{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 260 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 261 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 262 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 269 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 } {CSTYLE "Grey Emphasis" -1 270 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT 259 69 "The Remez algorithm for construc ting minimax rational approximations:" }}{PARA 3 "" 0 "" {TEXT 263 58 "Version I - minimax error obtained by an iterative method " }{TEXT 264 1 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C. , Canada" }}{PARA 0 "" 0 "" {TEXT -1 20 "Version: 25.3.2007\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 56 "lo ad interpolation and function approximation procedures" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 270 10 "fcnapprx.m" } {TEXT -1 37 " contains the code for the procedure " }{TEXT 0 7 "critpt s" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 123 "It can be read into a Maple session by a command similar to th e one that follows, where the file path gives its location. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "read \"K:\\\\Maple/procdrs/f cnapprx.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 56 "The calculation of a minimax rational approximation for " }{XPPEDIT 18 0 "ln(1+x);" "6#-%#lnG6#,&\"\"\"F'%\"xGF'" }{TEXT -1 4 " on " }{XPPEDIT 18 0 "[0, 1];" "6#7$\"\"!\"\"\"" }{TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 57 "We consider the problem of finding the rational function " } {XPPEDIT 18 0 "r(x)=p(x)/q(x)" "6#/-%\"rG6#%\"xG*&-%\"pG6#F'\"\"\"-%\" qG6#F'!\"\"" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#% \"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" } {TEXT -1 17 " each have degree" }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 21 ", which approximates " }{XPPEDIT 18 0 "f(x) = ln(1+x);" "6#/-%\"fG6#%\"xG-%#lnG6#,&\"\"\"F,F'F," }{TEXT -1 45 " with minimax a bsolute error in the interval " }{XPPEDIT 18 0 "[0, 1];" "6#7$\"\"!\" \"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 14 "Suppose that " } {XPPEDIT 18 0 "r(x) = (a[0]+a[1]*x)/(b[0]+b[1]*x);" "6#/-%\"rG6#%\"xG* &,&&%\"aG6#\"\"!\"\"\"*&&F+6#F.F.F'F.F.F.,&&%\"bG6#F-F.*&&F46#F.F.F'F. F.!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "First we select 4 numbers, " }{XPPEDIT 18 0 "x[1],x[ 2],x[3],x[4]" "6&&%\"xG6#\"\"\"&F$6#\"\"#&F$6#\"\"$&F$6#\"\"%" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "0 = x[1];" "6#/\"\"!&%\"xG6#\"\"\"" } {XPPEDIT 18 0 "`` " 0 "" {MPLTEXT 1 0 442 "a := table(): b := table():\nunassign('mu');\nf := x ->ln(x+1);\nr := x -> (a[0]+a[1]*x)/(1+b[1]*x);\nc := [0.,.2,.7,1.];\n mu1 := 0;\neps := Float(1,1-Digits);\nfor i to 25 do\n eqns := seq(a [0]+a[1]*c[i]=(f(c[i])-(-1)^i*mu)+\n (f(c[i])-(-1)^i*mu1)*b[1]*c[ i],i=1..nops(c));\n soln := solve(\{eqns\});\n newmu := subs(soln, mu);\n if abs(newmu-mu1)%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%# lnG6#,&9$\"\"\"F1F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6 #%\"xG6\"6$%)operatorG%&arrowGF(*&,&&%\"aG6#\"\"!\"\"\"*&&F/6#F2F29$F2 F2F2,&F2F2*&&%\"bGF5F2F6F2F2!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG7&$\"\"!F'$\"\"#!