{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 "Blue emphasis" -1 263 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 1 14 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 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 270 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 } {CSTYLE "Grey Emphasis" -1 271 "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 264 66 "Version II - minimax error obtained by solving a rational equation" } {TEXT 265 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 "load interpolation and function approximation procedures" }}{PARA 0 " " 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 271 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$6#\"\"%" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "0 = x[1];" "6#/\"\"!&%\"xG6#\"\"\"" } {XPPEDIT 18 0 "`` " 0 "" {MPLTEXT 1 0 215 "a := table( ): b := table():\nunassign('mu');\nf := x->ln(x+1);\nr := x -> (a[0]+a [1]*x)/(1+b[1]*x);\nc := [0.,.2,.7,1.];\neqns := seq(r(c[i])-f(c[i])=( -1)^(i+1)*mu,i=1..nops(c)-1);\nsoln := solve(\{eqns\},\{a[0],a[1],b[1] \});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operator G%&arrowGF(-%#lnG6#,&9$\"\"\"F1F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,&&%\"aG6#\"\"!\"\"\" *&&F/6#F2F29$F2F2F2,&F2F2*&&%\"bGF5F2F6F2F2!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG7&$\"\"!F'$\"\"#!\"\"$\"\"(F*$\"\"\"F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqnsG6%/,$*&$\"\"\"\"\"!F*&%\"aG6#F +F*F*%#muG/,&*&,&F,F**&$\"\"#!\"\"F*&F-6#F*F*F*F*,&F*F**&F5F*&%\"bGF9F *F*F7F*$\"+ob@B=!#5F7,$F/F7/,&*&,&F,F**&$\"\"(F7F*F8F*F*F*,&F*F**&FGF* F%%solnG< %/&%\"aG6#\"\"!%#muG/&F(6#\"\"\",$*($\"+9dG9d!#=F/,($\"+^0cYg\"\")!\" \"*&$\"+W,6kj\"\"*F/F+F/F/*&$\"++++]<\"#5F/)F+\"\"#F/F/F/,&$\"+Vp1$[$F *F/*&$\"+++++?F/F/F+F/F/F9F9/&%\"bGF.,$*($\"+H9dG9!\"*F/,&*&$\"+++++qF /F/F+F/F/$\"+x>(\\2\"F*F9F/FDF9F9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 38 "Now we substitute the expressions for " } {XPPEDIT 18 0 "a[0],a[1]" "6$&%\"aG6#\"\"!&F$6#\"\"\"" }{TEXT -1 5 " a nd " }{XPPEDIT 18 0 "b[1]" "6#&%\"bG6#\"\"\"" }{TEXT -1 13 " in terms \+ of " }{XPPEDIT 18 0 "mu" "6#%#muG" }{TEXT -1 56 " into the last equati on to give an equation of the form " }{XPPEDIT 18 0 "M(mu) = 0;" "6#/- %\"MG6#%#muG\"\"!" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "M(mu);" "6#- %\"MG6#%#muG" }{TEXT -1 27 " is a rational function of " }{XPPEDIT 18 0 "mu" "6#%#muG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 96 "The ze ros of the numerator and denominator are then determined and sorted in order of magnitude." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 221 "lasteq := r(c[4])-f(c[4])+mu;\nlasteq := normal(subs(soln,lasteq));\nM_numer := numer(lasteq):\nmuvals := sort ([fsolve(M_numer,mu)],(_X,_Y)->evalb(abs(_X)%'lasteqG,(*&,&&%\"aG6#\"\"!\"\"\"*&$F,F+F,&F)6#F,F,F, F,,&F,F,*&F.F,&%\"bGF0F,F,!\"\"F,$\"+1=ZJp!#5F5%#muGF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'lasteqG*&,(*&$\"+\"*QseF\"\"\"F*%#muGF*!\"\"*&$ \"+++++;\"\"#F*)F+F0F*F*$\")_)[N#\"\"!F*F*,&$\"+'>T(=]F4F,*&$\"+++++!) F*F*F+F*F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'muvalsG7$$\"+=P#)y& )!#8$\"+3Oi:%)notmuvalG$\"+& \\EMF'!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Now we pick the value for " }{XPPEDIT 18 0 "mu" "6#%#muG" } {TEXT -1 66 " of smallest magnitude that is not also a zero of the den ominator." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "mu1 := muvals[1];\nsolne := subs(mu=mu1,soln);\nr1 := unapply(subs(solne,r(x)),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$mu 1G$\"+=P#)y&)!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&solneG<%/&%\"bG 6#\"\"\"$\"+rcEUT!#5/&%\"aG6#\"\"!$\"+=P#)y&)!#8/&F0F)$\"+a.'>y*F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r1Gf*6#%\"xG6\"6$%)operatorG%&arrow GF(*&,&$\"+=P#)y&)!#8\"\"\"*&$\"+a.'>y*!#5F19$F1F1F1,&F1F1*&$\"+rcEUTF 5F1F6F1F1!\"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot(r1(x)-f(x),x=0..1,color=COLOR(RGB,.7 ,0,.9));" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'C URVESG6#7fo7$$\"\"!F)$\"3#******zrB)y&)!#@7$$\"3WmmmT&)G\\a!#?$\"37qvQ \")yA*R(F,7$$\"3ILLL3x&)*3\"!#>$\"3QBS84&QWF'F,7$$\"3$*****\\ilyM;F6$ \"3q'o**zN/H?&F,7$$\"3emmm;arz@F6$\"3G\"e61Z5J=%F,7$$\"3.++D\"y%*z7$F6 $\"3MN_:rtXFDF,7$$\"3[LL$e9ui2%F6$\"3R;^R:QZ;5F,7$$\"33++voMrU^F6$!3S? 8F^)*f*=&!#A7$$\"3nmmm\"z_\"4iF6$!3))3NS'4c/*=F,7$$\"3emmmm6m#G(F6$!3= ,#*4*z<]6$F,7$$\"3[mmmT&phN)F6$!3;**z'*[!3A>%F,7$$\"36++v=ddC%*F6$!3mZ ,9A*fm7&F,7$$\"3CLLe*=)H\\5!#=$!3qVXGqY5KfF,7$$\"3')***\\(=JN[6Fdo$!3S =kw3@]qlF,7$$\"3gmm\"z/3uC\"Fdo$!3e4HK!eP36(F,7$$\"3%)***\\7LRDX\"Fdo$ !3cnp;wIoUzF,7$$\"3/LLekGhe:Fdo$!3qy$QLODNB)F,7$$\"3]mm\"zR'ok;Fdo$!3X '*f)HbuwV)F,7$$\"3#)*****\\2`vr\"Fdo$!3dNtF-v\"*3&)F,7$$\"39LL3_(>/x\" Fdo$!3!QdPtuF3c)F,7$$\"3Xmm;HkGB=Fdo$!3G6;mgW5%f)F,7$$\"3w***\\i5`h(=F do$!3?MDZT:V4')F,7$$\"3KL3x\")H`I>Fdo$!3H0S*z>urg)F,7$$\"3gm;HdG\"\\)> Fdo$!3WFPVpYM(e)F,7$$\"3!**\\7Gt#HR?Fdo$!3eDPqV*Q1b)F,7$$\"3WLLL3En$4# Fdo$!3!)ol!>SPx\\)F,7$$\"33++Dc#o%*=#Fdo$!35OO\"y*3ND>>)F,7$$\"3\")*****\\K]4]#Fdo$!3+r-4CL;^wF,7$$\"3$** ****\\PAvr#Fdo$!3d&fXW)*eQ$pF,7$$\"3)******\\nHi#HFdo$!3R')=i$>c'3hF,7 $$\"3jmm\"z*ev:JFdo$!3]SbR%HD)o_F,7$$\"3?LLL347TLFdo$!3+'p_@g\"3&=%F,7 $$\"3,LLLLY.KNFdo$!3tG-hE?7:KF,7$$\"3w***\\7o7Tv$Fdo$!3')*fXdWV(\\?F,7 $$\"3'GLLLQ*o]RFdo$!3yt(>V0\"z-5F,7$$\"3A++D\"=lj;%Fdo$\"3U.8,Ns'4W\"F R7$$\"31++vV&RY2aFdo$ \"3%[WT!*[&)[!fF,7$$\"39mm;zXu9cFdo$\"3UVIyXX5=mF,7$$\"3l******\\y))Ge Fdo$\"3;)4x$*RCND(F,7$$\"3'*)***\\i_QQgFdo$\"3a:$)*et=vw(F,7$$\"3@*** \\7y%3TiFdo$\"3U?'[?k:o:)F,7$$\"35****\\P![hY'Fdo$\"3Cl/VDfad%)F,7$$\" 3)em;/risc'Fdo$\"3+'[Y$)*exX&)F,7$$\"3kKLL$Qx$omFdo$\"3*z^#)o!*)4/')F, 7$$\"3mlm;z)Qjx'Fdo$\"3o/sDJ,tK')F,7$$\"3!)*****\\P+V)oFdo$\"3>(pEy^#* fi)F,7$$\"3WK$ek`H@)pFdo$\"3%*>WF![O))e)F,7$$\"3?mm\"zpe*zqFdo$\"3RkBA [)><_)F,7$$\"3%)*****\\#\\'QH(Fdo$\"3EY\\#Qtz'o#)F,7$$\"3GKLe9S8&\\(Fd o$\"3C`/:\"z^Y*yF,7$$\"3R***\\i?=bq(Fdo$\"3gnK%Q6&pftF,7$$\"3\"HLL$3s? 6zFdo$\"3%\\#ezinu\"p'F,7$$\"3a***\\7`Wl7)Fdo$\"3*Rn!Ro?XOeF,7$$\"3#pm mm'*RRL)Fdo$\"3A " 0 "" {MPLTEXT 1 0 88 "e := x -> r1(x)-f(x);\nd := unapply(diff(e(x),x),x);\nsolve(d(x),x );\nc := sort([0.,%,1.]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"eGf*6 #%\"xG6\"6$%)operatorG%&arrowGF(,&-%#r1G6#9$\"\"\"-%\"fGF/!\"\"F(F(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dGf*6#%\"xG6\"6$%)operatorG%&a rrowGF(,(*&$\"+a.'>y*!#5\"\"\",&F1F1*&$\"+rcEUTF0F19$F1F1!\"\"F1*($\"+ rcEUTF0F1,&$\"+=P#)y&)!#8F1*&F.F1F6F1F1F1F2!\"#F7*&F1F1,&F6F1F1F1F7F7F (F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+zAU'*=!#5$\"+D^(*4oF%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG7&$\"\"!F'$\"+zAU'*=!#5$\"+D^(*4 oF*$\"\"\"F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "We can repeat the process using these new critical points ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 319 "eqns := seq(r(c[i])-f(c[i])=(-1)^(i+1)*mu,i=1..nops( c)-1):\nsoln := solve(\{eqns\},\{a[0],a[1],b[1]\}):\nlasteq := r(c[4]) -f(c[4])+mu;\nlasteq := normal(subs(soln,lasteq));\nlasteq1 := numer(l asteq):\nmuvals := sort([fsolve(lasteq1,mu)],(_X,_Y)->evalb(abs(_X)%'lasteqG,(*&,&&%\"aG6#\"\"!\"\"\"* &$F,F+F,&F)6#F,F,F,F,,&F,F,*&F.F,&%\"bGF0F,F,!\"\"F,$\"+1=ZJp!#5F5%#mu GF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'lasteqG*&,(*&$\"+mxhAu\"\"! \"\"\"%#muGF+!\"\"*&$\"+wr3tUF+F+)F,\"\"#F+F+$\"++Aodj!\"$F+F+,&$\"+N \\$oC\"F*F-*&$\"+)eVl8#F+F+F,F+F+F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%'muvalsG7$$\"+=$Rzg)!#8$\"+dTXG%)notmuvalG$\"+qqvNe!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "mu2 := muvals[1];\nsolne := subs(mu=mu2,soln);\nr2 := unapply(subs (solne,r(x)),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$mu2G$\"+=$Rzg)! #8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&solneG<%/&%\"aG6#\"\"!$\"+=$R zg)!#8/&%\"bG6#\"\"\"$\"+eb8UT!#5/&F(F1$\"+))**z\"y*F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r2Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,&$\"+= $Rzg)!#8\"\"\"*&$\"+))**z\"y*!#5F19$F1F1F1,&F1F1*&$\"+eb8UTF5F1F6F1F1! \"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "We can again compute the critical points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "e := x -> r2 (x)-f(x);\nd := unapply(diff(e(x),x),x);\nsolve(d(x),x);\nc := sort([0 .,%,1.]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"eGf*6#%\"xG6\"6$%)ope ratorG%&arrowGF(,&-%#r2G6#9$\"\"\"-%\"fGF/!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(*&$\"+) )**z\"y*!#5\"\"\",&F1F1*&$\"+eb8UTF0F19$F1F1!\"\"F1*($\"+eb8UTF0F1,&$ \"+=$Rzg)!#8F1*&F.F1F6F1F1F1F2!\"#F7*&F1F1,&F6F1F1F1F7F7F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+\"zA#)*=!#5$\"+zRB4oF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG7&$\"\"!F'$\"+\"zA#)*=!#5$\"+zRB4oF*$ \"\"\"F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "A third iteration produces little change within the accuracy possi ble using 10 digit arithmetic." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 400 "eqns := seq(r(c[i])-f(c[i]) =(-1)^(i+1)*mu,i=1..nops(c)-1):\nsoln := solve(\{eqns\},\{a[0],a[1],b[ 1]\}):\nlasteq := r(c[4])-f(c[4])+mu;\nlasteq := normal(subs(soln,last eq));\nlasteq1 := numer(lasteq):\nmuvals := sort([fsolve(lasteq1,mu)], (_X,_Y)->evalb(abs(_X)%'lasteqG,(*&,&&%\"aG6#\"\"!\"\"\"*&$F,F+F,&F)6#F,F,F,F,,&F,F,*&F. F,&%\"bGF0F,F,!\"\"F,$\"+1=ZJp!#5F5%#muGF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'lasteqG*&,(*&$\"+)3![Mf\"\"!\"\"\"%#muGF+!\"\"*&$\"+ <%oWT$F+F+)F,\"\"#F+F+$\"++d1$3&!\"$F+F+,&$\"+;)z*o**F-F-*&$\"+4UB2%'muvalsG7$$\"+X:%zg)!# 8$\"+(HJ%H%)notmuvalG$\"+*Hd# Re!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$mu3G$\"+X:%zg)!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&solneG<%/&%\"aG6#\"\"!$\"+=$Rzg)!#8 /&%\"bG6#\"\"\"$\"+yc8UT!#5/&F(F1$\"+_+!=y*F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r3Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,&$\"+=$Rzg) !#8\"\"\"*&$\"+_+!=y*!#5F19$F1F1F1,&F1F1*&$\"+yc8UTF5F1F6F1F1!\"\"F(F( F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "e := x -> r3(x)-f(x);\nd := unapply(diff(e(x),x),x); \nsolve(d(x),x);\nc := sort([0.,%,1.]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"eGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%#r3G6#9$\"\"\"-%\"f GF/!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dGf*6#%\"xG6\"6$ %)operatorG%&arrowGF(,(*&$\"+_+!=y*!#5\"\"\",&F1F1*&$\"+yc8UTF0F19$F1F 1!\"\"F1*($\"+yc8UTF0F1,&$\"+=$Rzg)!#8F1*&F.F1F6F1F1F1F2!\"#F7*&F1F1,& F6F1F1F1F7F7F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+sCA)*=!#5$\" +nFB4oF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG7&$\"\"!F'$\"+sCA)*= !#5$\"+nFB4oF*$\"\"\"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 critical 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 "cri tpts: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 266 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 267 2 " " }{TEXT -1 21 " critpts( f, rng N) " }{TEXT 268 1 "\n" } {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 11 "Parameters:" }}{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 e valuates to a real floating point number." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 12 " rng - " } {TEXT 269 53 "the range x=a..b for the function to be approximated." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 12 " N - " }{TEXT 270 65 "the number of subintervals 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 t o find all the critical points of f on the interval [a,b] by dividing \+ the interval into N subintervals, and searching through these interval s to find those in which the derivative changes sign. For any such sub intervals the associated critical point is located accurately by solvi ng 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, pla ce the cursor anywhere after the prompt [ > and press [Enter].\nYou c an 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::algebraic,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 i f not type(val,numeric) then\n error \"non-numeric value\"\n en d if; \n for i to N do\n newx := xx + h;\n newval := d(n ewx);\n if not type(newval,numeric) then\n error \"non-nu meric value\"\n end if;\n if signum(0,val,0)<>signum(0,newva l,0) then\n c := traperror(fsolve(d(x),x=xx..newx));\n \+ if c=lasterror or not type(c,numeric) then\n error \"faile d to locate critical point in the interval %1\",x..newx;\n end if;\n cpts := cpts,c;\n end if;\n xx := newx;\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 the works heet." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 56 "The calculation of a minimax rational app roximation 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#-%\"pG 6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"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-%$expG6#F'" }{TEXT -1 45 " with minimax absolute err or in the interval " }{XPPEDIT 18 0 "[-1, 1];" "6#7$,$\"\"\"!\"\"F%" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " } {XPPEDIT 18 0 "p(x) = a[0]+a[1]*x+a[2]*x^2+` . . . `+a[m]*x^m;" "6#/-% \"pG6#%\"xG,,&%\"aG6#\"\"!\"\"\"*&&F*6#F-F-F'F-F-*&&F*6#\"\"#F-*$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#%\"x G,,&%\"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#,(%\"mG\"\"\" %\"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 0 239 "m := 3;\nn := 3;\nmn := m+n:\na := table(): b := tab le():\nvars := \{seq(a[i],i=0..m),seq(b[i],i=1..n)\}:\nr_numer := conv ert([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\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"nG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"\"!\"\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"xG6\"6$%)operatorG%&arro wGF(*&,*&%\"aG6#\"\"!\"\"\"*&&F/6#F2F29$F2F2*&&F/6#\"\"#F2)F6F:F2F2*&& F/6#\"\"$F2)F6F?F2F2F2,*F2F2*&&%\"bGF5F2F6F2F2*&&FDF9F2F;F2F2*&&FDF>F2 F@F2F2!\"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Set up the critical point s, and the system of the first " }{XPPEDIT 18 0 "n+m+1" "6#,(%\"nG\"\" \"%\"mGF%F%F%" }{TEXT -1 11 " equations " }{XPPEDIT 18 0 "r(x[k])-f(x[ k]) = (-1)^(k-1)*mu" "6#/,&-%\"rG6#&%\"xG6#%\"kG\"\"\"-%\"fG6#&F)6#F+! \"\"*&),$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 0 318 "Digits := 10:\nunassign('mu');\nf := exp;\ncrit := [seq(evalf(cos(Pi*(mn+2-i)/(mn+1))),i=1..mn+2)];\nF := seq(evalf(f (crit[i])),i=1..mn+2):\nR := seq(subs(x=crit[i],r(x)),i=1..mn+2):\neqn s := \{\}:\ns := -1:\nfor i to mn+1 do\n s := -s;\n eq := normal(F [i]-R[i]=s*mu,expanded);\n print(eq);\n eqns := eqns union \{eq\} \nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"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#/*&,2$\"+7WzyO!#5\"\"\"*&$\"+7WzyOF(F)&%\"bG6#F )F)!\"\"*&F&F)&F.6#\"\"#F)F)*&$\"+7WzyOF(F)&F.6#\"\"$F)F0&%\"aG6#\"\"! F0*&$F)F>F)&FF)F-F)F0F2F)* &$F)F>F)F8F)F0F0%#muG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,2$\"+$Qf< 1%!#5\"\"\"*&$\"+`(=&fOF(F)&%\"bG6#F)F)!\"\"*&$\"+nC6(H$F(F)&F.6#\"\"# F)F)*&$\"+(o&fqHF(F)&F.6#\"\"$F)F0&%\"aG6#\"\"!F0*&$\"+y')o4!*F(F)&F>F /F)F)*&$\"+2!\\u6)F(F)&F>F5F)F0*&$\"+U)oNJ(F(F)&F>F;F)F)F),*$F)F@F)*&$ \"+y')o4!*F(F)F-F)F0*&$\"+2!\\u6)F(F)F4F)F)*&$\"+U)oNJ(F(F)F:F)F0F0,$% #muGF0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,2$\"+,Rqg`!#5\"\"\"*&$\" +8UMULF(F)&%\"bG6#F)F)!\"\"*&$\"+Iv\"R3#F(F)&F.6#\"\"#F)F)*&$\"+G8I*H \"F(F)&F.6#\"\"$F)F0&%\"aG6#\"\"!F0*&$\"+=!)*[B'F(F)&F>F/F)F)*&$\"+H`R ()QF(F)&F>F5F)F0*&$\"+W8vBCF(F)&F>F;F)F)F),*$F)F@F)*&$\"+=!)*[B'F(F)F- F)F0*&$\"+H`R()QF(F)F4F)F)*&$\"+W8vBCF(F)F:F)F0F0%#muG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/*&,2$\"+$\\#)\\+)!#5\"\"\"*&$\"+xhF\"y\"F(F)&%\"b G6#F)F)!\"\"*&$\"+yBrjR!#6F)&F.6#\"\"#F)F)*&$\"+z(*3?))!#7F)&F.6#\"\"$ F)F0&%\"aG6#\"\"!F0*&$\"+N$4_A#F(F)&F@F/F)F)*&$\"+&ec:&\\F4F)&F@F6F)F0 *&$\"+$*\\#=5\"F4F)&F@F=F)F)F),*$F)FBF)*&$\"+N$4_A#F(F)F-F)F0*&$\"+&ec :&\\F4F)F5F)F)*&$\"+$*\\#=5\"F4F)FA\\7!\"*\"\"\"*&$\"+*Q!yzF!#5F)&%\"bG6#F)F )F)*&$\"+sKf&='!#6F)&F/6#\"\"#F)F)*&$\"+))RUw8F4F)&F/6#\"\"$F)F)&%\"aG 6#\"\"!!\"\"*&$\"+N$4_A#F-F)&F?F0F)FB*&$\"+&ec:&\\F4F)&F?F6F)FB*&$\"+$ *\\#=5\"F4F)&F?FF/F)FA*&$\"+H`R()QF3F)&F>F5F)FA*&$\"+W8vBCF3F)&F>F;F) FAF),*$F)F@F)*&$\"+=!)*[B'F3F)F-F)F)*&$\"+H`R()QF3F)F4F)F)*&$\"+W8vBCF 3F)F:F)F)FA,$%#muGFA" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,2$\"+'H()> Y#!\"*\"\"\"*&$\"+2R<=AF(F)&%\"bG6#F)F)F)*&$\"+Lc])*>F(F)&F.6#\"\"#F)F )*&$\"+e8f+=F(F)&F.6#\"\"$F)F)&%\"aG6#\"\"!!\"\"*&$\"+y')o4!*!#5F)&F=F /F)F@*&$\"+2!\\u6)FDF)&F=F4F)F@*&$\"+U)oNJ(FDF)&F=F:F)F@F),*$F)F?F)*&$ \"+y')o4!*FDF)F-F)F)*&$\"+2!\\u6)FDF)F3F)F)*&$\"+U)oNJ(FDF)F9F)F)F@%#m uG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "So lve the system of equations for the coefficients in terms of " } {XPPEDIT 18 0 "mu" "6#%#muG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "soln := solve(eqns ,vars);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%solnG<)/&%\"bG6#\"\"$,$* ($\"$+#\"\"!\"\"\",**&$\"+s$4'\\L\"#^F0)%#muGF*F0F0*&$\"+BZ8RY\"#]F0)F 7\"\"#F0!\"\"*&$\"+Flj7T\"#ZF0F7F0F>$\"+NI9ca\"#TF0F0,**&$\"+cO[vA\"#_ F0FF >/&%\"aG6#F=*&,,*&$\"+\\ar(f'\"#`F0F6F0F>*&$\"+#4z@%HFgnF0F*&$\"+ TS,$*eF;F0F7F0F0$\"+Swo'=\"\"#XF0*&$\"+p$4'\\LFgnF0)F7\"\"%F0F0F0FFF>/ &FV6#F/,$*($\"+++++!)!#>F0,,*&$\"+FHvY5\"#iF0FdoF0F0*&$\"+\">O'**=FbpF 0F6F0F>*&$\"+'*o6*&$\"+>oc.*)\"#fF0F7F0F>$\"+pnRT:\"#bF> F0FFF>F>/&FV6#F0,$*($\"+++++]!#5F0,,*&$\"+X(=#*p'FgnF0FdoF0F0*&$\"+_)y E1(FgnF0F6F0F>*&$\"+<%)e**GFgnF0F$\"+;VG:7FS F>F0FFF>F>/&F(FW,$*($F=F/F0,**&$\"+&o/[n\"FgnF0F6F0F0*&$\"+'[l`9(FJF0F RU$H#F;F0F7F0F0$\"+#)p.zj\"#WF>F0FFF>F>/&FVF),$*($F=F/F0,, *&$\"+'))GZq%FgnF0F6F0F>*&$\"+\"fjgX&FJF0F*&$\"+f+AKW\"#\\F0F7F0F 0$\"+xzgGZ\"#VF0*&$F4FgnF0FdoF0F0F0FFF>F0/&F(Fcq*&,**&$\"+t$4'\\LFgnF0 F6F0F0*&$\"+fk*\\#pF;F0F7F0F0*&$\"+MHIP;FgnF0F$\"+D;vaiF`oF>F0FFF >" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "Sub stitute the expressions involving mu for the coefficients in the last \+ equation " }{XPPEDIT 18 0 "r(x[m+n+2])-f(x[m+n+2]) = (-1)^(m+n+1)*mu" "6#/,&-%\"rG6#&%\"xG6#,(%\"mG\"\"\"%\"nGF-\"\"#F-F--%\"fG6#&F)6#,(F,F- F.F-F/F-!\"\"*&),$F-F6,(F,F-F.F-F-F-F-%#muGF-" }{TEXT -1 40 " to give \+ an equation for mu of the form " }{XPPEDIT 18 0 "M(mu) = 0;" "6#/-%\"M G6#%#muG\"\"!" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "M(mu);" "6#-%\"M G6#%#muG" }{TEXT -1 27 " is a rational function of " }{XPPEDIT 18 0 "m u" "6#%#muG" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "lasteq := (F[mn+2]-R[mn+2])- (-1)^(mn+3)*mu;\nlasteq := sort(normal(subs(soln,lasteq)));\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'lasteqG,($\"+G=G=F!\"*\"\"\"*&,*&% \"aG6#\"\"!F)*&$F)F/F)&F-6#F)F)F)*&F1F)&F-6#\"\"#F)F)*&F1F)&F-6#\"\"$F )F)F),*F)F)*&F1F)&%\"bGF3F)F)*&F1F)&F?F6F)F)*&F1F)&F?F:F)F)!\"\"FD%#mu GF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'lasteqG*&,,*&$\"++k\"='e\"\" (\"\"\")%#muG\"\"%F+F+*&$\"+SCZWMF*F+)F-\"\"$F+F+*&$\"+%*)oIn$\"\"&F+) F-\"\"#F+!\"\"*&$\"+Wb=!z\"F+F+F-F+F:$\"%1A\"\"!F+F+,**&$\"++#34$HF*F+ F2F+F+*&$\"+0O+b&*\"\"'F+F8F+F+*&$\"+SCiT^F.F+F-F+F:$\"*Za:i$F@F:F:" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The sol utions of the equation " }{XPPEDIT 18 0 "M(mu)=0" "6#/-%\"MG6#%#muG\" \"!" }{TEXT -1 49 " are the values of mu for which the numerator of " }{XPPEDIT 18 0 "M(mu)" "6#-%\"MG6#%#muG" }{TEXT -1 32 " is zero and th e denominator of " }{XPPEDIT 18 0 "M(mu)" "6#-%\"MG6#%#muG" }{TEXT -1 13 " is non-zero." }}{PARA 0 "" 0 "" {TEXT -1 47 "The solutions are so rted in order of magnitude." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "M_numer := numer(lasteq):\nmuvals \+ := sort([fsolve(M_numer,mu)],(_X,_Y)->evalb(abs(_X)%'muvalsG7&$\"+jXI&!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Now we pick the value for " }{XPPEDIT 18 0 "mu" "6#%#muG" } {TEXT -1 66 " of smallest magnitude that is not also a zero of the den ominator." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "mu := muvals[1];\nsolne := subs(e=mu,soln);\nr1 := un apply(subs(solne,r(x)),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#muG$ \"+jX%&solneG<)/&%\"aG6#\" \"\"$\"++D)>.&!#5/&%\"bG6#\"\"#$\"+WNC>)*!#6/&F(6#\"\"$$\"+_Z`#\\)!#7/ &F0F)$!+m*3!o\\F-/&F(F1$\"+zr)Q,\"F-/&F(6#\"\"!$\"+I,++5!\"*/&F0F8$!+# eJF(zF<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r1Gf*6#%\"xG6\"6$%)opera torG%&arrowGF(*&,*$\"+I,++5!\"*\"\"\"*&$\"++D)>.&!#5F19$F1F1*&$\"+zr)Q ,\"F5F1)F6\"\"#F1F1*&$\"+_Z`#\\)!#7F1)F6\"\"$F1F1F1,*F1F1*&$\"+m*3!o\\ F5F1F6F1!\"\"*&$\"+WNC>)*!#6F1F:F1F1*&$\"+#eJF(zF?F1F@F1FFFFF(F(F(" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " } {XPPEDIT 18 0 "r[1](x)" "6#-&%\"rG6#\"\"\"6#%\"xG" }{TEXT -1 45 " is t he resulting rational approximation for " }{XPPEDIT 18 0 "exp(x)" "6#- %$expG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-1,1] " "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 61 ", we can calculate the critical \+ points of the error function " }{XPPEDIT 18 0 "r[1](x)-exp(x)" "6#,&-& %\"rG6#\"\"\"6#%\"xGF(-%$expG6#F*!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "e := x \+ -> r1(x)-exp(x);\nnewcrit := critpts(r1(x)-exp(x),x=-1..1,50);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"eGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,&-%#r1G6#9$\"\"\"-%$expGF/!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(newcritG7*$!\"\"\"\"!$!+?1v%y)!#5$!+EQv)\\&F+$!+8vvc 5F+$\"+HF2bMF+$\"+1/-#4(F+$\"+BXj,$*F+$\"\"\"F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 150 "The following picture co mpares the location of the original guesses for the critical points an d the actual critical points of the current error curve." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 283 "ords 1 := seq([[crit[i],0],[crit[i],e(crit[i])]],i=1..mn+2):\nords2 := seq( [[newcrit[i],0],[newcrit[i],e(newcrit[i])]],i=1..mn+2):\nplot([r1(x)-f (x),ords1,ords2],x=-1..1,\n color=[COLOR(RGB,.7,0,.9),COLOR(RGB,0,.8, 0)$(mn+2),COLOR(RGB,1,.4,0)$(mn+2)],\n thickness=[1$(mn+3),2$(mn+2) ]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "65-%'CURVES G6%7`t7$$!\"\"\"\"!$!3B[p)3q*=E7!#C7$$!3/MLe9r]X**!#=$!3'o/HR+qOn*!#D7 $$!3%pmm\"HU,\"*)*F1$!3K;%RACA'fsF47$$!3()***\\PM@l$)*F1$!33:*yL=MP,&F 47$$!3!RLL$e%G?y*F1$!3WK%\\^o#4IHF47$$!3u****\\(oUIn*F1$\"3O$Q$z')QtNx !#E7$$!3ommm;p0k&*F1$\"3#y;q!G'pl*QF47$$!3#HL3-)*G#p%*F1$\"3IlT<]\"o`< 'F47$$!3E++vV5Su$*F1$\"37o@=?c=v!)F47$$!3_m;H2Jdz#*F1$\"3\"f-%>+>tA'*F 47$$!3wKL$3:xdVH\"F-7$$!3&emmmwnMa)F1$\"3iaR`'4 +QB\"F-7$$!3:mmm\"4m(G$)F1$\"3E-^0'3hE4\"F-7$$!3)****\\i&[3:\")F1$\"31 $H_U\"yyB*)F47$$!3\"QLL3i.9!zF1$\"3G!>ggH9W]'F47$$!3'3++Dw$H.xF1$\"3? \"e8V+Q(GSF47$$!3\"ommT!R=0vF1$\"3\\p]l7!\\[W\"F47$$!3tKL$3i_+I(F1$!3, '3Hs%H aF1$!3C(pXF'Rhp9F-7$$!3]***\\7)*G;K&F1$!39K@kk<'yX\"F-7$$!3]LL$3x&y8_F 1$!3[\"=Zib]gV\"F-7$$!3]nmTgD%f5&F1$!3KqG.PYX/9F-7$$!3Q+++]$*4)*\\F1$! 3okI!o*yRj8F-7$$!3E++++t_\"y%F1$!3ARENVv!RD\"F-7$$!39+++]_&\\c%F1$!3#4 CDvn)H66F-7$$!33+++]zCcVF1$!31%z.fs-NY*F47$$!31+++]1aZTF1$!3y;t^5UC&e( F47$$!3SLL3FW,eRF1$!3%eS-gn*R=dF47$$!3umm;/#)[oPF1$!3hf_4Q%>St$F47$$!3 <++v$>BJa$F1$!3ASay\"y)4r7F47$$!3hLLL$=exJ$F1$\"3qeyzvZ'fC\"F47$$!3!QL L$eW%o7$F1$\"3#QCM.#y3vLF47$$!3*RLLLtIf$HF1$\"3kz\"*[B)QCY&F47$$!3CnmT &o_Qr#F1$\"3O-wV1<)3z(F47$$!3]++]PYx\"\\#F1$\"3'\\:)p0B[h**F47$$!3QnmT Nz>&H#F1$\"3[J?DX&G6<\"F-7$$!3EMLLL7i)4#F1$\"3Kl^X7[rE8F-7$$!3c****\\P 'psm\"F1$\"3KiN7&=\"*oe\"F-7$$!3s*****\\F&*=Y\"F1$\"3M9nm;pEm;F-7$$!3' )****\\74_c7F1$\"3k!*\\Tk_N9E\\H?\"F1$\"3@<\"RW4#f@$\"3(ffPD)49G:))F__l$\"3%HFd'>&oy r\"F-7$$!3)3LLL3x%z#)F__l$\"3$eS\"e#fP$4)H)eZF47$$\"3ilm\"z\\1A-\"F 1$\"3dE88H0#3C#F47$$\"3GKLLe\"*[H7F1$!3?#R\\&f%p2f$FI7$$\"3ylm;HCjV9F1 $!3om!RvJ*pwIF47$$\"3I*******pvxl\"F1$!3CI*o_&o;odF47$$\"3g)***\\7JFn= F1$!3bHsZ;4x<$)F47$$\"3#z****\\_qn2#F1$!3'**oAw]EF2\"F-7$$\"3=)**\\P/q %zAF1$!3wZB.SE?(G\"F-7$$\"3U)***\\i&p@[#F1$!31**[nN&>$y9F-7$$\"3L)**\\ (=GB2FF1$!3WIk>gdYd;F-7$$\"3B)****\\2'HKHF1$!3mL&*o:C:'z\"F-7$$\"3uJL$ 3UDX8$F1$!3!yA8B-.?)=F-7$$\"3ElmmmZvOLF1$!3z*=)fZ*Gz#>F-7$$\"3yJLe9bt! 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HHB=&[S>8!#C$\"5F1de(y(*yqd'!#A" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 28 "We again pick the value for " }{XPPEDIT 18 0 "mu" "6#%#muG" }{TEXT -1 34 " which has the smallest magnitude." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 424 "mu := muvals[1];\nsolne := subs(e=mu,soln):\nr2 := unapply(subs (solne,r(x)),x);\ne := x -> r2(x)-exp(x);\nnewcrit := critpts(r2(x)-ex p(x),x=-1..1,50);\nords1 := seq([[crit[i],0],[crit[i],e(crit[i])]],i=1 ..mn+2):\nords2 := seq([[newcrit[i],0],[newcrit[i],e(newcrit[i])]],i=1 ..mn+2):\nplot([e(x),ords1,ords2],x=-1..1,\n color=[COLOR(RGB,.7,0,.9 ),COLOR(RGB,0,.8,0)$(mn+2),COLOR(RGB,1,.4,0)$(mn+2)],\n thickness=[ 1$(mn+3),2$(mn+2)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#muG$\"59iH& )p@)f*R:!#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r2Gf*6#%\"xG6\"6$%)o peratorG%&arrowGF(*&,*$\"5P_W`FJ,++5!#>\"\"\"*&$\"5=-*yc*>(Hh.&!#?F19$ F1F1*&$\"5=#[Q%ep10;5F5F1)F6\"\"#F1F1*&$\"5QjxE;d@]I&)!#AF1)F6\"\"$F1F 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++]iS\"*3))4)*F-$!+$R.DN#F`cl7$$\"5+++D1k2/P)*F-$\"-:EuEt@D)Fa_lF[]n7$Fjbn$\"/aA\"*puf:F-F]anFaan-F$6%7$7$$ \"5Y-FX='p(*3_$F-F[]n7$Fccn$!.@3?0?a\"F`clF]anFaan-F$6%7$7$$\"50A233<3 T8qF-F[]n7$F\\dn$\".pXR+ca\"F`clF]anFaan-F$6%7$7$$\"5(*\\/'4v\"G9S#*F- F[]n7$Fedn$!.6eh7Jb\"F`clF]anFaan-F$6%Fi`nF]anFaan-%+AXESLABELSG6$Q\"x 6\"Q!F`en-%%VIEWG6$;F(Fh[n%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "For the \+ next iteration the tacit assumption is made that the denominator of th e rational expression " }{XPPEDIT 18 0 "M(mu)" "6#-%\"MG6#%#muG" } {TEXT -1 19 " used to calculate " }{XPPEDIT 18 0 "mu" "6#%#muG" } {TEXT -1 31 ", is not zero for the value of " }{XPPEDIT 18 0 "mu" "6#% #muG" }{TEXT -1 10 " selected." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 889 "unassign('mu');\nfor i to m n+2 do crit[i] := newcrit[i] end do:\nF := seq(evalf(f(crit[i])),i=1.. mn+2):\nR := seq(subs(x=crit[i],r(x)),i=1..mn+2):\neqns := \{\}:\ns := -1:\nfor i to mn+1 do\n s := -s;\n eq := normal(F[i]-R[i]=s*mu,ex panded);\n eqns := eqns union \{eq\}\nend do:\nsoln := solve(eqns,va rs):\nlasteq := (F[mn+2]-R[mn+2])-(-1)^(mn+3)*mu:\nlasteq := sort(norm al(subs(soln,lasteq))):\nM_numer := numer(lasteq):\nmu := op(1,sort([f solve(M_numer,mu)],(_X,_Y)->evalb(abs(_X) r3(x)-exp(x) ;\nnewcrit := critpts(r3(x)-exp(x),x=-1..1,50);\nords1 := seq([[crit[i ],0],[crit[i],e(crit[i])]],i=1..mn+2):\nords2 := seq([[newcrit[i],0],[ newcrit[i],e(newcrit[i])]],i=1..mn+2):\nplot([e(x),ords1,ords2],x=-1.. 1,\n color=[COLOR(RGB,.7,0,.9),COLOR(RGB,0,.8,0)$(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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$6%7$7$F($F*F*F'-F]\\n6&F_\\nF*$\"\")F)F*Fd\\n-F$6%7$7$$!5S6'\\/XIf%)p )F-F[]n7$Fd]n$\"/9iQ@m]:F-F\\]nFd\\n-F$6%7$7$$!5\\u4daHVzq_F-F[]n7$F]^ n$!/@iQ@m]:F-F\\]nFd\\n-F$6%7$7$$!5\\ak*)H#Q>@D)Fa_lF[]n7$Ff^nFg]nF\\] nFd\\n-F$6%7$7$$\"5Y-FX='p(*3_$F-F[]n7$F]_n$!.B'Q@m]:F`clF\\]nFd\\n-F$ 6%7$7$$\"50A233<3T8qF-F[]n7$Ff_nFj[nF\\]nFd\\n-F$6%7$7$$\"5(*\\/'4v\"G 9S#*F-F[]n7$F]`n$!.C'Q@m]:F`clF\\]nFd\\n-F$6%7$7$Fh[nF[]nFg[nF\\]nFd\\ n-F$6%Fi\\n-F]\\n6&F_\\nFi[n$\"\"%F)F*-Fe\\n6#\"\"#-F$6%7$7$$!5GIV.]-_ 4&p)F-F[]n7$Fcan$\"/$)Fa_lF[]n7$Febn$\"/ 8Pk)y1b\"F-Fh`nF\\an-F$6%7$7$$\"5IU\")[xR8_:NF-F[]n7$F^cn$!.Lb%en]:F`c lFh`nF\\an-F$6%7$7$$\"5*o()QOq:2K,(F-F[]n7$Fgcn$\".9h " 0 "" {MPLTEXT 1 0 110 "ec \+ := map(abs@(r3-f),newcrit):\nemin := min(op(ec));\nemax := max(op(ec)) ;\nemax/emin:\nevalf(%,15);\nDigits := 10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eminG$\".@'Q@m]:!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%emaxG$\"/8Pk)y1b\"!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0$*) >'y5++\"!#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(*&$\"+Iu#R'\\F,F(F-F(!\"\"*&$\"+nu v*z*!#6F(F1F(F(*&$\"+xTdVzF6F(F7F(F?F?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "r(x)=(1.0000001 30+.5036066042*x+.1016019444*x^2+.8530021958e-2*x^3)/(1.-.4963927430*x +.9799757467e-1*x^2-.7943574177e-2*x^3)" "6#/-%\"rG6#%\"xG*&,*-%&Float G6$\"+I,++5!\"*\"\"\"*&-F+6$\"+Ug1O]!#5F/F'F/F/*&-F+6$\"+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 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 89 "Code template for a pplying version 2 of Remez algorithm to find a rational approximation \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1809 "f := exp;\nx0 := -1.;\nx1 := 1.;\nm := 3;\nn := 3;\nmn := m+ n:\nDigits := 20:\na := table(): b := table():\nvars := \{seq(a[i],i=0 ..m),seq(b[i],i=1..n)\}:\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);\nnewcrit := [seq(evalf((x0+x1)/2+(x1-x0)/ 2*cos(Pi*(mn+2-i)/(mn+1))),i=1..mn+2)];\neps := Float(1,-8):\nemax := \+ Float(1,Digits):\nemin := 1.:\nfor i to 10 while emax/emin -1. > eps d o \n unassign('mu');\n for i to mn+2 do crit[i] := newcrit[i] end do;\n F := seq(evalf(f(crit[i])),i=1..mn+2);\n R := seq(subs(x=cr it[i],r(x)),i=1..mn+2);\n eqns := \{\};\n s := -1;\n for i to mn +1 do\n s := -s;\n eq := normal(F[i]-R[i]=s*mu,expanded);\n \+ eqns := eqns union \{eq\}\n end do;\n soln := solve(eqns,vars );\n lasteq := (F[mn+2]-R[mn+2])-(-1)^(mn+3)*mu;\n lasteq := sort( normal(subs(soln,lasteq)));\n M_numer := numer(lasteq);\n mu := op (1,sort([fsolve(M_numer,mu)],(_X,_Y)->evalb(abs(_X) rnew(x)-f(x);\n newcrit := critpts(e(x),x= x0..x1,50);\n print(``); print(`critical points:`);\n print(newcri t); 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=[C OLOR(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)]));\n ec := map(abs@e,newcrit);\n emin := min(op(ec));\n print(`minimum absolute error -> `,emin); \n emax := max(op(ec));\n print(`maximum absolute error -> `,emax) ;\n print(`ratio -> `,emax/emin);print(``);\nend do:\neval(rnew);\nc onvert(rnew(x),horner);\nemax/emin:\nevalf(%,15);\nDigits := 10:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%$expG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$!\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #x1G$\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"bG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"rGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,*&%\"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 12 " " 1 "" {XPPMATH 20 "6#>%(newcritG7*$!5+++++++++5!#>$!5CE\">C!z')o4!*!# ?$!5^INte=!)*[B'F+$!5F/WJcR$4_A#F+$\"5F/WJcR$4_A#F+$\"5^INte=!)*[B'F+$ \"5CE\">C!z')o4!*F+$\"5+++++++++5F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #f*6#%\"xG6\"6$%)operatorG%&arrowGF&*&,*$\"5Qb%*)R,8+++\"!#>\"\"\"*&$ \"5Rg!eB5O!*>.&!#?F/9$F/F/*&$\"5::$ydH&3*Q,\"F3F/)F4\"\"#F/F/*&$\"5c$) *G(*4^(e#\\)!#AF/)F4\"\"$F/F/F/,*F/F/*&$\"5!fIl-'36+o\\F3F/F4F/!\"\"*& $\"5**>!H.os,#>)*!#@F/F8F/F/*&$\"5)e0y#G3clszF=F/F>F/FDFDF&F&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%;minimum~absolute~error~->~G$\"/OyV&) >H7!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;maximum~absolute~error~->~ G$\".F)z/5J>!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%*ratio~->~G$\"5O@& QEV0C5d\"!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#f*6#%\"xG6\"6$%)operatorG%&arrowGF&*&,*$\"5'oXl^78+++ \"!#>\"\"\"*&$\"5!ef>il/Fh.&!#?F/9$F/F/*&$\"5Q_>@+?$\\g,\"F3F/)F4\"\"# F/F/*&$\"5!*=*32*)yz/`)!#AF/)F4\"\"$F/F/F/,*F/F/*&$\"5)>s6QJXmQ'\\F3F/ F4F/!\"\"*&$\"5SXuVlr1VzF=F/F>F/FDFDF&F& F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;minimum~absolute~error~->~G$\" /!ojdm)R:!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;maximum~absolute~err or~->~G$\"/P(Ga=Oc\"!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%*ratio~->~ G$\"52Yp$\\7pCa,\"!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"xG6\"6$%)operatorG%&arrowGF&*&,*$\"5$ )*Hp<+8+++\"!#>\"\"\"*&$\"5^Dh)3L6mg.&!#?F/9$F/F/*&$\"5w)Q;\"**y%>g,\" F3F/)F4\"\"#F/F/*&$\"5sER=69D-I&)!#AF/)F4\"\"$F/F/F/,*F/F/*&$\"5c%4xj( et#R'\\F3F/F4F/!\"\"*&$\"5i;hRA2rv*z*!#@F/F8F/F/*&$\"5X0F/FDFDF&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;minimum~absolute~er ror~->~G$\".8lCi1b\"!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;maximum~a bsolute~error~->~G$\"/?[b'y1b\"!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %*ratio~->~G$\"5)R,V5>e5++\"!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"xG6\"6$%)operatorG%&arrowGF &*&,*$\"5j.>o,I,++5!#>\"\"\"*&$\"5p$fzaA/mg.&!#?F/9$F/F/*&$\"5N$*[(QRW >g,\"F3F/)F4\"\"#F/F/*&$\"5E$RwZ$e>-I&)!#AF/)F4\"\"$F/F/F/,*F/F/*&$\"5 sGM%R)Hu#R'\\F3F/F4F/!\"\"*&$\"5'zv#[omuv*z*!#@F/F8F/F/*&$\"5u\\T$>uF/FDFDF&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;minimum~abs olute~error~->~G$\"/n'R0p1b\"!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%; maximum~absolute~error~->~G$\".$)R0p1b\"!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%*ratio~->~G$\"5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"xG6\"6$%) operatorG%&arrowGF&*&,*$\"5j.>o,I,++5!#>\"\"\"*&$\"5p$fzaA/mg.&!#?F/9$ F/F/*&$\"5N$*[(QRW>g,\"F3F/)F4\"\"#F/F/*&$\"5E$RwZ$e>-I&)!#AF/)F4\"\"$ F/F/F/,*F/F/*&$\"5sGM%R)Hu#R'\\F3F/F4F/!\"\"*&$\"5'zv#[omuv*z*!#@F/F8F /F/*&$\"5u\\T$>uF/FDFDF&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&$\"5j.>o,I,++5!#>\"\"\"*&,&$\"5p$fzaA/mg.&!#?F(*&,&$\"5N$*[( QRW>g,\"F-F(*&$\"5E$RwZ$e>-I&)!#AF(%\"xGF(F(F(F6F(F(F(F6F(F(F(,&F(F(*& ,&$!5sGM%R)Hu#R'\\F-F(*&,&$\"5'zv#[omuv*z*!#@F(*&$\"5u\\T$>u " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 61 "Comparison of polynomial and rational minimax a pproximations " }}{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+.5 036066042*x+.1016019444*x^2+.8530021958e-2*x^3)/(1.-.4963927430*x+.979 9757467e-1*x^2-.7943574177e-2*x^3)" "6#/-%\"rG6#%\"xG*&,*-%&FloatG6$\" +I,++5!\"*\"\"\"*&-F+6$\"+Ug1O]!#5F/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$\"+I u#R'\\F4F/F'F/!\"\"*&-F+6$\"+nuv*z*!#6F/*$F'F:F/F/*&-F+6$\"+xTdVzF?F/* $F'FAF/FJFJ" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 164 "for exp(x ) on the interval [-1,1] can be evaluated using nested multiplication \+ for the numerator and denominator with 6 additions, 6 multiplications \+ and 1 division." }}{PARA 0 "" 0 "" {TEXT -1 36 "The maximum absolute e rror is about " }{XPPEDIT 18 0 "1.55*`.`*10^(-7);" "6#*(-%&FloatG6$\"$ b\"!\"#\"\"\"%\".GF))\"#5,$\"\"(!\"\"F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The degree 7 " } {TEXT 260 18 "minimax polynomial" }{TEXT -1 15 " approximation " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "p(x)=.9999998014+.99 99998235*x+.5000063462*x^2+.1666686052*x^3+.4163501497e-1*x^4" "6#/-% \"pG6#%\"xG,,-%&FloatG6$\"+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%*$F- \"\"(F%F%" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 84 "can be evalu ated using nested multiplication with 7 additions and 7 multiplication s." }}{PARA 0 "" 0 "" {TEXT -1 36 "The maximum absolute error is about " }{XPPEDIT 18 0 "2*`.`*10^(-7);" "6#*(\"\"#\"\"\"%\".GF%)\"#5,$\"\"( !\"\"F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 141 "If division \+ can be performed as economically as multiplication, the rational appro ximation performs better than the polynomial approximation." }}{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 }