{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 "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times " 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis " -1 262 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 2 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 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 0 1 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 "Text Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Map le 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 "Time s" 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 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" -1 257 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" -1 258 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 -1 66 "Procedures for evaluating the hyp erbolic sine and cosine functions" }}{PARA 0 "" 0 "" {TEXT -1 37 "by P eter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Vers ion: 25.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 a nd function approximation procedures including: " }{TEXT 0 5 "remez" } }{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 262 10 "fcnapp rx.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 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 24 "load numerical functions" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 262 8 "numfcn.m" }{TEXT -1 72 " contains the code for the alternative mathematical functions including " } {TEXT 0 5 "sinh_" }{TEXT -1 5 " and " }{TEXT 0 5 "cosh_" }{TEXT -1 1 " ." }}{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 giv es its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "read \+ \"K:\\\\Maple/procdrs/numfcn.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "load numerical integration proced ures and data" }}{PARA 0 "" 0 "" {TEXT -1 18 "The Maple m-files " } {TEXT 262 6 "intg.m" }{TEXT -1 5 " and " }{TEXT 262 8 "gkdata.m" } {TEXT -1 67 " contain the code and data for the numerical integration \+ procedure " }{TEXT 0 8 "quad/Int" }{TEXT -1 25 " used in this workshee t. " }}{PARA 0 "" 0 "" {TEXT -1 122 "They can be read into a Maple ses sion by commands similar to those that follow, where the file paths gi ve their location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "read \+ \"K:\\\\Maple/procdrs/intg.m\";\nread \"K:\\\\Maple/procdrs/gkdata.m\" ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "Ar bitrary precision procedures to evaluate " }{XPPEDIT 18 0 "sinh(x)" "6 #-%%sinhG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cosh(x)" "6#-%%c oshG6#%\"xG" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "sinhAP, coshAP: implemen tation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 15 "The procedures " }{TEXT 0 6 "sinhAP" }{TEXT -1 5 " and " }{TEXT 0 6 "coshAP" }{TEXT -1 26 " use the Maclaurin series " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sinh(x)=x+x^3/3!+x^5/ 5!+x^7/7!+` . . . `" "6#/-%%sinhG6#%\"xG,,F'\"\"\"*&F'\"\"$-%*factoria lG6#F+!\"\"F)*&F'\"\"&-F-6#F1F/F)*&F'\"\"(-F-6#F5F/F)%(~.~.~.~GF)" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "cosh(x)=1+x^2/2!+x^4/4!+x^6/6!+` . . \+ . `" "6#/-%%coshG6#%\"xG,,\"\"\"F)*&F'\"\"#-%*factorialG6#F+!\"\"F)*&F '\"\"%-F-6#F1F/F)*&F'\"\"'-F-6#F5F/F)%(~.~.~.~GF)" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 5 "when " }{XPPEDIT 18 0 "abs(x) " 0 "" {MPLTEXT 1 0 1622 "sinhAP := proc(xx::realcons)\n \+ local t,val,x,z,term,eps,k,maxit,saveDigits,sum;\n \n # Increase p recision\n saveDigits := Digits;\n Digits := Digits+2;\n x := ev alf(xx);\n\n if abs(x)>0.28782313662425571050 then\n # Use form ula sinh(x)=(exp(x)-exp(-x))/2\n t := `evalf/exp_`(x);\n val := (t-1/t)*0.5;\n return evalf[saveDigits](val);\n else\n \+ # Use Maclaurin series\n Digits := saveDigits+length(saveDigits) ;\n z := evalf(x);\n eps := Float(1,-saveDigits);\n max it := Digits*2;\n term := x;\n sum := term;\n z := x*x; \n for k from 2 to maxit by 2 do\n term := term*z/(k*(k+1 ));\n sum := sum+term;\n if abs(term)<=eps*abs(sum) th en break end if;\n end do;\n\n return evalf[saveDigits](sum) ;\n end if;\nend proc: # of sinhAP\n\ncoshAP := proc(xx::realcons)\n local t,val,x,z,term,eps,k,maxit,saveDigits,sum;\n \n # Increas e precision\n saveDigits := Digits;\n Digits := Digits+2;\n x := evalf(xx);\n\n if abs(x)>0.28782313662425571050 then\n # Use f ormula sinh(x)=(exp(x)-exp(-x))/2\n t := `evalf/exp_`(x);\n \+ val := (t+1/t)*0.5;\n return evalf[saveDigits](val);\n else\n \+ # Use Maclaurin series\n Digits := saveDigits+length(saveDigi ts);\n x := evalf(xx);\n eps := Float(1,-saveDigits);\n \+ maxit := Digits*2;\n term := 1.0;\n sum := term;\n z : = x*x;\n for k from 1 to maxit by 2 do\n term := term*z/( k*(k+1));\n sum := sum+term;\n if abs(term)<=eps*abs(s um) then break end if;\n end do;\n return evalf[saveDigits]( sum);\n end if;\nend proc: # of coshAP" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "Testing the procedures sinhAP, coshAP" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "D igits := 50:\nxx := 0.25:\nsinhAP(xx);\nevalf(evalf(sinh(xx),Digits+2) );\ncoshAP(xx);\nevalf(evalf(cosh(xx),Digits+2));\nDigits := 10:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"SmFuGa(>b!z0U0:DT\"zIo\"3oJ7ED!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"SmFuGa(>b!z0U0:DT\"zIo\"3oJ7ED!#] " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"SQ&ydEl\"Gby._ " 0 "" {MPLTEXT 1 0 129 "Digits := 50:\nxx := 0.35:\nsinhAP(xx);\nevalf(evalf (sinh(xx),Digits+2));\ncoshAP(xx);\nevalf(evalf(cosh(xx),Digits+2));\n Digits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"SZ2^tLiK#)p\\a9[(y dp!>FP%H(*=d$!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"SZ2^tLiK#)p\\a9 [(ydp!>FP%H(*=d$!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"SD>Trk50Xv`[ !=3EJT`)f:>y(=1\"!#\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"SD>Trk50Xv `[!=3EJT`)f:>y(=1\"!#\\" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "plot( ['sinhAP'(x),'coshAP'(x)],x=-2..2,color=[red,green],thickness=2);" }} {PARA 13 "" 1 "" {GLPLOT2D 277 333 333 {PLOTDATA 2 "6'-%'CURVESG6$7S7$ $!\"#\"\"!$!+3/'oi$!\"*7$$!+$Q6G\">F-$!+&==AJ$F-7$$!+M!\\p$=F-$!+]r-fI F-7$$!+))Qj^<8#F-7$$!+oU)*=9F-$!+b.[X>F-7$$!+Ta7M8F-$!+GGkm '***F-$!+2Zhu6F-7$$!*]5*H\"*F-$!+p6AX5F-7$$!*I\"3&H)F-$!+ Afjz#*!#57$$!*Twp`(F-$!+%Q'4r#)F^p7$$!*P;bj'F-$!+[lGLrF^p7$$!*Zh=(eF-$ !+x0::iF^p7$$!*G\\N)\\F-$!+R)4C>&F^p7$$!*ZUs>%F-$!+.&p:K%F^p7$$!*GRXL$ F-$!+k\"zmR$F^p7$$!*$=/8DF-$!+NodRDF^p7$$!*U&*el\"F-$!+:KZj;F^p7$$!)Wn (o)F-$!+&*pg)p)!#67$$!(eV(>F-$!+$GfV(>!#77$$\")f`@')F-$\"+V1AK')F\\s7$ $\"*nZ)H;F-$\"+[J2P;F^p7$$\"*Ky*eCF-$\"+1S$Q[#F^p7$$\"*S^bJ$F-$\"+J>jw LF^p7$$\"*0TN:%F-$\"+[H+uUF^p7$$\"*7RV'\\F-$\"+pRxq^F^p7$$\"*:#fkeF-$ \"+!H$f1iF^p7$$\"*`4Nn'F-$\"+]p+!=(F^p7$$\"*],s`(F-$\"+$\\)Qr#)F^p7$$ \"*zM)>$)F-$\"+vPV8$*F^p7$$\"*qfa<*F-$\"+B4#=0\"F-7$$\"*1O0)**F-$\"+v* *>s6F-7$$\"+#G2A3\"F-$\"+\\)phI\"F-7$$\"+$)G[k6F-$\"+N24Y9F-7$$\"+7yh] 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functions " } {TEXT 0 4 "sinh" }{TEXT -1 5 " and " }{TEXT 0 4 "cosh" }{TEXT -1 17 " \+ are faster than " }{TEXT 0 5 "sinAP" }{TEXT -1 5 " and " }{TEXT 0 6 "c oshAP" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "st := time():\nfor i to 500 do sinhAP(ra nd()*Float(1,-11)) end do:\ntime()-st;\nst := time():\nfor i to 500 do sinh(rand()*Float(1,-11)) end do:\ntime()-st;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$M(!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$>$!\" $" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "st := time():\nDigits := 50:\nfor i to 100 do sinhAP (rand()*Float(1,-11)) end do:\ntime()-st;\nst := time():\nfor i to 100 do sinh(rand()*Float(1,-11)) end do:\ntime()-st;\nDigits := 10:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%3;!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$$H!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "Fixed prec ision procedures to evaluate " }{XPPEDIT 18 0 "sinh(x)" "6#-%%sinhG6#% \"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cosh(x)" "6#-%%coshG6#%\"xG " }{TEXT -1 4 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 112 "In this section we construct fixed preci sion versions of the hyperbolic sine and cosine functions which can us e " }{TEXT 259 34 "hardware floating point arithmetic" }{TEXT -1 59 " \+ and so run faster than the arbitrary precision procedures " }{TEXT 0 6 "sinhAP" }{TEXT -1 5 " and " }{TEXT 0 6 "coshAP" }{TEXT -1 2 ". " }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "A polynomial approximation for " }{XPPEDIT 18 0 "sinh(x)" "6#-%%sinhG6#%\"xG" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 23 "W e choose a polynomial " }{XPPEDIT 18 0 "s(x)" "6#-%\"sG6#%\"xG" } {TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "sinh(x)" "6#-%%sinhG6#% \"xG" }{TEXT -1 79 " on the interval [-0.34665, 0.34665] which is slig htly wider than the interval " }{XPPEDIT 18 0 "[-ln(2)/2, ln(2)/2];" " 6#7$,$*&-%#lnG6#\"\"#\"\"\"F)!\"\"F+*&-F'6#F)F*F)F+" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "First we find a rational approximation for the function " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([(sinh(x)-x)/(x^3), x <> 0],[1/6, x = 0]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$*&,&-%%sinhG6 #F'\"\"\"F'!\"\"F1*$F'\"\"$F20F'\"\"!7$*&F1F1\"\"'F2/F'F6" }{TEXT -1 4 ". \n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "evalf(remez((s inh(x)-x)/x^3,x=-0.34665..0.34665,8,\n 'maxerr',errtype=absolute,w eight=x^2+1e-30,type=even,info=true),30):\nq := unapply(%,x);\nmaxerr; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,iteration~4G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%G--------------------------------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Uerr or~graph~drawn~for~the~right~half~of~the~intervalG" }}{PARA 13 "" 1 " " {GLPLOT2D 400 300 300 {PLOTDATA 2 "61-%'CURVESG6%7fs7$$\"\"!F)%%FAIL G7$$\"D+++++++]P%[@0'31OO#!#Q$!Dk$Qq7*oD8[w=FphC9_&!#e7$$\"D++++++++vo H/@<7ss%F.$!D$fm\\\"[N_haQ^e1&f_3A!#d7$$\"D+++++++]7`Wc\"e#=34(F.$!D#z \"[Wq,\\_2B.19y=!p\\F77$$\"D++++++++]Pf3UMCWX*F.$!DDO0e;f83Z\"[t]([,M$ ))F77$$\"D+++++++]i!*GJ;lj\"=9!#P$!DU@?kQ-e!RREyd%)>D()>!#c7$$\"D+++++ +++](=<%)o[)3*=FE$!D,rJ='4.Kh![=*GNqBKNFH7$$\"D++++++++D\"yDE.tKOGFE$! 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" 0 "" {MPLTEXT 1 0 74 "evalf(evalf(numapprox[infnorm](1-h(x)/sinh(x),x=-ln(2 )/2..ln(2)/2),25),5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&*>E!#B" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The max imum relative error in using " }{XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"xG " }{TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "sinh(x);" "6#-%%sinh G6#%\"xG" }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[-ln(2)/2, \+ ln(2)/2];" "6#7$,$*&-%#lnG6#\"\"#\"\"\"F)!\"\"F+*&-F'6#F)F*F)F+" } {TEXT -1 11 " is about " }{XPPEDIT 18 0 "2.6*`. `*10^(-19);" "6#*(-%& FloatG6$\"#E!\"\"\"\"\"%#.~GF))\"#5,$\"#>F(F)" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "xx := evalf(ln(2)/2,20);\nevalf(evalf(h(xx),20),17);\nevalf(evalf( sinh(xx),20),17);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"5raE(*z- ftlM!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"2wtKf!R`NN!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"2wtKf!R`NN!#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "We can test the accuracy of the approximation " }{XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"xG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "sinh(x);" "6#-%%sinhG6#%\"xG" }{TEXT -1 35 " wi th random numbers between 0 and " }{XPPEDIT 18 0 "ln(2)/2;" "6#*&-%#ln G6#\"\"#\"\"\"F'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 92 " The printout occurs only when the relative error is greater than or eq ual to the specified \"" }{TEXT 262 3 "eps" }{TEXT -1 3 "\".\n" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 385 "randomize():\neps := Float(5,-16);\nfor i from 1 to 100 do\n xx := evalf(rand()/Float(1,12)*.34657359027997265471,20);\n axx := ev alf(sinh(xx),20);\n hxx := evalf(h(xx),16);\n e := evalf(abs((axx- hxx)/axx),20);\n if e>=eps then \n printf(\" trial no. %d, x \+ = %.16f,\\n\",i,xx);\n printf(\" sinh(x) = %.16f, h(x) = %.16f, \+ rel error = %.2e\\n\\n\",axx,hxx,e); \n end if;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$epsG$\"\"&!#;" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 29, x = .3256558631269387," }}{PARA 6 "" 1 "" {TEXT -1 77 " sinh(x) = .3314425242146987, h(x) = .3314425242146989, \+ rel error = 6.23e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 " " {TEXT -1 39 " trial no. 44, x = .2320410863165294," }}{PARA 6 "" 1 "" {TEXT -1 77 " sinh(x) = .2341289999583006, h(x) = .2341289999583 004, rel error = 7.40e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 " " 1 "" {TEXT -1 39 " trial no. 49, x = .3247886694878718," }}{PARA 6 "" 1 "" {TEXT -1 77 " sinh(x) = .3305290635630198, h(x) = .33052906 35630200, rel error = 5.77e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 39 " trial no. 54, x = .1365416316808733," }}{PARA 6 "" 1 "" {TEXT -1 77 " sinh(x) = .1369662990061371, h(x) = . 1369662990061370, rel error = 5.19e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 " " }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 86, x = .020236363099091 1," }}{PARA 6 "" 1 "" {TEXT -1 77 " sinh(x) = .0202377442942006, h(x) = .0202377442942006, rel error = 5.63e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 89, x = .320804178 8440347," }}{PARA 6 "" 1 "" {TEXT -1 77 " sinh(x) = .3263351743232873 , h(x) = .3263351743232871, rel error = 5.68e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 97, x = .098 2245004535892," }}{PARA 6 "" 1 "" {TEXT -1 77 " sinh(x) = .0983825225 206093, h(x) = .0983825225206093, rel error = 5.65e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "A polynomial approximation for cosh(x) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 23 "We choose a polynom ial " }{XPPEDIT 18 0 "s(x)" "6#-%\"sG6#%\"xG" }{TEXT -1 16 " to approx imate " }{XPPEDIT 18 0 "cosh(x)" "6#-%%coshG6#%\"xG" }{TEXT -1 79 " on the interval [-0.34665, 0.34665] which is slightly wider than the int erval " }{XPPEDIT 18 0 "[-ln(2)/2, ln(2)/2];" "6#7$,$*&-%#lnG6#\"\"#\" \"\"F)!\"\"F+*&-F'6#F)F*F)F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 56 "First we find a rational approximation for the function " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([(co sh(x)-1)/(x^2)-1/(2*x^2), x <> 0],[1/24, x = 0]);" "6#/-%\"fG6#%\"xG-% *PIECEWISEG6$7$,&*&,&-%%coshG6#F'\"\"\"F2!\"\"F2*$F'\"\"#F3F2*&F2F2*&F 5F2*$F'F5F2F3F30F'\"\"!7$*&F2F2\"#CF3/F'F:" }{TEXT -1 2 ". 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R(**f;Q`i*))F\\rFb]nFf]n-F$6%7$7$$\"IUoCEy)))eG&)z3mAQ?E#zuLFisF(7$Fh_ n$!IW=3W<(fIkT7Xg+P]ynq>)F\\rFb]nFf]n-F$6%Fb\\nFb]nFf]n-%+AXESLABELSG6 $Q\"x6\"Q!Fc`n-%%VIEWG6$;F($\"0+++]m*pM!#:;$!0+g;Q`i*))!#O$\"0+g;Q`i*) )F_an" 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" "Curve 12" "Curve 13" }}}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-iterati on~14G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G-------------------------- ------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Fprovisional~polynom ial~approximation:G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,,$\"LSKVFR*)f) zF!pA$\\ommmmmmT!#W\"\"\"*&$\"L5HzV^\\'\\)fYYv0XM()))))))))Q\"!#XF')% \"xG\"\"#F'F'*&$\"L&o6z[-'\\\">9VP*p*fI)[te,[#!#ZF')F-\"\"%F'F'*&$\"Ly ]Ce`_\\8!>W&GF)G/:<_sbF!#\\F')F-\"\"'F'F'*&$\"L%RK(R\\d$Q<$eti#)3gE*3_ ;#4#!#^F')F-\"\")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#$\"Ix*e$>9C\\X0!3N'G5T$z`lJ%!#d" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%Scritical~points~in~the~right~half~of~the~interval:G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6($\"L)3=)>8PH!Qd>\\Hc>!fA*H=/7!#V$\"LcP)QVO:T>LS\\r $)HQQX/bE?F%$\"LsI:6-C)=`(3Y**RVWw,\\'eWEF%$\"LBZM%e@[J[i]PvE%4e7avg4$ F%$\"LxV**y!Ri;N\"z(R.(f?*)R9VP*p*fI)[te,[#!#RF')F-\"\"%F'F'*&$\"Da_\\8!>W &GF)G/:<_sbF!#TF')F-\"\"'F'F'*&$\"D\\d$Q<$eti#)3gE*3_;#4#!#VF')F-\"\") F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0minimax~error:~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"2jJyv7B)3$)!#Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%\"qGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,,$\"DR *)f)zF!pA$\\ommmmmmT!#O\"\"\"*&$\"D^\\'\\)fYYv0XM()))))))))Q\"!#PF0)9$ \"\"#F0F0*&$\"DDg\\\">9VP*p*fI)[te,[#!#RF0)F6\"\"%F0F0*&$\"Da_\\8!>W&G F)G/:<_sbF!#TF0)F6\"\"'F0F0*&$\"D\\d$Q<$eti#)3gE*3_;#4#!#VF0)F6\"\")F0 F0F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"D0u:Hrb0#yG;$yv7B)3$)!# c" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The polynomial " }{XPPEDIT 18 0 "s(x)=1+x^2/2+q(x)*x^4" "6#/-%\"sG6#%\"xG ,(\"\"\"F)*&F'\"\"#F+!\"\"F)*&-%\"qG6#F'F)*$F'\"\"%F)F)" }{TEXT -1 14 " approximates " }{XPPEDIT 18 0 "cosh(x)" "6#-%%coshG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 38 "The relative error graph is d rawn for " }{XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"xG" }{TEXT -1 11 ", whi ch is " }{XPPEDIT 18 0 "s(x)" "6#-%\"sG6#%\"xG" }{TEXT -1 72 " with th e coefficients rounded to 20 digits and arranged in nested form." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 395 "s:=x->1+x^2/2+(.41666666666666684932269027798598939e-1+.138888888 88873445057546465984964951e-2*x^2+.24801587348830599699374314191496025 e-4*x^4+.27557252171504288272854419013495254e-6*x^6+.20921652089266008 826273583173835749e-8*x^8)*x^4:\nh := unapply(evalf(convert(s(x),horne r),20),x);\nevalf(plot((1-h(x)/cosh(x)),x=-ln(2)/2..ln(2)/2,\n \+ color=COLOR(RGB,.4,0,.9)),25);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&$\"\"\" \"\"!F.*&,&$\"5+++++++++]!#?F.*&,&$\"5K\\ommmmmmT!#@F.*&,&$\"5e]Wt)))) )))))Q\"!#AF.*&,&$\"5*p*fI)[te,[#!#CF.*&,&$\"5t#)G/:<_sbF!#EF.*&$\"5E) 3gE*3_;#4#!#GF.)9$\"\"#F.F.F.FMF.F.F.FMF.F.F.FMF.F.F.FMF.F.F.FMF.F.F(F (F(" }}{PARA 13 "" 1 "" {GLPLOT2D 451 291 291 {PLOTDATA 2 "6&-%'CURVES G6#7ex7$$!:_h3Zls*z-ftlM!#D$\"%_vF*7$$!:VH^9U$Q1aW,hMF*$\"%9dF*7$$!:M( R>)=%zK0IHcMF*$\"%FSF*7$$!:DlO\\&\\?fc:d^MF*$\"%\"[#F*7$$!:;Lz;s:cy5]o W$F*$\"%m5F*7$$!:2,A%)[E?\"f'G@W$F*$!#A!#C7$$!:)*ok^DP%Q5sSPMF*$!$Q\"F F7$$!:*ot!>-[[;w&oKMF*$!$T#FF7$$!:z/]')ye7HJkzU$F*$!$M$FF7$$!:iSN@K!3W :9_=MF*$!$*[FF7$$!:Vw?c&=!pz^y!4MF*$!$1'FF7$$!:D71\"*QB(\\?cj*R$F*$!$# pFF7$$!:1[\"fA\\a-BF>!R$F*$!$Y(FF7$$!:(Qo2ckObD)\\2Q$F*$!$v(FF7$$!:p>i &*)z=3GpIrLF*$!$#yFF7$$!:]bZI_451.k=O$F*$!$q(FF7$$!:L\"H`c5$QJ8@CN$F*$ !$T(FF7$$!:'HO]BTZ>Q``LLF*$!$U'FF7$$!:gMu/><^Ka\\YJ$F*$!$2&FF7$$!:qI\" e0pD09r@)H$F*$!$p$FF7$$!:!o#)o?mR&[o%y\"G$F*$!$A#FF7$$!:!H_zNj`lbANlKF *$!#sFF7$$!:,>-40wck#)>*[KF*$\"$T(F*7$$!:5:4gw:esR([KKF*$\"%P@F*7$$!:@ 6;6[bf!o\\0;KF*$\"%7MF*7$$!:J2Bi>&4')QDi*>$F*$\"%kXF*7$$!:U.I8\"\\Bm4, >$=$F*$\"%cbF*7$$!:z!=Z)=Xm\"R,BYJF*$\"%jrF*7$$!::e8cYbq'o,F4JF*$\"%&* yF*7$$!:]-**[.eYg]:3$F*$\"%#*yF*7$$!:_NbFulu\")>5B2$F*$\"%*z(F*7$$!:?CE8)3n#H@ IQ0$F*$\"%puF*7$$!:(Gr*)>g(ywA]`.$F*$\"%')pF*7$$!:a7*pt\"z#4Xxu;IF*$\" %\\jF*7$$!:?7,vK#o]i_9)*HF*$\"%(f&F*7$$!:'=JI\"[&3#*zFazHF*$\"%[ZF*7$$ !:^60^j)[L(HS4'HF*$\"%@QF*7$$!:=62*)y\"*[Z\"yLUHF*$\"%OGF*7$$!:$3\"4F% \\H;K`tBHF*$\"%7=F*7$$!:\\56l4)pd\\G80HF*$\"$x(F*7$$!:958.D,\"*pOIl)GF *$!#EFF7$$!:f2K\\*evv)H;!oGF*$!$F\"FF7$$!:.0^&R0T_IA]\\GF*$!$C#FF7$$!: [-qT=l!Hi\"))4$GF*$!$;$FF7$$!:$***)yG)>dS4uC\"GF*$!$/%FF7$$!:#[p-=\"H! 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" 0 "" {MPLTEXT 1 0 74 "evalf(evalf(numapprox[infnor m](1-h(x)/cosh(x),x=-ln(2)/2..ln(2)/2),25),5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%]#)!#D" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The maximum relative error in using " }{XPPEDIT 18 0 "r(x)" "6#-%\"rG6#%\"xG" }{TEXT -1 16 " to approximate " } {XPPEDIT 18 0 "cosh(x);" "6#-%%coshG6#%\"xG" }{TEXT -1 17 " in the int erval " }{XPPEDIT 18 0 "[-ln(2)/2, ln(2)/2];" "6#7$,$*&-%#lnG6#\"\"#\" \"\"F)!\"\"F+*&-F'6#F)F*F)F+" }{TEXT -1 12 " is about " }{XPPEDIT 18 0 "8*`. `*10^(-22);" "6#*(\"\")\"\"\"%#.~GF%)\"#5,$\"#A!\"\"F%" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "xx := evalf(ln(2)/2,20);\nevalf(evalf(h(xx),20), 18);\nevalf(evalf(cosh(xx),20),18);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#xxG$\"5raE(*z-ftlM!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3H@)z< \+ " 0 "" {MPLTEXT 1 0 385 "randomize():\neps := Float(4,-16);\nfor i fro m 1 to 100 do\n xx := evalf(rand()/Float(1,12)*.34657359027997265471 ,20);\n axx := evalf(cosh(xx),20);\n hxx := evalf(h(xx),16);\n \+ e := evalf(abs((axx-hxx)/axx),20);\n if e>=eps then \n printf( \" trial no. %d, x = %.16f,\\n\",i,xx);\n printf(\" cosh(x) = \+ %.16f, h(x) = %.16f, rel error = %.2e\\n\\n\",axx,hxx,e); \n end if; \nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$epsG$\"\"%!#;" }} {PARA 6 "" 1 "" {TEXT -1 38 " trial no. 9, x = .0806403652656465," } }{PARA 6 "" 1 "" {TEXT -1 79 " cosh(x) = 1.0032531966078475, h(x) = 1 .0032531966078470, rel error = 4.70e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 10, x = .19928349666950 98," }}{PARA 6 "" 1 "" {TEXT -1 79 " cosh(x) = 1.0199227595297926, h( x) = 1.0199227595297930, rel error = 4.18e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 12, x = .057 1678699369798," }}{PARA 6 "" 1 "" {TEXT -1 79 " cosh(x) = 1.001634527 7627494, h(x) = 1.0016345277627490, rel error = 4.16e-16" }}{PARA 6 " " 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 20, x = .0971657138772195," }}{PARA 6 "" 1 "" {TEXT -1 79 " cosh(x) = 1.00 47243031374545, h(x) = 1.0047243031374540, rel error = 4.49e-16" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial n o. 28, x = .1815572834309726," }}{PARA 6 "" 1 "" {TEXT -1 79 " cosh( x) = 1.0165268467943285, h(x) = 1.0165268467943290, rel error = 4.70e- 16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " t rial no. 31, x = .3384798926665904," }}{PARA 6 "" 1 "" {TEXT -1 79 " \+ cosh(x) = 1.0578333273253076, h(x) = 1.0578333273253080, rel error = \+ 4.08e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 35, x = .0284444317886373," }}{PARA 6 "" 1 "" {TEXT -1 79 " cosh(x) = 1.0004045701264444, h(x) = 1.0004045701264440, rel \+ error = 4.12e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 43, x = .1790642074375932," }}{PARA 6 "" 1 " " {TEXT -1 79 " cosh(x) = 1.0160748784818435, h(x) = 1.01607487848184 40, rel error = 4.66e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 " " 1 "" {TEXT -1 39 " trial no. 49, x = .0991392490620429," }}{PARA 6 "" 1 "" {TEXT -1 79 " cosh(x) = 1.0049183217210115, h(x) = 1.004918 3217210120, rel error = 4.79e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 39 " trial no. 57, x = .0627630771312823," }}{PARA 6 "" 1 "" {TEXT -1 79 " cosh(x) = 1.0019702485656875, h(x) = \+ 1.0019702485656880, rel error = 4.61e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 59, x = .218026010140 6181," }}{PARA 6 "" 1 "" {TEXT -1 79 " cosh(x) = 1.0238619702184885, \+ h(x) = 1.0238619702184890, rel error = 4.82e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 61, x = .017 6965276776329," }}{PARA 6 "" 1 "" {TEXT -1 79 " cosh(x) = 1.000156587 6323664, h(x) = 1.0001565876323660, rel error = 4.14e-16" }}{PARA 6 " " 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 62, x = .2849745103850402," }}{PARA 6 "" 1 "" {TEXT -1 79 " cosh(x) = 1.04 08807782744215, h(x) = 1.0408807782744220, rel error = 4.64e-16" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial n o. 64, x = .0580347738190377," }}{PARA 6 "" 1 "" {TEXT -1 79 " cosh( x) = 1.0016844901916625, h(x) = 1.0016844901916630, rel error = 4.53e- 16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " t rial no. 73, x = .2425588258569849," }}{PARA 6 "" 1 "" {TEXT -1 79 " \+ cosh(x) = 1.0295619056489455, h(x) = 1.0295619056489450, rel error = \+ 4.56e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 74, x = .1855675359076633," }}{PARA 6 "" 1 "" {TEXT -1 79 " cosh(x) = 1.0172671198806165, h(x) = 1.0172671198806170, rel \+ error = 4.72e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 75, x = .3306775148583085," }}{PARA 6 "" 1 " " {TEXT -1 79 " cosh(x) = 1.0551738331208256, h(x) = 1.05517383312082 60, rel error = 4.13e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 " " 1 "" {TEXT -1 39 " trial no. 77, x = .1470453295400122," }}{PARA 6 "" 1 "" {TEXT -1 79 " cosh(x) = 1.0108306587283055, h(x) = 1.010830 6587283060, rel error = 4.47e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 39 " trial no. 87, x = .1769880794699477," }}{PARA 6 "" 1 "" {TEXT -1 79 " cosh(x) = 1.0157033179291225, h(x) = \+ 1.0157033179291220, rel error = 4.86e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 91, x = .281236647319 7532," }}{PARA 6 "" 1 "" {TEXT -1 79 " cosh(x) = 1.0398083753039144, \+ h(x) = 1.0398083753039140, rel error = 4.12e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "sinh1 6, cosh16" }}{PARA 0 "" 0 "" {TEXT -1 158 "Here is the code for the fi xed precision versions of the hyperbolic sine and cosine functions, wh ich can be evaluated with hardware floating point arithmetic." }} {PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 45 ": These procedures ma ke use of the procedure " }{TEXT 0 5 "exp16" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1080 " sinh16 := proc(xx)\n local x,z,a1,a2,a3,a4,a5,val,k,t; \n \n x : = evalf(xx); \n\n if abs(x)<.3465735902799727 then\n # coeffici ents of polynomial approximation\n a1 := .16666666666666676612;\n a2 := .83333333333216038084e-2;\n a3 := .198412698861928154 44e-3;\n a4 := .27557244143919119361e-5;\n a5 := .2510905924 7008562146e-7;\n\n # evaluate the approximation\n z := x*x; \n (1+(a1+(a2+(a3+(a4+a5*z)*z)*z)*z)*z)*x;\n else\n t := e xp16(x);\n (t-1/t)*0.5;\n end if;\nend proc: # sinh16\n\ncosh16 := proc(xx)\n local x,z,a1,a2,a3,a4,a5,a6,t; \n \n x := evalf(x x); \n\n if abs(x)<.3465735902799727 then\n # coefficients of \+ polynomial approximation\n a1 := .5;\n a2 := .41666666666666 684932e-1;\n a3 := .13888888888873445058e-2;\n a4 := .248015 87348830599699e-4;\n a5 := .27557252171504288273e-6;\n a6 := .20921652089266008826e-8;\n\n # evaluate the approximation\n \+ z := x*x;\n 1+(a1+(a2+(a3+(a4+(a5+a6*z)*z)*z)*z)*z)*z;\n else \n t := exp16(x);\n (t+1/t)*0.5;\n end if;\nend proc: # co sh16" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "Testing th e procedures sinh16, cosh16" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "We can use Maple's built-in functions to check \+ the accuracy of the procedures " }{TEXT 0 6 "sinh16" }{TEXT -1 5 " and " }{TEXT 0 6 "cosh16" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "evalf(plot(1-'sinh16'(x)/sinh(x),x=0..1,color=blue),2 5);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVES G6#7^y7$$\":ILLLLL$3x1h6o!#G$!#Y!#C7$$\":mmmmmm;a8ABO\"!#F$!$%=F-7$$\" :***********\\7.K[V?F1$!$:%F-7$$\":KLLLLLL3FWYs#F1$!$Q(F-7$$\":)****** *****\\iSmp3%F1$!%d;F-7$$\":lmmmmmm;a)G\\aF1$!%VHF-7$$\":(************ \\7G$R<)F1$!%#f'F-7$$\":LLLLLLL$3x&)*3\"!#E$!&[;\"F-7$$\":+++++++Dc'yM ;FP$!&[d#F-7$$\":mmmmmmmmT:(z@FP$!&\\Y%F-7$$\":************\\7y%*z7$FP $!&zk)F-7$$\":LLLLLLLe9ui2%FP$!'N[8F-7$$\":************\\(oMrU^FP$!'^) )=F-7$$\":mmmmmmm;z_\"4iFP$!'mIBF-7$$\":lmmmmmm\"zp!fu'FP$!'9([#F-7$$ \":lmmmmmmm;hEG(FP$!'9'e#F-7$$\":kmmmmm;/EQ5b(FP$!'d6EF-7$$\":kmmmmmmT N:%>yFP$!'')>EF-7$$\":kmmmmm;zW#z(3)FP$!'S5EF-7$$\":lmmmmmm;aphN)FP$!' o#e#F-7$$\":(***********\\(=ddC%*FP$!'w(G#F-7$$\":LLLLLL$e*=)H\\5!#D$! 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\n else 'sinh_'(x)\n end if\n elif type(x,'r ealcons') then\n if x=0 then 0\n elif x='infinity' then 'inf inity'\n else 'sinh_'(x)\n end if\n else 'sinh_'(x)\n en d if;\nend proc: # of sinh_\n\ncosh_ := proc(x::algebraic)\n local n ,t,u;\n\n if nargs <> 1 then\n error \"expecting 1 argument, go t %1\", nargs;\n end if;\n if type(x,'float') then evalf('cosh_'(x ))\n elif (type(x,'numeric') or type(x,'radnum')) \n \+ and signum(0,x,0)<0 then cosh_(-x)\n elif type(x,`*`) and \+ type(op(1,x),'numeric') \n and signum(0,op(1,x),0)< 0 then cosh_(-x)\n elif type(x,'function') and nops(x)=1 then\n \+ n := op(0,x);\n t := op(1,x);\n if n='arcsinh_' or n='arcsi nh' then sqrt(1+t^2)\n elif n='arccosh_' or n='arccosh' then t\n \+ elif n='arctanh_' or n='arctanh' then 1/sqrt(1-t^2)\n elif n ='ln_' or n='ln' then\n u := t/2+1/(2*t);\n if type(u, radnum) then normal(u) else u end if; \n else 'cosh_'(x)\n e nd if\n elif type(x,'realcons') then\n if x=0 then 1\n eli f x='infinity' then 'infinity'\n else 'cosh_'(x)\n end if\n \+ else 'cosh_'(x)\n end if;\nend proc: # of cosh_\n\n`evalf/sinh_` : = proc(xx)\n local t,val,x,z,s,k,term,eps,maxit,saveDigits,hfDigits, sum;\n\n if not type(xx,realcons) then return 'sinh_'(xx) end if;\n \n # Use the fixed precision procedure for low precision values\n \+ hfDigits := trunc(evalhf(Digits));\n if Digits<=hfDigits then\n \+ x := evalf(xx,hfDigits+2);\n if abs(x).2878231366 then\n # Use formula sinh(x)=(exp(x)-exp(-x))/2\n t := `evalf/exp_`(x);\n Digits := Digits+1;\n val := (t-1/t)*0.5;\n return eva lf[saveDigits](val);\n else\n if saveDigits<58 then\n # use the function exp(x)-1\n if saveDigits<27 then \+ \n s := expm25(x);\n val := s-0.5*s^2/(s+1);\n \+ return evalf[saveDigits](val);\n elif saveDigits<58 \+ then\n s := expm55(x);\n val := s-0.5*s^2/(s+1); \n return evalf[saveDigits](val);\n end if;\n \+ else\n # Use Maclaurin series\n Digits := saveDigits+l ength(saveDigits)+1;\n x := evalf(xx);\n eps := Float( 1,-saveDigits);\n maxit := Digits*2;\n term := x;\n \+ sum := term;\n z := x*x;\n for k from 2 to maxit by 2 do\n term := term*z/(k*(k+1));\n sum := su m+term;\n if abs(term)<=eps*abs(sum) then break end if;\n \+ end do;\n end if;\n\n return evalf[saveDigits](sum); \n end if;\nend proc: # of `evalf/sinh_`\n\n`evalf/cosh_` := proc(xx )\n local t,val,x,z,s,k,term,eps,maxit,saveDigits,sum,hfDigits;\n\n \+ if not type(xx,realcons) then return 'cosh_'(xx) end if;\n\n # Use the fixed precision procedure for low precision values\n hfDigits : = trunc(evalhf(Digits));\n if Digits<=hfDigits then\n x := eval f(xx,hfDigits+2);\n if abs(x)0.28782313662425571050 then\n \+ # Use formula sinh(x)=(exp(x)-exp(-x))/2\n t := `evalf/exp_`(x); \n Digits := Digits+1;\n val := (t+1/t)*0.5;\n return e valf[saveDigits](val);\n else\n if saveDigits<58 then\n \+ # use the function exp(x)-1\n if saveDigits<27 then \+ \n s := expm25(x);\n val := 1+0.5*s^2/(s+1);\n return evalf[saveDigits](val);\n elif saveDigits<5 8 then\n s := expm55(x);\n val := 1+0.5*s^2/(s+1 );\n return evalf[saveDigits](val);\n end if;\n \+ else\n # Use Maclaurin series\n Digits := saveDigits +length(saveDigits)+1;\n x := evalf(xx);\n eps := Floa t(1,-saveDigits);\n maxit := Digits*2;\n term := 1.0; \n sum := term;\n z := x*x;\n for k from 1 to \+ maxit by 2 do\n term := term*z/(k*(k+1));\n sum \+ := sum+term;\n if abs(term)<=eps*abs(sum) then break end if ;\n end do;\n return evalf[saveDigits](sum);\n en d if;\n end if;\nend proc: # of `evalf/cosh_`\n\nexpm55 := proc(x)\n local a1,a2,a3,a4,a5,a6,a7,b1,b2,b3,b4,b5,b6,z,n,d,t;\n \n a1 := \n .166666666666666666666666666666666666666666666666666666666571565; \n a2 := \n .3395050330308604661409457824891090536919169436836132754 03380137e-2;\n a3 := \n .1984104683589237072835798643959118061664937 47671723846797302008e-4;\n a4 := \n .4472954114528217502960803062487 26901656997154800468190597343065e-7;\n a5 := \n.40339403153976306703 3041400818864128288431392911408985198279373e-10;\n a6 := \n.12248225 9583600568922888945476478139244263303786334208506378068e-13;\n a7 := \n.571764694596822796978120873215841239524458473955271551257567659e-1 8;\n b1 := \n .37036968648518294635123413616013209888181683287683462 6466591020e-1;\n b2 := \n .33950369499859564289204608207903708977276 5573239209990946031469e-3;\n b3 := \n .11501973029918209733294425612 5175514189676603556261971226439331e-5;\n b4 := \n .15974636518154811 4758598142014828683416955083052207093458714985e-8;\n b5 := \n.849232 569219197625336843378312648323130436490088335129238683053e-12;\n b6 \+ := \n.120076286825632969873337000713027499696163087057058585410372308e -15;\n \n # evaluate the approximation\n z := x*x;\n n := (a1 +(a2+(a3+(a4+(a5+(a6+a7*z)*z)*z)*z)*z)*z)*z;\n d := 1+(b1+(b2+(b3+(b 4+(b5+b6*z)*z)*z)*z)*z)*z;\n t := x-n/d; \n x-x*t/(t-2);\nend pr oc: # of expm55\n\nexpm25 := proc(x)\n local a1,a2,a3,b1,b2,b3,z,n,d ,t; \n\n a1 := .166666666666666666666665472751;\n a2 := .25639955 8152201038014309336620e-2;\n a3 := .624256766847658759948139608267e- 5;\n b1 := .320506401557987289458437423472e-1;\n b2 := .1748073451 15441647383751465420e-3;\n b3 := .115582674221396338415384591432e-6; \n \n # evaluate the rational approximation\n z := x*x ;\n n := (a1+(a2+a3*z)*z)*z;\n d := 1+(b1+(b2+b3*z)*z)*z;\n t := x-n/d;\n x-x*t/(t-2); \nend proc: # of expm25\n\nsinh16 := pro c(xx)\n local x,z,a1,a2,a3,a4,a5,val,k,t; \n \n x := evalf(xx); \+ \n\n if abs(x)<.3465735902799727 then\n # coefficients of polyn omial approximation\n a1 := .16666666666666676612;\n a2 := . 83333333333216038084e-2;\n a3 := .19841269886192815444e-3;\n \+ a4 := .27557244143919119361e-5;\n a5 := .25109059247008562146e-7 ;\n\n # evaluate the approximation\n z := x*x;\n (1+(a1 +(a2+(a3+(a4+a5*z)*z)*z)*z)*z)*x;\n else\n t := exp16(x);\n \+ (t-1/t)*0.5;\n end if;\nend proc: # sinh16\n\ncosh16 := proc(xx)\n local x,z,a1,a2,a3,a4,a5,a6,t; \n \n x := evalf(xx); \n\n if abs(x)<.3465735902799727 then\n # coefficients of polynomial app roximation\n a1 := .5;\n a2 := .41666666666666684932e-1;\n \+ a3 := .13888888888873445058e-2;\n a4 := .24801587348830599699 e-4;\n a5 := .27557252171504288273e-6;\n a6 := .209216520892 66008826e-8;\n\n # evaluate the approximation\n z := x*x;\n \+ 1+(a1+(a2+(a3+(a4+(a5+a6*z)*z)*z)*z)*z)*z;\n else\n t := e xp16(x);\n (t+1/t)*0.5;\n end if;\nend proc: # cosh16" }}} {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 39 "Examples are given in the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 11 "sinh_,cosh_" }{TEXT -1 10 ": examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Here are some numerical examples." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "Digits := 10 0:\nxx := 0.56789;\nsinh_(xx);\nsinh(xx);\ncosh_(xx);\ncosh(xx);\nDigi ts := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"&*yc!\"&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_q6+t\\$p#4Mfi&3$z`7(R$=*)p*ec370[; kMDS7Qf@(oS&>Q(e5.X@**4*)f!$+\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" _q6+t\\$p#4Mfi&3$z`7(R$=*)p*ec370[;kMDS7Qf@(oS&>Q(e5.X@**4*)f!$+\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_qy)ox)**yGRu&yF;Q`Ku`'\\Sn.&4d[b%) *\\8us&e/]26kbYw?,%H\\*Hc;\"!#**" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"_qy)ox)**yGRu&yF;Q`Ku`'\\Sn.&4d[b%)*\\8us&e/]26kbYw?,%H\\*Hc;\"!#** " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "Digits := 100:\nxx := evalf(sqrt(2)/10,Digits+5):\ns inh_(xx);\nsinh(xx);\ncosh_(xx);\ncosh(xx);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_q]oL'*>=3kf(=ID<\")o!e,()R)pl(eC$>8 $y/=3kf(=ID<\")o!e,()R)pl(eC$>8$y/_iprk4C2YFq5(QvQkuV>>p)3F%fyC#[t7Sf! 3UG(Qr8pu\"yxm,55!#**" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_q\">_iprk 4C2YFq5(QvQkuV>>p)3F%fyC#[t7Sf!3UG(Qr8pu\"yxm,55!#**" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Some special cases can be handled.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "Digits := 20:\nyy := 3/7;\nxx := convert(arcsinh(yy),ln);\nsinh_(xx);\nevalf(%) ;\nsinh_(evalf(xx,Digits+2));\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#yyG#\"\"$\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #xxG-%#lnG6#,&#\"\"$\"\"(\"\"\"*&F+!\"\"\"#e#F,\"\"#F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"$\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" 5Vr&G9dG9dG%!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5Vr&G9dG9dG%!#? " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "Digits := 20:\nyy := 12/11;\nxx := convert(arccosh(y y),ln);\ncosh_(xx);\nevalf(%);\ncosh_(evalf(xx,Digits+2));\nDigits := \+ 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#yyG#\"#7\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG-%#lnG6#,&#\"#7\"#6\"\"\"*&\"$@\"!\"\"\"%$y# #F,\"\"#F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#7\"#6" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"5\"444444444\"!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5\"444444444\"!#>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "We solve the equation " }{XPPEDIT 18 0 "sinh(x)^ 2 = cosh(x);" "6#/*$-%%sinhG6#%\"xG\"\"#-%%coshG6#F(" }{TEXT -1 22 " \+ using the functions " }{TEXT 0 5 "sinh_" }{TEXT -1 5 " and " }{TEXT 0 5 "cosh_" }{TEXT -1 2 ". 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r\"*FE$\"+\\lH0;F27$$\"+lxGp$*FE$\"+CTLU;F27$$\"+!oK0e*FE$\"+kV(Ro\"F2 7$$\"+<5s#y*FE$\"+Q5.E " 0 "" {MPLTEXT 1 0 38 "Int( cosh_(sinh_(x)),x=0..1);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #-%$IntG6$-%&cosh_G6#-%&sinh_G6#%\"xG/F,;\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+lP<<7!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Int(cosh_(sinh_(x)),x=0..1); \nquad(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%&cosh_G6#-%&s inh_G6#%\"xG/F,;\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+lP< <7!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Int(cosh(sinh(x)),x=0..1);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%%coshG6#-%%sinhG6#%\"xG/F,;\"\"!\"\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+lP<<7!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }