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$F[gm$FbdmFdbmF\\dmF`dm-F$6%7$7$$\"3J+++Z8fTrF]oF[cm7$FcgmFefmF\\dmF`d m-F$6%7$7$$\"39+++S[Yf#*F]oF[cm7$FjgmF^gmF\\dmF`dm-F$6%7$7$$FccmFdbmF[ cm7$Fahm$!\"#FdbmF\\dmF`dm-%%TEXTG6$7$$!$-\"Fdhm$\"#:FdhmQ#x16\"-Ffhm6 $7$$!#$*Fdhm$!#:FdhmQ#x2F^im-Ffhm6$7$$!#rFdhmF[imQ#x3F^im-Ffhm6$7$$!#P FdhmFdimQ#x4F^im-Ffhm6$7$$FccmFdhmF[imQ#x5F^im-Ffhm6$7$$\"#RFdhmFdimQ# x6F^im-Ffhm6$7$$\"#sFdhmF[imQ#x7F^im-Ffhm6$7$$\"#$*FdhmFdimQ#x8F^im-Ff hm6$7$$\"$-\"FdhmF[imQ#x9F^im-%*AXESTICKSG6$FdbmFdbm-%+AXESLABELSG6%Q \"xF^imQ\"yF^im%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$;$!$0\"Fdhm$ \"$0\"Fdhm;FchmF^gm" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu rve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve \+ 19" "Curve 20" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "The phenomena of oscillation of the error curve is called " }{TEXT 260 12 "equal ripple" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Example " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 32 "The minimax polynom ial of degree" }{XPPEDIT 18 0 "``<=7" "6#1%!G\"\"(" }{TEXT -1 20 " whi ch approximates " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F% " }{TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "p(x)=.9999998014+.9999998235*x+.5000063462*x^2+.1666686052*x^3+.416 3501497e-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$\"+_g om;F-F.*$F'\"\"$F.F.*&-F*6$\"+(\\,N;%!#6F.*$F'\"\"%F.F." }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` +.8327356910e -2*x^5+.1439272379e-2*x^6+.2054080265e-3*x^7" "6#,*%!G\"\"\"*&-%&Float G6$\"+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 46 "where the coefficients are given to 10 digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 202 "p := x - > .9999998014+.9999998235*x+.5000063462*x^2+\n .1666686052*x^3+.41635 01497e-1*x^4+.8327356910e-2*x^5+\n .1439272379e-2*x^6+.2054080265e-3* x^7;\nplot(p(x)-exp(x),x=-1..1,color=COLOR(RGB,.7,0,.9));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,2$\"+ 9!)******!#5\"\"\"*&$\"+N#)******F/F09$F0F0*&$\"+iM1+]F/F0)F4\"\"#F0F0 *&$\"+_gom;F/F0)F4\"\"$F0F0*&$\"+(\\,N;%!#6F0)F4\"\"%F0F0*&$\"+5pNF$)! #7F0)F4\"\"&F0F0*&$\"+zBFR9FHF0)F4\"\"'F0F0*&$\"+l-3a?!#8F0)F4\"\"(F0F 0F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 369 190 190 {PLOTDATA 2 "6&-%'CU RVESG6#7iu7$$!\"\"\"\"!$!3u`A^C%*e)*>!#C7$$!3-n;HdNvs**!#=$!35__!*[Rlo ;F-7$$!3/MLe9r]X**F1$!3]_uN.k+d8F-7$$!3/,](=ng#=**F1$!3C(4+x2nI1\"F-7$ 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "e := x -> p(x)-exp(x);\nd := unapply(diff(e(x),x),x);\nplot(d(x),x=-1..1,color=COLOR(RGB,.6,.4,0)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"eGf*6#%\"xG6\"6$%)operatorG%& arrowGF(,&-%\"pG6#9$\"\"\"-%$expGF/!\"\"F(F(F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"dGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,2$\"+N#)***** *!#5\"\"\"*&$\"+#p7++\"!\"*F09$F0F0*&$\"+c\"e++&F/F0)F5\"\"#F0F0*&$\"+ *f+am\"F/F0)F5\"\"$F0F0*&$\"+b%yO;%!#6F0)F5\"\"%F0F0*&$\"+uUjN')!#7F0) F5\"\"&F0F0*&$\"+'=cyV\"FIF0)F5\"\"'F0F0-%$expG6#F5!\"\"F(F(F(" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7ip7 $$!\"\"\"\"!$\"3)*=Ywb!\\_C\"!#A7$$!3/MLe9r]X**!#=$\"3.?COz0+66F-7$$!3 %pmm\"HU,\"*)*F1$\"3o(\\F\"e2v^)*!#B7$$!3()***\\PM@l$)*F1$\"3!z$f\"[hf Un)F97$$!3!RLL$e%G?y*F1$\"3!==SaYiTd(F97$$!3u****\\(oUIn*F1$\"3=\"o#eV *RJf&F97$$!3ommm;p0k&*F1$\"3iMXjbyd$)QF97$$!3E++vV5Su$*F1$\"3C$*Gx'*=@ '[\"F97$$!3wKL$3>Nq9VGF97$$ 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3#z****\\_qn2#F1$\"3)*o')=Z0(yj\"F97$$\"3=)**\\P/q%zAF1$\"3>+TtrH%\\i \"F97$$\"3U)***\\i&p@[#F1$\"3gDBZt]%pc\"F97$$\"3L)**\\(=GB2FF1$\"3QV%> !H)e,X\"F97$$\"3B)****\\2'HKHF1$\"3u*R4%*3k-G\"F97$$\"3ElmmmZvOLF1$\"3 #pE[:sKna)FX7$$\"3i******\\2goPF1$\"3#3#Q.7=MpEFX7$$\"3UKL$eR<*fTF1$!3 3^(*)>&*=nI$FX7$$\"3m******\\)Hxe%F1$!3I`Kej3Y#y*FX7$$\"3ckm;H!o-*\\F1 $!3C16vZ%f)*\\\"F97$$\"3nJL$3A_1?&F1$!3eN<,$[b-r\"F97$$\"3y)***\\7k.6a F1$!3/ZsDNw#G'=F97$$\"3IKLe9as;cF1$!3%o**G:m)zX>F97$$\"3#emmmT9C#eF1$! 3G$H?'=lPa>F97$$\"3WKLeRq\"F97$$\"3Ymm\"z\\%[gkF1$!3koZ&zwy?W\"F97$$\"3%QLLL$*zym'F1$! 3%Gz+qE[H4\"F97$$\"3wKLL3N1#4(F1$!3AMl>w'*[.8FX7$$\"3Nmm;HYt7vF1$\"3)R \"R?iUjW5F97$$\"3Y*******p(G**yF1$\"3'GYj`%yr:@F97$$\"3)HLL3U/37)F1$\" 3gg)H+#4P7EF97$$\"3]mmmT6KU$)F1$\"3MvKCZ0/KHF97$$\"3w**\\iSH&=R)F1$\"3 y/@HKw@qHF97$$\"3-LLeRZQT%)F1$\"3I-9')QYz$*HF97$$\"3Gm;aQl\"4\\)F1$\"3 !yp%o=ib,IF97$$\"3a****\\P$[/a)F1$\"3q=Ys4qB#*HF97$$\"3!GLek8!)**e)F1$ \"3k)Qq\"*Q?X'HF97$$\"32mmTN>^R')F1$\"3C(oUL?Pq\"HF97$$\"3K**\\PMP/*o) F1$\"35?Js![k$[GF97$$\"3fKLLLbdQ()F1$\"3o&\\6T7Cqv#F97$$\"3amm\"zW?)\\ *)F1$\"3+9.M&3#*e2#F97$$\"3[++]i`1h\"*F1$\"34+Qs\"3:u@)FX7$$\"3Y++DJ&f @E*F1$!37>fQ=RI([#F[v7$$\"3Y++++PDj$*F1$!3/bJf&zHe0\"F97$$\"3Y++voyMk% *F1$!3CY]zq>%4H#F97$$\"3W++]P?Wl&*F1$!3k*)*y&Q " 0 "" {MPLTEXT 1 0 307 "n := 40;\nh := 2./n;\nxx := -1.;\ncritpts := xx:\nval := d(xx);\nfor i to n do\n newx := xx + h;\n newval := d (newx);\n if signum(val)<>signum(newval) then\n critpts := crit pts,fsolve(d(x),x=xx..newx);\n end if;\n xx := newx;\n val := ne wval;\nend do:\ncritpts := [critpts,1.];\nevalf[15](map(e,critpts));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"+++++]!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#xxG$!\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$valG$\"'DX7!#5 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(critptsG7+$!\"\"\"\"!$!+j7\"*=# *!#5$!+)3!#:$\"*ATz *>F&$!*1y%)*>F&$\"*Ml\")*>F&$!)>B)*>!#9$\")yC)*>F/$!)AV)*>F/$\")M'y*>F /$!)bt)*>F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 78 "The calculation of a minimax \+ polynomial .. introduction to the Remez algorithm" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 50 "We consider the problem of finding the polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\" pG6#%\"xG" }{TEXT -1 10 " of degree" }{XPPEDIT 18 0 "``<=2" "6#1%!G\" \"#" }{TEXT -1 19 " that approximates " }{XPPEDIT 18 0 "f(x) = sqrt(x) ;" "6#/-%\"fG6#%\"xG-%%sqrtG6#F'" }{TEXT -1 45 " with minimax absolute error in the interval " }{XPPEDIT 18 0 "[1/4,1]" "6#7$*&\"\"\"F%\"\"% !\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 31 "Suppose that t he polynomial is " }{XPPEDIT 18 0 "p(x)=a[0]+a[1]*x+a[2]*x^2" "6#/-%\" pG6#%\"xG,(&%\"aG6#\"\"!\"\"\"*&&F*6#F-F-F'F-F-*&&F*6#\"\"#F-*$F'F4F-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 "1/4=x[1]" "6#/*&\"\"\"F%\"\"%!\"\"&%\"xG6#F% " }{XPPEDIT 18 0 "`` " 0 "" {MPLTEXT 1 0 203 "a := table():\nunassign('mu');\np := x -> a[0]+a[1]*x+a[2]*x^2;\nc := [ 1/4,2/5,4/5,1.];\neqns := seq(p(c[i])-sqrt(c[i])=(-1)^(i+1)*mu,i=1..no ps(c));\nsolns := solve(\{eqns\});\np1 := unapply(subs(solns,p(x)),x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operatorG%&a rrowGF(,(&%\"aG6#\"\"!\"\"\"*&&F.6#F1F19$F1F1*&&F.6#\"\"#F1)F5F9F1F1F( F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG7&#\"\"\"\"\"%#\"\"#\"\" &#F(F+$F'\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqnsG6&/,*&%\"aG6 #\"\"!\"\"\"*&#F,\"\"%F,&F)6#F,F,F,*&#F,\"#;F,&F)6#\"\"#F,F,#F,F7!\"\" %#muG/,*F(F,*&#F7\"\"&F,F0F,F,*&#F/\"#DF,F5F,F,*&F?F9\"#5#F,F7F9,$F:F9 /,*F(F,*&#F/F?F,F0F,F,*&#F4FBF,F5F,F,*(F7F,F?F9F?FEF9F:/,*F(F,*&$F,F+F ,F0F,F,*&FQF,F5F,F,$F,F+F9FF" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sol nsG<&/&%\"aG6#\"\"!$\"+%*>UWE!#5/&F(6#\"\"#$!+'o1w)HF-/%#muG$\"+:*>%zM !#7/&F(6#\"\"\"$\"+\\!R3.\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# p1Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,($\"+%*>UWE!#5\"\"\"*&$\"+\\!R3 .\"!\"*F09$F0F0*&$\"+'o1w)HF/F0)F5\"\"#F0!\"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "The error curve alread y looks close to having the equal ripple property." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plot(p1(x)-s qrt(x),x=1/4..1,color=COLOR(RGB,.7,0,.9));" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7do7$$\"3++++++++D!#= $\"3Y****\\i)>%zM!#?7$$\"3.+D1k'p3a#F*$\"3P3,NmM_1IF-7$$\"31+]7G$RYIF*$!3!\\J:\"F-7$$\"3?+]i:FrEJF*$! 3-$4YtwV?i\"F-7$$\"3:]i!*yJ%o?$F*$!3WhU-T6#z*>F-7$$\"35+v=UO(pG$F*$!3: *y\"=#GaPK#F-7$$\"3'**\\i!R[EhLF*$!3%p*zJS4%Re#F-7$$\"3\")*\\Pf.cbV$F* $!3j:bl:1H1GF-7$$\"33+vV)\\/%*e$F*$!3e>#=,@gh:$F-7$$\"3y*\\P%['f*oOF*$ !3ydYa2hX$G$F-7$$\"3-+vV)z9&[PF*$!39\"G/WrhtP$F-7$$\"38+D19[\"y#QF*$!3 Dy$=)3%y*RMF-7$$\"3C+voH[62RF*$!3'3YVwxxPZ$F-7$$\"3SD\"Gjt**y%RF*$!3#o $RS!y*f![$F-7$$\"3/](oHk%o))RF*$!3%)4^:xff![$F-7$$\"3lu$4'\\&p%HSF*$!3 !p1(3eG-uMF-7$$\"3#)***\\iXa-2%F*$!3Aa_@R/8hMF-7$$\"31+v=#>,@9%F*$!3#e 1;*=zyBMF-7$$\"3I+]7Gz%R@%F*$!3qC0cif()oLF-7$$\"3')***\\Pu7dP%F*$!3u_t 0j6C(=$F-7$$\"3B++D\"yT\"QXF*$!3,F4km`pMHF-7$$\"3%)***\\iDsYp%F*$!3rxT =.78OEF-7$$\"3D+vVBp\"o$[F*$!3BiDz*oflK#F-7$$\"3/++D\"oSe+&F*$!3/DG1+2 i?>F-7$$\"3/+++vf-\\^F*$!3#p))zgwi?b\"F-7$$\"3&**\\P4^%e:`F*$!3/Ze\"oz \"G.6F-7$$\"3y****\\Pq,jaF*$!3]/fWye;ZpFL7$$\"3h*\\Pf)QxCcF*$!3@vH_R'H @T#FL7$$\"3k*\\7yl/)ydF*$\"3a\"*p\"z]Vd*=FL7$$\"3B+]P4'>&RfF*$\"3ihEs8 h:=jFL7$$\"3!3](o/h5(3'F*$\"3G-V3#*ySE5F-7$$\"3%***\\7y!)HYiF*$\"3`1)= RmnXV\"F-7$$\"3%**\\P%)z`;T'F*$\"38_\\#4!f/M=F-7$$\"3!**\\i!RkfblF*$\" 3yWB;f-Qd@F-7$$\"3;+]PM%e5r'F*$\"3SK8t!omqZ#F-7$$\"3G++]()emroF*$\"35E )Q]4n5x#F-7$$\"3y**\\(o%*)yGqF*$\"3&pocKWO\">IF-7$$\"3)**\\Pfe83=(F*$ \"3$*yb'H')*[=KF-7$$\"3Y+]7G5h\\tF*$\"3C=WN^%z\"*Q$F-7$$\"3Z****\\PIG, vF*$\"3O7Ibn1*Q\\$F-7$$\"3C**\\PfTD#e(F*$\"3%[u*G[.$*HNF-7$$\"37++D\"G DKm(F*$\"3e\")GZi7i^NF-7$$\"3M\\PM_rfOxF*$\"3'****eF$fbeNF-7$$\"3m*\\P M-p*4yF*$\"3c=!)3O;5`NF-7$$\"3/]PfeQ=!*yF*$\"3FC*34uME`$F-7$$\"3U++v$p )RqzF*$\"3eV-vfMn'\\$F-7$$\"3\\*\\P4^]87)F*$\"3'y)RoC%zdQ$F-7$$\"35+vo a'Q\"z#)F*$\"3CX-?Aq\\2KF-7$$\"3o***\\iS0MV)F*$\"3AS)ylKw#pHF-7$$\"3k* \\P%)R3\\f)F*$\"3;`2jiX!)\\EF-7$$\"3[+++v\\X]()F*$\"3)f5!4Mq`@J!e\"485F-7$$\"3O++]()GB7#*F*$\"3I\"[VU.$>usFL 7$$\"3G+D\"yl,`H*F*$\"3=,$f26,mz$FL7$$\"3?+]7G/Py$*F*$\"3Coqn\")HX>**! #B7$$\"37+DcE\"oEX*F*$!3gX^oVC7'R$FL7$$\"3++++De'p_*F*$!3O'3b1Y:62(FL7 $$\"34](ozm#=1'*F*$!3F-7$$\"3;+D1k2/P)*F*$!350[(\\&3kQCF- 7$$\"3k]7.#Q?&=**F*$!3#G,4uK(fZHF-7$$\"\"\"\"\"!$!3UQ,++-UzMF--%&COLOR G6&%$RGBG$\"\"(!\"\"F[bl$\"\"*Fdbl-%+AXESLABELSG6$Q\"x6\"Q!F[cl-%%VIEW G6$;$\"+++++D!#5Fial%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 88 "The actual critical points are not very d ifferent from those given in the initial guess." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "d := unapply (diff(p1(x)-sqrt(x),x),x);\nsolve(d(x),x);\nc := sort([0.25,%,1]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,($\"+\\!R3.\"!\"*\"\"\"*&$\"+sL@vf!#5F09$F0!\"\"*&#F0\"\"#F0*&F0F0 *$F5#F0F9F6F0F6F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+c\\DTx!#5 $\"+VR?oRF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG7&$\"#D!\"#$\"+VR ?oR!#5$\"+c\\DTxF+\"\"\"" }}}{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 0 246 "eqns := seq(p(c[i])-sqrt(c[i])=(-1)^i*mu,i=1..nops(c )):\nsolns := solve(\{eqns\}):\np2 := unapply(subs(solns,p(x)),x);\npl ot(p2(x)-sqrt(x),x=1/4..1,color=COLOR(RGB,.7,0,.9));\nd := unapply(dif f(p2(x)-sqrt(x),x),x):\nsolve(d(x),x):\nc := sort([0.25,%,1]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2Gf*6#%\"xG6\"6$%)operatorG%&arrow GF(,($\"+NszXE!#5\"\"\"*&$\"+m@GI5!\"*F09$F0F0*&$\"+6Qk$)HF/F0)F5\"\"# F0!\"\"F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6 &-%'CURVESG6#7do7$$\"3++++++++D!#=$\"3Z6+D\"=\"\\-N!#?7$$\"3.+D1k'p3a# F*$\"3*4ZV\"\\_8GIF-7$$\"31+]7G$RYIF*$!3u.Up&Hde=\"F-7$$\"3?+]i:FrEJF*$!3B`Ry*f)*)>;F-7$$\"3:]i!*yJ%o? $F*$!3'=h055F#)*>F-7$$\"35+v=UO(pG$F*$!3)f\\G$z/YEBF-7$$\"3'**\\i!R[Eh LF*$!3].Lz,q#))e#F-7$$\"3\")*\\Pf.cbV$F*$!39!G.yz7L\"GF-7$$\"33+vV)\\/ %*e$F*$!3)o;D!epYnJF-7$$\"3y*\\P%['f*oOF*$!3sBn)H#[!pH$F-7$$\"3-+vV)z9 &[PF*$!3C3h8NA!HR$F-7$$\"38+D19[\"y#QF*$!3k8s>0TbdMF-7$$\"3C+voH[62RF* $!3#Q5]'3)QL\\$F-7$$\"3SD\"Gjt**y%RF*$!3fuK(H[i6]$F-7$$\"3/](oHk%o))RF *$!3hqh_nr9-NF-7$$\"3lu$4'\\&p%HSF*$!3'>@*\\_$\\l\\$F-7$$\"3#)***\\iXa -2%F*$!3uAw!R0>Y[$F-7$$\"31+v=#>,@9%F*$!3G:;IP$R*[MF-7$$\"3I+]7Gz%R@%F *$!3EFlGs#\\cR$F-7$$\"3')***\\Pu7dP%F*$!3gZUBYk^F-7$$\"3/+++vf-\\^F *$!3+e6rqN@'f\"F-7$$\"3&**\\P4^%e:`F*$!3GVf7!z+)\\6F-7$$\"3y****\\Pq,j aF*$!3lr!f.y$[JuFL7$$\"3h*\\Pf)QxCcF*$!3%f1(R#)>X:HFL7$$\"3k*\\7yl/)yd F*$\"39)pYb-_iP\"FL7$$\"3B+]P4'>&RfF*$\"31DV&[U)z$y&FL7$$\"3!3](o/h5(3 'F*$\"3v!=*emD(yr*FL7$$\"3%***\\7y!)HYiF*$\"3ES(zM_i)y8F-7$$\"3%**\\P% )z`;T'F*$\"3S\\')\\NpUxXM>CF-7$$\"3G++]()emroF*$\"3&e?$4Pf/8FF-7$$\"3y**\\ (o%*)yGqF*$\"3KO3;81-hHF-7$$\"3)**\\Pfe83=(F*$\"3A(Gi,oo/;$F-7$$\"3Y+] 7G5h\\tF*$\"3g2ZbjzZJLF-7$$\"3Z****\\PIG,vF*$\"3vq!3/1pmV$F-7$$\"3C** \\PfTD#e(F*$\"3/,D#)y&RIZ$F-7$$\"37++D\"GDKm(F*$\"3'H&pX%H8^\\$F-7$$\" 3M\\PM_rfOxF*$\"3*oUqqrRC]$F-7$$\"3m*\\PM-p*4yF*$\"3IE*e/%)>u\\$F-7$$ \"3/]PfeQ=!*yF*$\"3Gp?].nZxMF-7$$\"3U++v$p)RqzF*$\"38*)z;y,4UMF-7$$\"3 \\*\\P4^]87)F*$\"3ZlvgGgTKLF-7$$\"35+voa'Q\"z#)F*$\"37?)H9\"=gbJF-7$$ \"3o***\\iS0MV)F*$\"3eC%)*oI2!>HF-7$$\"3k*\\P%)R3\\f)F*$\"323i07,W,EF- 7$$\"3[+++v\\X]()F*$\"34%=sHMUeA#F-7$$\"32+]i:Q_4*)F*$\"3)H$*>PjP)o8S#)oFL7$$\"3G+D\"yl,`H*F*$\"3r)otl(\\Z >MFL7$$\"3?+]7G/Py$*F*$!3zMS5R6>FE!#A7$$\"37+DcE\"oEX*F*$!3K4*p1:\"F- 7$$\"3=+v$4^*R&o*F*$!3mAG8xzn\"e\"F-7$$\"3i****\\P,Ah(*F*$!3nm.#Q,iQ,# F-7$$\"3;+D1k2/P)*F*$!3]:\"Ho$yWlCF-7$$\"3k]7.#Q?&=**F*$!3i1%yv#GcsHF- 7$$\"\"\"\"\"!$!34N*****f\"\\-NF--%&COLORG6&%$RGBG$\"\"(!\"\"F[bl$\"\" *Fdbl-%+AXESLABELSG6$Q\"x6\"Q!F[cl-%%VIEWG6$;$\"+++++D!#5Fial%(DEFAULT G" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" } }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG7&$\"#D!\"#$\"+F%HU(R!#5$\"+K8 sVxF+\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "One more iteration produces little change within the accuracy p ossible using 10 digit arithmetic." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 251 "a := table(): unassign(mu): p := x -> a[0]+a[1]*x+a[2]*x^2:\neqns := seq(p(c[i])-sqrt(c[i])=(-1)^ i*mu,i=1..nops(c)):\nsolns := solve(\{eqns\}):\np3 := unapply(subs(sol ns,p(x)),x);\nd := unapply(diff(p3(x)-sqrt(x),x),x):\nsolve(d(x),x):\n c := sort([0.25,%,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3Gf*6#% \"xG6\"6$%)operatorG%&arrowGF(,($\"+/qzXE!#5\"\"\"*&$\"+\\CGI5!\"*F09$ F0F0*&$\"+)pYO)HF/F0)F5\"\"#F0!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG7&$\"#D!\"#$\"+$4GU(R!#5$\"+#>9Pu(F+\"\"\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "An error estimate for t he minimax polynomial " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 76 "We can estimate the error obtained by \+ the minimax polynomial for a function " }{XPPEDIT 18 0 "phi(x);" "6#-% $phiG6#%\"xG" }{TEXT -1 16 " on an interval " }{XPPEDIT 18 0 "[a,b]" " 6#7$%\"aG%\"bG" }{TEXT -1 165 " without actually constructing it expli citly, provided that we have some polynomial whose error characteristi cs are close to those of the genuine minimax polynomial." }}{PARA 0 " " 0 "" {TEXT -1 42 "Indeed, suppose that we have a polynomial " } {XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 10 " of degree" } {XPPEDIT 18 0 "``<=n" "6#1%!G%\"nG" }{TEXT -1 20 ", and a sequence of \+ " }{XPPEDIT 18 0 "n+2" "6#,&%\"nG\"\"\"\"\"#F%" }{TEXT -1 8 " points \+ " }{XPPEDIT 18 0 "x[1],x[2],` . . . `,x[n+2]" "6&&%\"xG6#\"\"\"&F$6#\" \"#%(~.~.~.~G&F$6#,&%\"nGF&F)F&" }{TEXT -1 6 " with " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a= ``" "6#/%\"aG%!G" }{XPPEDIT 18 0 "x[1] < x[2];" "6#2&%\"xG6#\"\"\"&F%6#\"\"#" }{XPPEDIT 18 0 "`` " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "Proof of error estimate " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 26 "We need to establ ish that " }{XPPEDIT 18 0 "e[min];" "6#&%\"eG6#%$minG" }{TEXT -1 40 " \+ is a lower bound for the minimax error." }}{PARA 0 "" 0 "" {TEXT -1 27 "Assume that the inequality " }{XPPEDIT 18 0 "e[min] <= mu;" "6#1&% \"eG6#%$minG%#muG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "abs(abs(p[`#`]-ph i));" "6#-%$absG6#-F$6#,&&%\"pG6#%\"#G\"\"\"%$phiG!\"\"" }{TEXT -1 20 " is false, that is, " }{XPPEDIT 18 0 "mu < e[min];" "6#2%#muG&%\"eG6# %$minG" }{TEXT -1 62 ". We shall show that this assumption leads to a \+ contradiction." }}{PARA 0 "" 0 "" {TEXT -1 91 "We start by noting that a consequence of the assumption is that there must be a polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 11 " such that " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "mu <= abs(abs(p-phi)) ;" "6#1%#muG-%$absG6#-F&6#,&%\"pG\"\"\"%$phiG!\"\"" }{XPPEDIT 18 0 "`` " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "The calculation of a minimax polynomial for " }{XPPEDIT 18 0 "exp(x)" "6#-%$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 50 "We consider the p roblem of finding the polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#% \"xG" }{TEXT -1 10 " of degree" }{XPPEDIT 18 0 "`` <= n;" "6#1%!G%\"nG " }{TEXT -1 19 " that approximates " }{XPPEDIT 18 0 "f(x) = exp(x);" " 6#/-%\"fG6#%\"xG-%$expG6#F'" }{TEXT -1 45 " with minimax absolute erro r in the interval " }{XPPEDIT 18 0 "[-1, 1];" "6#7$,$\"\"\"!\"\"F%" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 31 "Suppose that the polynom ial is " }{XPPEDIT 18 0 "p(x) = a[0]+a[1]*x+a[2]*x^2+` . . . `+a[n]*x^ n;" "6#/-%\"pG6#%\"xG,,&%\"aG6#\"\"!\"\"\"*&&F*6#F-F-F'F-F-*&&F*6#\"\" #F-*$F'F4F-F-%(~.~.~.~GF-*&&F*6#%\"nGF-)F'F:F-F-" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "First we \+ select " }{XPPEDIT 18 0 "n+2" "6#,&%\"nG\"\"\"\"\"#F%" }{TEXT -1 10 " \+ numbers, " }{XPPEDIT 18 0 "x[1],x[2],x[3],` . . . `,x[n+2];" "6'&%\"xG 6#\"\"\"&F$6#\"\"#&F$6#\"\"$%(~.~.~.~G&F$6#,&%\"nGF&F)F&" }{TEXT -1 8 " with " }{XPPEDIT 18 0 "-1 = x[1];" "6#/,$\"\"\"!\"\"&%\"xG6#F%" } {XPPEDIT 18 0 "`` " 0 "" {MPLTEXT 1 0 276 "Dig its := 10:\nf := exp;\nn := 7;\na := table():\nunassign('mu'):\np := u napply(convert([seq(a[i]*x^i,i=0..n)],`+`),x);\nc := [seq(evalf(cos((n +2-k)*Pi/(n+1))),k=1..n+2)];\neqns := seq(p(c[i])-f(c[i])=(-1)^(i-1)*m u,i=1..n+2):\nsolns := solve(\{eqns\}):\np := unapply(subs(solns,p(x)) ,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%$expG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"p Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,2&%\"aG6#\"\"!\"\"\"*&&F.6#F1F19$ F1F1*&&F.6#\"\"#F1)F5F9F1F1*&&F.6#\"\"$F1)F5F>F1F1*&&F.6#\"\"%F1)F5FCF 1F1*&&F.6#\"\"&F1)F5FHF1F1*&&F.6#\"\"'F1)F5FMF1F1*&&F.6#\"\"(F1)F5FRF1 F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG7+$!\"\"\"\"!$!+D`zQ #*!#5$!+5y1rqF+$!+DV$o#QF+$F(F($\"+DV$o#QF+$\"+5y1rqF+$\"+D`zQ#*F+$\" \"\"F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operat orG%&arrowGF(,2$\"+3!)******!#5\"\"\"*&$\"+T#)******F/F09$F0F0*&$\"+xN 1+]F/F0)F4\"\"#F0F0*&$\"+0gom;F/F0)F4\"\"$F0F0*&$\"+]$)\\jT!#6F0)F4\" \"%F0F0*&$\"+u[OF$)!#7F0)F4\"\"&F0F0*&$\"+kLHR9FHF0)F4\"\"'F0F0*&$\"+k *QS0#!#8F0)F4\"\"(F0F0F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 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\"3Z+](oW7;@*F1$\"37bMO\"Q+#z>F-7$$\"3\"**\\i!*)f)oB*F1$\"3#\\'Q+xxE() >F-7$$\"3Y++DJ&f@E*F1$\"31#zjeX>(*)>F-7$$\"3-,vVtIV(G*F1$\"3;8`NX$oi)> F-7$$\"3Y+]i:mq7$*F1$\"35SBkl(>m(>F-7$$\"3\"**\\7y:!)zL*F1$\"3eQs-j8Zg >F-7$$\"3Y++++PDj$*F1$\"3]7(y+&H^P>F-7$$\"3Y++voyMk%*F1$\"3[%Q2(*z%4r< F-7$$\"3W++]P?Wl&*F1$\"3W#ff=E!yo9F-7$$\"3Q+D\"Gyh(>'*F1$\"3,*)HH?:'>C \"F-7$$\"3K+]7G:3u'*F1$\"3U\"*H_(=\"o]'*FC7$$\"3G+vVt7SG(*F1$\"3QFM'Qr _\"RjFC7$$\"3A++v=5s#y*F1$\"3ZD/!RY*3TCFC7$$\"3v]iS\"*3))4)*F1$\"3-nO< T\"oyd#FS7$$\"3;+D1k2/P)*F1$!3)**)y#eR5'*3#FC7$$\"3e\\(=nj+U')*F1$!3.! f@879sg%FC7$$\"35+]P40O\"*)*F1$!38fUL-1G,tFC7$$\"3k]7.#Q?&=**F1$!3)z2b 1?@y,\"F-7$$\"31+voa-oX**F1$!35y.CefXC8F-7$$\"3[\\PMF,%G(**F1$!3Uz%zd_ -2l\"F-7$$\"\"\"F*$!3a>lA[kC(*>F--%&COLORG6&%$RGBG$\"\"(F)F*$\"\"*F)-% +AXESLABELSG6$Q\"x6\"Q!F`cn-%%VIEWG6$;F(F`bn%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 24 "The following procedure " }{TEXT 0 7 "cri tpts" }{TEXT -1 46 " can be used to calculate the critical points." }} {PARA 0 "" 0 "" {TEXT -1 14 "The parameter " }{TEXT 269 1 "f" }{TEXT -1 37 " is the function being approximated, " }{TEXT 269 1 "p" }{TEXT -1 34 " is the polynomial approximation, " }{TEXT 269 1 "a" }{TEXT -1 5 " and " }{TEXT 269 1 "b" }{TEXT -1 41 " are the end points of the in terval, and " }{TEXT 269 1 "N" }{TEXT -1 157 " is a suitably large int eger giving the number of steps to be used in searching for an interva l over which the derivative of the error function changes sign." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 402 "critpts := proc(f,p,a,b,N)\n local h,xx,newx,i,cpts,d,val,newva l;\n h := evalf((b-a)/N);\n xx := evalf(a);\n cpts := xx;\n d \+ := D(f-p);\n val := d(xx);\n for i to N do\n newx := xx + h; \n newval := d(newx);\n if signum(val)<>signum(newval) then \n cpts := cpts,fsolve(d(x),x=xx..newx);\n end if;\n \+ xx := newx;\n val := newval;\n end do:\n [cpts,evalf(b)]\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "It is probably a good idea to increase precision for further calcu lations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Digits := 20;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'D igitsG\"#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 139 "We can check how close we are to the minimax polynomial by det ermining the critical points of the approximation, and comparing the m aximum " }{XPPEDIT 18 0 "e[max]" "6#&%\"eG6#%$maxG" }{TEXT -1 17 " and the minimum " }{XPPEDIT 18 0 "e[min]" "6#&%\"eG6#%$minG" }{TEXT -1 46 " of the associated absolute errors. The ratio " }{XPPEDIT 18 0 "e[ max]/e[min]" "6#*&&%\"eG6#%$maxG\"\"\"&F%6#%$minG!\"\"" }{TEXT -1 74 " approaches 1 as the Remez approximaton approaches the minimax polynom ial." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "c := critpts(f,p,-1,1,50);\nec := map(abs@(f-p),c): \nemin := min(op(ec));\nemax := max(op(ec));\nemax/emin:\nevalf[10](%) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"cG7+$!\"\"\"\"!$!5f$\\$\\$\\f r(>#*!#?$!5PtRn!=0^`+(F+$!5T[%on:kz@r$F+$\"5&oD/**31=tP\"!#@$\"5!=*43? ZWCZRF+$\"5&yY@w$Q%yP9(F+$\"5mrd@QUOAg#*F+$\"\"\"F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%eminG$\".%\\Nnt*)>!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%emaxG$\".umLFT+#!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+:CB25!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "We can repeat the process using the new critical points. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "p := unapply(convert([seq(a[i]*x^i,i=0..n)],`+`),x):\neqns := seq(p(c[i])-f(c[i])=(-1)^(i-1)*mu,i=1..n+2):\nsolns := solve(\{eqns\} ):\np := unapply(subs(solns,p(x)),x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,2$\"5`-,T)R,)******!#?\" \"\"*&$\"5@5%HS[B)******F/F09$F0F0*&$\"5$45FDCYj++&F/F0)F4\"\"#F0F0*&$ \"5qa`NE_gom;F/F0)F4\"\"$F0F0*&$\"5P6_$*)p\\,N;%!#@F0)F4\"\"%F0F0*&$\" 55wc3\\/pNF$)!#AF0)F4\"\"&F0F0*&$\"5Ku.cWzBFR9FHF0)F4\"\"'F0F0*&$\"5AK _$e#*G!3a?!#BF0)F4\"\"(F0F0F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "c := critpts(f,p,-1,1,50);\nec := map(abs@(f-p),c):\nemin := min( op(ec));\nemax := max(op(ec));\nemax/emin:\nevalf[10](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"cG7+$!\"\"\"\"!$!5oV3%*\\s$e*=#*!#?$!5p*f)) 4$3\"*)H+(F+$!5%pck%p;tr4PF+$\"5dt!e-@f$e%Q\"!#@$\"5r*ylKCr%3YRF+$\"5@ %f)=BUlXTrF+$\"5Iu*Rjzh;&f#*F+$\"\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eminG$\"/;]u'\\#)*>!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%% emaxG$\".6A&oD)*>!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+#f.++\"!\" *" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "One more iteration produces little change in the first 10 digits of the c oefficients . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "p := unapply(convert([seq(a[i]*x^i,i=0..n)], `+`),x):\neqns := seq(p(c[i])-f(c[i])=(-1)^(i-1)*mu,i=1..n+2):\nsolns \+ := solve(\{eqns\}):\np := unapply(subs(solns,p(x)),x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,2$\"5nhB 7)R,)******!#?\"\"\"*&$\"5%[kd[[B)******F/F09$F0F0*&$\"5qn[3XiM1+]F/F0 )F4\"\"#F0F0*&$\"5A)HMFA0'om;F/F0)F4\"\"$F0F0*&$\"5 " 0 "" {MPLTEXT 1 0 117 "c := critpts(f,p,-1,1,50);\nec := map(abs@(f -p),c):\nemin := min(op(ec));\nemax := max(op(ec));\nemax/emin:\nevalf [15](%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"cG7+$!\"\"\"\"!$!51&=] Ph#z&*=#*!#?$!5gS)o[e%*))H+(F+$!5b'Q)Q>T%=(4PF+$\"5bNx)**QdwXQ\"!#@$\" 5tGMD(HZ&3YRF+$\"5I=BC(pfc99(F+$\"5Y6F9$H?;&f#*F+$\"\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eminG$\".`(pFD)*>!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%emaxG$\".c(pFD)*>!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0],+++++\"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "A degree 7 minimax polynomial approximation for " } {XPPEDIT 18 0 "f(x) = exp(x);" "6#/-%\"fG6#%\"xG-%$expG6#F'" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F% " }{TEXT -1 51 " with the coefficients given to 10 digits is . . . " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "Digits := 10:\np := unapply(evalf(p(x)),x);\nplot(p(x)-f(x),x=-1 ..1,color=COLOR(RGB,.7,0,.9));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\" pGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,2$\"+9!)******!#5\"\"\"*&$\"+N#) ******F/F09$F0F0*&$\"+iM1+]F/F0)F4\"\"#F0F0*&$\"+_gom;F/F0)F4\"\"$F0F0 *&$\"+(\\,N;%!#6F0)F4\"\"%F0F0*&$\"+5pNF$)!#7F0)F4\"\"&F0F0*&$\"+zBFR9 FHF0)F4\"\"'F0F0*&$\"+l-3a?!#8F0)F4\"\"(F0F0F(F(F(" }}{PARA 13 "" 1 " " {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7iu7$$!\"\"\"\"!$!3 u`A^C%*e)*>!#C7$$!3-n;HdNvs**!#=$!35__!*[Rlo;F-7$$!3/MLe9r]X**F1$!3]_u N.k+d8F-7$$!3/,](=ng#=**F1$!3C(4+x2nI1\"F-7$$!3%pmm\"HU,\"*)*F1$!3m&*H .$G#oiy!#D7$$!3'GLekynP')*F1$!3!=(fM:#R0E&FC7$$!3()***\\PM@l$)*F1$!3qh P#o^)z=GFC7$$!3)omT5!\\F4)*F1$!3L*f@)G.K@`!#E7$$!3!RLL$e%G?y*F1$\"3'=! pDf`n/;FC7$$!3#om;HdNvs*F1$\"3oz24@o,\\aFC7$$!3u****\\(oUIn*F1$\"3/l6' )=Ou`()FC7$$!3xLL3-)\\&='*F1$\"3I&y`JPmc:\"F-7$$!3ommm;p0k&*F1$\"3oW@@ PCR*Q\"F-7$$!3#HL3-)*G#p%*F1$\"3!*zT2@bF&p\"F-7$$!3E++vV5Su$*F1$\"3#p! 40\"z.w)=F-7$$!3&R$3_vq)pK*F1$\"3%)fNXO#[h%>F-7$$!3_m;H2Jdz#*F1$\"35;S *R+'4#)>F-7$$!33*\\i!R\"f@B*F1$\"3W4FzO2@(*>F-7$$!3wKL$3F-7$$!3'fmT&Q75y!*F1$\"3S3L1?VQA>F-7$$!39***\\iId9(*)F1$\"3u'* yoU'3)zR va8q(FC7$$!3+mm;Hp6O%)F1$\"3eB\"z9ENBl%FC7$$!3:mmm\"4m(G$)F1$\"3!>DZEG \"3#e\"FC7$$!3iL$eRZD>A)F1$!3@PRcwNVB9FC7$$!3)****\\i&[3:\")F1$!3\\)Qj A^x&=VFC7$$!3Mm;aQUC3!)F1$!3uU.a_=&G0(FC7$$!3\"QLL3i.9!zF1$!3UN%\\Du4Z e*FC7$$!3Mnmm\"p[B!yF1$!37R4CP`?s6F-7$$!3'3++Dw$H.xF1$!339*=54(ej8F-7$ $!3RMLLL)QUg(F1$!3b[#Q\\FV5`\"F-7$$!3\"ommT!R=0vF1$!31&>J/ljMn\"F-7$$! 3tKL$3i_+I(F1$!3'3=pk!fD')=F-7$$!3u****\\P8#\\4(F1$!3QEx!>bdz)>F-7$$!3 ILL$3d%)=/(F1$!3q)G4(y#3m*>F-7$$!3%ommT!y%)))pF1$!3-*Q0$o=B)*>F-7$$!3R ++]P5\"e$pF1$!3iDs!H=uH*>F-7$$!3#RLL3FuF)oF1$!39m82Qx*4)>F-7$$!3-,+]P2 qwnF1$!3k,1PiUhP>F-7$$!3+nm;/siqmF1$!3m1\"Q@Cn'p=F-7$$!3uLL$e\\g\"fkF1 $!3qgs#G'\\6o;F-7$$!3[++](y$pZiF1$!3iW6nn$[=R\"F-7$$!3MnmT&Gu,.'F1$!3y o0On.\"y/\"F-7$$!33LLL$yaE\"eF1$!3VA*>\"faPAmFC7$$!3%)****\\([j5i&F1$! 3`:2OT_;NIFC7$$!3hmmm\">s%HaF1$\"3W\"e(*fo%[*)fFS7$$!3]***\\7)*G;K&F1$ \"3@5C!='GL?EFC7$$!3]LL$3x&y8_F1$\"3W&f,rBN#*f%FC7$$!3]nmTgD%f5&F1$\"3 3i\"4P*H6eOv!p\"F-7$$!31+++]1aZTF1$\"3!G&GiMebc=F-7$$!3Ym;aQvx_SF1$\"3cKQ')) o$>6>F-7$$!3SLL3FW,eRF1$\"3SeRbrwj_>F-7$$!3M+]i:8DjQF1$\"3e1/l9ny!)>F- 7$$!3umm;/#)[oPF1$\"3gsYcYZi&*>F-7$$!3'**\\i:&p97PF1$\"3-t'Hy#4;)*>F-7 $$!3=L$e*)p0el$F1$\"3k71\"yGTg*>F-7$$!3%p;ajWk%*f$F1$\"39`f]aII*)>F-7$ $!3<++v$>BJa$F1$\"3/o@$Q+$*z(>F-7$$!3F-7$$! 3hLLL$=exJ$F1$\"37V'*pc)*G))=F-7$$!3!QLL$eW%o7$F1$\"3)[,v\\dU&f&H#F1$\"3Ux22=:%*pvFC7$$!3 EMLLL7i)4#F1$\"3\\'*)R**yo[^%FC7$$!3#pm;aVXH)=F1$\"3w(*f\")\\%Hr/\"FC7 $$!3c****\\P'psm\"F1$!3%))GdhFd\"RCFC7$$!3s*****\\F&*=Y\"F1$!3'H[1gxq2 o&FC7$$!3')****\\74_c7F1$!3q/O$pD*4e()FC7$$!3ZmmT5VBU5F1$!3ib`HKThq6F- 7$$!3)3LLL3x%z#)!#>$!36f3$o1?.V\"F-7$$!3gKL$e9d;J'F_dl$!3klJ-$\\\"fJ;F -7$$!3KMLL3s$QM%F_dl$!35w-:B&)G#z\"F-7$$!3T,+]ivF@AF_dl$!3F$z4j'4x:>F- 7$$!3]^omm;zr)*!#@$!3U.u@.e>%)>F-7$$\"3o'H$3x\"yY_%!#?$!3S!*=)o:&o#*>F -7$$\"3&yK$3_Nl.5F_dl$!31-wDVaI(*>F-7$$\"3/E$ekGR[b\"F_dl$!3#Gj,F\\Y!) *>F-7$$\"3@CL$3-Dg5#F_dl$!37hT%*p^!\\*>F-7$$\"3e?Le*['R3KF_dl$!3]zwvr# )*p(>F-7$$\"3%pJL$ezw5VF_dl$!38V=O`6qV>F-7$$\"3L`mmmJ+IiF_dl$!39A#>g*> v\\=F-7$$\"3s*)***\\PQ#\\\")F_dl$!3\"eBYSe;>r\"F-7$$\"3ilm\"z\\1A-\"F1 $!3dR50L2C<:F-7$$\"3GKLLe\"*[H7F1$!3!HHqsL0*z7F-7$$\"3ylm;HCjV9F1$!3A. 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#8F,)F'\"\"(F,F," }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 18 "Code for pictures " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 23 "Maximum absolute error " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 990 "fn := x -> 0.8+x^3/50-x/5+0 .1*cos(3*x):\ngn := x -> 0.78+x^3/50-x/5+0.12*sin(2.8*x):\np1:= plot([ fn(x),gn(x)],x=-1..3,\n color=[COLOR(RGB,0,.7,0),COLOR(RGB,.2,0,.9)],t ickmarks=[0,0]):\np2 := plot([[[-1,0],[-1,pn(-1.)]],[[3,0],[3,pn(3.)]] ],\n color=black,linestyle=2):\np3 := plot([[-1,0],[3,0]],color=tan ,thickness=2):\nxx := 1.88:\np4 := plot([[xx,gn(xx)],[xx,fn(xx)]],colo r=COLOR(RGB,1,.4,0),thickness=2):\nt1 := plots[textplot]([3.3,.82,`y = (x)`],color=COLOR(RGB,.2,0,.9)):\nt2 := plots[textplot]([3.3,.65,`y = (x)`],color=COLOR(RGB,0,.7,0)):\nt3 := plots[textplot]([[-1,-0.05 ,`a`],[3,-0.05,`b`],\n [3.6,-0.02,`x`],[-.1,1.13,`y`],\n \+ [2.65,.51,`[a,b]`]],color=black):\nt4 := plots[textplot]([2.2,.55,` || f - h ||`],font=[SYMBOL,10],color=black):\nt5 := plots[textplot]([3 .33,.83,f],font=[SYMBOL,10],color=COLOR(RGB,.2,0,.9)):\nt6 := plots[te xtplot]([3.33,.66,h],font=[SYMBOL,10],color=COLOR(RGB,0,.7,0)):\nplots [display]([p1,p2,p3,p4,t1,t2,t3,t4,t5,t6],view=[-1.3..3.6,-0.1..1.2]); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 31 "Error gr aph and critical points" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 735 "phi := x -> (.9999998014+.99999982 35*x+.5000063462*x^2+\n .1666686052*x^3+.4163501497e-1*x^4+.832735691 0e-2*x^5+\n .1439272379e-2*x^6+.2054080265e-3*x^7-exp(x))*10^7:\ncrit :=[-1.,-.9218911263,-.7003117088,-.3709591521,.1384511295e-1,\n .3 945963204,.7141591347,.9259464840,1.]:\nords := seq([[crit[i],0],[crit [i],phi(crit[i])]],i=1..9):\nxaxis := [[-1.05,0],[1.05,0]]:\np1 := plo t(phi(x),x=-1.05..1.05,y=-2..2,color=COLOR(RGB,.7,0,.9)):\np2 := plot( [xaxis,ords],color=[black,COLOR(RGB,1,.4,0)$9],\n thickness=[1,2$9 ]):\nt1 := plots[textplot]([[-1.02,.15,x1],[-.93,-.15,x2],[-.71,.15,x3 ],\n [-.37,-.15,x4],[.01,.15,x5],[.39,-.15,x6],[.72,.15,x7],\n [ .93,-.15,x8],[1.02,.15,x9]]):\nplots[display]([p1,p2,t1],tickmarks=[0, 0],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}} {MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }