{VERSION 3 0 "SUN SPARC SOLARIS" "3.0" } {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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 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 "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 2 6 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 }0 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 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple O utput" 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 "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 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 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 215 "restart;read \"/hom e/mat331/www/Worksheets1/lsq_data.txt\";\npts:=line_pts():\nplot(pts,s tyle=point);\nepsilon:= (m,b, pt) -> (m*pt[1] + b - pt[2]);\n\nE :=(m ,b, pts) -> sum( ( epsilon(m, b, pts[i]))^2 , i=1..nops(pts));\n" }} {PARA 6 "" 1 "" {TEXT -1 82 "defined line_pts(), bad_line_pts(), quadr atic_pts(), cubic_pts(), and circle_pts()" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$777$$!1+++m$f0<%!#:$!1+++4A06`F*7$$!1+++y5rkVF *$!1+++?/=pRF*7$$\"1++++QXG)F*7$$\"1++++U\"*R\\F5$!1+++u#Q#)z( F*7$$\"1,++?;-%y)F*$!1+++(4/y^\"!#97$$\"1+++!p@'ycF*$!1+++5r7'=\"FL7$$ !1+++A@\"or#F*$!1+++D!f&*Q&F*7$$\"1+++gx[&y#F*$!1*****4E9d&)*F*7$$\"1+ ++]$3<_(F*$!1+++U(*fQ8FL7$$\"1*******3-![**F*$!1+++B`B!e\"FL7$$\"1+++S 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"6#<$/%\"mG$\"+f77BI!\"*/%\"bG$!+qs@7'*F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(plots);m;" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#7U%(animateG%*animate3dG%-animatecurveG%-changec oordsG%,complexplotG%.complexplot3dG%*conformalG%,contourplotG%.contou rplot3dG%*coordplotG%,coordplot3dG%-cylinderplotG%,densityplotG%(displ ayG%*display3dG%*fieldplotG%,fieldplot3dG%)gradplotG%+gradplot3dG%-imp licitplotG%/implicitplot3dG%(inequalG%-listcontplotG%/listcontplot3dG% 0listdensityplotG%)listplotG%+listplot3dG%+loglogplotG%(logplotG%+matr ixplotG%(odeplotG%'paretoG%*pointplotG%,pointplot3dG%*polarplotG%,poly gonplotG%.polygonplot3dG%4polyhedra_supportedG%.polyhedraplotG%'replot G%*rootlocusG%,semilogplotG%+setoptionsG%-setoptions3dG%+spacecurveG%1 sparsematrixplotG%+sphereplotG%)surfdataG%)textplotG%+textplot3dG%)tub eplotG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+f77BI!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "display(\{plot(pts,style=point),plo t(m*x+b,x=-8..8,color=blue)\});" 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{MPLTEXT 1 0 49 "yvalues:= [ seq( pts[i][2 ], i=1..nops(pts)) ];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(yvaluesG77 $\"+$)fHm]!\"*$\"+tQ<\"e\"!\")$!+M1KjLF+$!+bmbt:F+$\"+?u5l5F+$!+a2#[l$ F+$!+bfa;5F+$!+YK;t)*F($!+,4a,8F+$!+3!=dn\"F($\"+r!*yn7F+$!+o[%Rd#F+$! +JHt+HF+$!+E'R@I&F($\"+j\"e%R=F+$\"+:n)>x\"F+$\"*W!\\;p!#5$\"+#ppNi\"F +$!+$3u^Z\"F+$\"+$4asG'F($!+5Nc1CF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "fit[leastsquare[[x,y]]]([xvalues,yvalues]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,&$!+ms@7'*!\"*\"\"\"%\"xG$\"+c77BIF( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "\nm:='m':b:='b':Q :=(m, b, pts) -> sum( ( epsilon(m, b, pts[i]))^4 , i=1..nops(pts));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QGR6%%\"mG%\"bG%$ptsG6\"6$%)operat orG%&arrowGF*-%$sumG6$*$)-%(epsilonG6%9$9%&9&6#%\"iG\"\"%\"\"\"/F;;\" \"\"-%%nopsG6#F9F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "s implify(diff(Q(m,b,pts), m)) = 0;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/ ,6$!+KR->[!\"$\"\"\"*&%\"mGF()%\"bG\"\"#\"\"\"$\"+5I7z%)!\"'F,$!+]>\"= ?#!\"%F*$\"+i#)3%3%F'*$)F*F-F.$!+iMLN8F'*$)F,\"\"$F.$\"+(>YYT\"!\"(*$) F*F=F.$\"+TJ()Q " 0 "" {MPLTEXT 1 0 95 "solve( \+ \{ simplify(diff(Q(m,b,pts), m)) = 0,\n simplify(diff(Q(m,b,pts ), b)) = 0\}, \{m,b\});" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, compu tation interrupted" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "assig n(\{m = 2.022255731, b = -14.55234924\});" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 66 "plot( [pts,m*x+b],x=-10..10,color=[red,blue], style =[point,line]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 257 44 "Consider now polinomials of highe r degree..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "pts:=cubic_p ts():\nplot(pts,style=point); " }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURV ESG6$777$$\"1+++H^A!y\"!#:$\"1+++Hs>T&)F*7$$\"1+++8s4X:F*$\"1+++'p7lw& F*7$$\"1+++v\\tyDF*$\"1+++F!R^J'F*7$$\"1+++\\]>yGF*$\"1+++\\sZ.mF*7$$! 1+++q!Q\"=m!#;$\"1+++R/$)*>\"F?7$$!1++++9>Za!#<$!1+++sv%Gi%F?7$$\"1+++ zy0-:F*$\"1+++I6W3tF*7$$!1+++jsx]HF*$\"1+++([%*fO%!#97$$\"1+++>?:H@F*$ \"1+++I\\y`zF*7$$!1+++IsId`F?$\"1+++?e9FKFE7$$\"1+++YsUy:F*$\"1+++xnK^ kF*7$$\"1+++/lxJAF*$\"1+++,yrC!)F*7$$\"1+++!*)*ex:F?$\"1+++#Qw#3$*F?7$ $!1+++5Gu3!)F?$\"1+++&y%\\t6F*7$$!1+++I`muFF?$!1+++<$zI/&F?7$$!1+++)41 Dp#F*$\"1+++q[7'H$FR7$$!1+++]$oBz&F?$\"1+++]m/LZF?7$$\"1+++idc_7F*$\"1 +++()p\\)Q&F*7$$!1+++qYe+MF?$\"1+++$ywla$F?7$$!1+++6mIb9F*$\"1+++o(H9K 'F*7$$\"1+++xgqwCF*$\"1+++#z&H\"p'F*-%'COLOURG6&%$RGBG$\"#5!\"\"\"\"!F er-%&STYLEG6#%&POINTG-%+AXESLABELSG6$%!GF]s-%%VIEWG6$%(DEFAULTGFas" 2 426 426 426 5 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "n:=2;a:='a';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f:=(a,x)->sum(a[j]*x^j,j=0..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6$%\"aG%\"xG6\"6$%)operatorG%&arrowGF)-%$sumG6$* &&9$6#%\"jG\"\"\")9%F4F5/F4;\"\"!%\"nGF)F)F)" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "epsilon:= (a,pt) -> (f(a,pt[1])- p t[2]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(epsilonGR6$%\"aG%#ptG6 \"6$%)operatorG%&arrowGF),&-%\"fG6$9$&9%6#\"\"\"F5&F36#\"\"#!\"\"F)F)F )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "E :=(a, pts) -> sum( ( epsilon(a,pts[i]))^2 ,i=1..nops(pts));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"EGR6$%\"aG%$ptsG6\"6$%)operatorG%&arrowGF)-%$sumG6$ *$)-%(epsilonG6$9$&9%6#%\"iG\"\"#\"\"\"/F9;\"\"\"-%%nopsG6#F7F)F)F)" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "solve( \{ diff(E(a,pts),a [0]) = 0,\n diff(E(a,pts), a[1]) = 0, diff(E(a,pts),a[2])=0 \}, \{a[0],a[1],a[2]\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/&%\"aG6 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0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 117 "N ow let's consider interpolation of points that (up to a normally distr ibuted error) are expected to lie on a circle:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 63 "data:=circle_pts();\nplot(data,style=point,sca ling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "distance from one point to a circl e with center [a,b] radius r " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "alpha:= (a,b,r,pt) -> (pt[1] - a)^2 + (pt[2] -b)^2 -r^2;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "C := (a,b,r, data) ->sum( ( alpha(a,b,r, data[i]))^2,i=1..nops(data)); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "a:='a':b:='b':\nC(a,b,r,data);\nsimplify(diff(C(a, b,r,data),b) =0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "solve( \{diff(C(a,b,r,data),a) =0, diff(C(a,b,r,data),b) =0, diff(C(a,b,r,dat a),r) =0\}, \{a,b,r\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " r:='r'; a:='a'; b:='b';\nassign(\{r = , a = , b = \});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "cplot:=plot( [ a + r*cos(t),b+r*si n(t),t=0..2*Pi], scaling=constrained):\npplot:=plot(data,style=point,c olor=black):\nplots[display]([cplot,pplot]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "plot( [data,[ a + r*cos(t),b+r*sin(t),t=0..2*Pi] ], scaling=constrained,style=[point,line]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 40 "Now we discuss the robust approximation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "data:=bad_l ine_pts():" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot(data,sty le=point,color=black);" }}{PARA 13 "" 1 "" {INLPLOT "6'-%'CURVESG6#7T7 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{MPLTEXT 1 0 8 "?fscanf;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "fd:=fopen(`alpha.dat`,READ);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fdG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "fscanf(fd,`%e`);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#$\"(p99$!\"' " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "fclose(fd);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 195 "file2list:=proc(name)\n local l,f d;\n fd:=fopen(name,READ);\n l:=[];\n while not feof(fd) do\n l: =[op(l),op(fscanf(fd,`%e`))];\n od;\n fclose(fd);\nl:=[seq(l[i],i=1. .nops(l)-1)];\nRETURN(l);\nend;" }}{PARA 7 "" 1 "" {TEXT -1 42 "Warnin g, `i` in call to `seq` is not local" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#>%*file2listGR6#%%nameG6$%\"lG%#fdG6\"F+C(>8%-%&fopenG6$9$%%READG>8$ 7\"?(F+\"\"\"F8F+4-%%feofG6#F.>F57$-%#opG6#F5-F@6#-%'fscanfG6$F.%#%eG- %'fcloseGF<>F57#-%$seqG6$&F56#%\"iG/FQ;F8,&-%%nopsGFAF8!\"\"F8-%'RETUR NGFAF+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "file2list(`al pha.dat`);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&$\"#7!\"\"$\"$c#!\"#$\"(p99$!\"'$!\"%F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "71 0 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 }