\"\"$\"\"(F*$\"\"\"F'" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$mu1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ep sG$\"\"\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&/%#muG$\"+&yB)y&)!# 8/&%\"aG6#\"\"!F&/&F+6#\"\"\"$\"+^.'>y*!#5/&%\"bGF0$\"+ocEUTF4" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r1Gf*6#%\"xG6\"6$%)operatorG%&arrow GF(*&,&$\"+&yB)y&)!#8\"\"\"*&$\"+^.'>y*!#5F19$F1F1F1,&F1F1*&$\"+ocEUTF 5F1F6F1F1!\"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot(r1(x)-f(x),x=0..1,color=COLOR(RGB,.7 ,0,.9));" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'C URVESG6#7fo7$$\"\"!F)$\"3I+++&yB)y&)!#@7$$\"3WmmmT&)G\\a!#?$\"37Mj-KzA *R(F,7$$\"3ILLL3x&)*3\"!#>$\"3uFtlV&QWF'F,7$$\"3$*****\\ilyM;F6$\"3&3g ]mP/H?&F,7$$\"3emmm;arz@F6$\"3)HCIOZ5J=%F,7$$\"3.++D\"y%*z7$F6$\"3N$[N vMdu_#F,7$$\"3[LL$e9ui2%F6$\"3-x>%ewtk,\"F,7$$\"33++voMrU^F6$!3e+J?I1g *=&!#A7$$\"3nmmm\"z_\"4iF6$!3/:%F,7$$\"36++v=ddC%*F6$!3+(*HS0,mE^F,7 $$\"3CLLe*=)H\\5!#=$!3qu(Hy([5KfF,7$$\"3')***\\(=JN[6Fdo$!3S,y8QB]qlF, 7$$\"3gmm\"z/3uC\"Fdo$!3x%Hu3$y$36(F,7$$\"3%)***\\7LRDX\"Fdo$!3z0+eoLo UzF,7$$\"3/LLekGhe:Fdo$!3$Q#*pjnDNB)F,7$$\"3]mm\"zR'ok;Fdo$!3*R4af)[nP %)F,7$$\"3#)*****\\2`vr\"Fdo$!3W^=$\\%y\"*3&)F,7$$\"39LL3_(>/x\"Fdo$!3 mF]^*4G3c)F,7$$\"3Xmm;HkGB=Fdo$!3e5**>A[5%f)F,7$$\"3w***\\i5`h(=Fdo$!3 EH:@7>V4')F,7$$\"3KL3x\")H`I>Fdo$!32j0.yX<2')F,7$$\"3gm;HdG\"\\)>Fdo$! 3qCCge]M(e)F,7$$\"3!**\\7Gt#HR?Fdo$!3+#3T=MR1b)F,7$$\"3WLLL3En$4#Fdo$! 3e\"Q\\)3yt(\\)F,7$$\"33++Dc#o%*=#Fdo$!3Eu!z)>8>)F,7$$\"3\")*****\\K]4]#Fdo$!33Gz*>zj6l(F,7$$\"3$***** *\\PAvr#Fdo$!3\")p'[8[fQ$pF,7$$\"3)******\\nHi#HFdo$!3tJ:M;nl3hF,7$$\" 3jmm\"z*ev:JFdo$!3uJR#)Qe#)o_F,7$$\"3?LLL347TLFdo$!3')3VLq@3&=%F,7$$\" 3,LLLLY.KNFdo$!3[q)HKhA^@$F,7$$\"3w***\\7o7Tv$Fdo$!3A'z^>0W(\\?F,7$$\" 3'GLLLQ*o]RFdo$!3hS*ein\"z-5F,7$$\"3A++D\"=lj;%Fdo$\"3*y>A'f3'4W\"FR7$ $\"31++vV&RY2aFdo$\"38L*3@z% )[!fF,7$$\"39mm;zXu9cFdo$\"3Es?XVQ5=mF,7$$\"3l******\\y))GeFdo$\"3S\"R (o#pBND(F,7$$\"3'*)***\\i_QQgFdo$\"3md;4E!=vw(F,7$$\"3@***\\7y%3TiFdo$ \"3KbYEI\\\"o:)F,7$$\"35****\\P![hY'Fdo$\"3s/!*f7_ad%)F,7$$\"3)em;/ris c'Fdo$\"33KMU&=vda)F,7$$\"3kKLL$Qx$omFdo$\"3]@g4%>)4/')F,7$$\"3mlm;z)Q jx'Fdo$\"3\"y0n)=%HFj)F,7$$\"3!)*****\\P+V)oFdo$\"3*)*e\"31=*fi)F,7$$ \"3WK$ek`H@)pFdo$\"3cU`Kpd$))e)F,7$$\"3?mm\"zpe*zqFdo$\"31d\"p#Q\"><_) F,7$$\"3%)*****\\#\\'QH(Fdo$\"3KIhrE!z'o#)F,7$$\"3GKLe9S8&\\(Fdo$\"3Go 5`(3^Y*yF,7$$\"3R***\\i?=bq(Fdo$\"3%yJ!o9WpftF,7$$\"3\"HLL$3s?6zFdo$\" 3?Wavogu\"p'F,7$$\"3a***\\7`Wl7)Fdo$\"3FdJ\\!Q^k$eF,7$$\"3#pmmm'*RRL)F do$\"33v$R7p)yf[F,7$$\"3Qmm;a<.Y&)Fdo$\"3\"G;njNoSq$F,7$$\"3=LLe9tOc() Fdo$\"3EJ-$QPM(*R#F,7$$\"3Ym;H#e0I&))Fdo$\"3a\"=9%*)QJZ`ifP0c9F,7$$\"3 ImmmmxGp$*Fdo$!3^^q8r_G4BF,7$$\"3sK$eRA5\\Z*Fdo$!3c@hg^bGeKF,7$$\"3A++ D\"oK0e*Fdo$!37\")fc]E'yC%F,7$$\"3C+++]oi\"o*Fdo$!3?T\\\\>i/L_F,7$$\"3 A++v=5s#y*Fdo$!3MB.i'yWaD'F,7$$\"35+]P40O\"*)*Fdo$!396Wx%oVcR(F,7$$\" \"\"F)$!3IM/eEK#)y&)F,-%&COLORG6&%$RGBG$\"\"(!\"\"F)$\"\"*F^cl-%+AXESL ABELSG6$Q\"x6\"Q!Fecl-%%VIEWG6$;F(Fdbl%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "The actual critical point s are not very different from those given in the initial guess." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "e := x -> r1(x)-f(x);\nd := unapply(diff(e(x),x),x);\nsolve(d(x),x );\nc := sort([0.,%,1.]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"eGf*6 #%\"xG6\"6$%)operatorG%&arrowGF(,&-%#r1G6#9$\"\"\"-%\"fGF/!\"\"F(F(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dGf*6#%\"xG6\"6$%)operatorG%&a rrowGF(,(*&$\"+^.'>y*!#5\"\"\",&F1F1*&$\"+ocEUTF0F19$F1F1!\"\"F1*($\"+ ocEUTF0F1,&$\"+&yB)y&)!#8F1*&F.F1F6F1F1F1F2!\"#F7*&F1F1,&F6F1F1F1F7F7F (F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+2BU'*=!#5$\"+F^(*4oF%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG7&$\"\"!F'$\"+2BU'*=!#5$\"+F^(*4 oF*$\"\"\"F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "We can repeat the process using these new critical points ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "mu1 := 0;\neps := Float(1,1-Digits);\nfor i to 25 do \n eqns := seq(a[0]+a[1]*c[i]=(f(c[i])-(-1)^i*mu)+\n (f(c[i])-( -1)^i*mu1)*b[1]*c[i],i=1..nops(c));\n soln := solve(\{eqns\});\n n ewmu := subs(soln,mu);\n if abs(newmu-mu1)%$mu1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$epsG$\"\"\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&/&%\"aG6#\"\"!$\"+#RQzg)!#8/&F&6#\"\"\"$\"+7+!=y*!#5/&%\"bGF.$ \"+#fN@9%F2/%#muGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r2Gf*6#%\"xG 6\"6$%)operatorG%&arrowGF(*&,&$\"+#RQzg)!#8\"\"\"*&$\"+7+!=y*!#5F19$F1 F1F1,&F1F1*&$\"+#fN@9%F5F1F6F1F1!\"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "The critical points can be comp uted as before." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 88 "e := x -> r2(x)-f(x);\nd := unapply(diff(e(x), x),x):\nsolve(d(x),x):\nc := sort([0.,%,1.]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"eGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%#r2G6#9$\" \"\"-%\"fGF/!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG7&$\" \"!F'$\"+8EA)*=!#5$\"+gPB4oF*$\"\"\"F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "A third iteration produces little ch ange within the accuracy possible using 10 digit arithmetic." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "m u1 := 0;\neps := Float(1,1-Digits);\nfor i to 25 do\n eqns := seq(a[ 0]+a[1]*c[i]=(f(c[i])-(-1)^i*mu)+\n (f(c[i])-(-1)^i*mu1)*b[1]*c[i ],i=1..nops(c));\n soln := solve(\{eqns\});\n newmu := subs(soln,m u);\n if abs(newmu-mu1)%$mu1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%$epsG$\"\"\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&/%#muG$\"+)ySz g)!#8/&%\"aG6#\"\"\"$\"+=+!=y*!#5/&%\"bGF,$\"+5c8UTF0/&F+6#\"\"!F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r3Gf*6#%\"xG6\"6$%)operatorG%&arrow GF(*&,&$\"+)ySzg)!#8\"\"\"*&$\"+=+!=y*!#5F19$F1F1F1,&F1F1*&$\"+5c8UTF5 F1F6F1F1!\"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "e := x -> r3(x)-f(x);\nd := unapply(diff( e(x),x),x):\nsolve(d(x),x):\nc := sort([0.,%,1.]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"eGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%#r3G6#9$ \"\"\"-%\"fGF/!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG7&$ \"\"!F'$\"+IEA)*=!#5$\"+([L#4oF*$\"\"\"F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 69 "A utility routine for determining the cri tical points of a function: " }{TEXT 0 7 "critpts" }{TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 14 "critpts: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 265 18 "Calling Sequence:\n" }} {PARA 0 "" 0 "" {TEXT 266 2 " " }{TEXT -1 21 " critpts( f, rng N) " }{TEXT 267 1 "\n" }{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 11 "Para meters:" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT 23 9 " f - " }{TEXT -1 55 " an expression involving a single \+ variable, say x," }}{PARA 0 "" 0 "" {TEXT -1 80 " \+ which evaluates to a real floating point number." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 12 " rng - " }{TEXT 268 53 "the range x=a..b for the function to be a pproximated." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 12 " N - " }{TEXT 269 65 "the number of subin tervals used in the search for critical points" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 7 "critpts" }{TEXT -1 333 " attempts to find all the critical points of f on the interval [a ,b] by dividing the interval into N subintervals, and searching throug h these intervals to find those in which the derivative changes sign. \+ For any such subintervals the associated critical point is located acc urately by solving the equation f ' = 0 numerically using " }{TEXT 0 6 "fsolve" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 16 "How to activate: " }{TEXT -1 156 "\nTo make the procedures active open the subsection, \+ place the cursor anywhere after the prompt [ > and press [Enter].\nYo u can then close up the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "critpts: implementation" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 896 "critpts := proc(fx::algebra ic,rng::name=realcons..realcons,N::posint)\n local x,a,b,h,xx,newx,i ,cpts,d,val,newval,c;\n x := op(1,rng);\n a := evalf(op(1,op(2,rng )));\n b := evalf(op(2,op(2,rng)));\n h := evalf((b-a)/N);\n xx \+ := a;\n cpts := xx;\n d := unapply(diff(fx,x),x);\n val := d(xx) ;\n if not type(val,numeric) then\n error \"non-numeric value\" \n end if; \n for i to N do\n newx := xx + h;\n newval := d(newx);\n if not type(newval,numeric) then\n error \+ \"non-numeric value\"\n end if;\n if signum(0,val,0)<>signum (0,newval,0) then\n c := traperror(fsolve(d(x),x=xx..newx));\n if c=lasterror or not type(c,numeric) then\n error \"failed to locate critical point in the interval %1\",x..newx;\n \+ end if;\n cpts := cpts,c;\n end if;\n xx := new x;\n val := newval;\n end do:\n [cpts,evalf(b)]\nend proc:" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 42 "Examples are given later in t he worksheet." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 56 "The calculation of a minimax rati onal approximation for " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" } {TEXT -1 4 " on " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" } {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 57 "We consider the problem of finding the rational function " }{XPPEDIT 18 0 "r(x)=p(x)/q(x)" "6#/-%\"rG6#%\"xG*&-%\"pG6 #F'\"\"\"-%\"qG6#F'!\"\"" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "p(x) " "6#-%\"pG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "q(x)" "6#-%\"q G6#%\"xG" }{TEXT -1 17 " each have degree" }{XPPEDIT 18 0 "`` <= 3;" " 6#1%!G\"\"$" }{TEXT -1 20 ", that approximates " }{XPPEDIT 18 0 "f(x) \+ = exp(x);" "6#/-%\"fG6#%\"xG-%$expG6#F'" }{TEXT -1 45 " with minimax a bsolute error in the interval " }{XPPEDIT 18 0 "[-1, 1];" "6#7$,$\"\" \"!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }{XPPEDIT 18 0 "p(x) = a[0]+a[1]*x+a[2]*x^2+` . . . `+a[m]*x^m;" "6 #/-%\"pG6#%\"xG,,&%\"aG6#\"\"!\"\"\"*&&F*6#F-F-F'F-F-*&&F*6#\"\"#F-*$F 'F4F-F-%(~.~.~.~GF-*&&F*6#%\"mGF-)F'F:F-F-" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "q(x) = b[0]+b[1]*x+b[2]*x^2+` . . . `+b[n]*x^n" "6#/-% \"qG6#%\"xG,,&%\"bG6#\"\"!\"\"\"*&&F*6#F-F-F'F-F-*&&F*6#\"\"#F-*$F'F4F -F-%(~.~.~.~GF-*&&F*6#%\"nGF-)F'F:F-F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 12 "We may take " }{XPPEDIT 18 0 "b[0]=1" "6#/&%\"bG6#\" \"!\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 16 "First we select " }{XPPEDIT 18 0 "m+n+2" "6#,(%\"m G\"\"\"%\"nGF%\"\"#F%" }{TEXT -1 10 " numbers, " }{XPPEDIT 18 0 "x[1], x[2],x[3],` . . . `,x[m+n+2];" "6'&%\"xG6#\"\"\"&F$6#\"\"#&F$6#\"\"$%( ~.~.~.~G&F$6#,(%\"mGF&%\"nGF&F)F&" }{TEXT -1 8 " with " }{XPPEDIT 18 0 "-1 = x[1];" "6#/,$\"\"\"!\"\"&%\"xG6#F%" }{XPPEDIT 18 0 "`` " 0 "" {MPLTEXT 1 0 194 "m := 3;\n n := 3;\nmn := m+n:\na := table(): b := table():\nr_numer := convert([ seq(a[i]*x^i,i=0..m)],`+`):\nb[0] := 1;\nr_denom := convert([seq(b[i]* x^i,i=0..n)],`+`):\nr := unapply(r_numer/r_denom,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"n G\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"\"!\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"xG6\"6$%)operatorG%&arrow GF(*&,*&%\"aG6#\"\"!\"\"\"*&&F/6#F2F29$F2F2*&&F/6#\"\"#F2)F6F:F2F2*&&F /6#\"\"$F2)F6F?F2F2F2,*F2F2*&&%\"bGF5F2F6F2F2*&&FDF9F2F;F2F2*&&FDF>F2F @F2F2!\"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Set up the critical point s, and the system of the first " }{XPPEDIT 18 0 "n+m+1" "6#,(%\"nG\"\" \"%\"mGF%F%F%" }{TEXT -1 11 " equations " }{XPPEDIT 18 0 "r(x[k])-f(x[ k]) = (-1)^(k-1)*mu" "6#/,&-%\"rG6#&%\"xG6#%\"kG\"\"\"-%\"fG6#&F)6#F+! \"\"*&),$F,F2,&F+F,F,F2F,%#muGF," }{TEXT -1 5 " for " }{XPPEDIT 18 0 " k=1,` . . . `,n+m+1" "6%/%\"kG\"\"\"%(~.~.~.~G,(%\"nGF%%\"mGF%F%F%" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 503 "Digits := 10:\nunassign('mu');\nf := exp;\nr_de nom1 := convert([seq(b[i]*x^i,i=1..n)],`+`):\ncrit := [seq(evalf(cos(P i*(mn+2-i)/(mn+1))),i=1..mn+2)];\nmu1 := 0;\neps := Float(1,1-Digits); \nfor j to 25 do\n eqns := seq(subs(x=crit[i],r_numer)=(f(crit[i])-( -1)^i*mu)+\n (f(crit[i])-(-1)^i*mu1)*subs(x=crit[i],r_denom1),i=1 ..nops(crit));\n soln := solve(\{eqns\});\n newmu := subs(soln,mu) ;\n if abs(newmu-mu1)%\"fG%$expG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% %critG7*$!\"\"\"\"!$!+y')o4!*!#5$!+=!)*[B'F+$!+N$4_A#F+$\"+N$4_A#F+$\" +=!)*[B'F+$\"+y')o4!*F+$\"\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% $mu1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$epsG$\"\"\"!\"*" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#<*/&%\"aG6#\"\"!$\"+I,++5!\"*/&F&6#\" \"\"$\"+#4))>.&!#5/&%\"bG6#\"\"$$!+n$yE(z!#7/&F5F.$!+\"Q.!o\\F2/%#muG$ !+)Hl#H7!#;/&F&6#\"\"#$\"+D)*)Q,\"F2/&F&F6$\"+PSd#\\)F:/&F5FF$\"+7T@>) *!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r1Gf*6#%\"xG6\"6$%)operator G%&arrowGF(*&,*$\"+I,++5!\"*\"\"\"*&$\"+#4))>.&!#5F19$F1F1*&$\"+D)*)Q, \"F5F1)F6\"\"#F1F1*&$\"+PSd#\\)!#7F1)F6\"\"$F1F1F1,*F1F1*&$\"+\"Q.!o\\ F5F1F6F1!\"\"*&$\"+7T@>)*!#6F1F:F1F1*&$\"+n$yE(zF?F1F@F1FFFFF(F(F(" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "r[1](x)" "6#-&%\"rG6# \"\"\"6#%\"xG" }{TEXT -1 45 " is the resulting rational approximation \+ for " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 61 ", we can calculate the critical points of the error function " } {XPPEDIT 18 0 "r[1](x)-exp(x)" "6#,&-&%\"rG6#\"\"\"6#%\"xGF(-%$expG6#F *!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "e := x -> r1(x)-exp(x);\nnewcrit := critpts(r1(x)-exp(x),x=-1..1,50);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"eGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%#r1G6#9$\"\"\"-%$expGF/! \"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(newcritG7*$!\"\"\"\"! $!+\"p2Zy)!#5$!+i&=\"*\\&F+$!+0z-c5F+$\"+b,B`MF+$\"+:)>L4(F+$\"+aU^-$* F+$\"\"\"F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 150 "The following picture compares the location of the original gu esses for the critical points and the actual critical points of the cu rrent error curve." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 283 "ords1 := seq([[crit[i],0],[crit[i],e(crit[i ])]],i=1..mn+2):\nords2 := seq([[newcrit[i],0],[newcrit[i],e(newcrit[i ])]],i=1..mn+2):\nplot([r1(x)-f(x),ords1,ords2],x=-1..1,\n color=[COL OR(RGB,.7,0,.9),COLOR(RGB,0,.8,0)$(mn+2),COLOR(RGB,1,.4,0)$(mn+2)],\n \+ thickness=[1$(mn+3),2$(mn+2)]);" 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f/\"F-7$$\"3K+]7G:3u'*F1$!3+*e5y6?Xf)F47$$\"3G+vVt7SG(*F1$!3_#)e,aPdHj F47$$\"3A++v=5s#y*F1$!33jqHA-KJOF47$$\"3;+D1k2/P)*F1$!3gnm%RfQsk%FI7$$ \"35+]P40O\"*)*F1$\"3fa>Rr#[q?$F47$$\"3k]7.#Q?&=**F1$\"3\"Q9]s\\$RW_F4 7$$\"31+voa-oX**F1$\"3nyu&)[LpAuF47$$\"3[\\PMF,%G(**F1$\"3!G%)>.n(3Z(* F47$$\"\"\"F*$\"3F-Fa_nFe_n-F$6%7$7$$\"3%)*****\\\")>L4(F 1F\\[n7$F`bn$\"3/++++++]=F-Fa_nFe_n-F$6%7$7$$\"3a+++aU^-$*F1F\\[n7$Fib n$!3+++++++!\\\"F-Fa_nFe_n-F$6%F\\_nFa_nFe_n-%+AXESLABELSG6$Q\"x6\"Q!F dcn-%%VIEWG6$;F(Fiim%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "We can re peat the process using these new critical points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 412 "Digits := 2 0;\nfor i to mn+2 do crit[i] := newcrit[i] end do:\nmu1 := 0;\neps := \+ Float(1,1-Digits);\nfor i to 25 do\n eqns := seq(subs(x=crit[i],r_nu mer)=(f(crit[i])-(-1)^i*mu)+\n (f(crit[i])-(-1)^i*mu1)*subs(x=cri t[i],r_denom1),i=1..nops(crit));\n soln := solve(\{eqns\});\n newm u := subs(soln,mu);\n if abs(newmu-mu1)%'DigitsG\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$mu1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$epsG $\"\"\"!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r2Gf*6#%\"xG6\"6$%)op eratorG%&arrowGF(*&,*$\"5?VR$\\78+++\"!#>\"\"\"*&$\"5(fd7&f:m7O]!#?F19 $F1F1*&$\"5nCCt*p5\\g,\"F5F1)F6\"\"#F1F1*&$\"5ky[s:9kZI&)!#AF1)F6\"\"$ F1F1F1,*F1F1*&$\"56f]3)R)o'Q'\\F5F1F6F1!\"\"*&$\"5P>kN5#*)\\%*z*!#@F1F :F1F1*&$\"52#>[*4Kb1VzF?F1F@F1FFFFF(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "As before, we can compute the new \+ critical points and plot the error graph." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 346 "e := x -> r2(x)-exp( x);\nnewcrit := critpts(r2(x)-exp(x),x=-1..1,50);\nords1 := seq([[crit [i],0],[crit[i],e(crit[i])]],i=1..mn+2):\nords2 := seq([[newcrit[i],0] ,[newcrit[i],e(newcrit[i])]],i=1..mn+2):\nplot([e(x),ords1,ords2],x=-1 ..1,\n color=[COLOR(RGB,.7,0,.9),COLOR(RGB,0,.8,0)$(mn+2),COLOR(RGB,1 ,.4,0)$(mn+2)],\n thickness=[1$(mn+3),2$(mn+2)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"eGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%#r2G6#9 $\"\"\"-%$expGF/!\"\"F(F(F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(newc ritG7*$!\"\"\"\"!$!5O'3k)eqhZ)p)!#?$!5$)=dC*)HP\"3F&F+$!5ysrFo?<+`#)!# @$\"5IypRIU[u?NF+$\"5'e#*zT\"[/K8qF+$\"5n'H\"))>d.9S#*F+$\"\"\"F(" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "65-%'CURVESG6%7gt7 $$!\"\"\"\"!$!/uIR:))R:!#?7$$!5MLLLe9r]X**F-$!/t)*G\\WT7F-7$$!5nmmm;HU ,\"*)*F-$!.?Fa!y>'*F-7$$!5,+++vV8_O)*F-$!.o/)4^3qF-7$$!5MLLLLe%G?y*F-$ !.]uW]Wd%F-7$$!5nmmm\"HdNvs*F-$!.Q)Hj^6BF-7$$!5,+++](oUIn*F-$!-B92LP@F -7$$!5MLLL3-)\\&='*F-$\".q5KoZs\"F-7$$!5nmmmm;p0k&*F-$\".1Ky](4NF-7$$! 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urve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 " A third iteration should give the visual appearance of the minimax rat ional approximation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 398 "for i to mn+2 do crit[i] := newcrit[i] e nd do:\nmu1 := 0;\neps := Float(1,1-Digits);\nfor i to 25 do\n eqns \+ := seq(subs(x=crit[i],r_numer)=(f(crit[i])-(-1)^i*mu)+\n (f(crit[ i])-(-1)^i*mu1)*subs(x=crit[i],r_denom1),i=1..nops(crit));\n soln := solve(\{eqns\});\n newmu := subs(soln,mu);\n if abs(newmu-mu1)%$mu1G\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$epsG$\"\"\"!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r3Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,*$\"5& GZo<+8+++\"!#>\"\"\"*&$\"5(QDeiI6mg.&!#?F19$F1F1*&$\"5X_b&p)y%>g,\"F5F 1)F6\"\"#F1F1*&$\"5ZD`&)=7D-I&)!#AF1)F6\"\"$F1F1F1,*F1F1*&$\"5>D6,,ft# R'\\F5F1F6F1!\"\"*&$\"5]$[En%3rv*z*!#@F1F:F1F1*&$\"5Umay0%etN%zF?F1F@F 1FFFFF(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "The error graph shows that there is little change in the locati on of the critical points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 346 "e := x -> r3(x)-exp(x);\nnewcrit : = critpts(r2(x)-exp(x),x=-1..1,50);\nords1 := seq([[crit[i],0],[crit[i ],e(crit[i])]],i=1..mn+2):\nords2 := seq([[newcrit[i],0],[newcrit[i],e (newcrit[i])]],i=1..mn+2):\nplot([e(x),ords1,ords2],x=-1..1,\n color= [COLOR(RGB,.7,0,.9),COLOR(RGB,0,.8,0)$(mn+2),COLOR(RGB,1,.4,0)$(mn+2)] ,\n thickness=[1$(mn+3),2$(mn+2)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 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max(op(ec));\nemax/emin:\nevalf(%,1 5);\nDigits := 10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eminG$\"/w7&G i1b\"!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%emaxG$\".H^Gi1b\"!#>" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0!4+++++5!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf(rnew(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,*$\"+I,++5!\"*\"\"\"*&$\"+Ug1O]!#5F(%\"xGF(F (*&$\"+W%>g,\"F,F()F-\"\"#F(F(*&$\"+e>-I&)!#7F()F-\"\"$F(F(F(,*$F(\"\" !F(*&$\"+Iu#R'\\F,F(F-F(!\"\"*&$\"+nuv*z*!#6F(F1F(F(*&$\"+xTdVzF6F(F7F (F?F?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "r(x)=(1.000000130+.5036066042*x+.1016019444*x^2 +.8530021958e-2*x^3)/(1.-.4963927430*x+.9799757467e-1*x^2-.7943574177e -2*x^3)" "6#/-%\"rG6#%\"xG*&,*-%&FloatG6$\"+I,++5!\"*\"\"\"*&-F+6$\"+U g1O]!#5F/F'F/F/*&-F+6$\"+W%>g,\"F4F/*$F'\"\"#F/F/*&-F+6$\"+e>-I&)!#7F/ 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mu1 := 0;\n eps := Float(1,1-Digits);\n for j to 25 do \n eqns := seq(subs(x=crit[i],r_numer)=(f(crit[i])-(-1)^i*mu)+\n \+ (f(crit[i])-(-1)^i*mu1)*subs(x=crit[i],r_denom1),i=1..mn+2);\n \+ soln := solve(\{eqns\});\n newmu := subs(soln,mu);\n if \+ abs(newmu-mu1) rnew(x)-f(x);\n newcrit := critpts(e(x),x=x0..x1,50);\n print(``); print(`critical points:`);\n print(newcrit); print(``) ;\n ords1 := seq([[crit[i],0],[crit[i],e(crit[i])]],i=1..mn+2):\n \+ ords2 := seq([[newcrit[i],0],[newcrit[i],e(newcrit[i])]],i=1..mn+2):\n print(plot([e(x),ords1,ords2],x=x0..x1,\n color=[COLOR(RGB,.7,0 ,.9),COLOR(RGB,0,.8,0)$(mn+2),COLOR(RGB,1,.4,0)$(mn+2)],\n thicknes s=[1$(mn+3),2$(mn+2)]));\n ec := map(abs@e,newcrit);\n emin := min (op(ec));\n print(`minimum absolute error -> `,emin);\n emax := ma x(op(ec));\n print(`maximum absolute error -> `,emax);\n print(`ra tio -> `,emax/emin);print(``);\nend do:\neval(rnew);\nDigits := 10:" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%$expG" }}{PARA 11 "" 1 "" 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{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 61 "Comparison of polynomial and rational minimax approximations " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 16 "rational minimax" }{TEXT -1 15 " approximation " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "r(x)=(1.000000130+.5036066042*x+.1016019444*x^2+.853002 1958e-2*x^3)/(1.-.4963927430*x+.9799757467e-1*x^2-.7943574177e-2*x^3) " "6#/-%\"rG6#%\"xG*&,*-%&FloatG6$\"+I,++5!\"*\"\"\"*&-F+6$\"+Ug1O]!#5 F/F'F/F/*&-F+6$\"+W%>g,\"F4F/*$F'\"\"#F/F/*&-F+6$\"+e>-I&)!#7F/*$F'\" \"$F/F/F/,*-F+6$F/\"\"!F/*&-F+6$\"+Iu#R'\\F4F/F'F/!\"\"*&-F+6$\"+nuv*z *!#6F/*$F'F:F/F/*&-F+6$\"+xTdVzF?F/*$F'FAF/FJFJ" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "for " } {TEXT 270 6 "exp(x)" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 " [-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 131 " can be evaluated using n ested multiplication for the numerator and denominator with 6 addition s, 6 multiplications and 1 division." }}{PARA 0 "" 0 "" {TEXT -1 36 "T he maximum absolute error is about " }{XPPEDIT 18 0 "1.55*`.`*10^(-7); " "6#*(-%&FloatG6$\"$b\"!\"#\"\"\"%\".GF))\"#5,$\"\"(!\"\"F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The degree 7 " }{TEXT 260 18 "minimax polynomial" }{TEXT -1 15 " a pproximation " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "p(x )=.9999998014+.9999998235*x+.5000063462*x^2+.1666686052*x^3+.416350149 7e-1*x^4" "6#/-%\"pG6#%\"xG,,-%&FloatG6$\"+9!)******!#5\"\"\"*&-F*6$\" +N#)******F-F.F'F.F.*&-F*6$\"+iM1+]F-F.*$F'\"\"#F.F.*&-F*6$\"+_gom;F-F .*$F'\"\"$F.F.*&-F*6$\"+(\\,N;%!#6F.*$F'\"\"%F.F." }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` +.8327356910e-2*x^ 5+.1439272379e-2*x^6+.2054080265e-3*x^7" "6#,*%!G\"\"\"*&-%&FloatG6$\" +5pNF$)!#7F%*$%\"xG\"\"&F%F%*&-F(6$\"+zBFR9F+F%*$F-\"\"'F%F%*&-F(6$\"+ l-3a?!#8F%*$F-\"\"(F%F%" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 84 "can be evaluated using nested multiplication with 7 additions and \+ 7 multiplications." }}{PARA 0 "" 0 "" {TEXT -1 36 "The maximum absolut e error is about " }{XPPEDIT 18 0 "2*`.`*10^(-7);" "6#*(\"\"#\"\"\"%\" .GF%)\"#5,$\"\"(!\"\"F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 141 "If division can be performed as economically as multiplication, t he rational approximation performs better than the polynomial approxim ation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }