b etween a and b." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f(.5),f( .6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$!*2B(G5!#5$\"*S[&y))F%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "a := .5 : b := .6:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 39 ":Now calculate midpoint m and sign of " }{XPPEDIT 18 0 "f(m)*f(a)" "*&-%\"fG6#%\"mG\"\"\"-F$6#%\"aGF'" }{TEXT -1 23 ". If positve, update " }{XPPEDIT 18 0 "a = m" "/%\"aG% \"mG" }{TEXT -1 13 " else update " }{XPPEDIT 18 0 "b=m" "/%\"bG%\"mG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "m := .5*(a+b); f(m)*f(a) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG$\"#b!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+VeWbQ!#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "b := m;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG$\"#b!\"#" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "After 1 iteration of bisection, w e know a root is trapped between .5 and .55" }}{PARA 0 "" 0 "" {TEXT -1 32 "Iterating this a few times ---" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "for i from 1 to 5 do m := .5*(a+b):\n if f(m)*f(a)>0 then a:= m else b := m fi od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" mG$\"$D&!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG$\"%D^!\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG$\"&D1&!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG$\"'v$4&!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"mG$\"(v$4^!\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 266 7 "Newton:" }{TEXT -1 23 " We could start with " }{XPPEDIT 18 0 "x1 = .5" "/%#x1G$\"\"&!\"\"" }{TEXT -1 31 " and iterate Newton's method " }{XPPEDIT 18 0 "x1 = x1 - f(x1)/` f'`(x1)" "/%#x1G,&F#\"\"\"*&-%\"fG6#F#F%-%#f'G6#F#!\"\"F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "x1 := .5; for i from 1 to 5 do \n \+ x1 := x1-f(x1)/D(f)(x1) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G $\"\"&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"+v[.7^!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"+%[A56&!#5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#x1G$\"+.C-6^!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#x1G$\"+.C-6^!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"+.C-6^ !#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Convergence is obtained ra pidly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 7 "Secant:" }{TEXT -1 78 " we could start with x1 = .4 and x2 = .5 \+ and iterate the secant method " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x3 = x1 - f(x2)*(x2-x1)/(f(x2)-f(x1)); x1 = x2 ; x2 = x3" "C%/%#x3G,&%#x1G\"\"\"*(-%\"fG6#%#x2GF',&F,F'F&!\"\"F',&-F*6#F,F' -F*6#F&F.F.F./F&F,/F,F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "x1 := .4: x2 := .5: for i from 1 to 4 do \nx3 := x2 - f(x2)*(x2-x1)/ (f(x2)-f(x1)); \n x1 := x2 : x2 := x3: od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x3G$\"+SmaA^!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#x1G$\"\"&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x2G$\"+SmaA^!#5 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x3G$\"+)))=46&!#5" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#x1G$\"+SmaA^!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x2G$\"+)))=46&!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x3G$ \"+3B-6^!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"+)))=46&!#5" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x2G$\"+3B-6^!#5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#x3G$\"+.C-6^!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#x1G$\"+3B-6^!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x2G$\"+.C-6^ !#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "A little slower than Newto n, but faster than bisection." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "2. Find an interval on which the \+ function " }{XPPEDIT 18 0 "x^3 + 2*x + 5" ",(*$%\"xG\"\"$\"\"\"*&\" \"#F&F$F&F&\"\"&F&" }{TEXT -1 132 " is invertible. Describe, in pseud o code or C or Maple, a function routine which returns the values of \+ the inverse function there." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "The derivative \+ is " }{XPPEDIT 18 0 "3*x^2 + 2 " ",&*&\"\"$\"\"\"*$%\"xG\"\"#F%F%\"\"# F%" }{TEXT -1 316 " which is always positive, so the function is 1-1 on the entire x-axis. We could take any interval, but it is diff icult to use Newton's method exclusively near 5. One solution is to \+ use bisection for x between 0 and 10 and Newton's method outside this \+ interval. Here is a Maple word which does this. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 473 "inver := proc(x) \n local x1, x2, a, b, i, m;\nif n ot hastype(evalf(x),float) then RETURN('inver(x)') fi ;\nif x>0 and x \+ <10 then a:= -1.328268856: b:= 1.328268856: \nfor i from 1 to 10 do m := .5*(a+b): \n if (m^3+2*m+5.-x)*(a^3+2*a+5.-x)>0 then a:= m else \+ b := m fi od;\nRETURN(m) fi;\n if x >= 10 then x1 := 1.3; else x1 := - 20 fi;\n x2 := x1 - (x1^3+2*x1+5.-x)/(3*x1^2+2.):\nwhile abs((x2-x1)/ x2) > 10^(-6) do \nx1 := x2: \nx2 := x1 - (x1^3+2*x1+5.-x)/(3*x1^2+2. ):\nod; end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Plot the inverse function " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(inver,-5 ..15.);" }}{PARA 13 "" 1 "" {INLPLOT "6$-%'CURVESG6$7S7$$!\"&\"\"!$!+ Q!>u%=!\"*7$$!+s%H%F-$!+_?N&Q\"F-7$$ \"(]1!>F-$!+\"eucK\"F-7$$\"*]Z/N%F-$!+I0go7F-7$$\"*]$fC&)F-$!+w$\\6?\" F-7$$\"+'z6:B\"F-$!+un))Q6F-7$$\"+<=C#o\"F-$!+@&e51\"F-7$$\"+n#pS1#F-$ !+n\")=%))*!#57$$\"+j`A3DF-$!+FOO)*))F]q7$$\"+n(y8!HF-$!+(eCW'zF]q7$$ \"+j.tKLF-$!+P!e5x'F]q7$$\"+)3zMu$F-$!+$[?RZ&F]q7$$\"+#H_?<%F-$!+:*pa' QF]q7$$\"+!G;cc%F-$!+SGO,@F]q7$$\"+4#G,*\\F-$!+5^F%f#!#77$$\"*!o2Ja!\" )$\"+SGO,@F]q7$$\"*%Q#\\\"eFes$\"+8We8QF]q7$$\"*;*[HiFes$\"+x%\\,P&F]q 7$$\"*qvxl'Fes$\"+P!e5x'F]q7$$\"*`qn2(Fes$\"+&3RD\"zF]q7$$\"*cp@[(Fes$ \"+FOO)*))F]q7$$\"*3'HKzFes$\"+n\")=%))*F]q7$$\"*xanL)Fes$\"+rqCm5F-7$ $\"*v+'o()Fes$\"+un))Q6F-7$$\"*S<*f\"*Fes$\"+FzL17F-7$$\"*&)Hxe*Fes$\" +I0go7F-7$$\"*.o-***Fes$\"+\"eucK\"F-7$$\"+TO5T5Fes$\"+TD(HQ\"F-7$$\"+ U9C#3\"Fes$\"+b_oM9F-7$$\"+1*3`7\"Fes$\"+_j'f[\"F-7$$\"+$*zym6Fes$\"+# )3\"H`\"F-7$$\"+^j?47Fes$\"+&Q3(y:F-7$$\"+jMF^7Fes$\"+Jc9A;F-7$$\"+q(G **G\"Fes$\"+%3'[g;F-7$$\"+9@BM8Fes$\"+'RWFq\"F-7$$\"+`v&QP\"Fes$\"+Br9 R u%=F--%'COLOURG6&%$RGBG$\"#5!\"\"F*F*-%%VIEWG6$;F(Fgz%(DEFAULTG" 2 434 434 434 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 149 197 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "The inverse functio n can be graphed exactly with a parametric plot." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "plot([x^3+2*x+5.,x,x=-1.9..1.9]);" }}{PARA 13 "" 1 "" {INLPLOT "6#-%'CURVESG6$7S7$$!1************ec!#:$!1+++++++> F*7$$!1R;mB\"R[j%F*$!1mm;93<<=F*7$$!1_Sm\"F*7$$!1(3gn*ptF@F*$!1nmTdbY#e\"F*7$$!1T0\">u!4' Q\"F*$!1L$ezom7]\"F*7$$!1uhP!*z9;v!#;$!1n;zT\\)fU\"F*7$$!1x`\">'RuE\"F*7$$\"16;N$Q)oJ&*FI$!1+] i>=1(=\"F*7$$\"1[8tw!RTW\"F*$!1LL$)3WS/6F*7$$\"1Vt.QU(*Q=F*$!1nmT;(*fJ 5F*7$$\"1DVD \"3q%RF*$!1++D6=PMZFI7$$\"1uUzlx7RTF*$!1LLLV.Q()RFI7$$\"1?A%3V[YL%F*$! 1++D6B\"y;$FI7$$\"1.1s:[\"*3XF*$!1)**\\Pt*Q(Q#FI7$$\"1>ePM[F*$!1:L$ep!H`#)!#<7$$\"18\"oDr[i*\\F*$!1dmm; /kv=!#=7$$\"1^*o!G'eV;&F*$\"1QL$37f/>)F[s7$$\"1(z&e\"4$Q8`F*$\"1,+D\"H b$[:FI7$$\"1%fS0n`*zaF*$\"1LL$3SHgL#FI7$$\"13z'*3R?hcF*$\"1+++IQx\\JFI 7$$\"1]W?\"G41&eF*$\"1(***\\(*R'e%RFI7$$\"1#f'**=&>\"[gF*$\"1,+vo@7;ZF I7$$\"1Q%f203sG'F*$\"1++]UDOrbFI7$$\"1P:G0pyAlF*$\"1immcS$)RjFI7$$\"1D kd$\\#=*z'F*$\"1*****\\UT.;(FI7$$\"1c99&\\FX2(F*$\"1KL3_I%Q!zFI7$$\"1[ RZnmj0uF*$\"1-++:no;()FI7$$\"1sG.?-o[xF*$\"1lmTb#4:[*FI7$$\"1rM3$3vG9) F*$\"1+]P=p4G5F*7$$\"1mrm8iOm&)F*$\"1nm;R(ei5\"F*7$$\"1&RyEd=K0*F*$\"1 +](=#p3)=\"F*7$$\"1e_rX/?n&*F*$\"1LLL(=(*oE\"F*7$$\"1TV[Ys;95!#9$\"1LL em?\\Z8F*7$$\"1))zuU\"F*7$$\"1$fGm\"pDQ6Fex$\"1+++j Y'3]\"F*7$$\"1d.!)znA:7Fex$\"1nm\"p,T]e\"F*7$$\"1&o\\Czn(*G\"Fex$\"1LL L^$H.m\"F*7$$\"1&)3DQ-Zv8Fex$\"1+]()=CgSF*-%'COLOURG6&%$RGBG$\"#5!\"\"\" \"!Fb[l" 2 434 434 434 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 14 "Interpolation." }{TEXT -1 324 " What is data? What are nod es? What is polynomial interpolation? Know the Lagrange form of the \+ interpolating polynomial. What is a divided difference table? Know t he Newton form of the interpolation polynomial. What is the error in using the interpolating polynomial to estimate f(x). What are Cheb yshev nodes? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "3. Given the data x = [1,3,5,6] and y= [0,1,2,-2], w rite the Langrange form of the cubic polynomial which" }}{PARA 0 "" 0 "" {TEXT -1 52 "interpolates the data. Also write the Newton form." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "x := [1,3,5,6]; y := [0,1,2,-2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG7&\"\"\"\"\"$\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG7&\"\"!\"\"\"\"\"#!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "l[1] := t->(t-3)*(t-5)*(t-6)/((1-3)*(1-5)*(1-6)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"lG6#\"\"\":6#%\"tG6\"6$%)oper atorG%&arrowGF+,$*(,&9$F'!\"$F'F',&F2F'!\"&F'F',&F2F'!\"'F'F'#!\"\"\"# SF+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "l[2] := t->(t-1)*( t-5)*(t-6)/((3-1)*(3-5)*(3-6));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&% \"lG6#\"\"#:6#%\"tG6\"6$%)operatorG%&arrowGF+,$*(,&9$\"\"\"!\"\"F3F3,& F2F3!\"&F3F3,&F2F3!\"'F3F3#F3\"#7F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "l[3] := t->(t-1)*(t-3)*(t-6)/((5-1)*(5-3)*(5-6));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"lG6#\"\"$:6#%\"tG6\"6$%)operator G%&arrowGF+,$*(,&9$\"\"\"!\"\"F3F3,&F2F3!\"$F3F3,&F2F3!\"'F3F3#F4\"\") F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "l[4] := t->(t-1)*( t-3)*(t-5)/((6-1)*(6-3)*(6-5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&% \"lG6#\"\"%:6#%\"tG6\"6$%)operatorG%&arrowGF+,$*(,&9$\"\"\"!\"\"F3F3,& F2F3!\"$F3F3,&F2F3!\"&F3F3#F3\"#:F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "lag := y[1]*l[1] + y[2]*l[2]+y[3]*l[3]+y[4]*l[4];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$lagG,(&%\"lG6#\"\"#\"\"\"&F'6#\"\"$ F)&F'6#\"\"%!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "plot(la g,1..6);" }}{PARA 13 "" 1 "" {INLPLOT "6$-%'CURVESG6$7Y7$$\"\"\"\"\"!F *7$$\"+rd)*36!\"*$!+Z5\"3'=!#57$$\"+2P\"Q?\"F.$!+F#R,:$F17$$\"+SwX58F. $!+DNb`UF17$$\"+x%3yT\"F.$!+*H8],&F17$$\"+&4\\Y_\"F.$!+'=_pW&F17$$\"+C SqB;F.$!+P38wbF17$$\"+m'pis\"F.$!+lv&RX&F17$$\"+*>VB$=F.$!+zs8u]F17$$ \"+`l2Q>F.$!+=Y;gWF17$$\"+0j$o/#F.$!+K](eg$F17$$\"+_>jU@F.$!+%eaXo#F17 $$\"+j^Z]AF.$!+C7qy9F17$$\"+)=h(eBF.$!*)>_+6F17$$\"+Q[6jCF.$\"+Gn@P8F1 7$$\"+\\z(yb#F.$\"+eC>VFF17$$\"+b/cqEF.$\"+n.z1XF17$$\"+F.7$$\"+g4t.PF.$\"+%\\)y\"4#F.7$$\"+!Hst!QF.$\"+,sk9AF.7$$\"+ERW9RF. $\"+h^:DBF.7$$\"+KE>>SF.$\"+j/-:CF.7$$\"+#RU07%F.$\"+J5$G[#F.7$$\"+?S2 LUF.$\"+['pP`#F.7$$\"+$p)=MVF.$\"+8-kbDF.7$$\"+*=]@W%F.$\"+e8%=b#F.7$$ \"+]$z*RXF.$\"+r(HB_#F.7$$\"+kC$pk%F.$\"+b*\\'fCF.7$$\"+3qcZZF.$\"+H![ (pBF.7$$\"+/\"fF&[F.$\"+m:pTAF.7$$\"+0Ogb\\F.$\"+I:$33#F.7$$\"+nAFj]F. $\"+=1]s=F.7$$\"+&)*pp;&F.$\"+(Qy6j\"F.7$$\"+ye,t_F.$\"+Oq*4M\"F.7$$\" +fO=y`F.$\"+*=$y25F.7$$\"+$z-lU&F.$\"+e]T*Q)F17$$\"+E>#[Z&F.$\"+WP-*f' F17$$\"+26?IbF.$\"+/:+>WF17$$\"+(G!e&e&F.$\"+#p.$*4#F17$$\"+(37^j&F.$! (l#H'*F.7$$\"+&)Qk%o&F.$!+v&e#3CF17$$\"+9^XPdF.$!+X3%R+&F17$$\"+UjE!z& F.$!+g+0PxF17$$\"+FM\"3%eF.$!+lZT[5F.7$$\"+60O\"*eF.$!+e%*=O8F.7$$\"+c -oXfF.$!+pAFg;F.7$$\"\"'F*$!+++++?F.-%'COLOURG6&%$RGBG$\"#5!\"\"F*F*-% %VIEWG6$;F(Fa\\l%(DEFAULTG" 2 434 434 434 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "4. Show tha t the divided differences " }{XPPEDIT 18 0 "f*[x[1],x[2],x[3]]" "*& %\"fG\"\"\"7%&%\"xG6#\"\"\"&F'6#\"\"#&F'6#\"\"$F$" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "f*[x[1],x[3],x[2]]" "*&%\"fG\"\"\"7%&%\"xG6#\"\"\"& F'6#\"\"$&F'6#\"\"#F$" }{TEXT -1 12 " are equal." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "x := 'x': y := 'y': f := 'f':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "g := \+ (i,j,k) -> ((y[i]-y[j])/(x[i]-x[j])-(y[j]-y[k])/(x[j]-x[k]))/(x[i]-x [k]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG:6%%\"iG%\"jG%\"kG6\"6$ %)operatorG%&arrowGF**&,&*&,&&%\"yG6#9$\"\"\"&F36#9%!\"\"F6,&&%\"xGF4F 6&F=F8F:F:F6*&,&F7F6&F36#9&F:F6,&F>F6&F=FBF:F:F:F6,&F " 0 "" {MPLTEXT 1 0 18 "g(1,2,3)-g( 1,3,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&*&,&&%\"yG6#\"\"\"F+& F)6#\"\"#!\"\"F+,&&%\"xGF*F+&F2F-F/F/F+*&,&F,F+&F)6#\"\"$F/F+,&F3F+&F2 F7F/F/F/F+,&F1F+F:F/F/F+*&,&*&,&F(F+F6F/F+F;F/F+*&,&F6F+F,F/F+,&F:F+F3 F/F/F/F+F0F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify( \");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Yep. But let's check it 'by hand'. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "expr : = g(1,2,3)-g(1,3,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG,&*&,& *&,&&%\"yG6#\"\"\"F-&F+6#\"\"#!\"\"F-,&&%\"xGF,F-&F4F/F1F1F-*&,&F.F-&F +6#\"\"$F1F-,&F5F-&F4F9F1F1F1F-,&F3F-F " 0 "" {MPLTEXT 1 0 133 "expr :=su bs(\{y[1]-y[2]=y12,y[2]-y[3]=y23,y[1]-y[3]=y13,x[1]-x[2]=x12,\nx[1]-x[ 3]=x13,x[2]-x[3]=x23,x[3]-x[2]=x23,y[3]-y[2]=y23\},expr);" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG, &*&,&*&%$y12G\"\"\"%$x12G!\"\"F**&%$y23GF*%$x23GF,F,F*%$x13GF,F**&,&*& %$y13GF*F0F,F*F-F,F*F+F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Now clear some fractions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "e xpand(x13*x12*expr);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%$y12G\"\"\" *(%$x12GF%%$y23GF%%$x23G!\"\"F*%$y13GF**(%$x13GF%F(F%F)F*F%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "expand(x23*\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&%$x23G\"\"\"%$y12GF&F&*&%$x12GF&%$y23GF&!\" \"*&F%F&%$y13GF&F+*&%$x13GF&F*F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "collect(\",[x23,y23]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&%$y12G\"\"\"%$y13G!\"\"F'%$x23GF'F'*&,&%$x12GF)%$x13GF'F'% $y23GF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "But now notice that " }{XPPEDIT 18 0 "y1 2 - y13 = y32" "/,&%$y12G\"\"\"%$y13G!\"\"%$y32G" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "x13 - x12 = x32" "/,&%$x13G\"\"\"%$x12G!\"\"%$x32G" }{TEXT -1 11 ". Hence " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "subs(\{y12-y13=y32,x13-x12=x23\},\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%$y32G\"\"\"%$x23GF&F&*&F'F&%$y23GF&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "But now notice that " }{XPPEDIT 18 0 "y32 = -y23" "/%$y32G,$%$y23G!\"\"" }{TEXT -1 46 ", and so the expression reduces to 0. Voila!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 263 8 "Splines." }{TEXT -1 230 " What a re knots? What are interpolation conditions? Continuity conditions? \+ What is a linear spline? natural cubic spline? Be able to check tha t a piecewise polynomial function is a spline. What is a tridiago nal system?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "5. Find the quadratic spline q with knots 0,1,3 such that q(0) = 2 and " }{XPPEDIT 18 0 "`q'`(t) = 1+t" "/-%#q'G6#%\"tG,&\"\" \"\"\"\"F&F)" }{TEXT -1 15 " on [0,1] and " }{XPPEDIT 18 0 "`q'`(t) = 2-3*(t-1)" "/-%#q'G6#%\"tG,&\"\"#\"\"\"*&\"\"$F),&F&F)\"\"\"!\"\"F)F. " }{TEXT -1 11 " on [1,3]. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 184 "We can get this by integration. First check t hat q' is indeed a linear spline with knots 0,1,3. All we need to do \+ here is check that both rules agree at 1. They both give 2. So " }{XPPEDIT 18 0 "q(t) = t + t^2/2 + 2 " "/-%\"qG6#%\"tG,(F&\"\"\"*&F&\" \"#\"\"#!\"\"F(\"\"#F(" }{TEXT -1 14 "on [0,1] and " }{XPPEDIT 18 0 " q(t) = 2*t -3/2*(t-1)^2 + c" "/-%\"qG6#%\"tG,(*&\"\"#\"\"\"F&F*F**(\" \"$F*\"\"#!\"\",&F&F*\"\"\"F.\"\"#F.%\"cGF*" }{TEXT -1 88 " on [1,3] f or some c. Since q(1) = 3.5 by rule 1, c must be 1.5. So the splin e q is " }{XPPEDIT 18 0 "t+t^2 /2 + 2" ",(%\"tG\"\"\"*&F#\"\"#\"\"#!\" \"F$\"\"#F$" }{TEXT -1 27 " between 0 and 1 and q is " }{XPPEDIT 18 0 "2*t -3/2*(t-1)^2 + 1.5" ",(*&\"\"#\"\"\"%\"tGF%F%*(\"\"$F%\"\"#!\" \",&F&F%\"\"\"F*\"\"#F*$\"#:!\"\"F%" }{TEXT -1 17 " between 1 and 3." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "readlib(piecewise): q := \+ unapply(piecewise(t<1,t+t^2/2+2,2*t-3/2*(t-1)^2+1.5),t);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"qG:6#%\"tG6\"6$%)operatorG%&arrowGF(-%*piece wiseG6%29$\"\"\",(F0F1*$F0\"\"##F1F4F4F1,(F0F4*$,&F0F1!\"\"F1F4#!\"$F4 $\"#:F9F1F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "plot(q,0.. 3);" }}{PARA 13 "" 1 "" {INLPLOT "6$-%'CURVESG6$7S7$\"\"!$\"\"#F(7$$\" 1+++]i9Rl!#<$\"1WR=%[Hv1#!#:7$$\"1++vVA)GA\"!#;$\"1&R&GHawH@F17$$\"1++ ]Peui=F5$\"1=c-%pBO?#F17$$\"1++]i3&o]#F5$\"1&3%[#f1@G#F17$$\"1++voX*y9 $F5$\"1H[&zlNVO#F17$$\"1++vVTAUPF5$\"1F`3AOCWCF17$$\"1++v$*zhdVF5$\"1p AFsfqIDF17$$\"1++v$>fS*\\F5$\"1&RGb04Ti#F17$$\"1++v=$f%GcF5$\"1>\"*Q.P C@FF17$$\"1+++Dy,\"G'F5$\"1R)*32xNDGF17$$\"1++]7fFr&)e?HF17 $$\"1,++v4&G](F5$\"1$Hl7'*[<.$F17$$\"1+++Drc_\")F5$\"1a\"QhYyv9$F17$$ \"1+++D!*oy()F5$\"1y#)[2e>jKF17$$\"1++v$pnsM*F5$\"17\"og$QerLF17$$\"1+ +]siL-5F1$\"14iudVm/NF17$$\"1+++!R5'f5F1$\"1sg5#o!*Qh$F17$$\"1+]P/QBE6 F1$\"1@&*p[^cGPF17$$\"1+++:o?&=\"F1$\"1Lj^$Gh*=QF17$$\"1+]Pa&4*\\7F1$ \"1L\")pIp81RF17$$\"1+]7j=_68F1$\"1$pU%>\\ZxRF17$$\"1++vVy!eP\"F1$\"1j RO%Qo(RSF17$$\"1+](=WU[V\"F1$\"1\"Gm)fH0'3%F17$$\"1++DJ#>&)\\\"F1$\"1k 3hEjDCTF17$$\"1+]P>:mk:F1$\"1vcb\"4f5:%F17$$\"1+]iv&QAi\"F1$\"1div$)eq jTF17$$\"1++vtLU%o\"F1$\"1-qp:P>mTF17$$\"1+++bjm[NTF17$$\"1+]PMaKs=F1$\"1-PctLA.TF17$$\"1++D6W% )R>F1$\"16KYYvsaSF17$$\"1+++:K^+?F1$\"1Yb\"**Q'[**RF17$$\"1++]7,Hl?F1$ \"11YJ*o:$GRF17$$\"1+]P4w)R7#F1$\"14sq**H&H&QF17$$\"1++]x%f\")=#F1$\"1 !RS!RXtePF17$$\"1+]P/-a[AF1$\"1])QeW,)eOF17$$\"1+](=Yb;J#F1$\"1@-vZ3lU NF17$$\"1++]i@OtBF1$\"1&)*>4))QvT$F17$$\"1+]PfL'zV#F1$\"1ZCR%y=VF$F17$ $\"1+++!*>=+DF1$\"1f%>`(\\aCJF17$$\"1++DE&4Qc#F1$\"1EY\"4qo$fHF17$$\"1 +]P%>5pi#F1$\"1\\!3z@lNy#F17$$\"1+++bJ*[o#F1$\"1=7N%*))[6EF17$$\"1++Dr \"[8v#F1$\"1@*)3:.AF17$$\"1+]P/)fT(G F1$\"1$fZ-:2'z>F17$$\"1+]i0j\"[$HF1$\"1S9[19Oa " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "6. Find the natural cubic spline s with knots 0,1,2 which interpolates the the data 0,1,4. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "readlib(spline):s := unapply(spline([0,1,2] ,[0,1,4],t,cubic),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG:6#%\"t G6\"6$%)operatorG%&arrowGF(-%*piecewiseG6%29$\"\"\",&F0#F1\"\"#*$F0\" \"$F3,*F1F1F0#!\"&F4*$F0F4F6F5#!\"\"F4F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 13 "plot(s,0..2);" }}{PARA 13 "" 1 "" {INLPLOT "6$-%'CU RVESG6$7S7$\"\"!F(7$$\"1LLLL3VfV!#<$\"1@Rt')y&Q=#F,7$$\"1nmm\"H[D:)F,$ \"1/$3@rmL5%F,7$$\"1LLLe0$=C\"!#;$\"1?HA:o!\\I'F,7$$\"1LLL3RBr;F7$\"1L p>g#f&*e)F,7$$\"1mm;zjf)4#F7$\"1_66-/^&4\"F77$$\"1LL$e4;[\\#F7$\"1]o$R 1[]K\"F77$$\"1++]i'y]!HF7$\"1'pmr7E^d\"F77$$\"1LL$ezs$HLF7$\"1h2]xA@\\ =F77$$\"1++]7iI_PF7$\"1.F=Z+&F7$\"16'>(RLmEJF77$$\"1+++]Z/ NaF7$\"1TIuU1F?NF77$$\"1+++]$fC&eF7$\"18BS23]GRF77$$\"1LL$ez6:B'F7$\"1 ,RA/\"ecK%F77$$\"1mmm;=C#o'F7$\"17%p055I$[F77$$\"1mmmm#pS1(F7$\"1g9wqy b%H&F77$$\"1++]i`A3vF7$\"1`!3]WN/(eF77$$\"1mmmm(y8!zF7$\"1W'H5MvrT'F77 $$\"1++]i.tK$)F7$\"1'4'f]cDfqF77$$\"1++](3zMu)F7$\"18i'zjlQr(F77$$\"1n mm\"H_?<*F7$\"1([aWT\"4W%)F77$$\"1nm;zihl&*F7$\"1S9DBh7f\"*F77$$\"1LLL 3#G,***F7$\"1IrpH5F!)**F77$$\"1LLezw5V5!#:$\"1PuK9F'*)3\"F`s7$$\"1++v$ Q#\\\"3\"F`s$\"1\\Gg%ovE<\"F`s7$$\"1LL$e\"*[H7\"F`s$\"1(o?3@VwE\"F`s7$ $\"1+++qvxl6F`s$\"1z/ox/]q8F`s7$$\"1++]_qn27F`s$\"1l.o6-dv9F`s7$$\"1++ Dcp@[7F`s$\"1qS_p[?\"e\"F`s7$$\"1++]2'HKH\"F`s$\"1'=M+:GGq\"F`s7$$\"1n mmwanL8F`s$\"1gz'zL%y:=F`s7$$\"1+++v+'oP\"F`s$\"1QMsFR**R>F`s7$$\"1LLe R<*fT\"F`s$\"1T%)\\mOcb?F`s7$$\"1+++&)Hxe9F`s$\"1#)HHk`(\\=#F`s7$$\"1m m\"H!o-*\\\"F`s$\"1o)4*p%f%4BF`s7$$\"1++DTO5T:F`s$\"1%o1%)G\"=UCF`s7$$ \"1nmmT9C#e\"F`s$\"1Kg*)>\"*HuDF`s7$$\"1++D1*3`i\"F`s$\"1HO7`K)[r#F`s7 $$\"1LLL$*zym;F`s$\"1fKV\"GcA&GF`s7$$\"1LL$3N1#4F`s$\"1A[s-\"omq$F`s7$$\"1++v.Uac>F`s$\"1NJAWd%z%QF`s7$$\" \"#F($\"\"%F(-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%VIEWG6$;F(Fbz%(DEFAULT G" 2 366 366 366 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "7. Verify that " }{XPPEDIT 18 0 "Int(`s\"`(t) ,t=0..2)<=Int(2,t=0..2)" "1-%$IntG6$-%#s\"G6#%\"tG/F);\"\"!\"\"#-F$6$ \"\"#/F);F,\"\"#" }{TEXT -1 23 " for the spline s in 6." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Int(D(D(s))(t),t=0..2)=int(D(D(s))( t),t=0..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%*PIECEWISEG 6$7$,$%\"tG\"\"$2F,\"\"\"7$,&\"\"'F/F,!\"$%*otherwiseG/F,;\"\"!\"\"#F- " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Since the right hand integral is 4, it woiks." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 264 28 "Least squares approximation." }{TEXT -1 149 " W hat is it? Be able to set up using calculus (and solve for small dat a sets) the problem finding the least squares fit of a curve to some \+ data." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 113 "8. Find the polynomial of degree \+ no more than 2 which has the best least squares fit to the data in pro blem 3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 " x = [1,3,5,6] and y= [0,1,2,-2], " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "restart:x := [1,3,5,6]: y := [0,1,2,-2]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f := unapply(a*t^2+b*t+c,t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"tG6\"6$%)operatorG%&arrowG F(,(*&%\"aG\"\"\"9$\"\"#F/*&%\"bGF/F0F/F/%\"cGF/F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Phi := sum((f(x[i])-y[i])^2,i=1..nops(x)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$PhiG,**$,(%\"aG\"\"\"%\"bGF)% \"cGF)\"\"#F)*$,*F(\"\"*F*\"\"$F+F)!\"\"F)F,F)*$,*F(\"#DF*\"\"&F+F)!\" #F)F,F)*$,*F(\"#OF*\"\"'F+F)F,F)F,F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "eqns:=\{diff(Phi,a),diff(Phi,b),diff(Phi,c)\};" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqnsG<%,*%\"aG\"%1S%\"bG\"$Q(%\"cG \"$U\"\"#E\"\"\",*F'F*F)F,F+\"#I!\"#F.,*F'F,F)F0F+\"\")F1F." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "mat:=linalg[genmatrix](eqns, [a,b,c],flag);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$matG-%'MATRIXG6#7 %7&\"%1S\"$Q(\"$U\"!#E7&F+F,\"#I\"\"#7&F,F/\"\")F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "linalg[rref](mat);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7&\"\"\"\"\"!F)#!$r\"\"$)R7&F)F(F)#\"%<6F ,7&F)F)F(#!$F&\"$*>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "sol: =solve(eqns,\{a,b,c\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG<%/% \"aG#!$r\"\"$)R/%\"cG#!$F&\"$*>/%\"bG#\"%<6F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f := unapply(subs(sol,f(t)),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"tG6\"6$%)operatorG%&arrowGF(,(*$9$\"\" ##!$r\"\"$)RF.#\"%<6F2#!$F&\"$*>\"\"\"F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 93 "plots[display]([plot([seq([x[i],y[i]],i=1..nops(x)) ],style=point,symbol=box),\nplot(f,1..6)]);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6&7&7$$\"\"\"\"\"!F*7$$\"\"$F*F(7$$\"\"&F*$\"\"# F*7$$\"\"'F*$!\"#F*-%'COLOURG6&%$RGBG$\"#5!\"\"F*F*-%&STYLEG6#%&POINTG -%'SYMBOLG6#%$BOXG-F$6$7S7$F($!1\")f>Ryc8F!#;7$$\"1LL$3x&)*36!#:$!1?!) *Qe1RU'!#<7$$\"1n;H2P\"Q?\"FQ$\"1\"GDUT!pw5FM7$$\"1LLeRwX58FQ$\"1uMY;P mjFM7$$\"1L$eR-/Pi\"FQ$\"1$4I:4i+w(FM7$$\"1+]il'pis\"FQ$\"1X?cxCOi\"*F M7$$\"1L$e*)>VB$=FQ$\"1e(f%oPv^5FQ7$$\"1+]7`l2Q>FQ$\"1=Dx(o9s<\"FQ7$$ \"1nm;/j$o/#FQ$\"1)e2rtViH\"FQ7$$\"1ML3_>jU@FQ$\"1Jc94am#R\"FQ7$$\"1++ ]i^Z]AFQ$\"1*o,,%yx\"\\\"FQ7$$\"1++](=h(eBFQ$\"1;i6X=C\"e\"FQ7$$\"1++] P[6jCFQ$\"1'*4Q9K#zl\"FQ7$$\"1M$e*[z(yb#FQ$\"1\"GIRC]%>FQ7$$\"1+](=xpe=$FQ$\"1,'*)=FQ7$$\"1nm\"H28IH$FQ$\"1e!f5!* HY$>FQ7$$\"1n;zpSS\"R$FQ$\"1Qu49O>G>FQ7$$\"1LL3_?`(\\$FQ$\"1D(\\QhD>\" >FQ7$$\"1M$e*)>pxg$FQ$\"1H>SFQ$\"1&ou$GaD\"p\"FQ7$$\"1+]i!RU07%FQ$\"1ItcXTD@;FQ7$$\"1++v= S2LUFQ$\"1%=`7q)=L:FQ7$$\"1mmm\"p)=MVFQ$\"1p!>!)QwZW\"FQ7$$\"1++](=]@W %FQ$\"1Xksw#z1M\"FQ7$$\"1L$e*[$z*RXFQ$\"1/q%H+-xB\"FQ7$$\"1,+]iC$pk%FQ $\"1A'zKW5d6\"FQ7$$\"1m;H2qcZZFQ$\"1YC&eI.&>**FM7$$\"1,]7.\"fF&[FQ$\"1 EOeY*GG`)FM7$$\"1nm;/Ogb\\FQ$\"1_cl`i<&3(FM7$$\"1+]ilAFj]FQ$\"1ssG\"RN AZ&FM7$$\"1MLL$)*pp;&FQ$\"1LunH/hCQFM7$$\"1ML3xe,t_FQ$\"1CB*[k6T/#FM7$ $\"1n;HdO=y`FQ$\"1nO.)p!>H=FT7$$\"1,++D>#[Z&FQ$!1\\Gm8W66;FM7$$\"1nmT& G!e&e&FQ$!1wl=bL(fw$FM7$$\"1MLL$)Qk%o&FQ$!1pf/`.h#y&FM7$$\"1+]iSjE!z&F Q$!1qthoNlD!)FM7$$\"1,]P40O\"*eFQ$!1=CtN>BE5FQ7$F4$!1Qxa4>Qw7FQF8-%%VI EWG6$;F(F4%(DEFAULTG" 2 434 434 434 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 -4878 -12228 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Crummy fit." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 " " }} }}{MARK "15 0 0" 84 }{VIEWOPTS 1 1 0 1 1 1803 }