{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Red Emphasis" -1 259 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 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 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "Purple Emphasis" -1 266 "Times" 1 12 103 0 230 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Grey Emphasis" -1 267 "Times" 1 12 96 52 84 1 0 1 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 "Purple Emphasis" -1 269 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 270 "Times" 1 12 128 0 0 1 0 1 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 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Norm al" -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 O utput" -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 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 35 "Applying Simpson's rule adaptivel y " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Cana da" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 24.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 49 "load numerical integration procedures including: \+ " }{TEXT 0 5 "SPint" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file \+ " }{TEXT 267 6 "intg.m" }{TEXT -1 37 " contains the code for the proce dure " }{TEXT 0 5 "SPint" }{TEXT -1 1 " " }{TEXT -1 24 "used in this w orksheet. " }}{PARA 0 "" 0 "" {TEXT -1 122 "It can be read into a Mapl e session by a command similar to the one that follows, where the file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "read \"K:\\\\Maple/procdrs/intg.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 79 "A preliminary adaptive procedure \+ for numerical integration using Simpson's rule" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 35 "Using uniforml y spaced subdividing " }{TEXT 268 1 "x" }{TEXT -1 148 " coordinates fo r the intervals used by Simpson's rule is efficient if the function's \+ behaviour is similar across the whole interval of integration. " }} {PARA 0 "" 0 "" {TEXT -1 34 " If this is not the case, then an " } {TEXT 266 18 "adaptive procedure" }{TEXT -1 23 " can be more efficient ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "For \+ example consider the function " }{XPPEDIT 18 0 "f(x)=sqrt(x)/(x^2+1)*( sin((51+x)*exp(-3*x^2))+2)" "6#/-%\"fG6#%\"xG*(-%%sqrtG6#F'\"\"\",&*$F '\"\"#F,F,F,!\"\",&-%$sinG6#*&,&\"#^F,F'F,F,-%$expG6#,$*&\"\"$F,*$F'F/ F,F0F,F,F/F,F," }{TEXT -1 2 ". 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>#Hw42od7Fcs7$$\"3@LLe9rR37Fcs$\"3iyT\\dc)**F07$$\"3GLLLj$[kL\"F cs$\"3y\"fOh6)*4J*F07$$\"3?LLL`Q\"GT\"Fcs$\"3!p*p9trma%)F07$$\"3!***** \\s]k,:Fcs$\"31'HHq.uuv(F07$$\"39LLL`dF!e\"Fcs$\"3au(*[=NN%H(F07$$\"33 ++]sgam;Fcs$\"31Fu8R@ZyoF07$$\"3/++]Fcs$\"3**3g5*3?*QfF07$$\"3immm Tc-)*>Fcs$\"3!RN\\dtXRm&F07$$\"3Mmm;f`@'3#Fcs$\"3S))eg\\*=vR&F07$$\"3y ****\\nZ)H;#Fcs$\"33kH0(=G+=&F07$$\"3YmmmJy*eC#Fcs$\"3G56:y>1f\\F07$$ \"3')******R^bJBFcs$\"3DG\"e/x1\\u%F07$$\"3f*****\\5a`T#Fcs$\"3]W?;-vI [XF07$$\"3o****\\7RV'\\#Fcs$\"3Md%*\\j%*QpVF07$$\"3k*****\\@fke#Fcs$\" 3I2Pv&GAG=%F07$$\"3/LLL`4NnEFcs$\"3%4*>]hPEDSF07$$\"3#*******\\,s`FFcs $\"3yE#fr=.o'QF07$$\"3[mm;zM)>$GFcs$\"3!>U.*[.KJPF07$$\"3$*******pfa " 0 "" {MPLTEXT 1 0 1 ";" }}}{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 47 " oscillate quite \+ wildly over the interval from " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" } {TEXT -1 4 " to " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 47 " , and also its derivative tends to infinity as " }{TEXT 272 1 "x" } {TEXT -1 30 " tends to zero. However, when " }{TEXT 271 1 "x" }{TEXT -1 20 " is greater than 1, " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 75 " is quite \"well-behaved\", that is, its graph appears to be nice and smooth." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 141 "An adaptive procedure \"discovers\" where the function is ill-behaved and shortens the intervals used for the integration ru le in such regions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 43 "A simple procedure which does this follows." }}{PARA 0 "" 0 "" {TEXT -1 25 "The procedure is defined " }{TEXT 266 11 "recursi vely" }{TEXT -1 27 ", that is, it calls itself." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "If the global variable " }{TEXT 267 5 "_INFO" }{TEXT -1 11 " is set to " }{TEXT 267 4 "true" } {TEXT -1 74 ", then the first print statement is executed each time th at the procedure " }{TEXT 0 8 "adapsimp" }{TEXT -1 12 " is called. " } }{PARA 0 "" 0 "" {TEXT -1 115 "The second print statement is also exec uted each time that the error condition is satisfied, so that the proc edure " }{TEXT 0 8 "adapsimp" }{TEXT -1 102 " exits to return a numeri cal value corresponding to the area associated with the current subint erval. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 818 "adapsimp := proc(f,a,b)\n local fn,m,h,fact,fa,fb, fm,fam,fmb,area1,area2,eps;\n global _INFO;\n \n eps := 10^(2-Di gits);\n fn := evalf@f;\n \n m := 1/2*a+1/2*b;\n h := b-a;\n \+ fact := h/6;\n fa := fn(a);\n fb := fn(b);\n fm := fn(m);\n f am := fn(1/2*a+1/2*m);\n fmb := fn(1/2*m+1/2*b);\n \n # This is \+ where Simpson's Rule is applied.\n area1 := evalf((fa+4*fm+fb)*fact) ;\n area2 := evalf((fa+4*(fam+fmb)+2*fm+fb)*fact/2);\n if _INFO=tr ue then\n print(`interval:`,a..b,` ------ area1 = `,area1,`area2 = `,area2);\n end if;\n if abs((area2-area1))<=eps*abs(area2) the n\n if _INFO=true then\n print(`interval width = `,h,` \+ error OK`);\n end if;\n return area2\n else # subdivide \+ further\n adapsimp(f,a,m)+adapsimp(f,m,b) # recursive function ca lls\n end if \nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 8 "adapsimp" }{TEXT -1 11 ": examples " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "(i) " }{XPPEDIT 18 0 "Int(sin(x^2) ,x=0..1)" "6#-%$IntG6$-%$sinG6#*$%\"xG\"\"#/F*;\"\"!\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "adapsimp(x->sin(x^2),0.,1.);\nevalf(Int(sin(x^2),x=0. .1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ " 0 "" {MPLTEXT 1 0 63 "adapsimp(x->exp(cos(x)),0.,5.);\nevalf(Int(exp(cos(x) ),x=0..5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+g#GM=&!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+h#GM=&!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "(iii) " }{XPPEDIT 18 0 "I nt((cos(x)-1)/x,x = 1 .. 6);" "6#-%$IntG6$*&,&-%$cosG6#%\"xG\"\"\"F,! \"\"F,F+F-/F+;F,\"\"'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "Digits := 7:\n_INFO := \+ true;\nadapsimp(x->(1-cos(x))/x,1.,6.);\n_INFO := false;\nevalf(Int((1 -cos(x))/x,x=1..6));\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%&_INFOG%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G;$\" \"\"\"\"!$\"\"'F'%2~------~~area1~=~G$\"(fGB#!\"'%)area2~=~G$\"(1\"*># F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G;$\"\"\"\"\"!$\"(++] $!\"'%2~------~~area1~=~G$\"(E\"G;F*%)area2~=~G$\"(CEi\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G;$\"\"\"\"\"!$\"(++D#!\"'%2~---- --~~area1~=~G$\"(28(y!\"(%)area2~=~G$\"(S%pyF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G;$\"\"\"\"\"!$\"(+]i\"!\"'%2~------~~area1~ =~G$\"(4$=N!\"(%)area2~=~G$\"(c#=NF." }}{PARA 11 "" 1 "" {XPPMATH 20 " 6(%*interval:G;$\"\"\"\"\"!$\"(+DJ\"!\"'%2~------~~area1~=~G$\"(r4h\"! \"(%)area2~=~G$\"(o4h\"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%2interva l~width~=~G$\"'+DJ!\"'%-~~~~error~OKG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G;$\"(+DJ\"!\"'$\"(+]i\"F'%2~------~~area1~=~G$\"(%G2>! \"(%)area2~=~G$\"(#G2>F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%2interval ~width~=~G$\"'+DJ!\"'%-~~~~error~OKG" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6(%*interval:G;$\"(+]i\"!\"'$\"(++D#F'%2~------~~area1~=~G$\"(K6N%!\"( %)area2~=~G$\"(s5N%F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G; $\"(+]i\"!\"'$\"(+v$>F'%2~------~~area1~=~G$\"(**p6#!\"(%)area2~=~G$\" ('*p6#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%2interval~width~=~G$\"'+D J!\"'%-~~~~error~OKG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G;$ \"(+v$>!\"'$\"(++D#F'%2~------~~area1~=~G$\"(sSB#!\"(%)area2~=~G$\"(qS B#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%2interval~width~=~G$\"'+DJ!\" '%-~~~~error~OKG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G;$\"(+ +D#!\"'$\"(++]$F'%2~------~~area1~=~G$\"(G\\N)!\"(%)area2~=~G$\"(:PN)F -" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G;$\"(++D#!\"'$\"(+](G F'%2~------~~area1~=~G$\"(^DX%!\"(%)area2~=~G$\"(-DX%F-" }}{PARA 11 " " 1 "" {XPPMATH 20 "6(%*interval:G;$\"(++D#!\"'$\"(+Dc#F'%2~------~~ar ea1~=~G$\"(P\"eA!\"(%)area2~=~G$\"(O\"eAF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%2interval~width~=~G$\"'+DJ!\"'%-~~~~error~OKG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G;$\"(+Dc#!\"'$\"(+](GF'%2~ ------~~area1~=~G$\"(iV>#!\"(%)area2~=~G$\"(hV>#F-" }}{PARA 11 "" 1 " " {XPPMATH 20 "6%%2interval~width~=~G$\"'+DJ!\"'%-~~~~error~OKG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G;$\"(+](G!\"'$\"(++]$F'%2~ ------~~area1~=~G$\"(n6!R!\"(%)area2~=~G$\"(V6!RF-" }}{PARA 11 "" 1 " " {XPPMATH 20 "6%%2interval~width~=~G$\"'+]i!\"'%-~~~~error~OKG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G;$\"(++]$!\"'$\"\"'\"\"!%2 ~------~~area1~=~G$\"(+)4d!\"(%)area2~=~G$\"(!4ZdF." }}{PARA 11 "" 1 " " {XPPMATH 20 "6(%*interval:G;$\"(++]$!\"'$\"(++v%F'%2~------~~area1~= ~G$\"(LXr%!\"(%)area2~=~G$\"(y]r%F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 (%*interval:G;$\"(++]$!\"'$\"(+]7%F'%2~------~~area1~=~G$\"(9x\"H!\"(% )area2~=~G$\"(=x\"HF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%2interval~wi dth~=~G$\"'+]i!\"'%-~~~~error~OKG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(% *interval:G;$\"(+]7%!\"'$\"(++v%F'%2~------~~area1~=~G$\"(ltz\"!\"(%)a rea2~=~G$\"(%R(z\"F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G;$ \"(+]7%!\"'$\"(+vV%F'%2~------~~area1~=~G$\"(Xa.\"!\"(%)area2~=~G$\"(Y a.\"F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%2interval~width~=~G$\"'+DJ! \"'%-~~~~error~OKG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G;$\" (+vV%!\"'$\"(++v%F'%2~------~~area1~=~G$\"(*[>w!\")%)area2~=~G$\"(0&>w F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%2interval~width~=~G$\"'+DJ!\"'% -~~~~error~OKG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G;$\"(++v %!\"'$\"\"'\"\"!%2~------~~area1~=~G$\"(eD.\"!\"(%)area2~=~G$\"(US.\"F ." }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G;$\"(++v%!\"'$\"(+]P& F'%2~------~~area1~=~G$\"('pj#)!\")%)area2~=~G$\"(STE)F-" }}{PARA 11 " " 1 "" {XPPMATH 20 "6(%*interval:G;$\"(++v%!\"'$\"(+D1&F'%2~------~~ar ea1~=~G$\"(E%f^!\")%)area2~=~G$\"(N%f^F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%2interval~width~=~G$\"'+DJ!\"'%-~~~~error~OKG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G;$\"(+D1&!\"'$\"(+]P&F'%2~------~~area1~ =~G$\"(5Z5$!\")%)area2~=~G$\"(CZ5$F-" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6%%2interval~width~=~G$\"'+DJ!\"'%-~~~~error~OKG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G;$\"(+]P&!\"'$\"\"'\"\"!%2~------~~area1~=~ G$\"(Jn2#!\")%)area2~=~G$\"('=x?F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6( %*interval:G;$\"(+]P&!\"'$\"(+vo&F'%2~------~~area1~=~G$\"(xha\"!\")%) area2~=~G$\"(#>Y:F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%2interval~widt h~=~G$\"'+DJ!\"'%-~~~~error~OKG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%*i nterval:G;$\"(+vo&!\"'$\"\"'\"\"!%2~------~~area1~=~G$\"()35`!\"*%)are a2~=~G$\"(D-J&F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G;$\"(+ vo&!\"'$\"(]P%eF'%2~------~~area1~=~G$\"(]#yN!\"*%)area2~=~G$\"(e#yNF- " }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%2interval~width~=~G$\"']i:!\"'%-~ ~~~error~OKG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G;$\"(]P%e! \"'$\"\"'\"\"!%2~------~~area1~=~G$\"(y>t\"!\"*%)area2~=~GF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%2interval~width~=~G$\"']i:!\"'%-~~~~error~O KG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"(@s>#!\"'" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%&_INFOG%&falseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"(@s>#!\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "(iv) " }{XPPEDIT 18 0 "Int(sqrt(x),x = 1/100 .. 1);" "6# -%$IntG6$-%%sqrtG6#%\"xG/F);*&\"\"\"F-\"$+\"!\"\"F-" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "Digits := 6:\n_INFO := true;\nadapsimp(x->sqrt(x),0.01,1.);\n_I NFO := false;\nevalf(Int(sqrt(x),x=0.01..1));\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&_INFOG%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G;$\"\"\"!\"#$F&\"\"!%2~------~~area1~=~G$\" '>0l!\"'%)area2~=~G$\"'D$F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G;$\"\"\"!\"#$\"'](=(!\"(%2~------~~area1~=~ G$\"'@97F*%)area2~=~G$\"'X<7F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%*in terval:G;$\"\"\"!\"#$\"'v$4%!\"(%2~------~~area1~=~G$\"'T][!\")%)area2 ~=~G$\"'wa[F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G;$\"\"\"! \"#$\"')oa#!\"(%2~------~~area1~=~G$\"'eU?!\")%)area2~=~G$\"'+V?F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G;$\"\"\"!\"#$\"'Wt7G!\")%)area2~=~G$\"'A7GF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%2interval~width~=~G$\"'(oa\"!\"(%-~~~~error~OKG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G;$\"'v$4%!\"($\"'](=(F'%2~ ------~~area1~=~G$\"'/Ct!\")%)area2~=~G$\"'ICtF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%2interval~width~=~G$\"'v$4$!\"(%-~~~~error~OKG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G;$\"'](=(!\"($\"']P8!\"'%2 ~------~~area1~=~G$\"'Fw>F'%)area2~=~G$\"'Ow>F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%2interval~width~=~G$\"'](='!\"(%-~~~~error~OKG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G;$\"']P8!\"'$\"'+vDF'%2~-- ----~~area1~=~G$\"'$)\\a!\"(%)area2~=~G$\"'5]aF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%2interval~width~=~G$\"']P7!\"'%-~~~~error~OKG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G;$\"'+vD!\"'$\"'+]]F'%2~-- ----~~area1~=~G$\"'D@:F'%)area2~=~G$\"'M@:F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%2interval~width~=~G$\"'+vC!\"'%-~~~~error~OKG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6(%*interval:G;$\"'+]]!\"'$\"\"\"\"\"!%2 ~------~~area1~=~G$\"'\"RF%F'%)area2~=~G$\"';uUF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%2interval~width~=~G$\"'+]\\!\"'%-~~~~error~OKG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"'%*fm!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&_INFOG%&falseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"'+gm!\"'" }}}{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 26 "The preliminary procedure " }{TEXT 0 8 "adapsimp" } {TEXT -1 104 " in general involves repeat evaluations of the integrand at points where it has already been evaluated. " }}{PARA 0 "" 0 "" {TEXT -1 148 "A more efficient adaptive procedure, which avoids such r epeat evaluations, and so requires fewer function evaluations, is give n in the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "An adaptive version of Simpson's rule: " }{TEXT 0 5 "SPin t" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "SPint: usag e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 260 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 261 2 " " }{TEXT -1 20 " SPint( gx, rng ) " }{TEXT 262 1 "\n" }{TEXT -1 0 "" } }{PARA 257 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT 23 10 " fx - " }{TEXT -1 55 " \+ an expression involving a single variable, say x," }}{PARA 0 "" 0 "" {TEXT -1 85 " where g(x) evaluates to a real floating point number." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 12 " rng - " }{TEXT 263 61 "the range x=a..b for the definite integral to be aproximated." }} {PARA 0 "" 0 "" {TEXT -1 10 " " }}{PARA 257 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " } {TEXT 0 5 "SPint" }{TEXT -1 48 " attempts to find a numerical approxim ation for " }{XPPEDIT 18 0 "Int(gx,x = a .. b);" "6#-%$IntG6$%#gxG/%\" xG;%\"aG%\"bG" }{TEXT -1 196 " by using Simpson's rule. The rule can \+ be applied in compound form or adaptively. In the latter case the exte nt of the subdivision used is assessed locally by tracking the relativ e error locally." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 8 "Options:" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 19 "adaptive=true/false" }}{PARA 0 "" 0 "" {TEXT -1 112 "This option spec ifies whether an adaptive mechanism based on a fixed rule with \"nump oints\" nodes is to be used." }}{PARA 0 "" 0 "" {TEXT -1 22 "In the ad aptive mode, " }{TEXT 0 5 "SPint" }{TEXT -1 182 " attempts to ensure t hat the answer is accurate to the number of digits given according to \+ the current setting of \"Digits\".\nAdaptive quadrature is performed i f the option is not set." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 127 "maxdepth=n\nThis option can be used to restrict t he maximum depth of subdivision to produce sub-intervals of width no l ess than " }{XPPEDIT 18 0 "1/2^n" "6#*&\"\"\"F$)\"\"#%\"nG!\"\"" } {TEXT -1 32 " of the original interval width." }}{PARA 0 "" 0 "" {TEXT -1 41 "The default value is \"maxdepth=2*Digits\"." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "factor=n" }}{PARA 0 "" 0 "" {TEXT -1 177 "In the non-adaptive mode the Simpson's rule ca n be applied on \"factor\" equal subintervals which span the original \+ interval of integration x = a to b. The results are then added." }} {PARA 0 "" 0 "" {TEXT -1 38 "This is the compound form of the rule." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "info=tru e/false" }}{PARA 0 "" 0 "" {TEXT -1 71 "The option \"info=true'' gives the total number of function evaluations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 15 "How to activate" }{TEXT 256 2 ":\n" }{TEXT -1 154 "To make the pro cedure active open the subsection, place the cursor anywhere after the prompt [ > and press [Enter].\nYou can then close up the subsection. " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "SPint: implementation" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6048 "SPint := proc(alg_expr,eq)\n local Options,adapt,mthd,x,rs,val ,ct,x0,x1,L,h,y0,y1,x01,y01,\n yi,s,i,xi,prntflg,vals,area,sign,t ,SPadapt,saveDigits,\n eps,maxdpth,currentmaxbisect,fact,j,p;\n\n if nargs<2 then\n error \"at least 2 arguments are required; th e basic syntax is: 'SPint(f(x),x=a..b)'.\"\n end if;\n \n # get \+ the requested options, but first give defaults\n adapt := true;\n \+ maxdpth := Digits*2;\n fact := 1;\n prntflg := false;\n if nargs >2 then\n Options:=[args[3..nargs]];\n if not type(Options,l ist(equation)) then\n error \"each optional argument must be a n equation\"\n end if;\n if hasoption(Options,'adaptive','ad apt','Options') then\n if adapt<>true then adapt := false end \+ if; \n end if;\n if hasoption(Options,'maxdepth','maxdpth',' Options') then\n if not type(maxdpth,posint) then\n \+ error \"\\\"maxdepth\\\" must be a positive integer\"\n end i f;\n end if;\n if hasoption(Options,'factor','fact','Options ') then\n if not type(fact,posint) then\n error \" \\\"factor\\\" must be a positive integer\"\n end if;\n e nd if;\n if hasoption(Options,'info','prntflg','Options') then\n \+ if prntflg<>true then prntflg := false end if;\n end if; \n if nops(Options)>0 then\n error \"%1 is not a valid op tion for %2\",op(1,Options),procname;\n end if;\n end if;\n\n# \+ SPadapt performs recursion for adaptive integration\n\nSPadapt := proc (x0,x1,x2,y0,y1,y2,bisectionlevel)\n local L,x01,x12,y01,y12,area1,a rea2;\n\n if prntflg then\n if bisectionlevel > currentmaxbisec t then\n currentmaxbisect := bisectionlevel;\n print(` deepest bisection level --> `,bisectionlevel);\n if bisectionl evel >= maxdpth then\n print(`bisection level has been rest ricted to `,maxdpth);\n end if;\n end if;\n end if;\n \+ if x0=x2 then return 0 end if;\n\n L := evalf(x2-x0);\n h := L/6; \n\n x01 := evalf((x0+x1)/2);\n y01 := traperror(evalf(eval(subs(x =x01,alg_expr))));\n if y01=lasterror or not type(y01,numeric) then \n error \"evaluation failed at %1\",evalf(x01,saveDigits);\n e nd if;\n\n x12 := evalf((x1+x2)/2);\n y12 := traperror(evalf(eval( subs(x=x12,alg_expr))));\n if y12=lasterror or not type(y01,numeric) then\n error \"evaluation failed at %1\",evalf(x12,saveDigits); \n end if;\n if prntflg then ct := ct + 2 end if;\n # This is wh ere Simpson's Rule is applied.\n area1 := evalf((y0+4*y1+y2)*h);\n \+ area2 := evalf((y0+4*(y01+y12)+2*y1+y2)*h/2);\n if abs((area2-area1 ))<=eps*abs(area1) or bisectionlevel>=maxdpth then\n return area2 ;\n else \n return SPadapt(x0,x01,x1,y0,y01,y1,bisectionlevel+1 )+\n SPadapt(x1,x12,x2,y1,y12,y2,bisectionlevel+1);\n \+ end if; \nend proc:\n\n # now do the quadrature\n if not type(alg _expr,algebraic) then \n error \"the 1st argument, %1, is invalid ..it should be an algebraic expression in a single variable\",alg_exp r;\n end if; \n if not type(eq,`=`) then \n error \"the 2nd a rgument, %1, is invalid ..it should be an equation of the form 'x=a..b ' to give the required interval for the integral to be estimated\",eq; \n end if;\n x := op(1,eq);\n if not type(x,symbol) then\n \+ error \"the 2nd argument equation left side, %1, should be the indepen dent variable\",x;\n end if;\n if not type(indets(alg_expr,name) m inus \{x\},set(realcons)) then\n error \"the 1st argument, %1, mu st depend only on the variable %2\",alg_expr,x;\n end if;\n \n r s := op(2,eq);\n if not type(rs,realcons..realcons) then\n erro r \"the 2nd argument equation right side, %1, should be a range of rea l values\",rs;\n end if;\n\n ct := 0;\n\n # increase precision f or the computation\n saveDigits := Digits;\n Digits := min(trunc(D igits*4/3),Digits+5);\n\n x0 := evalf(op(1,rs));\n x1 := evalf(op( 2,rs));\n if x0=x1 then return 0 end if;\n sign := 1;\n if x1 `,ct);\n end if;\n\n Digits := saveDigits;\n if sign > 0 then\n return evalf(val) ;\n else\n return evalf(-val);\n end if;\nend proc:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 5 "SPint" }{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 14 " The procedure " }{TEXT 0 5 "SPint" }{TEXT -1 60 " can be used to apply Simpson's rule adaptively to evaluate " }{XPPEDIT 18 0 "Int(sin(x),x \+ = Pi/4 .. 3*Pi/4);" "6#-%$IntG6$-%$sinG6#%\"xG/F);*&%#PiG\"\"\"\"\"%! \"\"*(\"\"$F.F-F.F/F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 543 "f := x -> sin(x): # fun ction\na := evalf(Pi/4): # lower limit of integral\nb := evalf(3*Pi/4) : # upper limit of integral\nclr := grey: # color for shading\npp := p lot([0,f(x)],x=a..b,adaptive=false,numpoints=20):\nu := op(1,op(1,pp)) : v := op(1,op(2,pp)):\np1 := plots[polygonplot]([seq([u[i],v[i],v[i+1 ],u[i+1]],i=1..19)],\n color=clr,style= patchnogrid):\np2 := plot([[[a,0],[a,f(a)]],[[b,0],[b,f(b)]]],color=bl ack):\np3 := plot(f(x),x=0..Pi,thickness=2): # adjust plot range\nplot s[display]([p1,p2,p3],labels=[`x`,`y`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 367 257 257 {PLOTDATA 2 "6(-%)POLYGONSG677&7$$\"+N;)R&y!#5$ \"\"!F,7$F($\"+8y1rqF*7$$\"+6T'*=()F*$\"+\"=/bl(F*7$F1F+7&F5F07$$\"+R2 er%*F*$\"+>7f<\")F*7$F8F+7&F7$$\"+V$)*p6\"FA$\"+(>')y)*)F*7$FGF+7&FKFF7$$\"+cSz,7FA$\"+en(oK*F*7$ FNF+7&FRFM7$$\"+P0T!G\"FA$\"+o[L\"e*F*7$FUF+7&FYFT7$$\"+]L\"=O\"FA$\"+ uTU#y*F*7$FfnF+7&FjnFen7$$\"+b-+Y9FA$\"+82BA**F*7$F]oF+7&FaoF\\o7$$\"+ sr\"*H:FA$\"+Pck\"***F*7$FdoF+7&FhoFco7$$\"+NgB;;FA$\"+Tzn*)**F*7$F[pF +7&F_pFjo7$$\"+PdE#p\"FA$\"+xlJE**F*7$FbpF+7&FfpFap7$$\"+ur&yx\"FA$\"+ \\QR'y*F*7$FipF+7&F]qFhp7$$\"+e+!Q'=FA$\"+0j!Qd*F*7$F`qF+7&FdqF_q7$$\" +^>iY>FA$\"+:!\\?I*F*7$FgqF+7&F[rFfq7$$\"+k@$=-#FA$\"+f1'****)F*7$F^rF +7&FbrF]r7$$\"+#Qk76#FA$\"+L%zYd)F*7$FerF+7&FirFdr7$$\"+-_-(=#FA$\"+/# Q1;)F*7$F\\sF+7&F`sF[s7$$\"+YI:vAFA$\"+?EG?wF*7$FcsF+7&FgsFbs7$$\"+!\\ %>cBFAF.7$FjsF+-%'COLOURG6&%$RGBG$\")=THv!\")FatFat-%&STYLEG6#%,PATCHN OGRIDG-%'CURVESG6$7$7$$\"3C+++N;)R&y!#=F+7$F]u$\"3U+++8y1rqF_u-F^t6&F` tF,F,F,-Fit6$7$7$$\"3-+++!\\%>cB!#$\"3v9-&QTFC%oFdv7$$\"3)\\$px*G*f!G\"F_u$\"3q>Km$*>5x7F_u7$ $\"3+5@exGm]>F_u$\"3/`P()fcJQ>F_u7$$\"3[99!=3o^i#F_u$\"3/)Q8J`>^f#F_u7 $$\"35!\\D0[nkH$F_u$\"3'yFM)o\")3PKF_u7$$\"37\"=Za&z%)=RF_u$\"3b2`lK+J >QF_u7$$\"3edXa()oGjXF_u$\"33g-l[Vb1WF_u7$$\"3W%3**Hbm(H_F_u$\"3OWs$HJ 6Y*\\F_u7$$\"37PRr4)3T*eF_u$\"3g4tHMSrebF_u7$$\"3y\"[)yykYxlF_u$\"32tC Ii;N8hF_u7$$\"3[s'ocGo$zrF_u$\"3iG>vb;KylF_u7$$\"3UQ0;gr'p&yF_u$\"3S'e 5DeyJ2(F_u7$$\"3vFMt?$[t`)F_u$\"3Cf#G%e3SPvF_u7$$\"3u\"p30k@I>*F_u$\"3 C$=wEj'y^zF_u7$$\"3#*R7\\HeV)y*F_u$\"3=)oD&\\m_)H)F_u7$$\"3#G[))*)3W' \\5F[v$\"3Y4SV'zgCn)F_u7$$\"30'[@XS@'46F[v$\"3oX3_YFIb*)F_u7$$\"3G9w$3 G*Qz6F[v$\"3=c,/>?tV#*F_u7$$\"3>%3*3tc9T7F[v$\"3#*zOG!*[bh%*F_u7$$\"3E y0DBA!*38F[v$\"3'*))o/b+VIn(**F_u7$$\"3=hw\"ymX#p:F[v$\"274'Gxz)*****F[v7$$\"3mwM!*4(4&Q;F [v$\"39vB0ZK3x**F_u7$$\"3CDL:iU!))p\"F[v$\"3[oG=d;==**F_u7$$\"3o(R1/.C Rw\"F[v$\"3Vz\\$>Q(39)*F_u7$$\"32Id)H7*>J=F[v$\"3?VTu'[jGm*F_u7$$\"3Gf b7wY,(*=F[v$\"3Sm0JN*4EZ*F_u7$$\"3cM5'zg%pg>F[v$\"3ADf%RFx%\\#*F_u7$$ \"3**oJ8:.SJ?F[v$\"3?af@Y>%y&*)F_u7$$\"3)**p!zPD$\\4#F[v$\"3?%>'G5ccd' )F_u7$$\"3k-G%)H)F_u7$$\"3wak3@YBCAF[v$\"3cJr!4) G)*RzF_u7$$\"39/bF_u7$$\"3H .6)e58)4IF[v$\"3M.OA\"o%)RJ\"F_u7$$\"3Kt2RXCLtIF[v$\"3ls_wI6s?oFdv7$$ \"3!)***\\/l#fTJF[v$\"3pawpOMzRJ!#E-F^t6&F`t$\"#5!\"\"F+F+-%*THICKNESS G6#\"\"#-%+AXESLABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F+$\" +aEfTJFAFdfl" 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" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "SPint(sin(x),x=Pi/4.. 3*Pi/4,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~Simpson' s~rule~adaptivelyG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisect ion~level~-->~G\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bise ction~level~-->~G\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bi section~level~-->~G\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~ bisection~level~-->~G\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepes t~bisection~level~-->~G\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deep est~bisection~level~-->~G\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnu mber~of~function~evaluations~-->~G\"$d#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+iN@99!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 87 "The last value is correct to 10 digits, as can be seen \+ from the following calculation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Int(sin(x),x=Pi/4..3*Pi/4); \nvalue(%);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$ sinG6#%\"xG/F);,$%#PiG#\"\"\"\"\"%,$F-#\"\"$F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+iN@99!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 173 "For a smooth function there is not much of an advantag e in using the adaptive method over an iterative method which subdivid es evenly over the whole interval of integration." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "simp(sin(x), x=Pi/4..3*Pi/4,iterate=true,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~with~2~intervals~--->~~~G$\"07VgzPuT\"!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~with~4~intervals~--->~ ~~G$\"0*)oU*QS99!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation ~with~8~intervals~--->~~~G$\"0f:SHDUT\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation~with~16~intervals~--->~~~G$\"0hvIH9UT\" !#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation~with~32~interva ls~--->~~~G$\"0p-!3O@99!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happrox imation~with~64~intervals~--->~~~G$\"0LC_c8UT\"!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%Iapproximation~with~128~intervals~--->~~~G$\"0G^Dc8U T\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iapproximation~with~256~int ervals~--->~~~G$\"0B%QiN@99!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iap proximation~with~512~intervals~--->~~~G$\"0ztBc8UT\"!#9" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+iN@99!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 4 "Note" }{TEXT -1 23 ": By using the option \"" }{TEXT 267 14 "adaptive=false" }{TEXT -1 21 "\" with the procedure " }{TEXT 0 5 " SPint" }{TEXT -1 128 " we can obtain a numerical approximation for the definite integral by essentially the same method as that used by the \+ procedure " }{TEXT 0 7 "simpson" }{TEXT -1 8 " in the " }{TEXT 0 7 "st udent" }{TEXT -1 30 " package and by the procedure " }{TEXT 0 4 "simp " }{TEXT -1 27 " (with the default option \"" }{TEXT 267 13 "iterate=f alse" }{TEXT -1 17 "\" ). The option \"" }{TEXT 267 6 "factor" }{TEXT -1 75 "\" then controls the number of intervals on which Simpson's rul e is applied." }}{PARA 0 "" 0 "" {TEXT -1 219 "Since a single applicat ion of Simpson's rule requires three function values, the total number of intervals into which the whole interval of integration is subdivid ed, to give the evaluation points for the function, is \"" }{TEXT 267 8 "2*factor" }{TEXT -1 3 "\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "SPint(sin(x),x=Pi/4..3*Pi/4, adaptive=false,factor=75,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%Husing~Simpson's~rule~with~150~intervalsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~evaluations~-->~G\"$^\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+iN@99!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "simp(sin(x),x=Pi/4..3 *Pi/4,iterate=false,intervals=150,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+iN@99!\"*" }}}{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 21 "We use the pro cedure " }{TEXT 0 5 "SPint" }{TEXT -1 13 " to evaluate " }{XPPEDIT 18 0 "Int(sin(x*sqrt(x)),x = 0 .. 2);" "6#-%$IntG6$-%$sinG6#*&%\"xG\"\"\" -%%sqrtG6#F*F+/F*;\"\"!\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 564 "f := x -> sin(x*s qrt(x)); # function\na := 0: # lower limit of integral\nb := 2: # uppe r limit of integral\nclr := COLOR(RGB,.75,.75,.9): # color for shading \npp := plot([0,f(x)],x=a..b,adaptive=false,numpoints=30):\nu := op(1 ,op(1,pp)): v := op(1,op(2,pp)):\np1 := plots[polygonplot]([seq([u[i], v[i],v[i+1],u[i+1]],i=1..29)],\n color= clr,style=patchnogrid):\np2 := plot([[[a,0],[a,f(a)]],[[b,0],[b,f(b)]] ],color=black):\np3 := plot(f(x),x=0..2.1,thickness=2): # adjust plot \+ range\nplots[display]([p1,p2,p3],labels=[`x`,`y`],tickmarks=[4,4]);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arro wGF(-%$sinG6#*&9$\"\"\"-%%sqrtG6#F0F1F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 404 292 292 {PLOTDATA 2 "6)-%)POLYGONSG6A7&7$$\"\"!F)F(F'7$$ \"+b'4c@(!#6$\"+Vv7Q>F-7$F+F(7&F0F*7$$\"+.tQ\\8!#5$\"+F2\"[&\\F-7$F3F( 7&F8F27$$\"+$oVa0#F5$\"+tQG0$*F-7$F;F(7&F?F:7$$\"+i-=mFF5$\"+8:t\\9F57 $FBF(7&FFFA7$$\"+M)QNZ$F5$\"+d2#H.#F57$FIF(7&FMFH7$$\"+z2NHTF5$\"+89\\ 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"6$%=deepest~bisection~level~-->~G\"\"'" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%=deepest~bisection~level~-->~G\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~-->~G\"\")" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~-->~G\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~-->~G\"#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~-->~G\"#6" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~-->~G\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~-->~G\"#8 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~-->~G\"# 9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~-->~G\" #:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~-->~G \"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~-->~ G\"#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~--> ~G\"#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~-- >~G\"#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~- ->~G\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Hbisection~level~has~been ~restricted~to~G\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~fu nction~evaluations~-->~G\"%T<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+! e,0?\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The last value agrees with the value given by Maple's numerical integration via " }{TEXT 0 5 "evalf" }{TEXT -1 5 " and " }{TEXT 0 3 " Int" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Int(sin(x*sqrt(x)),x=0..2);\nevalf(evalf( %,15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$sinG6#*$)%\"xG# \"\"$\"\"#\"\"\"/F+;\"\"!F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!e, 0?\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The iterative application of Simpson's rule via " }{TEXT 0 4 "simp " }{TEXT -1 42 " requires more function evaluations than " }{TEXT 0 5 "SPint" }{TEXT -1 51 " to obtain a result which is correct to 10 dig its. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "simp(sin(x*sqrt(x)),x=0..2,iterate=true,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~with~2~intervals~--- >~~~G$\"0(['Q*=lC7!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximati on~with~4~intervals~--->~~~G$\"0(=\\-=#e?\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~with~8~intervals~--->~~~G$\"0f?Jk=6?\"! #9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation~with~16~interval s~--->~~~G$\"02XFR!f+7!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroxi mation~with~32~intervals~--->~~~G$\"0)Q)o,;0?\"!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%Happroximation~with~64~intervals~--->~~~G$\"0KeJ0/0? \"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iapproximation~with~128~inte rvals~--->~~~G$\"02;B,-0?\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iap proximation~with~256~intervals~--->~~~G$\"0gvhl,0?\"!#9" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%Iapproximation~with~512~intervals~--->~~~G$\"0& z]$f,0?\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Japproximation~with~1 024~intervals~--->~~~G$\"0(yW#e,0?\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Japproximation~with~2048~intervals~--->~~~G$\"0%Q\\!e,0?\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Japproximation~with~4096~intervals~ --->~~~G$\"0[[,e,0?\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Japproxim ation~with~8192~intervals~--->~~~G$\"0W(3!e,0?\"!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+!e,0?\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 112 "Using Simpson's rule with a fixed number of 2038 intervals gives the same result as that given by the procedur e " }{TEXT 0 5 "SPint" }{TEXT -1 22 " in the adaptive mode." }}{PARA 0 "" 0 "" {TEXT -1 139 "This suggests that the bisections in the first computation have been performed to a fairly uniform depth over the in terval of integration. " }}{PARA 0 "" 0 "" {TEXT -1 51 "This calculati on can be performed in various ways. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "evalf(evalf(student[simp son](sin(x*sqrt(x)),x=0..2,2038),15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!e,0?\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "simp(sin(x*sqrt(x)),x=0..2,interval s=2038);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!e,0?\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "SP int(sin(x*sqrt(x)),x=0..2,adaptive=false,factor=1019,info=true);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%Iusing~Simpson's~rule~with~2038~inter valsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~evaluati ons~-->~G\"%R?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!e,0?\"!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 21 "We use the procedure " }{TEXT 0 5 "SPint" }{TEXT -1 13 " to evaluate " }{XPPEDIT 18 0 "Int(sqrt(1-x^2),x = -1 .. 1);" "6#- %$IntG6$-%%sqrtG6#,&\"\"\"F**$%\"xG\"\"#!\"\"/F,;,$F*F.F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 403 "f := x -> sqrt(1-x^2); # function\na := -1: # lower \+ limit of integral\nb := 1: # upper limit of integral\nclr := wheat: # \+ color for shading\np1 := plot(f(x),x=a..b,filled=true,color=clr,style= patchnogrid):\np2 := plot([[[a,0],[a,f(a)]],[[b,0],[b,f(b)]]],color=bl ack):\np3 := plot(f(x),x=-1.1..1.1,thickness=2,numpoints=100): # adjus t plot range\nplots[display]([p1,p2,p3],labels=[`x`,`y`],scaling=const rained);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)oper atorG%&arrowGF(-%%sqrtG6#,&\"\"\"F0*$)9$\"\"#F0!\"\"F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 524 269 269 {PLOTDATA 2 "6)-%)POLYGONSG6ao7&7$$! \"\"\"\"!F*7$F($F*F*7$$!3-n;HdNvs**!#=$\"3aIk@d'**oP(!#>7$F.F*7&F4F-7$ $!3/MLe9r]X**F0$\"3^xD,?$RD/\"F07$F7F*7&F;F67$$!3/,](=ng#=**F0$\"3W*\\ #GuA(fF\"F07$F>F*7&FBF=7$$!3%pmm\"HU,\"*)*F0$\"3=cuB4yNs9F07$FEF*7&FIF D7$$!3()***\\PM@l$)*F0$\"3gMpy'*3z+=F07$FLF*7&FPFK7$$!3!RLL$e%G?y*F0$ \"3iA8`Ih^w?F07$FSF*7&FWFR7$$!3u****\\(oUIn*F0$\"3nSQYJr=ODF07$FZF*7&F hnFY7$$!3ommm;p0k&*F0$\"3Y%G&e[NT?HF07$F[oF*7&F_oFjn7$$!3E++vV5Su$*F0$ \"3'eGkq$fY\"[$F07$FboF*7&FfoFao7$$!3wKL$3(eMbF07$F^qF*7&FbqF]q7$$!3\"QLL3i.9!zF0$\"3^7**H-%f#HhF07$FeqF*7&Fiq Fdq7$$!3\"ommT!R=0vF0$\"3^qi^Nc\\3mF07$F\\rF*7&F`rF[r7$$!3u****\\P8#\\ 4(F0$\"3ifl'faLr/(F07$FcrF*7&FgrFbr7$$!3+nm;/siqmF0$\"37dnjs%HaF0$\"3e&R<6Bo wR)F07$F_tF*7&FctF^t7$$!3Q+++]$*4)*\\F0$\"3]Y1e+6Nh')F07$FftF*7&FjtFet 7$$!39+++]_&\\c%F0$\"3S:_+\"3ds*))F07$F]uF*7&FauF\\u7$$!31+++]1aZTF0$ \"3i'GE3]N$*4*F07$FduF*7&FhuFcu7$$!3umm;/#)[oPF0$\"33y-$=zZFE*F07$F[vF *7&F_vFju7$$!3hLLL$=exJ$F0$\"3:IF>)f#eL%*F07$FbvF*7&FfvFav7$$!3*RLLLtI f$HF0$\"3kU7=H\\If&*F07$FivF*7&F]wFhv7$$!3]++]PYx\"\\#F0$\"3)G16\">%yX o*F07$F`wF*7&FdwF_w7$$!3EMLLL7i)4#F0$\"3i8>'>!*4tx*F07$FgwF*7&F[xFfw7$ $!3c****\\P'psm\"F0$\"35Fh>K5.g)*F07$F^xF*7&FbxF]x7$$!3')****\\74_c7F0 $\"3JpBi\"oV2#**F07$FexF*7&FixFdx7$$!3)3LLL3x%z#)F3$\"3%4o%3*=mc'**F07 $F\\yF*7&F`yF[y7$$!3KMLL3s$QM%F3$\"3#oX(\\%3h0***F07$FcyF*7&FgyFby7$$! 3]^omm;zr)*!#@$\"2sF&QF^******!#<7$FjyF*7&F`zFiy7$$\"3%pJL$ezw5VF3$\"3 )*Q2K>Vq!***F07$FczF*7&FgzFbz7$$\"3s*)***\\PQ#\\\")F3$\"3s^4RW'Rn'**F0 7$FjzF*7&F^[lFiz7$$\"3GKLLe\"*[H7F0$\"3+'Q!p2+8C**F07$Fa[lF*7&Fe[lF`[l 7$$\"3I*******pvxl\"F0$\"3]s:1(pJ;')*F07$Fh[lF*7&F\\\\lFg[l7$$\"3#z*** *\\_qn2#F0$\"3;GKN_W(>y*F07$F_\\lF*7&Fc\\lF^\\l7$$\"3U)***\\i&p@[#F0$ \"3!QaP4mWqo*F07$Ff\\lF*7&Fj\\lFe\\l7$$\"3B)****\\2'HKHF0$\"3+\\tky/Ug &*F07$F]]lF*7&Fa]lF\\]l7$$\"3ElmmmZvOLF0$\"3%e0ez0!)oU*F07$Fd]lF*7&Fh] lFc]l7$$\"3i******\\2goPF0$\"3wC!*\\-?qi#*F07$F[^lF*7&F_^lFj]l7$$\"3UK L$eR<*fTF0$\"3CILE@Ro$4*F07$Fb^lF*7&Ff^lFa^l7$$\"3m******\\)Hxe%F0$\"3 -\"f%eX^`&)))F07$Fi^lF*7&F]_lFh^l7$$\"3ckm;H!o-*\\F0$\"35DIn2b'em)F07$ F`_lF*7&Fd_lF__l7$$\"3y)***\\7k.6aF0$\"3a'=*zS\"f&4%)F07$Fg_lF*7&F[`lF f_l7$$\"3#emmmT9C#eF0$\"3H&p3sIf,8)F07$F^`lF*7&Fb`lF]`l7$$\"33****\\i! *3`iF0$\"3;c[JlKx.yF07$Fe`lF*7&Fi`lFd`l7$$\"3%QLLL$*zym'F0$\"3o4V`oXZ_ uF07$F\\alF*7&F`alF[al7$$\"3wKLL3N1#4(F0$\"3!Q)Q?)e4+0(F07$FcalF*7&Fga lFbal7$$\"3Nmm;HYt7vF0$\"3+_Th$[5**f'F07$FjalF*7&F^blFial7$$\"3Y****** *p(G**yF0$\"3_pyiCh)>8'F07$FablF*7&FeblF`bl7$$\"3]mmmT6KU$)F0$\"3I@,y! 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When using " }{TEXT 0 5 "SPint" }{TEXT -1 151 " in the \+ adaptive mode, to calculate this integral, the subdivision level becom es very deep in the neighbourhood of the singularities of the derivati ve." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "SPint(sqrt(1-x^2),x=-1..1,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@using~Simpson's~rule~adaptivelyG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~-->~G\"\"#" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~-->~G\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~-->~G\"\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~-->~G\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~-->~G\"\"' " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~-->~G\" \"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~-->~G \"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~--> ~G\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~- ->~G\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~ -->~G\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level ~-->~G\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~leve l~-->~G\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~lev el~-->~G\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~le vel~-->~G\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~l evel~-->~G\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~ level~-->~G\"#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection ~level~-->~G\"#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisectio n~level~-->~G\"#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisecti on~level~-->~G\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Hbisection~leve l~has~been~restricted~to~G\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnu mber~of~function~evaluations~-->~G\"% " 0 "" {MPLTEXT 1 0 55 "Int(sqrt(1-x^2),x=-1..1);\nv alue(%);\nevalf(evalf(%,15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$In tG6$*$,&\"\"\"F(*$)%\"xG\"\"#F(!\"\"#F(F,/F+;F-F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#!\"\"%#PiG\"\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Fjzq:!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 231 "If we apply Simpson's rule iteratively w ith equal intervals so that subdivision occurs evenly over the whole i nterval of integration, then many more function evaluations are needed to obtain a result which is correct to 10 digits. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "simp(sqrt( 1-x^2),x=-1..1,iterate=true,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~with~2~intervals~--->~~~G$\"0LLLLLLL\"!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~with~4~intervals~--->~ ~~G$\"0f7<(Q.)[\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximatio n~with~8~intervals~--->~~~G$\"0[tud(zT:!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation~with~16~intervals~--->~~~G$\"0r()[e%fg: !#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation~with~32~interva ls~--->~~~G$\"0I#\\M))>n:!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happr oximation~with~64~intervals~--->~~~G$\"0!o%*3h_p:!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iapproximation~with~128~intervals~--->~~~G$\"0FS!Qv Mq:!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iapproximation~with~256~int ervals~--->~~~G$\"0zn%4xjq:!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iap proximation~with~512~intervals~--->~~~G$\"0/slDS2d\"!#9" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%Japproximation~with~1024~intervals~--->~~~G$\"0 c`Y]w2d\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Japproximation~with~2 048~intervals~--->~~~G$\"021*=$*yq:!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Japproximation~with~4096~intervals~--->~~~G$\"0Z%=\\Qzq:!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%Japproximation~with~8192~intervals~-- ->~~~G$\"0gO3X&zq:!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Kapproximati on~with~16384~intervals~--->~~~G$\"0^)4~~~G$\"0)4I~~~G$\"0,$3)G'zq:!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%L approximation~with~131072~intervals~--->~~~G$\"0=3JJ'zq:!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Lapproximation~with~262144~intervals~--->~~ ~G$\"0\"f&>K'zq:!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Lapproximation ~with~524288~intervals~--->~~~G$\"01%3Djzq:!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Mapproximation~with~1048576~intervals~--->~~~G$\"0-!>E jzq:!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Mapproximation~with~209715 2~intervals~--->~~~G$\"01\"eEjzq:!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%Mapproximation~with~4194304~intervals~--->~~~G$\"0L>nK'zq:!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Fjzq:!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT -1 21 "We use the procedure " }{TEXT 0 5 "SPint" }{TEXT -1 13 " to evalua te " }{XPPEDIT 18 0 "Int(1+sin(x^2)*exp(-x/5),x = 0 .. 10)" "6#-%$IntG 6$,&\"\"\"F'*&-%$sinG6#*$%\"xG\"\"#F'-%$expG6#,$*&F-F'\"\"&!\"\"F5F'F' /F-;\"\"!\"#5" }{TEXT -1 2 ". 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"6$%=deepest~bisection~level~-->~G\"\"&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~-->~G\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~-->~G\"\"( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~-->~G\" \")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~-->~G \"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~--> ~G\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~-- >~G\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~- ->~G\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%=deepest~bisection~level~ -->~G\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Dnumber~of~function~eval uations~-->~G\"&D0\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+-8r_5!\") " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The \+ last value agrees with the value given by Maple's numerical integratio n via " }{TEXT 0 5 "evalf" }{TEXT -1 5 " and " }{TEXT 0 3 "Int" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Int(1+sin(x^2)*exp(-x/5),x=0..10);\nevalf(%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,&\"\"\"F'*&-%$sinG6#*$)%\"x G\"\"#F'F'-%$expG6#,$F.#!\"\"\"\"&F'F'/F.;\"\"!\"#5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+-8r_5!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 48 "The iterative application of Simpson's ru le via " }{TEXT 0 4 "simp" }{TEXT -1 111 " manages to obtain a result \+ which is correct to 10 digits with fewer function evaluations than was required by " }{TEXT 0 5 "SPint" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "simp(1+sin(x ^2)*exp(-x/5),x=0..10,iterate=true,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~with~2~intervals~--->~~~G$\"0LQ^=)=h&*! #9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapproximation~with~4~intervals ~--->~~~G$\"0n'426\"ed*!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Gapprox imation~with~8~intervals~--->~~~G$\"06h&)GsvE\"!#8" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%Happroximation~with~16~intervals~--->~~~G$\"0b%oTr#p 1\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happroximation~with~32~inte rvals~--->~~~G$\"0gvdj&fW5!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Happ roximation~with~64~intervals~--->~~~G$\"0I*zB4w^5!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iapproximation~with~128~intervals~--->~~~G$\"0AKlO% o_5!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Iapproximation~with~256~int ervals~--->~~~G$\"0+3<**4F0\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%I approximation~with~512~intervals~--->~~~G$\"0RWWA6F0\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Japproximation~with~1024~intervals~--->~~~G$\" 0k(4(H6F0\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Japproximation~with ~2048~intervals~--->~~~G$\"0\"Rd,8r_5!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Japproximation~with~4096~intervals~--->~~~G$\"0p_=I6F0\"!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%Japproximation~with~8192~intervals~-- ->~~~G$\"04q=I6F0\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+-8r_5!\" )" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "T asks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 53 "Calculate num erical values for the definite integral " }{XPPEDIT 18 0 "Int(sqrt(x), x = 1/100 .. 1);" "6#-%$IntG6$-%%sqrtG6#%\"xG/F);*&\"\"\"F-\"$+\"!\"\" F-" }{TEXT -1 56 " correct to 10 digits using each of the two procedu res " }{TEXT 0 4 "simp" }{TEXT -1 5 " and " }{TEXT 0 5 "SPint" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 41 "______________________________ ___________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 41 "_________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "f (x)=sqrt(x)/(x^2+1)*(sin((51+x)*exp(-3*x^2))+2)" "6#/-%\"fG6#%\"xG*(-% %sqrtG6#F'\"\"\",&*$F'\"\"#F,F,F,!\"\",&-%$sinG6#*&,&\"#^F,F'F,F,-%$ex pG6#,$*&\"\"$F,*$F'F/F,F0F,F,F/F,F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 126 "(a) Calculate numerical values for the following two d efinite integrals correct to 10 digits using each of the two procedure s " }{TEXT 0 4 "simp" }{TEXT -1 5 " and " }{TEXT 0 5 "SPint" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "(i) " }{XPPEDIT 18 0 "Int(f(x) ,x = 1 .. 2);" "6#-%$IntG6$-%\"fG6#%\"xG/F);\"\"\"\"\"#" }{TEXT -1 10 " (ii) " }{XPPEDIT 18 0 "Int(f(x),x = 0 .. 2);" "6#-%$IntG6$-%\"fG 6#%\"xG/F);\"\"!\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 162 "(b) Comment on the number of function evaluations required by eac h of the two methods for the two integrals in part (a). ______________ ___________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 41 "_______________________________________ __" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)=arctan(ln(sqrt(x)+1))" "6#/-%\"fG6#%\"xG-%'arctanG6#-%#lnG6 #,&-%%sqrtG6#F'\"\"\"F2F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 126 "(a) Calculate numerical values for the following two definite \+ integrals correct to 10 digits using each of the two procedures " } {TEXT 0 4 "simp" }{TEXT -1 5 " and " }{TEXT 0 5 "SPint" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "(i) " }{XPPEDIT 18 0 "Int(f(x),x = 1 \+ .. 2);" "6#-%$IntG6$-%\"fG6#%\"xG/F);\"\"\"\"\"#" }{TEXT -1 10 " (i i) " }{XPPEDIT 18 0 "Int(f(x),x = 0 .. 2);" "6#-%$IntG6$-%\"fG6#%\"xG /F);\"\"!\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 121 "(b) C omment on the number of function evaluations required by each of the t wo methods for the two integrals in part (a). " }}{PARA 0 "" 0 "" {TEXT -1 41 "_________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 41 "__ _______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }} {PARA 0 "" 0 "" {TEXT -1 44 "Calculate numerical values for the integr al " }{XPPEDIT 18 0 "Int(ln(x^(3/4)+1),x=0..2) " "6#-%$IntG6$-%#lnG6#, &)%\"xG*&\"\"$\"\"\"\"\"%!\"\"F.F.F./F+;\"\"!\"\"#" }{TEXT -1 54 " cor rect to10 digits using each of the two procedures " }{TEXT 0 4 "simp" }{TEXT -1 5 " and " }{TEXT 0 5 "SPint" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 41 "_________________________________________" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 41 "__ _______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5" }} {PARA 0 "" 0 "" {TEXT -1 69 "The following subsection contains the cod e for a numerical procedure " }{TEXT 267 2 "Yn" }{TEXT -1 39 " which e valuates the inverse function " }{XPPEDIT 18 0 "g^(-1)*``(x);" "6#*&) %\"gG,$\"\"\"!\"\"F'-%!G6#%\"xGF'" }{TEXT -1 17 " of the function " } {XPPEDIT 18 0 "g(x) = x*cosh(x);" "6#/-%\"gG6#%\"xG*&F'\"\"\"-%%coshG6 #F'F)" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "cosh(x)" "6#-%%coshG6#% \"xG" }{TEXT -1 46 " is the hyperbolic cosine function defined by " } {XPPEDIT 18 0 "cosh(x)=(exp(x)+exp(-x))/2" "6#/-%%coshG6#%\"xG*&,&-%$e xpG6#F'\"\"\"-F+6#,$F'!\"\"F-F-\"\"#F1" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 48 "(a) Calculate numerical values for the integral " } {XPPEDIT 18 0 "Int(g^(-1)*``(x),x = 0 .. 4);" "6#-%$IntG6$*&)%\"gG,$\" \"\"!\"\"F*-%!G6#%\"xGF*/F/;\"\"!\"\"%" }{TEXT -1 54 " correct to10 di gits using each of the two procedures " }{TEXT 0 4 "simp" }{TEXT -1 5 " and " }{TEXT 0 5 "SPint" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 28 "(b) Calculate the value of " }{XPPEDIT 18 0 "4*g^(-1)*``(4)-Int(g (x),x = 0 .. g^(-1)*``(4));" "6#,&*(\"\"%\"\"\")%\"gG,$F&!\"\"F&-%!G6# F%F&F&-%$IntG6$-F(6#%\"xG/F3;\"\"!*&)F(,$F&F*F&-F,6#F%F&F*" }{TEXT -1 23 " correct to 10 digits. " }}{PARA 0 "" 0 "" {TEXT -1 83 " Why i s the value for this expression the same as that of the integral in (a )? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "code for th e procedure " }{TEXT 0 2 "Yn" }{TEXT -1 37 " which gives a numerical i nverse for " }{XPPEDIT 18 0 "g(x)=x*cosh(x)" "6#/-%\"gG6#%\"xG*&F'\"\" \"-%%coshG6#F'F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2719 "Yn := proc(x::realcons)\n \+ local xx,eps,saveDigits,doY,val,p,q,maxit;\n \n if x=0 then retu rn 0. end if;\n\n doY := proc(x,eps,maxit)\n local csh,snh,s,t, u,v,h,i; \n # set up a starting approximation\n if x<.84 and x>-.84 then\n s := .4107111272*x+1.119421855/(x+2.357150354/ \n (x+.1556014159/(x+.6762074010/\n (x+.18730359 98e-1/(x+.4010065025/x)))));\n else\n if x>0 then\n \+ if x<100 then\n s := (-.7378955849e-1+(1.363864106 +(.3000905586+\n (.6430435726e-2+.3314657469e-5*x)*x)* x)*x)/\n (1.+(.9792814862+(.1062945123+.1357287434e-2* x)*x)*x)\n elif x-100 then\n s := (-.73789558 49e-1+(-1.363864106+(.3000905586+\n (-.6430435726e-2+.33 14657469e-5*x)*x)*x)*x)/\n (-1.+(.9792814862+(-.10629451 23+.1357287434e-2*x)*x)*x)\n elif x>-Float(1,8) then\n \+ s := ln(-x);\n s := ln(s)-s-(-.8567377032+.140 4294456e-5*x)/\n (-1.+(.1721594185e-5+.7399085925e-1 6*x)*x)\n elif x>-Float(1,20) then\n s := ln( -x);\n s := ln(s)-s+(-.8060973972+.1096137486e-11*x)/\n \+ (1.+(-.1415475176e-11+.3823119413e-33*x)*x)\n \+ else\n s := ln(-x);\n s := ln(s)-s-.7 5;\n end if;\n end if;\n end if;\n # solv e the equation y*cosh(y)=x for y by Halley's method \n for i to m axit do\n csh := cosh(s);\n snh := sinh(s);\n \+ t := s*csh-x;\n u := csh+s*snh;\n v := 2*snh+s*csh;\n \+ h := t/(u-1/2*v*t/u);\n s := s-h;\n if abs(h)< =eps*abs(s) then break end if;\n end do;\n s;\n end proc; \n\n p := ilog10(Digits);\n q := Float(Digits,-p);\n maxit := tr unc((p+(.02331061386+.1111111111*q))*2.095903274)+2;\n saveDigits := Digits;\n Digits := Digits+min(iquo(Digits,3),5);\n xx := evalf(x );\n eps := Float(3,-saveDigits-1);\n if Digits<=trunc(evalhf(Digi ts)) and \n ilog10(xx)trunc(evalhf(DBL_MIN_10_EXP)) then\n val := eval hf(doY(xx,eps,maxit))\n else\n val := doY(xx,eps,maxit)\n end if;\n evalf[saveDigits](val);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 269 4 "Note" }{TEXT -1 13 " : When using " }{TEXT 267 5 "Yn(x)" }{TEXT -1 20 " as an argument for \+ " }{TEXT 0 4 "plot" }{TEXT -1 2 ", " }{TEXT 0 4 "simp" }{TEXT -1 4 " o r " }{TEXT 0 5 "SPint" }{TEXT -1 35 ", it must be enclosed in quotes a s " }{TEXT 0 7 "'Yn(x)'" }{TEXT -1 3 " or" }{TEXT 0 8 " 'Yn'(x)" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "_________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 41 "__ _______________________________________" }}{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 }