{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 "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 265 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 1 12 0 0 0 0 0 0 0 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 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 269 "Times" 1 12 96 52 84 1 0 1 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 "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 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 "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 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 "Ti mes" 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 "T imes" 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 22 "The Remez algorithm: " }}{PARA 3 "" 0 "" {TEXT 264 90 " .. standard and non-standard error curves for rational approximations to an even function" }}{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 interpolation and function approxi mation procedures including: " }{TEXT 0 5 "remez" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 269 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 c an be read into a Maple session by a command similar to the one that f ollows, where the file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "read \"K:\\\\Maple/procdrs/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 38 "Standard and non-standard error curves" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 41 "Given a minimax polynomial app roximation " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 10 " of degree" }{XPPEDIT 18 0 "``<=n" "6#1%!G%\"nG" }{TEXT -1 27 " for a con tinuous function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 16 " on an interval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 56 ", the number of critical points, that is, the values of " } {TEXT 263 1 "x" }{TEXT -1 88 " for which the minimax error function at tains its maximum magnitude, is always at least " }{XPPEDIT 18 0 "n+2 " "6#,&%\"nG\"\"\"\"\"#F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }{XPPEDIT 18 0 "r(x)=p(x)/q(x)" "6#/-%\"rG6#%\"x G*&-%\"pG6#F'\"\"\"-%\"qG6#F'!\"\"" }{TEXT -1 21 " is an (irreducible) " }{TEXT 260 30 "minimax rational approximation" }{TEXT -1 27 " for a continuous function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" } {TEXT -1 8 ", where " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 11 " has degree" }{XPPEDIT 18 0 "``<=m" "6#1%!G%\"mG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 11 " has degr ee" }{XPPEDIT 18 0 "``<=n" "6#1%!G%\"nG" }{TEXT -1 6 ", and " } {XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 16 " is non-zero in \+ " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 74 "One might conjecture that the number of critical p oints would be at least " }{XPPEDIT 18 0 "m+n+2" "6#,(%\"mG\"\"\"%\"nG F%\"\"#F%" }{TEXT -1 55 ", and for the end points of the approximation interval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 121 " to be among them. Although this does usually happen, the number of criti cal points can actually be greater or less than " }{XPPEDIT 18 0 "m+n+ 2" "6#,(%\"mG\"\"\"%\"nGF%\"\"#F%" }{TEXT -1 48 ". When the number of \+ critical points is exactly " }{XPPEDIT 18 0 "m+n+2" "6#,(%\"mG\"\"\"% \"nGF%\"\"#F%" }{TEXT -1 37 ", the error function is said to be a " } {TEXT 260 20 "standard error curve" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 99 "When the error curve is non-standard, the use of Remez al gorithm is more difficult than otherwise. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "The examples below show one sp ecial situation involving minimax rational approximations for even fun ctions in which non-standard error curves arise." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "A m inimax rational approximation for " }{XPPEDIT 18 0 "cos(Pi/4*x)" "6#-% $cosG6#*(%#PiG\"\"\"\"\"%!\"\"%\"xGF(" }{TEXT -1 4 " on " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 1 " " }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 49 "Suppose we wish to find a rational approximation " }{XPPEDIT 18 0 "r(x) = p(x)/q (x)" "6#/-%\"rG6#%\"xG*&-%\"pG6#F'\"\"\"-%\"qG6#F'!\"\"" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "cos(Pi/4*x)" "6#-%$cosG6#*(%#PiG\"\"\"\"\"%!\" \"%\"xGF(" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-1,1]" "6 #7$,$\"\"\"!\"\"F%" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "p(x)" "6#-% \"pG6#%\"xG" }{TEXT -1 18 " has degree 4 and " }{XPPEDIT 18 0 "q(x)" " 6#-%\"qG6#%\"xG" }{TEXT -1 14 " has degree 2." }}{PARA 0 "" 0 "" {TEXT -1 63 "We could attempt to do this by specifing exactly these de grees." }}{PARA 0 "" 0 "" {TEXT -1 89 "The error curve throughout the \+ computation has 9 critical points, and so is non-standard." }}{PARA 0 "" 0 "" {TEXT -1 63 "This makes the calculation problematical, but it \+ does succeeed." }}{PARA 0 "" 0 "" {TEXT -1 59 "If we ignore the \"smal l\" coefficients of the odd powers of " }{TEXT 279 1 "x" }{TEXT -1 41 ", the resulting rational approximation is" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "(1.000000024-.2874648358*x^2+.9393339036e-2* x^4)/(1.+.2096107963e-1*x^2);" "6#*&,(-%&FloatG6$\"+C+++5!\"*\"\"\"*&- F&6$\"+e$[Y(G!#5F**$%\"xG\"\"#F*!\"\"*&-F&6$\"+O!RLR*!#7F**$F1\"\"%F*F *F*,&-F&6$F*\"\"!F**&-F&6$\"+jz5'4#!#6F**$F1F2F*F*F3" }{TEXT -1 2 ". \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 265 12 "Calculation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "var := evalf((log10(2)+6)/1 0);\nremez(cos(Pi/4*x),x=-1..1,[4,2],variation=var,info=true):\nr := u napply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$varG$\"+'**H5I'!#5 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]pRemez~algorithm:~calculating~mi nimax~error~estimate~by~solving~a~rational~equationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%D--~minimising~the~ABSOLUTE~error~--G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Wstandar d~Chebyshev~points~for~initial~critical~points:G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6*$!2++++++++\"!#;$!27>C!z')o4!*!#<$!2bL(e=!)*[B'F($!2U9j &R$4_A#F($\"2U9j&R$4_A#F($\"2bL(e=!)*[B'F($\"27>C!z')o4!*F($\"2+++++++ +\"F%" }}{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#*&,,$\"5P(eng\"\\+++5!#>\"\"\"*&$\"5\"p_V()odpm(Q!#MF(%\"xGF(!\" \"*&$\"5H@Ko.:08uG!#?F()F-\"\"#F(F.*&$\"5UCOoC9smQ6F,F()F-\"\"$F(F(*&$ \"5cD!\\;lkT%y$*!#AF()F-\"\"%F(F(F(,($F(\"\"!F(*&$\"5=C.+RkM\"z)QF,F(F -F(F.*&$\"5o)4AnFeJ85#!#@F(F3F(F(F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%goal~for~relative~difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"20WJh*******\\!#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,error~graphG" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "65-%'CURVESG6%7cs7$$!\"\"\"\"! 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#6F-F.F-F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 53 "The calculation fails if we specify the degree 5 f or " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 11 " and 3 for \+ " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 83 " because the e rror curve does not have the required number of 10 critical points. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 268 22 "Attempted calculation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "var := evalf((log10(2)+6)/10);\nre mez(cos(Pi/4*x),x=-1..1,[5,3],variation=var,\n method=ratsolve);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$varG$\"+'**H5I'!#5" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%Xtrying~another~candidate~for~the~minimax~error ~estimateG" }}{PARA 8 "" 1 "" {TEXT -1 110 "Error, (in remez_ratsolve) all tentative rational approximations have a singularity in the inter val -1. .. 1.\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "remez(cos(Pi /4*x),x=-1..1,[5,3],variation=var,\n method=itersolve);" }}{PARA 8 "" 1 "" {TEXT -1 117 "Error, (in remez_itersolve) error curve fails \+ to oscillate sufficiently; try different degrees, or increase \"Digits \"\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 163 "Because of these difficulties, it is convenient to hav e a modified version of the Remez algorithm to construct a rational mi max approximation for an even function." }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 5 "remez" }{TEXT -1 60 " can be set up to construct a rational minmax approximation " }{XPPEDIT 18 0 "r(x)=p(x) /q(x)" "6#/-%\"rG6#%\"xG*&-%\"pG6#F'\"\"\"-%\"qG6#F'!\"\"" }{TEXT -1 8 " for an " }{TEXT 260 4 "even" }{TEXT -1 10 " function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 16 " on an interval " } {XPPEDIT 18 0 "[-a,a]" "6#7$,$%\"aG!\"\"F%" }{TEXT -1 22 ", where the \+ numerator " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 15 " has (maximum) " }{TEXT 260 11 "even degree" }{TEXT -1 37 " m and the deno minator has (maximum) " }{TEXT 260 11 "even degree" }{TEXT -1 72 " n i n a more efficient manner than is done above by setting the option \" " }{TEXT 269 9 "type=even" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 90 "At each stage in applying the Remez algorithm, approximate valu es for the coefficients in " }{XPPEDIT 18 0 "r(x)" "6#-%\"rG6#%\"xG" } {TEXT -1 60 " are obtained along with the estimate for the minimax err or " }{XPPEDIT 18 0 "mu" "6#%#muG" }{TEXT -1 49 "\004 by solving the n on-linear system of equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "r(x[k])-f(x[k]) = (-1)^(k-1)*mu,k = 1,2,` . . . `,m/2+n /2+1;" "6'/,&-%\"rG6#&%\"xG6#%\"kG\"\"\"-%\"fG6#&F)6#F+!\"\"*&),$F,F2, &F+F,F,F2F,%#muGF,/F+F,\"\"#%(~.~.~.~G,(*&%\"mGF,F9F2F,*&%\"nGF,F9F2F, F,F," }{TEXT -1 4 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "0 = x[1];" "6#/\"\"!&%\"xG6#\"\"\"" }{XPPEDIT 18 0 "``< x[2]" "6#2%!G&%\"xG6#\"\"#" }{XPPEDIT 18 0 "``<` . . .`" "6#2%!G%'~.~. ~.G" }{TEXT -1 1 "<" }{XPPEDIT 18 0 "x[m/2+n/2+2] = a;" "6#/&%\"xG6#,( *&%\"mG\"\"\"\"\"#!\"\"F**&%\"nGF*F+F,F*F+F*%\"aG" }{TEXT -1 63 " are \+ the current approximations for the critical points in the " }{TEXT 260 15 "right-hand half" }{TEXT -1 17 " of the interval." }}{PARA 0 " " 0 "" {TEXT -1 31 "The resulting approximation is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(1.000000024-.2874648358*x^2+.939333 9035e-2*x^4)/(1.+.2096107964e-1*x^2)" "6#*&,(-%&FloatG6$\"+C+++5!\"*\" \"\"*&-F&6$\"+e$[Y(G!#5F**$%\"xG\"\"#F*!\"\"*&-F&6$\"+N!RLR*!#7F**$F1 \"\"%F*F*F*,&-F&6$F*\"\"!F**&-F&6$\"+kz5'4#!#6F**$F1F2F*F*F3" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 42 "which is essentially the same \+ as before. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 270 12 "Calculation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "var := evalf((log 10(2)+6)/10);\nremez(cos(Pi/4*x),x=-1..1,[4,2],variation=var,type=even ,info=true);\nr := unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% $varG$\"+'**H5I'!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]pRemez~algori thm:~calculating~minimax~error~estimate~by~solving~a~rational~equation G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%D--~minimising~the~ABSOLUTE~erro r~--G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\\qstandard~Chebyshev~points~for~initial~critical~poin ts~that~lie~in~the~right~half~of~the~interval:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'$\"\"!F$$\"/1lBV$o#Q!#9$\"/b'=\"y1rqF'$\"/H6D`zQ#*F'$\" /++++++5!#8" }}{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 11 "" 1 "" {XPPMATH 20 "6#*&,($\"2v;,T-+++\"!#;\"\"\"*&$\"2X8.uN[Y(G!#F()F.\"\"%F(F(F(,&$F(\"\"!F(*&$\"2,$p8nz5 '4#!#=F(F-F(F(F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%\\ n65C!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+U!=T.%!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%5relative~difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/\\'\\ \"o#Qn\"!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>goal~for~relative~dif ference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/98'*******\\!#?" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%Uerror~graph~drawn~for~the~right~half~of~the~intervalG" }}{PARA 13 " " 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6/-%'CURVESG6%7hp7$$\"\"!F)$ !3E>uh]n65C!#D7$$\"3Um;/mv%44\"!#>$!3%HR])y#e3S#F,7$$\"3%GL$3K^*==#F0$ !3m0(ocm\\JP#F,7$$\"31\\7.wF7JJF0$!3&Rg([ZA=MBF,7$$\"3)f;z*>/N!3%F0$!3 wsn/z&4;G#F,7$$\"3=K$eW%>O:iF0$!39bQ_!=d_6#F,7$$\"3%HL37rDXO)F0$!3oE-! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 275 22 "Attempted calculation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "var := evalf((log10(2 )+6)/10);\nremez(cosh(x),x=-1..1,[5,3],variation=var,info=true);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$varG$\"+'**H5I'!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]pRemez~algorithm:~calculating~minimax~error~esti mate~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~poin ts~for~initial~critical~points:G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6,$! 4+++++++++\"!#=$!4TQ3fy?EpR*!#>$!4`.y*=JWWgwF($!4+++++++++&F($!4%\\.$p mx\"[O " 0 "" {MPLTEXT 1 0 66 "remez(cosh(x ),x=-1..1,[5,3],variation=var,\n method=itersolve);" }}{PARA 8 "" 1 "" {TEXT -1 117 "Error, (in remez_itersolve) error curve fails to os cillate sufficiently; try different degrees, or increase \"Digits\"\n " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 163 "Because of these difficulties, it is convenient to hav e a modified version of the Remez algorithm to construct a rational mi max approximation for an even function." }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 5 "remez" }{TEXT -1 60 " can be set up to construct a rational minmax approximation " }{XPPEDIT 18 0 "r(x)=p(x) /q(x)" "6#/-%\"rG6#%\"xG*&-%\"pG6#F'\"\"\"-%\"qG6#F'!\"\"" }{TEXT -1 8 " for an " }{TEXT 260 4 "even" }{TEXT -1 10 " function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 16 " on an interval " } {XPPEDIT 18 0 "[-a,a]" "6#7$,$%\"aG!\"\"F%" }{TEXT -1 22 ", where the \+ numerator " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 15 " has (maximum) " }{TEXT 260 11 "even degree" }{TEXT -1 37 " m and the deno minator has (maximum) " }{TEXT 260 11 "even degree" }{TEXT -1 72 " n i n a more efficient manner than is done above by setting the option \" " }{TEXT 269 9 "type=even" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 "" {TEXT -1 90 "At each stage in applying the Remez algorithm, approximate valu es for the coefficients in " }{XPPEDIT 18 0 "r(x)" "6#-%\"rG6#%\"xG" } {TEXT -1 60 " are obtained along with the estimate for the minimax err or " }{XPPEDIT 18 0 "mu" "6#%#muG" }{TEXT -1 49 "\004 by solving the n on-linear system of equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "r(x[k])-f(x[k]) = (-1)^(k-1)*mu,k = 1,2,` . . . `,m/2+n /2+1;" "6'/,&-%\"rG6#&%\"xG6#%\"kG\"\"\"-%\"fG6#&F)6#F+!\"\"*&),$F,F2, &F+F,F,F2F,%#muGF,/F+F,\"\"#%(~.~.~.~G,(*&%\"mGF,F9F2F,*&%\"nGF,F9F2F, F,F," }{TEXT -1 4 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "0 = x[1];" "6#/\"\"!&%\"xG6#\"\"\"" }{XPPEDIT 18 0 "``< x[2]" "6#2%!G&%\"xG6#\"\"#" }{XPPEDIT 18 0 "``<` . . .`" "6#2%!G%'~.~. ~.G" }{TEXT -1 1 "<" }{XPPEDIT 18 0 "x[m/2+n/2+2] = a;" "6#/&%\"xG6#,( *&%\"mG\"\"\"\"\"#!\"\"F**&%\"nGF*F+F,F*F+F*%\"aG" }{TEXT -1 63 " are \+ the current approximations for the critical points in the " }{TEXT 260 15 "right-hand half" }{TEXT -1 17 " of the interval." }}{PARA 0 " " 0 "" {TEXT -1 31 "The resulting approximation is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(-1.000000170-.4676651374*x^2-.25528 48875e-1*x^4)/(-1.+.3232948562e-1*x^2)" "6#*&,(-%&FloatG6$\"+q,++5!\"* !\"\"*&-F&6$\"+u8lwY!#5\"\"\"*$%\"xG\"\"#F0F**&-F&6$\"+v)[Gb#!#6F0*$F2 \"\"%F0F*F0,&-F&6$F0\"\"!F**&-F&6$\"+i&[HB$F8F0*$F2F3F0F0F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 40 "which is essentially the same as before." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 276 12 "Calculation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "var := evalf((log 10(2)+6)/10);\nremez(cosh(x),x=-1..1,[4,2],variation=var,type=even,max iterations=20,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$varG$ \"+'**H5I'!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]pRemez~algorithm:~c alculating~minimax~error~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#%\\qstandard~Chebyshev~points~for~initial~critical~points~that~l ie~in~the~right~half~of~the~interval:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'$\"\"!F$$\"/1lBV$o#Q!#9$\"/b'=\"y1rqF'$\"/H6D`zQ#*F'$\"/++++++5! #8" }}{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 11 "" 1 "" {XPPMATH 20 "6#*&,($\"2gF)Gq,++5!#;!\"\"*&$\"2gL#Gt8lwY!#<\"\"\")%\"xG\"\"#F-F( *&$\"2lv&4F)[Gb#!#=F-)F/\"\"%F-F(F-,&$F-\"\"!F(*&$\"2mo&\\&f[HB$F4F-F. 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