{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 "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 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 265 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 262 266 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE " " -1 267 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {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 29 "The Remez algorithm: examples" } }{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 68 "load interp olation and function approximation procedures including: " }{TEXT 0 5 "remez" }}{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 5 "remez" }{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 the one that follows, where the file path gives its locatio n. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "read \"K:\\\\Maple/p rocdrs/fcnapprx.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "load root-finding procedures" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 271 7 "roots.m" }{TEXT -1 37 " \+ contains the code for the procedure " }{TEXT 0 6 "secant" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 122 "It can \+ be read into a Maple session by a command similar to the one that foll ows, where the file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\\\Maple/procdrs/roots.m\";" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "Example \+ 1 .. a pathological example" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 38 "We investigate an example involv ing a " }{TEXT 260 24 "non-standard error curve" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 22 "The rational function " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "r(x)=.4966935736/(1.-.2508590619*x) " "6#/-%\"rG6#%\"xG*&-%&FloatG6$\"+Od$p'\\!#5\"\"\",&-F*6$F.\"\"!F.*&- F*6$\"+>1f3DF-F.F'F.!\"\"F7" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 " approximates " }{XPPEDIT 18 0 "Gamma(x);" "6#-%&GammaG6#%\"xG" }{TEXT -1 17 " on the interval " } {XPPEDIT 18 0 "[2,3]" "6#7$\"\"#\"\"$" }{TEXT -1 29 " with minimax abs olute error." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 267 12 "Calcula tion " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "remez(GAMMA(x),x=2..3,[0,1],'maxerr','minerr',info=5) :\nr := unapply(%,x);\nmaxerr;\nminerr;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]pRemez~algorithm:~calculating~minimax~error~estimate~by~solvin g~a~rational~equationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%D--~minimis 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "evalf(fsolve(er1(x)=.74687745552221 474e-2,x=1.9..2),20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5d+E*[#4Hz ]>!#>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 196 "Alternatively, we can explicitly construct the rational approxima tion along with the associated interval arriving at the desired ration al approximation and the associated interval simultaneously. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "R := u -> remez(GAMMA(x),x=u..3,[0,1]);\nS := (rx,u) -> subs(x=u,G AMMA(x)-rx)+subs(x=3,GAMMA(x)-rx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"RGf*6#%\"uG6\"6$%)operatorG%&arrowGF(-%&remezG6%-%&GAMMAG6#%\"xG/F 2;9$\"\"$7$\"\"!\"\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG f*6$%#rxG%\"uG6\"6$%)operatorG%&arrowGF),&-%%subsG6$/%\"xG9%,&-%&GAMMA G6#F2\"\"\"9$!\"\"F8-F/6$/F2\"\"$F4F8F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "secant('S(R(u),u)' ,u=1.9..2,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximati on~1~~->~~~G$\".]]j\"pm>!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7appro ximation~2~~->~~~G$\".OU4#G\\>!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% 7approximation~3~~->~~~G$\".<;n.=&>!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\".G7E-6&>!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~5~~->~~~G$\".A=O#z]>!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~6~~->~~~G$\".`!4Lz]>!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~7~~->~~~G$\".e!GIz]>!#7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~8~~->~~~G$\".c#4Hz]>!# 7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~9~~->~~~G$\".X#4H z]>!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4Hz]>!\"*" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "evalf (secant('S(R(u),u)',u=1.95079290924..1.95079290925,info=true),15);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\"5!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\" 5R+E*[#4Hz]>!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~ ->~~~G$\"5r+E*[#4Hz]>!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0$*[#4H z]>!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "We can check that the associated minimax rational approximation fo r " }{XPPEDIT 18 0 "Gamma(x)" "6#-%&GammaG6#%\"xG" }{TEXT -1 17 " on t he interval " }{XPPEDIT 18 0 "[a,3]" "6#7$%\"aG\"\"$" }{TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 ".4966935736/(1.- .2508590619*x);" "6#*&-%&FloatG6$\"+Od$p'\\!#5\"\"\",&-F%6$F)\"\"!F)*& -F%6$\"+>1f3DF(F)%\"xGF)!\"\"F3" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 270 12 "Cal culation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "remez(GAMMA(x),x=1.95079290924893..3,[0,1],'maxerr', 'minerr',info=true):\nr := unapply(%,x);\nmaxerr;\nminerr;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]pRemez~algorithm:~calculating~minimax~erro r~estimate~by~solving~a~rational~equationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%D--~minimising~the~ABSOLUTE~error~--G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Wstandard~ Chebyshev~points~for~initial~critical~points:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\".[#4Hz]>!#7$\".CYX'RvCF%$\".++++++$F%" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,iterati on~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G--------------------------- -----------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Dprovisional~rational ~approximation:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&$\"1)**pr$)\\ \\+&!#;\"\"\",&F(!\"\"*&$\"170(R#[U(\\#F'F(%\"xGF(F(F*F*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Zv=%!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\".VzQW8y)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5relative~differenc e:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\".6(HQ,(4#!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>goal~for~relative~difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%,error~graphG" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6+-%'CURVESG6%7Z7$$\"3'****[#4Hz]>! 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"6#$\"&)ou!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"r Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&$\"+Od$p'\\!#5\"\"\",&$F1\"\"! !\"\"*&$\"+>1f3DF0F19$F1F1F5F5F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"+cXxou!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+bXxou!#7" }}} {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 64 "Example 2 .. approximating a function whi ch is slow to evaluate " }}{PARA 0 "" 0 "" {TEXT 265 8 "Question" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f( x)" "6#-%\"fG6#%\"xG" }{TEXT -1 28 " be the function defined by " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)=PIECEWISE([Int(1 /GAMMA(t),t = 0 .. x)/x^2 ,x<>0 ],[1/2 ,x=0 ])" "6#/-%\"fG6#%\"xG-%*PI ECEWISEG6$7$*&-%$IntG6$*&\"\"\"F1-%&GAMMAG6#%\"tG!\"\"/F5;\"\"!F'F1*$F '\"\"#F60F'F97$*&F1F1F;F6/F'F9" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "Find a rational function of the form " }{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 18 " has degree 7 and " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" } {TEXT -1 30 " has degree 6, to approximate " }{XPPEDIT 18 0 "f(x);" "6 #-%\"fG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0,4] " "6#7$\"\"!\"\"%" }{TEXT -1 29 " with minimax absolute error." }} {PARA 0 "" 0 "" {TEXT -1 77 "Find the maximum difference between the e xtreme values of the error function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "f := \+ x -> Int(1/GAMMA(t),t = 0 .. x)/(x^2);\nplot(f(x),x=-5.4..10);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,$-%$IntG6$*&\"\"\"F1-%&GAMMAG6#%\"tG!\"\"/F5;\"\"!9$*$F:!\"#F(F(F( " }}{PARA 13 "" 1 "" {GLPLOT2D 581 202 202 {PLOTDATA 2 "6%-%'CURVESG6$ 7^r7$$!#a!\"\"$!+/(4KL\"!#57$$!+R--z`!\"*$!+r@%fI)!#67$$!+y//e`F1$!+I= q=NF47$$!+<21P`F1$\"+!*4')>5F47$$!+c43;`F1$\"+,.u,`F47$$!+&>,^H&F1$\"+ -\">1K*F47$$!+M97u_F1$\"+%yzrI\"F-7$$!+t;9`_F1$\"+%)>@b;F-7$$!+8>;K_F1 $\"+@%*)f(>F-7$$!+!R-->&F1$\"+8#*yNDF-7$$!+pGC[^F1$\"+'*[&y)HF-7$$!+ZL G1^F1$\"+R3HNLF-7$$!+EQKk]F1$\"+)[nFe$F-7$$!+r]\"y-&F1$\"+Q_R@PF-7$$!+ :jI\"*\\F1$\"+eOX$z$F-7$$!+Q>0t\\F1$\"+j941QF-7$$!+gvza\\F1$\"+SP-/QF- 7$$!+#=Vl$\\F1$\"+dB%zy$F-7$$!+/))G=\\F1$\"+:abePF-7$$!+$Hr_%[F1$\"+:c ]ANF-7$$!+#y`Ax%F1$\"+fdNKJF-7$$!+az8!p%F1$\"+h'ywc#F-7$$!+E@-3YF1$\"+ bt))H>F-7$$!+)H1f_%F1$\"+d-^u7F-7$$!+q/zVWF1$\"+tCIzkF47$$!+D.8hVF1$\" +tDf%G)!#77$$!+\"=q%yUF1$!+%F1$!+%)3Uc2\" F-7$$!+)y!3%y$F1$!+qlr/5F-7$$!+(>O:j$F1$!+Oeu3kF47$$!+1;**yMF1$!+@(fD1 #F47$$!+=0/@LF1$\"+];P3@F47$$!+I%*3jJF1$\"+w5Mm^F47$$!+4GT\"3$F1$\"+q8 ^lhF47$$!+*=O(**HF1$\"+fvETnF47$$!+yy*)eHF1$\"+n\\xsoF47$$!+o&f!=HF1$ \"+UO4/pF47$$!+e7AxGF1$\"+&y*zRoF47$$!+ZHQOGF1$\"+txa&o'F47$$!+#e`Nn#F 1$\"+'*))e=`F47$$!+5#F1$!+N$QY$=F47$$!+'))=#G?F1$!+'=J!=AF47$$! +ivL\"*>F1$!+'3\"\\>BF47$$!+PiXa>F1$!+))e3dBF47$$!+7\\d<>F1$!+YdfGBF47 $$!+)e$p!)=F1$!+jmCKAF47$$!+W]h9'z$3'F-$\"+e!ftd&F-7$$\"+\\j?ikF -$\"+#[Gqd&F-7$$\"*];1%oF1$\"+(f=Id&F-7$$\"+?$o4n(F-$\"+tj!>b&F-7$$\"+ S,K,&)F-$\"+aw$\\^&F-7$$\"+g>nJ$*F-$\"+9:WjaF-7$$\"+yB?;5F1$\"+$[m()R& F-7$$\"+Q1Mu6F1$\"+@e)RC&F-7$$\"+(*)yCL\"F1$\"+#zj`0&F-7$$\"+k-[i;F1$ \"+'pp#*e%F-7$$\"+MX_l>F1$\"+Q.&*>TF-7$$\"+?()R#H#F1$\"+9`#Qh$F-7$$\"+ K\"H>j#F1$\"+'*Q.?JF-7$$\"+a8\\FHF1$\"+13EJFF-7$$\"+^mqYKF1$\"+o&=/O#F -7$$\"+)G([wNF1$\"+o_\"4.#F-7$$\"+.L6**QF1$\"+fOjdbKeF^jl$\"+Y( G_B)F47$$\"*Q1D9'F^jl$\"+,%e@V(F47$$\"*/)\\mkF^jl$\"+O[#*4nF47$$\"*5fK y'F^jl$\"+iz\")*4'F47$$\"*ey[6(F^jl$\"+P*oaa&F47$$\"*bnUV(F^jl$\"+$z-( z]F47$$\"*!*))3w(F^jl$\"+>\"z8m%F47$$\"*m0[3)F^jl$\"+VJY&H%F47$$\"*`^C Q)F^jl$\"+\\O)e*RF47$$\"*G(eB()F^jl$\"+0a]*o$F47$$\"*;.(G!*F^jl$\"+U]O WMF47$$\"*L?SN*F^jl$\"+U)e*3KF47$$\"*P!Rl'*F^jl$\"+=!Rb+$F47$$\"#5\"\" !$\"+T!px!GF4-%'COLOURG6&%$RGBG$F^_mF*$F__mF__mFg_m-%+AXESLABELSG6$Q\" x6\"Q!F\\`m-%%VIEWG6$;F(F]_m%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 " f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 111 " is very slow to evaluate when \+ using more than about 14 digit precision, so it is advantageous and co nstruct a " }{TEXT 260 30 "preliminary numerical function" }{TEXT -1 54 " to evaluate the function to about 15 digit precision." }}{PARA 0 "" 0 "" {TEXT -1 159 "( With 14 digit precision, or less, Maple 8 uses fast NAG numerical integration routines which peform calculations usi ng hardware floating point arithmetic. )\n" }}{PARA 0 "" 0 "" {TEXT -1 48 "This can be done by using the special procedure " }{TEXT 0 12 " local_taylor" }{TEXT -1 35 " to construct a procedure based on " } {TEXT 260 34 "local Taylor series approximations" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "f := x -> Int(1/GAMMA(t),t = 0 .. x)/(x^2):\nfn := evalf(local_taylor(f(x),x =-.2..4.2,errtype=relative,info=true),18);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Yfinding~degree~13~local~Taylor~polynomial~approximati onsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Owhile~attempting~to~control~ the~relative~errorG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*step~..~1G%.~ ~~centre~..~G$!\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*step~..~2 G%.~~~centre~..~G$!7wlBNUpV>tK8!#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6% %*step~..~3G%.~~~centre~..~G$\"8%GR@*)3[<&fcc\"!#C" }}{PARA 11 "" 1 " " {XPPMATH 20 "6%%*step~..~4G%.~~~centre~..~G$\"8-3j&)fx='*3$=_!#C" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%%*step~..~5G%.~~~centre~..~G$\"8(yA(3d c?l+)e;!#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*step~..~6G%.~~~centre~ ..~G$\"8B%\\ciD-)3%[vO!#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*step~.. ~7G%.~~~centre~..~G$\"8(p)*RZIY\\X>Qe!#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*step~..~8G%.~~~centre~..~G$\"8$z7v`@3?#Q8^)!#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*step~..~9G%.~~~centre~..~G$\"8'=!)>FrfI-8x5!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~10G%.~~~centre~..~G$\"8Ja% zUsP?;X*H\"!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~11G%.~~~ce ntre~..~G$\"8Q8JtFv^INI_\"!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+ste p~..~12G%.~~~centre~..~G$\"8sh*4skUdby]!#A" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~14G%.~~~centre~..~G$\"8[f/qS :mq[AB#!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~15G%.~~~centre ~..~G$\"8@\\1)QyX[8f/D!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~.. ~16G%.~~~centre~..~G$\"8gfCi-tkeV*zF!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~17G%.~~~centre~..~G$\"8Mpu**[\\1+P2/$!#A" }}{PARA 11 " " 1 "" {XPPMATH 20 "6%%+step~..~18G%.~~~centre~..~G$\"8aT'\\,2uY'*y*H$ !#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~19G%.~~~centre~..~G$ \"8<(HB.(=dbq:c$!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~20G%. ~~~centre~..~G$\"8=3[-UE%yb\\HQ!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6% %+step~..~21G%.~~~centre~..~G$\"8ud^q%[^*4Bw5%!#A" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "Digits := 18 :\nxx := evalf(sqrt(5));\nevalf(f(xx));\nevalf[18](fn(xx));\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"3q*y*\\xz1OA!#<" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3gK'*\\igd*p$!#=" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"3hK'*\\igd*p$!#=" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 14 "The values of " }{XPPEDIT 18 0 "f(x) " "6#-%\"fG6#%\"xG" }{TEXT -1 33 " and the numerical approximation " } {TEXT 271 2 "fn" }{TEXT -1 70 " agree ( for the most part) to 18 digit s, but the numerical procedure " }{TEXT 271 2 "fn" }{TEXT -1 25 " eval uates more rapidly. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "seq(evalf[15](evalf[18](fn(0.01+i*0.49)) ),i=0..8);\nseq(evalf[15](evalf[18](f(0.01+i*0.49))),i=0..8);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6+$\"0%)49Zw!>]!#:$\"0q*\\(\\Npb&F%$\"0# H\"HT;0U&F%$\"0/jP7Hk&[F%$\"0mx#[%QH6%F%$\"0X[xPkXO$F%$\"0zxYt9Mq#F%$ \"0/%*4#p&3;#F%$\"0rzc#>#Qt\"F%" }}{PARA 12 "" 1 "" {XPPMATH 20 "6+$\" 0%)49Zw!>]!#:$\"0q*\\(\\Npb&F%$\"0#H\"HT;0U&F%$\"0/jP7Hk&[F%$\"0mx#[%Q H6%F%$\"0X[xPkXO$F%$\"0zxYt9Mq#F%$\"0/%*4#p&3;#F%$\"0rzc#>#Qt\"F%" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The numer ical procedure " }{TEXT 271 2 "fn" }{TEXT -1 63 " can be used to const ruct a minimax rational approximation for " }{XPPEDIT 18 0 "f(x)" "6#- %\"fG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0,4]" "6#7$\"\"!\"\"%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "r := remez(fn,0..4,[7,6],'ma xerr','minerr',info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]pRemez ~algorithm:~calculating~minimax~error~estimate~by~solving~a~rational~e quationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%D--~minimising~the~ABSOLU TE~error~--G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%Wstandard~Chebyshev~points~for~initial~critical~poin ts:G" }}{PARA 12 "" 1 "" {XPPMATH 20 "61$\"\"!F$$\"7ujfy_jjvT9]!#B$\"8 zFvuh^>kA1)>F'$\"86$e#QSR1NqLO%F'$\"8**\\*QH`#G'R?IvF'$\"9Z[!fP)[w@DBK 6F'$\"9>A9>rt3K\"e\\b\"F'$\"9+++++++++++?F'$\"9\"yd3)GE\"z'=/XCF'$\"9` ^4C;^ByuwnGF'$\"9,]51nur.'zpC$F'$\"9*oT<'fg$\\'HmjNF'$\"9@sCDQ[!etP>!Q F'$\"9EOS@ZOOCe&)\\RF'$\"9+++++++++++SF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,iteration~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G--------------------------------------G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%Dprovisional~rational~approximation: G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&,2$\":F0F()F,\"\"$F(F3*&$\"<@\\QD$3jl#Q)QCYw!#HF ()F,\"\"%F(F(*&$\"aH-&G&36F3f!R6 )!#JF()F,\"\"'F(F(*&$\"<9U%G*z\\+qsV)='Q#!#KF()F,\"\"(F(F3F(,0$F(\"\"! 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