{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 12 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 1 12 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "Geneva" 1 14 0 0 0 1 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 257 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 258 "Geneva" 1 12 0 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Geneva" 1 12 0 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Geneva" 1 10 0 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 265 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Geneva" 1 10 0 0 0 1 2 2 2 0 0 0 0 0 0 1 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "M aple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "R3 Fo nt 0" -1 256 1 {CSTYLE "" -1 -1 "Monaco" 1 9 0 0 255 1 2 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Monaco" 1 9 255 0 0 1 2 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 3" -1 258 1 {CSTYLE "" -1 -1 "Geneva" 1 14 0 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 4" -1 259 1 {CSTYLE "" -1 -1 "Geneva" 1 10 0 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 261 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 27 "\011\011\011\011\011Leas t Squares Parabola" }}}{EXCHG {PARA 260 "" 0 "" {TEXT 257 465 "Solving an over-determined linear system A x = b is generally impossible, bu t finding an x0 so that A x0 is as close as possible to b leads to the concept of a least squares solution. One can see that such an x0 mus t satisfy the normal equations\n\011\011\011\011\011A_tr A x0 = A_tr \+ b \nwhere A_tr is the transpose of the matrix A. Generically this sys tem will have a unique solution. We use this worksheet to study this \+ in the case of fitting a set of points to a parabola.\n\n" }{TEXT 258 7 "Problem" }{TEXT 263 2 ": " }{TEXT 259 106 "Find the best fitting pa rabola y = ax^2 +bx+c through the following points:\011(1,2),(1,3),(2 ,4),(3,5),(4,6)" }}{PARA 261 "" 0 "" {TEXT 260 2 "\n " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "pointlist := [[1,2],[1,3],[2,4],[3,5],[4,6]];" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*pointlistG7'7$\"\"\"\"\"#7$F'\"\"$ 7$F(\"\"%7$F*\"\"&7$F,\"\"'" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 261 10 "Solution: " }{TEXT 262 119 "The over-determined system here is\n\011 \0111a +1b+c=2\n\011\0111a +1b+c=3\n\011\0114a +2b+c=4\n\011\0119a +3b +c=5\n\011 16a+4b +c=6\nIn matrix form Ax=" }{TEXT 264 3 "B:\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with (linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and trace have been red efined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "A:= array([[1,1,1],[1,1,1],[4,2,1],[9,3,1],[16,4,1]]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7'7%\"\"\"F*F*F)7%\"\"%\"\"#F* 7%\"\"*\"\"$F*7%\"#;F,F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "x:= vector ([a, b, c]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG-%'v ectorG6#7%%\"aG%\"bG%\"cG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "B:= vector([2,3,4,5,6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG -%'vectorG6#7'\"\"#\"\"$\"\"%\"\"&\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalm( A &* x = B);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%'vectorG6#7',(%\"aG\"\"\"%\"bGF*%\"cGF*F(,(F)\"\"%*&\"\"#F*F+F*F* F,F*,(F)\"\"**&\"\"$F*F+F*F*F,F*,(F)\"#;*&F.F*F+F*F*F,F*-F%6#7'F0F4F. \"\"&\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 27 "The normal equatio ns Nx=m. " }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "N:= eva lm ( transpose(A) &* A );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG-%' matrixG6#7%7%\"$b$\"$,\"\"#J7%F+F,\"#67%F,F.\"\"&" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 31 "m:= evalm ( transpose(A) &* B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG-%'vectorG6#7%\"$i\"\"#_\"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "x0:= evalm (inverse(N) &* m);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G-%'vectorG6#7%#!\"&\"#R#\"#qF+# \"#6\"#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "parabola := x - > x0[1] * x^2 + x0[2] * x + x0[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%)parabolaGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(*&&%#x0G6#\"\"\"F1)9$ \"\"#F1F1*&&F/6#F4F1F3F1F1&F/6#\"\"$F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot (\{pointlist, parabola\});" }}{PARA 13 "" 1 "" {GLPLOT2D 269 199 199 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$!#5\"\"!$!3 )>p2Bp2B*H!#;7$$!3!pmmm\"p0k&*!#<$!3B&)H.7xr/GF-7$$!3uKL$3+o2Z#F-7$$!3/nmm\"4m(G$)F1$!3E*o# >,\"H'*H#F-7$$!3OLL$3i.9!zF1$!3Hz(e)=y*R8#F-7$$!3fmm;/R=0vF1$!3aF7?,+i %)>F-7$$!3k++]P8#\\4(F1$!360e()R)*=M=F-7$$!3Kmm;/siqmF1$!3A4&HNYaJo\"F -7$$!3Q****\\(y$pZiF1$!39$p/lJ(>P:F-7$$!3jKLL$yaE\"eF1$!3Ub4gDx%=R\"F- 7$$!3s%HaF1$!3S\"yV+KVyE\"F-7$$!3]******\\$*4)*\\F1$!3+TCEk&[F8 \"F-7$$!3o******\\_&\\c%F1$!3-h^&4()**=+\"F-7$$!3%)******\\1aZTF1$!37( QB*zka.))F17$$!3Imm;/#)[oPF1$!3izf&y*[]QxF17$$!3%HLLL=exJ$F1$!3%*Q(z/< <+_'F17$$!3lKLLL2$f$HF1$!3W?:(oy`&GbF17$$!3%)****\\PYx\"\\#F1$!3$p=1oA !GAWF17$$!3gLLLL7i)4#F1$!3+[p&*fYC&[$F17$$!3o)***\\P'psm\"F1$!3&Gr@K[k F]#F17$$!3?****\\74_c7F1$!3VzS\")*4c:h\"F17$$!3M:LL$3x%z#)!#=$!3'zj_?E /zF(Fer7$$!3()HLL3s$QM%Fer$\"3Z'o$=BK)*HU!#>7$$!3]^omm;zr)*!#?$\"3u]jX =vA%G)Fer7$$\"3fVLLezw5VFer$\"3QW!=zUdgf\"F17$$\"3-.++v$Q#\\\")Fer$\"3 !Q;N5_'pBAF17$$\"3%\\LL$e\"*[H7F1$\"3%zsJ#)zG\"fGF17$$\"3=++++dxd;F1$ \"3-eF1gFJpMF17$$\"3e+++D0xw?F1$\"3_E4yo]u?SF17$$\"35,+]i&p@[#F1$\"3LZ !p2%pV6XF17$$\"3++++vgHKHF1$\"3!G:GW[&*o+&F17$$\"3ElmmmZvOLF1$\"3>([cf jtxS&F17$$\"3%4+++v+'oPF1$\"3[7`6HZ\\*y&F17$$\"3UKL$eR<*fTF1$\"3)pE,Dp $4%4'F17$$\"3K-++])Hxe%F1$\"3Wd$zgUt@Q'F17$$\"3!fmm\"H!o-*\\F1$\"3![NU '=GV@^/oF17$$\"3#emmmT9C#eF1$\"33%HWh2A /&pF17$$\"33****\\i!*3`iF1$\"3-/ypqDocqF17$$\"3;NLLL*zym'F1$\"357eCBO1 9rF17$$\"3'eLL$3N1#4(F1$\"3%*4wL,&=r7(F17$$\"3,pm;HYt7vF1$\"3]YKE5/]%4 (F17$$\"37-+++xG**yF1$\"3b^WB'GBX-(F17$$\"3gpmmT6KU$)F1$\"35>$*z)y,s*o F17$$\"3qNLLLbdQ()F1$\"3CK;>`loSnF17$$\"3[++]i`1h\"*F1$\"3oNdh*3h%HlF1 7$$\"3A-+]P?Wl&*F1$\"3Wl`(Q*RU%G'F17$$\"#5F*$\"3Mv*eV(*eV(fF1-%'COLOUR G6&%$RGBG$Fiz!\"\"$F*F*Fb[l-F$6$7'7$$\"\"\"F*$\"\"#F*7$Fg[l$\"\"$F*7$F i[l$\"\"%F*7$F\\\\l$\"\"&F*7$F_\\l$\"\"'F*-F][l6&F_[lFb[lF`[lFb[l-%+AX ESLABELSG6$Q!6\"F\\]l-%%VIEWG6$;F(Fhz%(DEFAULTG" 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 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "12 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }