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-1 361 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 362 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Co urier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple P lot" -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 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 1 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 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 "Normal" -1 260 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 "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 32 "Newton's method for root-finding " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada " }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 5.4.2007" }}{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 40 "load root-finding procedures including: " }{TEXT 0 6 " newton" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-fil e " }{TEXT 304 7 "roots.m" }{TEXT -1 38 " contains the code for the pr ocedures " }{TEXT 0 6 "newton" }{TEXT -1 5 " and " }{TEXT 0 11 "newton _step" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 122 "It can be read into a Maple session by a command similar to the one that follows, where the file path gives its location. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\\\maple/procdrs/r oots.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 "The iterative formula for Newton's method " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose tha t " }{XPPEDIT 18 0 "x = r" "6#/%\"xG%\"rG" }{TEXT -1 27 " is a root of an equation " }{XPPEDIT 18 0 "f(x) = 0;" "6#/-%\"fG6#%\"xG\"\"!" } {TEXT -1 8 ", where " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 24 " is differentiable near " }{XPPEDIT 18 0 "x = r;" "6#/%\"xG%\"r G" }{TEXT -1 30 ". Suppose that we don't know " }{TEXT 277 1 "r" } {TEXT -1 47 ", but that we can obtain a rough approximation " } {XPPEDIT 18 0 "x = a;" "6#/%\"xG%\"aG" }{TEXT -1 13 " to the root " } {TEXT 278 1 "r" }{TEXT -1 138 " by means of inspecting a graph, or by \+ some other means. The following geometrical idea often leads to a bett er approximation to the root." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 40 "Consider the tangent line to the graph \+ " }{XPPEDIT 18 0 "y = f(x);" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 "``(a,f(a));" "6#-%!G6$%\"aG-%\"fG6#F&" } {TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 33 "This tangent line has e quation " }{XPPEDIT 18 0 "y-f(a) = `f '`(a)*(x-a);" "6#/,&%\"yG\"\" \"-%\"fG6#%\"aG!\"\"*&-%$f~'G6#F*F&,&%\"xGF&F*F+F&" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 56 "The point of intersection of this tangent line with the " }{TEXT 279 1 "x" }{TEXT -1 19 " axis is given by " } {XPPEDIT 18 0 "x = a-f(a)/`f '`(a);" "6#/%\"xG,&%\"aG\"\"\"*&-%\"fG6#F &F'-%$f~'G6#F&!\"\"F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "We take this as the next approximation " }{TEXT 341 1 "b" }{TEXT -1 13 " to the root " }{TEXT 280 1 "r" }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 14 "Starting with " }{XPPEDIT 18 0 "x[1]" "6#&%\"xG6#\"\"\"" }{TEXT -1 80 " we can repeat the process to obtain a further, hopefull y better, approximation " }{XPPEDIT 18 0 "x[2]" "6#&%\"xG6#\"\"#" } {TEXT -1 10 " given by " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x[2] = x[1]-f(x[1])/`f '`(x[1]);" "6#/&%\"xG6#\"\"#,&&F%6#\"\"\" F+*&-%\"fG6#&F%6#F+F+-%$f~'G6#&F%6#F+!\"\"F7" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 67 "The process can be repeated to obtain a s equence of approximations " }{XPPEDIT 18 0 "x[0],x[1],x[2],x[3],` . . \+ . `;" "6'&%\"xG6#\"\"!&F$6#\"\"\"&F$6#\"\"#&F$6#\"\"$%(~.~.~.~G" } {TEXT -1 20 " where, in general, " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x[n+1] = x[n]-f(x[n])/`f '`(x[n]);" "6#/&%\"xG6#,&%\"n G\"\"\"F)F),&&F%6#F(F)*&-%\"fG6#&F%6#F(F)-%$f~'G6#&F%6#F(!\"\"F8" } {TEXT -1 2 ". " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{TEXT 281 12 "____ ________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 20 "This is the general " }{TEXT 267 17 "iterative for mula" }{TEXT -1 47 " used to obtain the sequence of approximations " } }{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x[0],x[1],x[2],x[3], ` . . . `,x[n],x[n+1],` . . . `" "6*&%\"xG6#\"\"!&F$6#\"\"\"&F$6#\"\"# &F$6#\"\"$%(~.~.~.~G&F$6#%\"nG&F$6#,&F3F)F)F)F0" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 13 "for the root " }{TEXT 283 1 "r" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 133 "The iterative formula can be \+ applied successively until some member of the sequence generated gives the root to the desired accuracy." }}{PARA 0 "" 0 "" {TEXT -1 39 "The sequence of approximations usually " }{TEXT 267 23 "converges quite r apidly" }{TEXT -1 21 " to the desired root " }{TEXT 282 1 "r" }{TEXT -1 15 " provided that " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" } {TEXT -1 55 " is sufficiently close the root, and that the function " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 44 " is sufficientl y well-behaved near the root." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 56 "Introductory example of root-finding by Newton's method " }}{PARA 0 "" 0 "" {TEXT -1 17 "For example, let " }{XPPEDIT 18 0 "f (x) = x^2-2;" "6#/-%\"fG6#%\"xG,&*$F'\"\"#\"\"\"F*!\"\"" }{TEXT -1 16 ", and take the " }{XPPEDIT 18 0 "x = 2" "6#/%\"xG\"\"#" }{TEXT -1 52 " as a starting approximation for the computation of " }{XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " }{XPPEDIT 18 0 "`f '`(x) = 2*x;" "6#/-%$f~'G6#% \"xG*&\"\"#\"\"\"F'F*" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "`f '`(2 ) = 4;" "6#/-%$f~'G6#\"\"#\"\"%" }{TEXT -1 27 ". The tangent to the cu rve " }{XPPEDIT 18 0 "y = x^2-2;" "6#/%\"yG,&*$%\"xG\"\"#\"\"\"F(!\"\" " }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 "``(2,2)" "6#-%!G6$\"\"# F&" }{TEXT -1 13 " is the line " }{XPPEDIT 18 0 "y = 4*x - 6" "6#/%\"y G,&*&\"\"%\"\"\"%\"xGF(F(\"\"'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 20 "This line meets the " }{TEXT 342 1 "x" }{TEXT -1 12 " a xis where " }{XPPEDIT 18 0 "x = 3/2;" "6#/%\"xG*&\"\"$\"\"\"\"\"#!\"\" " }{TEXT -1 50 ", which we can take as our next approximation for " } {XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "pl ot([x^2-2,4*x-6],x=1.2..2.3,-0.5..3,color=[red,blue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$\"1+++++++7 !#:$!1,++++++c!#;7$$\"1LLepo(RA\"F*$!1R)\\sA1)=]F-7$$\"1nTg:!R[C\"F*$! 1$\\'H_#eP]%F-7$$\"1L$32o+$o7F*$!1'yZJ$Q89RF-7$$\"1L$e\\'y\">H\"F*$!19 ],-B[4LF-7$$\"1nT&3!GU:8F*$!1%zH*\\&Gmp#F-7$$\"1L3F&)[@P8F*$!1B651Nc=@ F-7$$\"1+vVE$z(f8F*$!1*yJR$=+5:F-7$$\"1L3x.b6$Q\"F*$!1#QvGK]\"*p)!#<7$ $\"1+voToP19F*$!1m#4l\"zT5AFV7$$\"1nm\"p)RII9F*$\"1G7(*)\\\\pd%FV7$$\" 1L$e%H!z8X\"F*$\"1MT^r3,l5F-7$$\"1++vNX5v9F*$\"1EA!R\"RLf0*F-7$$\"1nTN&*)3hs\"F*$\"1 Xz*F-7$$\"1L$e90d%\\i<=F*$\"11.%[Q \\PI\"F*7$$\"1++]jw8d)=N**Q\"F*7$$\"1+]()yBAk=F*$\"1k$)*y2D` Z\"F*7$$\"1+v$fK>l)=F*$\"1zc8n^&*e:F*7$$\"1+]7%Gw7\">F*$\"1n'eU.xHl\"F *7$$\"1mm;7:_L>F*$\"1t7,Qa]QF*$\"1CS+ex\"4$=F*7$$\"1L 3xcazy>F*$\"1\"*otf9j:>F*7$$\"1++vT^K-?F*$\"1eTGtfI4?F*7$$\"1nTgTZYC?F *$\"1D%)***[d%)4#F*7$$\"1+vo-qgZ?F*$\"1*)[XPWp#>#F*7$$\"1mm\"HzK-2#F*$ \"1Tv'o\"Q'eG#F*7$$\"1+vV)*)>R4#F*$\"15D2T0]%Q#F*7$$\"1LLL'RLn6#F*$\"1 [G:r-c![#F*7$$\"1L$eH\\j+9#F*$\"1AI*Qvr)zDF*7$$\"1nTg//?j@F*$\"1J'z/*f VzEF*7$$\"1++]B3Y%=#F*$\"1(H1%*3p=x#F*7$$\"1m;ziw#)3AF*$\"1&p8Rk>*yGF* 7$$\"1LLLa;iIAF*$\"1x2ykHnvHF*7$$\"1+v$\\feQD#F*$\"1TO(fcy)zIF*7$$\"1+ D17$*4wAF*$\"1&Qr$y!G1=$F*7$$\"1+++++++BF*$\"1*************G$F*-%'COLO URG6&%$RGBG$\"*++++\"!\")\"\"!F`[l-F$6$7S7$F($!1+++++++7F*7$F/$!1nmm@D 4/6F*7$F4$!1LLePRk?5F*7$F9$!1qmmrF(zE*F-7$F>$!1lmm,aGB$)F-7$FC$!1NL$e' z3$Q(F-7$FH$!1sm;*e/9^'F-7$FM$!1++]Up#)3cF-7$FR$!1lm;\\)z`n%F-7$FX$!1+ +]Kj#\\u$F-7$Fgn$!1JLLB0%yy#F-7$F\\o$!1omm@)Q[%>F-7$Fao$!1W+++d=e**FV7 $Ffo$!1<0++]:!H%!#=7$F[p$\"1E*****p0Tv)FV7$F`p$\"1ML$3&fK4$4q#F-7$Fjp$\"1gmm'Q_4a$F-7$F_q$\"1****\\(z&4=XF-7$Fdq$\"1jmm'GL IQ&F-7$Fiq$\"1)***\\(z1?L'F-7$F^r$\"1++]#*RlNsF-7$Fcr$\"1nmmT]^y\")F-7 $Fhr$\"1rm;9eNW!*F-7$F]s$\"1GLLe?Gy**F-7$Fbs$\"1LL3&*o$[4\"F*7$Fgs$\"1 ++DWKGz6F*7$F\\t$\"1ML$[h([q7F*7$Fat$\"1+++a1rk8F*7$Fft$\"1****\\:&*)o X\"F*7$F[u$\"1++v.t2Y:F*7$F`u$\"1++]O^5X;F*7$Feu$\"1mmm[g3MF*7$Fdv$\"1+++n0I4?F*7$Fiv$\"1mmTm*ey4#F*7 $F^w$\"1,+v5!G/>#F*7$Fcw$\"1mmmr6$4G#F*7$Fhw$\"1++v$fzcP#F*7$F]x$\"1LL L&eLpY#F*7$Fbx$\"1LL$=(RDgDF*7$Fgx$\"1nmT=;!Gl#F*7$F\\y$\"1*****RHVyt# F*7$Fay$\"1lm;^1JNGF*7$Ffy$\"1MLL$F*-Fjz6&F\\[lF`[lF`[lF][l-%+AXES LABELSG6$Q\"x6\"%!G-%%VIEWG6$;$\"#7!\"\"$\"#BFfel;$!\"&Ffel$\"\"$F`[l " 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 22 "Thus we have computed " }{XPPEDIT 18 0 "x[1] = 3/2; " "6#/&%\"xG6#\"\"\"*&\"\"$F'\"\"#!\"\"" }{TEXT -1 17 " by the formula " }{XPPEDIT 18 0 "x[1] = x[0]-f(x[0])/`f '`(x[0]);" "6#/&%\"xG6#\"\" \",&&F%6#\"\"!F'*&-%\"fG6#&F%6#F+F'-%$f~'G6#&F%6#F+!\"\"F7" }{TEXT -1 9 " , where " }{XPPEDIT 18 0 "x[0] = 2;" "6#/&%\"xG6#\"\"!\"\"#" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 43 "We can now repeat the p rocess by computing " }{XPPEDIT 18 0 "x[2];" "6#&%\"xG6#\"\"#" }{TEXT -1 21 " using the formula " }{XPPEDIT 18 0 "x[2] = x[1]-f(x[1])/`f ' `(x[1]);" "6#/&%\"xG6#\"\"#,&&F%6#\"\"\"F+*&-%\"fG6#&F%6#F+F+-%$f~'G6# &F%6#F+!\"\"F7" }{TEXT -1 9 " , where " }{XPPEDIT 18 0 "x[1] = 3/2;" " 6#/&%\"xG6#\"\"\"*&\"\"$F'\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "This gives " }{XPPEDIT 18 0 "x[2];" "6#&%\"xG6#\"\"# " }{TEXT -1 3 " =" }{XPPEDIT 18 0 "17/12;" "6#*&\"#<\"\"\"\"#7!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 53 "The general iterative f ormula for Newton's method is " }}{PARA 258 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x[n+1] = x[n]-f(x[n])/`f '`(x[n]);" "6#/&%\"xG6#,&%\"nG \"\"\"F)F),&&F%6#F(F)*&-%\"fG6#&F%6#F(F)-%$f~'G6#&F%6#F(!\"\"F8" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 36 "which will give a seque nce of values" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x[0] , x[1], x[2], x[3], ` . . . `, x[n], x[n+1], ` . . . `" "6*&%\"xG6#\" \"!&F$6#\"\"\"&F$6#\"\"#&F$6#\"\"$%(~.~.~.~G&F$6#%\"nG&F$6#,&F3F)F)F)F 0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 23 "converging to the ro ot " }{XPPEDIT 18 0 "r=sqrt(2)" "6#/%\"rG-%%sqrtG6#\"\"#" }{TEXT -1 17 " of the equation " }{XPPEDIT 18 0 "g(x)=0" "6#/-%\"gG6#%\"xG\"\"! " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "We can automate this process as follows." }}{PARA 0 "" 0 "" {TEXT 267 4 "Note" }{TEXT -1 16 ": The procedure " }{TEXT 268 38 "n extapprox := x -> evalf(x-f(x)/Df(x))" }{TEXT -1 21 ", given below, wh ere " }{TEXT 268 3 "Df " }{TEXT -1 48 "is the derivative f ' of f, is \+ used to evaluate " }{XPPEDIT 18 0 "x-f(x)/`f '`(x)" "6#,&%\"xG\"\"\"*& -%\"fG6#F$F%-%$f~'G6#F$!\"\"F-" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 220 "f := x -> x ^2-2;\nDf := D(f); # Df is the derivative f' of f\nnextapprox := x -> \+ evalf(x - f(x)/Df(x));\nx0 := 2;\nx1 := nextapprox(x0);\nx2 := nextapp rox(x1);\nx3 := nextapprox(x2);\nx4 := nextapprox(x3);\nx5 := nextappr ox(x4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)opera torG%&arrowGF(,&*$)9$\"\"#\"\"\"F1F0!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DfGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&\"\"#\"\" \"9$F/F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+nextapproxGf*6#% \"xG6\"6$%)operatorG%&arrowGF(-%&evalfG6#,&9$\"\"\"*&-%\"fG6#F0F1-%#Df GF5!\"\"F8F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"+++++:!\"*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#x2G$\"+nmm;9!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#x3G$\"+'o:UT\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x4G$\"+i N@99!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x5G$\"+iN@99!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The last \+ two values give " }{XPPEDIT 18 0 "sqrt(2)" "6#-%%sqrtG6#\"\"#" }{TEXT -1 23 " correct to 10 digits. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "sqrt(2);\nevalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*$\"\"##\"\"\"F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+iN@99!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 49 "More examples of root-finding by Newton's method " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT -1 13 "The equation " }{XPPEDIT 18 0 "exp(-x)=x^3" "6#/-%$expG6#,$%\"xG!\"\"*$F(\"\"$" } {TEXT -1 39 " has exactly one real number solution. " }}{PARA 0 "" 0 " " {TEXT -1 51 "This can be seen from the fact that the two graphs " } {XPPEDIT 18 0 "y=exp(-x)" "6#/%\"yG-%$expG6#,$%\"xG!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=x^3" "6#/%\"yG*$%\"xG\"\"$" }{TEXT -1 38 " h ave a single point of intersection. " }}{PARA 0 "" 0 "" {TEXT -1 38 "I n the following picture the graph of " }{XPPEDIT 18 0 "y=exp(-x)" "6#/ %\"yG-%$expG6#,$%\"xG!\"\"" }{TEXT -1 15 " is plotted in " }{TEXT 268 3 "red" }{TEXT -1 20 " while the graph of " }{XPPEDIT 18 0 "y=x^3" "6# /%\"yG*$%\"xG\"\"$" }{TEXT -1 15 " is plotted in " }{TEXT 257 5 "green " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "plot([exp(-x),x^3],x=-0.4..1.5,y=-.3..2,color =[red,COLOR(RGB,0,.8,0)]); " }}{PARA 13 "" 1 "" {GLPLOT2D 397 333 333 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$!3A+++++++S!#=$\"3Mq7k(pC=\\\"!#<7$$!3 iLL$32aee$F*$\"3A_dMqKIJ9F-7$$!3*om\"H7z]DKF*$\"35Ngt8]k!Q\"F-7$$!3hLL ep4E?GF*$\"3Ns-B>L\"eK\"F-7$$!3XLL3(yFBT#F*$\"3C'>!f&G$\"3ad!o_28t3\"F-7$$!3Q2+D\")44`V FT$\"3WmN()yA\\W5F-7$$!3a'ommT/A?#!#?$\"3H+)\\5j/A+\"F-7$$\"3%Qmm\"zT, ?MFT$\"3/KNUE2yj'*F*7$$\"3d)****\\>F*$\"3!=uwYL@JD)F*7$$\"3OLL$esH\"[BF*$\"3u]3!)=s=2zF*7$$\"3F LLL.e'3r#F*$\"3]S'Gnr/bi(F*7$$\"3O**\\P%49G8$F*$\"3hN!R*3;W5tF*7$$\"3r KLLG)4j]$F*$\"3`*pP)QeVUqF*7$$\"3I**\\PWQ4;RF*$\"3K)Q&fg5ofnF*7$$\"3=+ ]7L^I1VF*$\"3Z6+c$p!*4]'F*7$$\"3SLL3x'\\Mr%F*$\"3V`4caAiTiF*7$$\"3eL$3 _YNt3&F*$\"3>6R*HglD,'F*7$$\"3/mm\"zz@1\\&F*$\"3M<\"oOR6\\x&F*7$$\"3Am ;/cH_4fF*$\"3u]voDl*z`&F*7$$\"3!)**\\iXw/)G\\F*7$$ \"3e(**\\()*>$HZ(F*$\"3X\"Hsg')okt%F*7$$\"3%)**\\P%3h!eyF*$\"3UVY,!o@v b%F*7$$\"3U***\\7F\"o&G)F*$\"3!4,uo+TnO%F*7$$\"3WILLGq\"*p')F*$\"3tYLL wx8-UF*7$$\"3k)***\\72BLSF*7$$\"3el;/E:#>X*F*$\"3%)H4& G#*[g)QF*7$$\"39++]dLMe)*F*$\"3r9d\"=$yFJPF*7$$\"31L3xia2C5F-$\"3'4Xs[ R$G\"f$F*7$$\"37+v=f%[S1\"F-$\"31a?.N.c]MF*7$$\"3ILLep$HJ5\"F-$\"3sWL] _.J=LF*7$$\"3l*\\P4YVS9\"F-$\"3e_-FqXG&=$F*7$$\"3wmmm$f[M=\"F-$\"3%\\+ F>%)3A1$F*7$$\"3am;HLguB7F-$\"328q`PHETHF*7$$\"3;L3x*y4PE\"F-$\"3)G$H# =zOg#GF*7$$\"3:++]JBV+8F-$\"3WG*4U%*RTs#F*7$$\"3:L$e%30_U8F-$\"3?)HuS: l=h#F*7$$\"3MmmmvY;!Q\"F-$\"3-$)y/w7P:DF*7$$\"3/+vV47I?9F-$\"3H\"zyJ@7 kT#F*7$$\"3u*\\iN*pre9F-$\"3qpt*eEW`K#F*7$$\"3++++++++:F-$\"3Y)H%[,;IJ AF*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fb[lFa[l-F$6$7S7$F($!3_,+++++ +kFT7$F/$!3kilAgk\"3h%FT7$F4$!3Gqto$f'ybLFT7$F9$!3\\iI(zX*>VAFT7$F>$!3 tr9&\\-7QS\"FT7$FC$!3k@uI1ACw!)Fin7$FH$!3\"z!RT0p9IVFin7$FM$!3Y6?i-EV2 >Fin7$FR$!3ycA&)>rxle!#@7$FX$!3A%f\\*)pY)[#)!#A7$Fgn$!3MH,J@S+o5!#D7$F ]o$\"3Omxw`&=-+%Fe]l7$Fbo$\"31Tx!*4KH\\UFa]l7$Fgo$\"3?L^dv\\Au:Fin7$F \\p$\"3gF%Qme@_z$Fin7$Fap$\"37DqH8D=xqFin7$Ffp$\"3O:wl)Q\"p%H\"FT7$F[q $\"3'=L!Qt#f@*>FT7$F`q$\"3qTOkezquIFT7$Feq$\"32#=x)[/t5VFT7$Fjq$\"39U+ ds'Rc+'FT7$F_r$\"3'*\\!y?)es&)zFT7$Fdr$\"3!H[g8jor/\"F*7$Fir$\"3B2p#HQ _mJ\"F*7$F^s$\"3qWCA&y`_l\"F*7$Fcs$\"3U4[<$)3vj?F*7$Fhs$\"3%[.FX<\\)pC F*7$F]t$\"3e*yfZqgZ'HF*7$Fbt$\"3u*o`_ll7a$F*7$Fgt$\"3W5`#=SPK<%F*7$F\\ u$\"37z(RM)QG_[F*7$Fau$\"3W>vS0&G$)o&F*7$Ffu$\"3qeJ7Dl&p^'F*7$F[v$\"3I L.(oObl[(F*7$F`v$\"3c$R2#4_BW%)F*7$Fev$\"3U088G;-\"e*F*7$Fjv$\"3_*\\Xo AzR2\"F-7$F_w$\"38hG&Gt9Z?\"F-7$Fdw$\"3:O9)*e>RU8F-7$Fiw$\"3YCJ;FYO(\\ \"F-7$F^x$\"3aKP)pgzul\"F-7$Fcx$\"3[,EM+iiK=F-7$Fhx$\"3H)Ry,1(4=?F-7$F ]y$\"3+fot\\E>*>#F-7$Fby$\"3rdo))Hqq>CF-7$Fgy$\"3'\\4#HPH,HEF-7$F\\z$ \"3a1Uli/6lGF-7$Faz$\"3)3oj#Q'QR5$F-7$Ffz$\"3+++++++vLF--%&COLORG6&F][ lFa[l$\"\")!\"\"Fa[l-%+AXESLABELSG6$Q\"x6\"Q\"yFfel-%%VIEWG6$;$!\"%Fae l$\"#:Fael;$!\"$Fael$\"\"#Fb[l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "The solution of " }{XPPEDIT 18 0 "exp(-x)=x^3" "6#/-%$ expG6#,$%\"xG!\"\"*$F(\"\"$" }{TEXT -1 27 " is a zero of the function \+ " }{XPPEDIT 18 0 "f(x)=exp(-x)-x^3" "6#/-%\"fG6#%\"xG,&-%$expG6#,$F'! \"\"\"\"\"*$F'\"\"$F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "f := x -> exp(-x)-x^3:\n 'f(x)'=f(x);\nplot(f(x),x=-0.4..1.1,color=magenta);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"fG6#%\"xG,&-%$expG6#,$F'!\"\"\"\"\"*$)F'\"\"$F.F -" }}{PARA 13 "" 1 "" {GLPLOT2D 333 348 348 {PLOTDATA 2 "6&-%'CURVESG6 #7S7$$!3A+++++++S!#=$\"3Sq7k(pCeb\"!#<7$$!3'*****\\(oUIn$F*$\"3sRitw5R $\\\"F-7$$!3M+]7y)e&)Q$F*$\"3t3KlZ'\\AW\"F-7$$!3]++D\"3F'oIF*$\"3A()Q! f**\\!)Q\"F-7$$!3%)***\\(oXdYFF*$\"3W7d&3/*zO8F-7$$!3I+]i:F0ECF*$\"3s9 ,;8X%))G\"F-7$$!3L+]7Gz))G@F*$\"3!>G_;_&*oC\"F-7$$!3K+]7.5>@=F*$\"32hm z([(z07F-7$$!3>+]7./(H]\"F*$\"3/]![b\\ub;\"F-7$$!3I+]iS.x&=\"F*$\"3.c$ \\o&3cF6F-7$$!3G+++v3\"\\f)!#>$\"3#p1.cz&Q!4\"F-7$$!3$=++vVT5s&Fhn$\"3 g@DsTe1f5F-7$$!3c*****\\7Xd[#Fhn$\"3tJYzKV=D5F-7$$\"3)z'****\\iNGw!#?$ \"3@x$y!>A+C**F*7$$\"3=+++D^W$*QFhn$\"3Io(eOKZvh*F*7$$\"3?(**\\(o%Qjt' Fhn$\"3qE`nTu\\X$*F*7$$\"3&)****\\i8o65F*$\"3x!*e2ngXF!*F*7$$\"3#)**** **\\>0)H\"F*$\"3mvE%ym$zg()F*7$$\"3o**\\(=-p6j\"F*$\"3qdFWN!=:X)F*7$$ \"3N*****\\2Mg#>F*$\"3))>pqB>kw\")F*7$$\"35+](=xZ&\\AF*$\"3U*[%fDdoryF *7$$\"3;+]i:$4wb#F*$\"3))zS9n!ofd(F*7$$\"3C++v=#R!zGF*$\"37h&\\se'pfsF *7$$\"3E+]P4A@uJF*$\"3%e#pXb'=/'pF*7$$\"3k***\\i:'f#\\$F*$\"3D1bZ!Rlgi 'F*7$$\"3m**\\(of2L#QF*$\"3es9VzX\"QE'F*7$$\"3c**\\7yG>6TF*$\"3B&o-'3& 3U$fF*7$$\"35++voo6AWF*$\"3ogv!ys&QhbF*7$$\"3D*****\\xJLu%F*$\"3sg&=&) G(yb^F*7$$\"3M***\\P*ydd]F*$\"3,jSGu:!ot%F*7$$\"3s**\\(=`r@vzFF*7$$\"34**\\(o/Q*>mF*$\"3/ Ho.wZ7dAF*7$$\"3k++](Q(zSpF*$\"3i>RdeKj^;F*7$$\"3v)*\\(=-,FC(F*$\"3kcC 6(z)\\Z5F*7$$\"3)***\\P4tFevF*$\"3kPs<)R#e$y$Fhn7$$\"39****\\73\"o'yF* $!3?=l**G'[(\\JFhn7$$\"33**\\(oz;)*=)F*$!3C$y!H2,O%3\"F*7$$\"3r+++]*44 ])F*$!3we-\"o6i%p=F*7$$\"3\"****\\7jZ!>))F*$!3A?k&RIJ\">FF*7$$\"3l+](= (4bM\"*F*$!3Wo%)RbE^5OF*7$$\"3\\+++vdYC%*F*$!3Q,3T)zHTZ%F*7$$\"3?++Dc3 uc(*F*$!3qP.YI$\\%=bF*7$$\"34+++lJR05F-$!3+C;+M;m.lF*7$$\"37+v=-*zq.\" F-$!3\"RR%\\K8F4wF*7$$\"3!**\\7G:3u1\"F-$!36fK&fKOEs)F*7$$\"33+++++++6 F-$!354#>I;*G\")**F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fb[lF^[l-%+A XESLABELSG6$Q\"x6\"Q!Fg[l-%%VIEWG6$;$!\"%!\"\"$\"#6F_\\l%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 55.000000 45.000000 0 0 "Curve 1" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 84 "This zero can be calculated numerically by applying Newton's me thod to the function " }{XPPEDIT 18 0 "f(x)=exp(-x)-x^3" "6#/-%\"fG6#% \"xG,&-%$expG6#,$F'!\"\"\"\"\"*$F'\"\"$F-" }{TEXT -1 39 " with (for ex ample) the starting value " }{XPPEDIT 18 0 "x[0] = 0;" "6#/&%\"xG6#\" \"!F'" }{TEXT -1 1 "." }{XPPEDIT 18 0 "8=4/5" "6#/\"\")*&\"\"%\"\"\"\" \"&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " } {XPPEDIT 18 0 "`f '`(x) = -exp(-x)-3*x^2;" "6#/-%$f~'G6#%\"xG,&-%$expG 6#,$F'!\"\"F-*&\"\"$\"\"\"*$F'\"\"#F0F-" }{TEXT -1 9 " so that " } {XPPEDIT 18 0 "`f '`(1) = -exp(-1)-3;" "6#/-%$f~'G6#\"\"\",&-%$expG6#, $F'!\"\"F-\"\"$F-" }{TEXT -1 1 " " }{TEXT 346 1 "~" }{TEXT -1 1 " " } {XPPEDIT 18 0 "-3;" "6#,$\"\"$!\"\"" }{TEXT -1 12 ".367879441. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "f := x -> exp(-x)-x^3:\n'f(x)'=f(x);\nDf := D(f):\na := 1;\nDiff( 'f(x)',x)=Df(x);\nEval(Diff('f(x)',x),x=a)=Df(a);\n``=evalf(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&-%$expG6#,$F'!\"\"\" \"\"*$)F'\"\"$F.F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6#%\"xGF*,&-%$expG6#, $F*!\"\"F0*&\"\"$\"\"\")F*\"\"#F3F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%%EvalG6$-%%DiffG6$-%\"fG6#%\"xGF-/F-\"\"\",&-%$expG6#!\"\"F4\"\"$F 4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$!+T%zyO$!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The tangent to the curve " }{XPPEDIT 18 0 "y = f(x);" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 20 " at the point w here " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 14 " is the line : " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y-f(a) = `f '`( a)*(x-a);" "6#/,&%\"yG\"\"\"-%\"fG6#%\"aG!\"\"*&-%$f~'G6#F*F&,&%\"xGF& F*F+F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = f(a)+`f '`(a)*(x-a);" " 6#/%\"yG,&-%\"fG6#%\"aG\"\"\"*&-%$f~'G6#F)F*,&%\"xGF*F)!\"\"F*F*" } {TEXT -1 2 ". " }}{PARA 259 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 13 " this gives: " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/exp(1)-1+(-1/exp(1)-3)*(x-1) " "6#/%\"yG,(*&\"\"\"F'-%$expG6#F'!\"\"F'F'F+*&,&*&F'F'-F)6#F'F+F+\"\" $F+F',&%\"xGF'F'F+F'F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = -(1/exp(1 )+3)*x+2+2/exp(1);" "6#/%\"yG,(*&,&*&\"\"\"F)-%$expG6#F)!\"\"F)\"\"$F) F)%\"xGF)F-\"\"#F)*&F0F)-F+6#F)F-F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "The tangent line at the point" }{XPPEDIT 18 0 "``(1,f(1));" "6#-%!G6$\"\"\"-%\"fG6#F&" } {TEXT -1 18 " is approximately " }}{PARA 258 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y = 2.735758882-3.367879441*x" "6#/%\"yG,&-%&FloatG6$\" +#))edt#!\"*\"\"\"*&-F'6$\"+T%zyO$F*F+%\"xGF+!\"\"" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 28 "This tangent line meets the " }{TEXT 347 1 "x" }{TEXT -1 12 " axis where " }{XPPEDIT 18 0 "x = (2+2/exp(1)) /(1/exp(1)+3);" "6#/%\"xG*&,&\"\"#\"\"\"*&F'F(-%$expG6#F(!\"\"F(F(,&*& F(F(-F+6#F(F-F(\"\"$F(F-" }{XPPEDIT 18 0 "``=(2*exp(1)+2)/(1+3*exp(1)) " "6#/%!G*&,&*&\"\"#\"\"\"-%$expG6#F)F)F)F(F)F),&F)F)*&\"\"$F)-F+6#F)F )F)!\"\"" }{TEXT -1 1 " " }{TEXT 348 1 "~" }{TEXT -1 75 " 0.8123090301 , which we can take as our next approximation for the zero of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "f := x -> exp(-x)-x^3:\na := 1;\ny=f(a)+D(f)(a)*(x-a);\ncollect(%,x);\nevalf [11](%);\n`x-intercept`=solve(subs(y=0,%),x);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG, (-%$expG6#!\"\"\"\"\"F*F)*&,&F&F)\"\"$F)F*,&%\"xGF*F*F)F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,(*&,&-%$expG6#!\"\"F+\"\"$F+\"\"\"%\" xGF-F-*&\"\"#F-F(F-F-F0F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,&* &$\",7WzyO$!#5\"\"\"%\"xGF*!\"\"$\",B))edt#F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%,x-interceptG$\"+,.4B\")!#5" }}}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The following picture sho ws the graph of " }{XPPEDIT 18 0 "y = exp(-x)-x^3" "6#/%\"yG,&-%$expG6 #,$%\"xG!\"\"\"\"\"*$F*\"\"$F+" }{TEXT -1 4 " in " }{TEXT 343 7 "magen ta" }{TEXT -1 55 " and the tangent line to this curve at the point whe re " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 4 " in " }{TEXT 269 5 "brown" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "f := x -> exp(-x)-x^3:\n'f( x)'=f(x);\na := 1.;\ng(x)=f(a)+D(f)(a)*(x-a);\nplot([f(x),f(a)+D(f)(a) *(x-a)],x=0.7..1.1,color=[magenta,brown]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&-%$expG6#,$F'!\"\"\"\"\"*$)F'\"\"$F.F- " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG$\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,&$\"+#))edt#!\"*\"\"\"*&$\"+T%zyO$F +F,F'F,!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 390 338 338 {PLOTDATA 2 "6& -%'CURVESG6$7S7$$\"3a**************p!#=$\"3m'49z.`e`\"F*7$$\"3Wmmm;')= (3(F*$\"3Q5/NpV(HO\"F*7$$\"3+LL$e'40jrF*$\"3]Fc@0'H-@\"F*7$$\"3Wmmm6hO [sF*$\"3H>0a$pJe.\"F*7$$\"3xmmm\"yYUL(F*$\"3U(ev$HrOu&)!#>7$$\"3-LL$eF >(>uF*$\"3%\\OO\"o&)=qnFA7$$\"3kmm;>K'*)\\(F*$\"3W/`)>>Y:2&FA7$$\"3Q** **\\Kd,\"e(F*$\"3%\\@[yU[gG$FA7$$\"3gmm;fX(em(F*$\"3TF&[#y]e59FA7$$\"3 o****\\U7Y]xF*$!3]g4t1#e\\)[!#?7$$\"3$QLLL/pu$yF*$!3I(f!)oE8KZ#FA7$$\" 3ommmhb59zF*$!37UojjH'zC%FA7$$\"3#*******H,Q+!)F*$!3g^%3GV5hF'FA7$$\"3 w******\\*3q3)F*$!3#4([?P6`)QOA\"F*7$$\"31LLLj$[kL)F*$!3q$*)z&*f+*[9F*7$ $\"34MLL`Q\"GT)F*$!3ma\\>h)GEk\"F*7$$\"3M++]s]k,&)F*$!3_Ur,O,Pr=F*7$$ \"3#HLLLvv-e)F*$!3;ifCD0#p2#F*7$$\"3)4++D2Ylm)F*$!3!>UetN.eI#F*7$$\"3q ++]sqMF*7$$\"3!)****\\nZ)H;*F*$!31yh.\\&*H$p$F*7$$\"3M nmmJy*eC*F*$!3$34;E+lq$RF*7$$\"3v+++S^bJ$*F*$!3eaj'[r%F*7$$\"3.+++0 TN:%*F*$!3]dG*Q'>KYWF*7$$\"3y++]7RV'\\*F*$!3/_p;4)3`p%F*7$$\"3_+++:#fk e*F*$!3O0&>m$)*F*$!3U&4EJYXKw&F*7$$\"3F,++qfa<**F*$!3 wT4C^*=a/'F*7$$\"3eLL$eg`!)***F*$!3)GTfTV^YJ'F*7$$\"3)****\\#G2A35!#<$ !3IzJ]/)y**f'F*7$$\"3XLLL)G[k,\"F`w$!3M5Tuq+$G)oF*7$$\"3.++D\"yh]-\"F` w$!3C!f!=V16$=(F*7$$\"3xmmm)fdL.\"F`w$!3/_Opv\"yjZ(F*7$$\"3$omm,FT=/\" F`w$!3UlIb@mW!y(F*7$$\"3QLLe#pa-0\"F`w$!3_*y-;m*='3)F*7$$\"3;+++ad)z0 \"F`w$!3Kk@:C\"\\3P)F*7$$\"3ULL$GUYo1\"F`w$!3mF[!4s8:q)F*7$$\"3)omm1^r Z2\"F`w$!3^'y>H_\"H,!*F*7$$\"35++D28A$3\"F`w$!3Adjn%3V^K*F*7$$\"3+++vS )384\"F`w$!3EhVLt(H#R'*F*7$$\"33+++++++6F`w$!354#>I;*G\")**F*-%'COLOUR G6&%$RGBG$\"*++++\"!\")$\"\"!Fb[lF^[l-F$6$7S7$F($\"3[)****HtKCy$F*7$F. $\"3Y.\"RM)>z)[$F*7$F3$\"3ABZLPnHLKF*7$F8$\"3)=sw;hlf%HF*7$F=$\"3?5#F*7$FM$\"3f(p:EzTc#=F* 7$FR$\"3sVw*[\\Z)R:F*7$FW$\"3)f$p6V(p\\D\"F*7$Fgn$\"3!=DP[fz$>'*FA7$F \\o$\"30n2f\\SNQqFA7$Fao$\"3RH4))*fIF8%FA7$Ffo$\"37J+7\"Qw^@\"FA7$F[p$ !3A\"[1$=xU'f\"FA7$F`p$!3IW@2(fL'\\TFA7$Fep$!33PVGQBk&=(FA7$Fjp$!3'>D8 g(*Rvv*FA7$F_q$!3S3RA]$F*7$F\\t$!3Q\\m&3S ![\"y$F*7$Fat$!3#*yQc))Q'*pSF*7$Fft$!3MaW'**p)=_VF*7$F[u$!34YRs;dDDYF* 7$F`u$!3y*\\$=#3]%G\\F*7$Feu$!3D'3jYq$)3?&F*7$Fju$!3gOCXYZw\"\\&F*7$F_ v$!3%G2PP?Y`v&F*7$Fdv$!3x3a`d.^VgF*7$Fiv$!3s,?5p.l9jF*7$F^w$!3UFAj1)p! )f'F*7$Fdw$!3eQZ<6T;voF*7$Fiw$!3]WrEohDlrF*7$F^x$!30jf](H\\YW(F*7$Fcx$ !3JTkzD\"p.t(F*7$Fhx$!3CN=l\\Is8!)F*7$F]y$!3+V[nxe4u#)F*7$Fby$!3Aoxk@D ^s&)F*7$Fgy$!3?&zna#*>%R))F*7$F\\z$!3i'>/u)))*R7*F*7$Faz$!3K$oMaExjR*F *7$Ffz$!33-++J]3*o*F*-F[[l6&F][l$\")#)eqkF`[l$\"))eqk\"F`[lF]el-%+AXES LABELSG6$Q\"x6\"Q!Fcel-%%VIEWG6$;$\"\"(!\"\"$\"#6F[fl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2 " }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 22 "The new approximation " }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6 #\"\"\"" }{TEXT -1 2 " " }{TEXT 344 1 "~" }{TEXT -1 46 " 0.8123090301 may be computed by the formula " }{XPPEDIT 18 0 "x[1] = x[0]-f(x[0]) /`f '`(x[0]);" "6#/&%\"xG6#\"\"\",&&F%6#\"\"!F'*&-%\"fG6#&F%6#F+F'-%$f ~'G6#&F%6#F+!\"\"F7" }{TEXT -1 9 " , where " }{XPPEDIT 18 0 "x[0] = 1; " "6#/&%\"xG6#\"\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "This process can be repeated to give " }{XPPEDIT 18 0 "x[2];" " 6#&%\"xG6#\"\"#" }{TEXT -1 19 " using the formula " }{XPPEDIT 18 0 "x[ 2] = x[1]-f(x[1])/`f '`(x[1]);" "6#/&%\"xG6#\"\"#,&&F%6#\"\"\"F+*&-%\" fG6#&F%6#F+F+-%$f~'G6#&F%6#F+!\"\"F7" }{TEXT -1 8 ", where " } {XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\"" }{TEXT -1 2 " " }{TEXT 345 1 "~" }{TEXT -1 16 " 0.8123090301. " }}{PARA 0 "" 0 "" {TEXT -1 5 "Th en " }{XPPEDIT 18 0 "x[3]" "6#&%\"xG6#\"\"$" }{TEXT -1 34 " can be com puted from the formula " }{XPPEDIT 18 0 "x[3] = x[2]-f(x[2])/`f '`(x[2 ])" "6#/&%\"xG6#\"\"$,&&F%6#\"\"#\"\"\"*&-%\"fG6#&F%6#F+F,-%$f~'G6#&F% 6#F+!\"\"F8" }{TEXT -1 13 ", and so on. " }}{PARA 0 "" 0 "" {TEXT -1 134 "The following Maple commands compute the first few \"Newton itera tes\" which provide successively better approximations for the zero of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 267 4 "Note" }{TEXT -1 16 ": The procedure " }{TEXT 268 38 "nextapprox := x -> evalf(x-f(x)/Df(x))" }{TEXT -1 8 ", where " } {TEXT 268 3 "Df " }{TEXT -1 48 "is the derivative f ' of f, is used to evaluate " }{XPPEDIT 18 0 "x-f(x)/`f '`(x)" "6#,&%\"xG\"\"\"*&-%\"fG6 #F$F%-%$f~'G6#F$!\"\"F-" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 224 "f := x -> exp(-x)-x^ 3:\nDf := D(f); # Df is the derivative f' of f\nnextapprox := x -> eva lf(x-f(x)/Df(x));\nx0 := 1;\nx1 := nextapprox(x0);\nx2 := nextapprox(x 1);\nx3 := nextapprox(x2);\nx4 := nextapprox(x3);\nx5 := nextapprox(x4 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+nextapproxGf*6#%\"xG6\"6$%)op eratorG%&arrowGF(-%&evalfG6#,&9$\"\"\"*&-%\"fG6#F0F1-%#DfGF5!\"\"F8F(F (F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G\"\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#x1G$\"+,.4B\")!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x2G$\"+!\\lFu(!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x3G$\"+i v%)Gx!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x4G$\"+\"fH)Gx!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x5G$\"+\"fH)Gx!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "The last two values gi ve the zero of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 23 " correct to 10 digits. " }}{PARA 0 "" 0 "" {TEXT -1 39 "The result ca n be checked by computing " }{XPPEDIT 18 0 "f(x[5])=``" "6#/-%\"fG6#&% \"xG6#\"\"&%!G" }{TEXT -1 17 "f(0.7728829591). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(f(x5)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"\"!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 144 "This result is not exact ly zero. However, if we change the last digit by 1 in either direction the value is further away from zero in each case." }}{PARA 0 "" 0 "" {TEXT -1 95 "This shows that 0.7728829591 is the best possible (most a ccurate) 10 digit value for the zero. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "Digits := 12:\nf(0.77288 29590);\nf(0.7728829591);\nf(0.7728829592);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$P$!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"$5\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!$:\"!#7" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "The solution of th e equation " }{XPPEDIT 18 0 "exp(-x)=x^3" "6#/-%$expG6#,$%\"xG!\"\"*$F (\"\"$" }{TEXT -1 39 " correct to 10 digits is 0.7728829591. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 0 " " 0 "" {TEXT 352 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 13 "The equation " }{XPPEDIT 18 0 "2*ln(x+1) = 1-x;" "6#/*&\" \"#\"\"\"-%#lnG6#,&%\"xGF&F&F&F&,&F&F&F+!\"\"" }{TEXT -1 39 " has exac tly one real number solution. " }}{PARA 0 "" 0 "" {TEXT -1 102 "(a) Co nstruct a Maple plot to illustrate this solution as the point of inter section of the two graphs " }{XPPEDIT 18 0 "y=2*ln(x+1)" "6#/%\"yG*&\" \"#\"\"\"-%#lnG6#,&%\"xGF'F'F'F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 " y=1-x" "6#/%\"yG,&\"\"\"F&%\"xG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "(b) Set up the function " }{XPPEDIT 18 0 "f(x) = 2*ln( x+1)-1+x;" "6#/-%\"fG6#%\"xG,(*&\"\"#\"\"\"-%#lnG6#,&F'F+F+F+F+F+F+!\" \"F'F+" }{TEXT -1 70 " in Maple and plot a graph to illustrate the sol ution of the equation " }{XPPEDIT 18 0 "2*ln(x+1) = 1-x;" "6#/*&\"\"# \"\"\"-%#lnG6#,&%\"xGF&F&F&F&,&F&F&F+!\"\"" }{TEXT -1 27 " as a zero o f the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 35 "(c) Set up a procedure to evaluate \+ " }{XPPEDIT 18 0 "x-f(x)/`f '`(x)" "6#,&%\"xG\"\"\"*&-%\"fG6#F$F%-%$f~ 'G6#F$!\"\"F-" }{TEXT -1 43 " and use it to calculate \"Newton iterate s\" " }{XPPEDIT 18 0 "x[0],x[1], ` . . . `" "6%&%\"xG6#\"\"!&F$6#\"\" \"%(~.~.~.~G" }{TEXT -1 7 ", with " }{XPPEDIT 18 0 "x[0]=0" "6#/&%\"xG 6#\"\"!F'" }{TEXT -1 10 ".4, where " }{XPPEDIT 18 0 "x[1] = x[0]-f(x[0 ])/`f '`(x[0]);" "6#/&%\"xG6#\"\"\",&&F%6#\"\"!F'*&-%\"fG6#&F%6#F+F'-% $f~'G6#&F%6#F+!\"\"F7" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "x[2] = x[1]-f( x[1])/`f '`(x[1]),` . . . `;" "6$/&%\"xG6#\"\"#,&&F%6#\"\"\"F+*&-%\"fG 6#&F%6#F+F+-%$f~'G6#&F%6#F+!\"\"F7%(~.~.~.~G" }{TEXT -1 43 ", etc., un til you can estimate the zero of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#% \"xG" }{TEXT -1 23 " correct to 10 digits. " }}{PARA 0 "" 0 "" {TEXT -1 67 "(d) Check that the result obtained in (c) is correct to 10 digi ts. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 351 8 "S olution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 0 "" 0 "" {TEXT -1 38 "In the following picture the graph of " } {XPPEDIT 18 0 "y = 2*ln(x+1);" "6#/%\"yG*&\"\"#\"\"\"-%#lnG6#,&%\"xGF' F'F'F'" }{TEXT -1 15 " is plotted in " }{TEXT 268 3 "red" }{TEXT -1 20 " while the graph of " }{XPPEDIT 18 0 "y = 1-x;" "6#/%\"yG,&\"\"\"F &%\"xG!\"\"" }{TEXT -1 15 " is plotted in " }{TEXT 257 5 "green" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "plot([2*ln(x+1),1-x],x=-0.4..1.4,y=-.3..2,color= [red,COLOR(RGB,0,.8,0)]); " }}{PARA 13 "" 1 "" {GLPLOT2D 397 333 333 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$!3A+++++++S!#=$!3W\")>`Z7l@5!#<7$$!3N+ ++D7l2OF*$!3o40RMkm\\*)F*7$$!3E++v`1FmKF*$!3l.\"*p.$>\"4zF*7$$!3K++]( \\_B)GF*$!3))QiP#ec,!oF*7$$!3E++]#[*)e\\#F*$!31K[trIoUdF*7$$!3[++veKE6 @F*$!3l8r!ff\")Hu%F*7$$!3I++v8bma$!3K \\F%[)HM'G\"F*7$$!3z)*****\\I*QJ#FY$!3!*HxNNw;#o%FY7$$\"3/'****\\F]Z8 \"FY$\"3k!)H#zb?nD#FY7$$\"3l(*****\\e5<]FY$\"3QtKa^Fh!z*FY7$$\"3\\&*** **\\FS:*)FY$\"3?'4joYD!3Gu*4)*4;a'F*7$$\"37++]iq%[D%F*$\"3o$*pU7!Q-4(F* 7$$\"3p***\\7la!4YF*$\"3u];f,&G6e(F*7$$\"3;****\\(Q:6*\\F*$\"3M-c\\-DX (4)F*7$$\"3u)**\\i6pzQ&F*$\"38*eNyp<+i)F*7$$\"3j)**\\PXJMt&F*$\"3Ks8aZ \\0k!*F*7$$\"3F****\\U-a1hF*$\"3'4K@]X1G`*F*7$$\"3e******H\")*>\\'F*$ \"3*ReCpT!e+5F-7$$\"3o****\\sM4poF*$\"3O+,5KhzX5F-7$$\"3D***\\ig_RB(F* $\"3c$[5wm#f)3\"F-7$$\"3i)***\\nk1RwF*$\"3qc8op?1N6F-7$$\"32)******GzI +)F*$\"39$RvQW:f<\"F-7$$\"33+++v1u\"R)F*$\"3c&4u*)=L'=7F-7$$\"3'*)**\\ ilDRu)F*$\"35a(oQGplD\"F-7$$\"3\"3++]'o&*G\"*F*$\"3KK9W?jB(H\"F-7$$\"3 )*)**\\iAT7\\*F*$\"3Dd)pv?gZL\"F-7$$\"3e***\\7xK*p)*F*$\"3WC$o-;XKP\"F -7$$\"3#*****\\(H ^s5[9F-7$$\"3&)*****R>4,5\"F-$\"3uf3/\"oyR[\"F-7$$\"3$****\\dr&GQ6F-$ \"3uQD?\"))3+_\"F-7$$\"3++]i;h9w6F-$\"3aZr%=&46b:F-7$$\"3#)*****H*e$4@ \"F-$\"3qLZw8=$oe\"F-7$$\"3y***\\F!*33D\"F-$\"3av6)yKzDi\"F-7$$\"3++++ )zrkG\"F-$\"3[J&4)f)>Sl\"F-7$$\"3'***\\i#)e\\C8F-$\"3-jSq\"Q1qo\"F-7$$ \"3y**\\P$y*)3O\"F-$\"3;N\"p'\\r2=$\"3++ +D[*)e\\7F-7$FC$\"3-+](eKE6@\"F-7$FH$\"3++]P^lYv6F-7$FM$\"3$***\\P?HaQ 6F-7$FR$\"3\"***\\P[kN+6F-7$FW$\"33+](3W#Hi5F-7$Fgn$\"3/++]I*QJ-\"F-7$ F\\o$\"3S++]s\\_'))*F*7$Fao$\"3y+++:%*G)\\*F*7$Ffo$\"3!******\\sf%3\"* F*7$F[p$\"3r*****\\e'yK()F*7$F`p$\"3G++v$QR;R)F*7$Fep$\"3e+++lB)f)zF*7 $Fjp$\"3O+++gwLUwF*7$F_q$\"3c++vtrfUsF*7$Fdq$\"3=,++5\"f())oF*7$Fiq$\" 3]++vtEa+lF*7$F^r$\"3a***\\7#)o38'F*7$Fcr$\"3!*****\\PH:XdF*7$Fhr$\"3I ++v[`%4R&F*7$F]s$\"3$3++Dh%))3]F*7$Fbs$\"3D,+v$)3.7YF*7$Fgs$\"3O,+DY&o lE%F*7$F\\t$\"3s++]d(fM*QF*7$Fat$\"3T+++q=+3NF*7$Fft$\"3J++]Fl!48$F*7$ F[u$\"3u++v$RZgw#F*7$F`u$\"3Q,+]KN$4O#F*7$Feu$\"3#>+++r?p*>F*7$Fju$\"3 \"******\\Kf#3;F*7$F_v$\"3/,+vVV2c7F*7$Fdv$\"3\">*****\\8V5()FY7$Fiv$ \"3/5+]Pxe(3&FY7$F^w$\"3I/+](Gs1I\"FY7$Fcw$!3M#*****\\(H4,5F*7$Fbx$!3G****\\dr&GQ\"F*7$Fgx$!3;+ +Dm6Yh)*****H*e$4@F*7$Fay$!3&y***\\F!*33DF*7$Ffy$!3++++!) zrkGF*7$F[z$!3a***\\i#)e\\C$F*7$F`z$!3t(**\\P$y*)3OF*7$Fez$!37******** ******RF*-%&COLORG6&F\\[lF`[l$\"\")!\"\"F`[l-%+AXESLABELSG6$Q\"x6\"Q\" yF`el-%%VIEWG6$;$!\"%F[el$\"#9F[el;$!\"$F[el$\"\"#Fa[l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "The solution of " }{XPPEDIT 18 0 "2*ln(x+1) = 1-x;" "6#/*&\"\"#\"\"\"-%#lnG6#,&%\"xGF&F&F&F&,&F&F&F+! \"\"" }{TEXT -1 27 " is a zero of the function " }{XPPEDIT 18 0 "f(x) \+ = 2*ln(x+1)-1+x;" "6#/-%\"fG6#%\"xG,(*&\"\"#\"\"\"-%#lnG6#,&F'F+F+F+F+ F+F+!\"\"F'F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "f := x -> 2*ln(x+1)-1+x:\n'f (x)'=f(x);\nplot(f(x),x=-0.2..1,color=magenta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,(*&\"\"#\"\"\"-%#lnG6#,&F'F+F+F+F+F+F+! \"\"F'F+" }}{PARA 13 "" 1 "" {GLPLOT2D 333 348 348 {PLOTDATA 2 "6&-%'C URVESG6#7S7$$!35+++++++?!#=$!3Q>%GE5(GY;!#<7$$!3,+++]TVQ$!3QW5&R7Q)48F-7$$!3&G++]s@%3uF@$!3hhA!*3#G!G7F-7$$ !3_-++DM5J]F@$!3!=c*=RDb`6F-7$$!3c.++D!G&pDF@$!39Y3)3idx2\"F-7$$!3sg.+ +DKwB!#@$!3*Q.GK&Hr+5F-7$$\"35(****\\FPQ^#F@$!3Y&oAoCk?D*F*7$$\"3L**** ***HrS7&F@$!3Og*y&44<)[)F*7$$\"3)e*****\\o;BuF@$!3:HUsL(pb#yF*7$$\"3)) *******QS6+\"F*$!3g)*3-&o#e!4(F*7$$\"3O******\\o-h7F*$!3=iO$*[(=PO'F*7 $$\"3')******4cZ6:F*$!3mOj*3\"zLtcF*7$$\"3H****\\xq!*QF*7$$\" 3#******\\\\$pPZF*$\"3_-&*HR&fT\\#F*7$$\"3-******>am%*\\F*$\"3a)QTPa`o 4$F*7$$\"3k*****\\JigC&F*$\"3?7aIscy!o$F*7$$\"3g)***\\P3\"F-7$$\"38+++?EdR()F*$\"3%G?'e`9V#*F*$\"3i&y0h;a MB\"F-7$$\"3G++]<#Rm\\*F*$\"3k+JbTz(\\G\"F-7$$\"3#)****\\A_ER(*F*$\"3) Q'\\Mdf(RL\"F-7$$\"\"\"\"\"!$\"3c!*)>6O%H'Q\"F--%'COLOURG6&%$RGBG$\"*+ +++\"!\")$FizFizF`[l-%+AXESLABELSG6$Q\"x6\"Q!Fh[l-%%VIEWG6$;$!\"#!\"\" Fgz%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "This zero can b e calculated numerically by applying Newton's method to the function \+ " }{XPPEDIT 18 0 "f(x) = 2*ln(x+1)-1+x;" "6#/-%\"fG6#%\"xG,(*&\"\"#\" \"\"-%#lnG6#,&F'F+F+F+F+F+F+!\"\"F'F+" }{TEXT -1 39 " with ( for examp le) the starting value" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x[0] = 0;" "6#/&%\"xG6#\"\"!F'" }{TEXT -1 1 "." }{XPPEDIT 18 0 "4 = 2/5;" "6#/\"\"%*&\"\"#\"\"\"\"\"&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 134 "The following Maple commands compute the first fe w \"Newton iterates\" which provide successively better approximations for the zero of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 267 4 "Note" }{TEXT -1 16 ": The proced ure " }{TEXT 268 38 "nextapprox := x -> evalf(x-f(x)/Df(x))" }{TEXT -1 8 ", where " }{TEXT 268 3 "Df " }{TEXT -1 48 "is the derivative f ' of f, is used to evaluate " }{XPPEDIT 18 0 "x-f(x)/`f '`(x)" "6#,&%\" xG\"\"\"*&-%\"fG6#F$F%-%$f~'G6#F$!\"\"F-" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 234 "f : = x -> 2*ln(x+1)-1+x;\nDf := D(f); # Df is the derivative f' of f\nnex tapprox := x -> evalf(x-f(x)/Df(x));\nDigits := 14:\nx0 := 0.4;\nx1 := nextapprox(x0);\nx2 := nextapprox(x1);\nx3 := nextapprox(x2);\nx4 := \+ nextapprox(x3);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(*&\"\"#\"\"\"-%#lnG6#,&9$F/F/F/ F/F/F/!\"\"F4F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DfGf*6#%\" xG6\"6$%)operatorG%&arrowGF(,&*&\"\"#\"\"\",&9$F/F/F/!\"\"F/F/F/F(F(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+nextapproxGf*6#%\"xG6\"6$%)oper atorG%&arrowGF(-%&evalfG6#,&9$\"\"\"*&-%\"fG6#F0F1-%#DfGF5!\"\"F8F(F(F (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$\"\"%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"/fH//k*p$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x2G$\"/O.l(Q:q$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#x3G$\"/;4V)Q:q$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x4G$\"/< 4V)Q:q$!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "The last two values both round to the same 10 digit value 0.370 1538843 to give the zero of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 23 " correct to 10 digits. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalf(x3);\nevalf(x4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+V)Q:q$!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+V)Q:q$!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 39 "The result can be checked by computing " }{XPPEDIT 18 0 "f(x[5])=``" "6#/-%\"fG6#&%\"xG6#\"\"&%!G" }{TEXT -1 17 "f(0.7728829591). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(f(x4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!\"&!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 144 "This result is not exactly zero. However, if we c hange the last digit by 1 in either direction the value is further awa y from zero in each case." }}{PARA 0 "" 0 "" {TEXT -1 97 "This shows t hat 0.3701538843 is the best possible (most accurate) 10 digit value f or the zero of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Digits := 14:\nf(.3701538842);\nf(.3701538843);\nf(.3 701538844);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!&co#!# 9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!%gA!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&OB#!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "The solution of the equation " }{XPPEDIT 18 0 "2* ln(x+1)=1-x" "6#/*&\"\"#\"\"\"-%#lnG6#,&%\"xGF&F&F&F&,&F&F&F+!\"\"" } {TEXT -1 39 " correct to 10 digits is 0.3701538843. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 5 "Note:" }{TEXT -1 57 " M aple can obtain an analytical soltion for the equation " }{XPPEDIT 18 0 "2*ln(x+1)=1-x" "6#/*&\"\"#\"\"\"-%#lnG6#,&%\"xGF&F&F&F&,&F&F&F+!\" \"" }{TEXT -1 1 " " }{TEXT 353 0 "" }{TEXT -1 76 "in terms of a specia l mathematical function called the Lambert W function. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "2*ln( x+1)=1-x;\nsolve(%);\nevalf[11](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/,$*&\"\"#\"\"\"-%#lnG6#,&%\"xGF'F'F'F'F',&F'F'F,!\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,&*&\"\"#\"\"\"-%)LambertWG6#,$*&#F&F%F&-%$expG6 #F&F&F&F&F&F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+V)Q:q$!#5" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 " }}{PARA 0 "" 0 "" {TEXT 350 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 13 "The equation " }{XPPEDIT 18 0 "2*exp(-x) = arctan(x);" "6 #/*&\"\"#\"\"\"-%$expG6#,$%\"xG!\"\"F&-%'arctanG6#F+" }{TEXT -1 39 " h as exactly one real number solution. " }}{PARA 0 "" 0 "" {TEXT -1 102 "(a) Construct a Maple plot to illustrate this solution as the point o f intersection of the two graphs " }{XPPEDIT 18 0 "y = 2*exp(-x);" "6# /%\"yG*&\"\"#\"\"\"-%$expG6#,$%\"xG!\"\"F'" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "y = arctan(x);" "6#/%\"yG-%'arctanG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "(b) Set up the function " } {XPPEDIT 18 0 "f(x) = 2*exp(-x)-arctan(x);" "6#/-%\"fG6#%\"xG,&*&\"\"# \"\"\"-%$expG6#,$F'!\"\"F+F+-%'arctanG6#F'F0" }{TEXT -1 70 " in Maple \+ and plot a graph to illustrate the solution of the equation " } {XPPEDIT 18 0 "2*exp(-x) = arctan(x);" "6#/*&\"\"#\"\"\"-%$expG6#,$%\" xG!\"\"F&-%'arctanG6#F+" }{TEXT -1 27 " as a zero of the function " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 35 "(c) Set up a procedure to evaluate " }{XPPEDIT 18 0 "x -f(x)/`f '`(x)" "6#,&%\"xG\"\"\"*&-%\"fG6#F$F%-%$f~'G6#F$!\"\"F-" } {TEXT -1 43 " and use it to calculate \"Newton iterates\" " }{XPPEDIT 18 0 "x[0],x[1], ` . . . `" "6%&%\"xG6#\"\"!&F$6#\"\"\"%(~.~.~.~G" } {TEXT -1 7 ", with " }{XPPEDIT 18 0 "x[0] = 1;" "6#/&%\"xG6#\"\"!\"\" \"" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "x[1] = x[0]-f(x[0])/`f '`(x [0]);" "6#/&%\"xG6#\"\"\",&&F%6#\"\"!F'*&-%\"fG6#&F%6#F+F'-%$f~'G6#&F% 6#F+!\"\"F7" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "x[2] = x[1]-f(x[1])/`f ' `(x[1]),` . . . `;" "6$/&%\"xG6#\"\"#,&&F%6#\"\"\"F+*&-%\"fG6#&F%6#F+F +-%$f~'G6#&F%6#F+!\"\"F7%(~.~.~.~G" }{TEXT -1 43 ", etc., until you ca n estimate the zero of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 23 " correct to 10 digits. " }}{PARA 0 "" 0 "" {TEXT -1 67 "( d) Check that the result obtained in (c) is correct to 10 digits. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 349 8 "Solution " }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 0 "" 0 "" {TEXT -1 38 "In the following picture the graph of " }{XPPEDIT 18 0 "y = exp(-x);" "6#/%\"yG-%$expG6#,$%\"xG!\"\"" }{TEXT -1 15 " is plo tted in " }{TEXT 268 3 "red" }{TEXT -1 20 " while the graph of " } {XPPEDIT 18 0 "y = arctan(x);" "6#/%\"yG-%'arctanG6#%\"xG" }{TEXT -1 15 " is plotted in " }{TEXT 257 5 "green" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "plot( [2*exp(-x),arctan(x)],x=-2..2.4,y=-1.2..2.4,color=[red,COLOR(RGB,0,.8, 0)]); " }}{PARA 13 "" 1 "" {GLPLOT2D 401 283 283 {PLOTDATA 2 "6&-%'CUR VESG6$7S7$$!\"#\"\"!$\"33Ihy>7\"yZ\"!#;7$$!3lmmm@D4/>!#<$\"32]sMNDmU8F -7$$!3@LLePRk?=F1$\"36/UF/o;N7F-7$$!3kmm;xszEo&oGY:B5F-7$$!3ILLe'z3$Q:F1$\"3wTv'RL;MJ*F17$$!3amm\" *e/9^9F1$\"3bYW<\"Ref`)F17$$!3')***\\Up#)3O\"F1$\"3))\\#[]$zE*z(F17$$! 3emm\"\\)z`n7F1$\"3n#>Xd%H>/rF17$$!3')***\\Kj#\\u6F1$\"3m=\"y*o3+tkF17 $$!3=LLL_Syy5F1$\"3mL=q,B?#)eF17$$!36mmm@)Q[%**!#=$\"3'oUz**[dmS&F17$$ !3=)*****p&=e**)F^o$\"3pKBl?(\\r\"\\F17$$!3&*)*****\\:!H/)F^o$\"3C,uu3 (=-Z%F17$$!3+******H%*eCrF^o$\"3MS6%Gw(*z2%F17$$!3#omm\"\\Sn!H'F^o$\"3 eqGD!p?[$F^o$\"3'G\"QV4R+LGF17$$!3(HLLLrmph#F^o$\" 3o]q5P[E)f#F17$$!3s(***\\-K*zm\"F^o$\"3[FX!*pU.jBF17$$!3sj***\\2gMk(!# >$\"3]%3C!)=j)e@F17$$\"31#pmmT]^y\"Far$\"3i`s,!y8Y'>F17$$\"3nom;9eNW5F ^o$\"3kUX#=ll;!=F17$$\"3!QLL$e?Gy>F^o$\"3!QP****f@5k\"F17$$\"3GLL$3&*o $[HF^o$\"3K()[EogI*[\"F17$$\"3g-+]UC$Gz$F^o$\"3E?$HI\\.(o8F17$$\"3cMLL [h([q%F^o$\"3MnJRk^R\\7F17$$\"3+-++Sl5ZcF^o$\"3ueMdY#\\q8\"F17$$\"3w++ +b^*)olF^o$\"3Akr*3*H#p.\"F17$$\"3k.+]PIxguF^o$\"3]+gnXGY%[*F^o7$$\"3k *****\\O^5X)F^o$\"33,7boRC!f)F^o7$$\"3Ykmm'[g3M*F^o$\"3o9?Vym*)eyF^o7$ $\"3B+++l@4H5F1$\"3yD(yGACm9(F^o7$$\"3eLL3F==:6F1$\"3m/F#)3K6dlF^o7$$ \"3b+++n0I47F1$\"3%>K$4G%=\"ofF^o7$$\"3UmmTm*eyH\"F1$\"3M$=apN=BY&F^o7 $$\"3W++v5!G/R\"F1$\"3Wf%)p5WPz\\F^o7$$\"3mmmmr6$4[\"F1$\"3]xBbXj^[XF^ o7$$\"3))***\\Pfzcd\"F1$\"3Fjfo)QQt8%F^o7$$\"3-MLL&eLpm\"F1$\"35mK#G!R ]wPF^o7$$\"3)QLL=(RDgF1$\"3%)=S*zA%G!)GF^o7$$\"3@nm;^1JN?F1$\"3^=DH byz7EF^o7$$\"35MLLP4 A7#=#F^o7$$\"3\"3+]#[sR/BF1$\"3e5?Tr*yj*>F^o7$$\"3!**************R#F1$ \"3C]#)yl!fV\"=F^o-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*Fb[l-F$6$7S7$F( $!3T!4%z<([r5\"F17$F/$!3%\\/'R^Y?(3\"F17$F5$!3i3=_QV_o5F17$F:$!3C(*=![ <\")e/\"F17$F?$!3@B-\"zwZ6-\"F17$FD$!33UK?otvV**F^o7$FI$!3m=y!3\"R9u'* F^o7$FN$!3_(y!QuB$3P*F^o7$FS$!3bN/knJTG!*F^o7$FX$!3qf`g>!=Zl)F^o7$Fgn$ !3%R2IMP\"zK#)F^o7$F\\o$!3z+40O[KEyF^o7$Fbo$!3EY5;b.%eK(F^o7$Fgo$!3w9f \\8U^tnF^o7$F\\p$!3)fx#><#)Q!>'F^o7$Fap$!3%oc.WX)=:cF^o7$Ffp$!31%[)>c \"eG([F^o7$F[q$!3o6A1$3JW>%F^o7$F`q$!3!e`&)*G\"=1N$F^o7$Feq$!3e%eCgRm& fDF^o7$Fjq$!3]6))))Hux_;F^o7$F_r$!3I;:UQqiGwFar7$Fer$\"3%oy)fC3'\\y\"F ar7$Fjr$\"3t+sQ7OeS5F^o7$F_s$\"3a!4StEkI&>F^o7$Fds$\"3aiZ6eH8nGF^o7$Fi s$\"3ND_u^a?DOF^o7$F^t$\"31Ulk>?g(R%F^o7$Fct$\"3^)>7!Q=nS^F^o7$Fht$\"3 *=UT]GK?\"eF^o7$F]u$\"3>=9ED&e)4kF^o7$Fbu$\"3\"p$z$zqak,(F^o7$Fgu$\"3I w-KR>J8vF^o7$F\\v$\"3#*H&)G(3Zt*zF^o7$Fav$\"3q%fquW#*zR)F^o7$Ffv$\"3I4 m@YQ_)z)F^o7$F[w$\"3a_xhI&RI9*F^o7$F`w$\"3IL!oBl$)HZ*F^o7$Few$\"3u:76P Ouo(*F^o7$Fjw$\"38k^g[+H05F17$F_x$\"3:]h9tuWI5F17$Fdx$\"3_-I!>;jT0\"F1 7$Fix$\"3%f\"on.u(e2\"F17$F^y$\"3)eGa-6,W4\"F17$Fcy$\"3ddBLxB696F17$Fh y$\"3E[:A=w\\I6F17$F]z$\"3%zH\"=BkzY6F17$Fbz$\"3W0%f%\\pOh6F17$Fgz$\"3 +N^42_+w6F1-%&COLORG6&F^[lFb[l$\"\")!\"\"Fb[l-%+AXESLABELSG6$Q\"x6\"Q \"yFcel-%%VIEWG6$;F($\"#CF^el;$!#7F^elFiel" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 16 "The solution of " }{XPPEDIT 18 0 "2*exp(- x) = arctan(x);" "6#/*&\"\"#\"\"\"-%$expG6#,$%\"xG!\"\"F&-%'arctanG6#F +" }{TEXT -1 27 " is a zero of the function " }{XPPEDIT 18 0 "f(x) = 2 *exp(-x)-arctan(x);" "6#/-%\"fG6#%\"xG,&*&\"\"#\"\"\"-%$expG6#,$F'!\" \"F+F+-%'arctanG6#F'F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "f := x -> 2*exp(-x)-arc tan(x):\n'f(x)'=f(x);\nplot(f(x),x=-0.2..2,color=magenta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&*&\"\"#\"\"\"-%$expG6#,$F'! \"\"F+F+-%'arctanGF&F0" }}{PARA 13 "" 1 "" {GLPLOT2D 328 329 329 {PLOTDATA 2 "6&-%'CURVESG6#7S7$$!35+++++++?!#=$\"3g?-K5\"F*$\"3f#pF_q_JM#F-7$$!3LLLLeQ ')Rj!#>$\"3cSr#4(o@%>#F-7$$!33KLL3qU;;F;$\"3!e_0)*4a([?F-7$$\"3oLL$3F-7$$\"3Amm;aq(HW(F;$\"3Ctg\"Ra_Ay\"F-7$$\"3/++vGle& >\"F*$\"3S$>1W5Icl\"F-7$$\"3)pm;a25Bm\"F*$\"3iIN]-k(*G:F-7$$\"3-++vLo` F@F*$\"3,`aqWB329F-7$$\"3&RLL$Q(zgg#F*$\"3'e*o\"oWGiG\"F-7$$\"3$omm\"* e!eFIF*$\"3y\"Q^=VqN=\"F-7$$\"3!3++]r!4-NF*$\"37i&ot[?A2\"F-7$$\"3U+++ D#\\&yRF*$\"39+U4*\\J'['*F*7$$\"3R+++&G0xV%F*$\"3]2***>PQcl)F*7$$\"3[m mTvHma[F*$\"3/`K.%Rr')y(F*7$$\"3eLLL)*fY]`F*$\"3_EgK@w$**z'F*7$$\"3sLL L$>w/x&F*$\"3w$fm3Eft*fF*7$$\"3!4+]()*y/fiF*$\"3$*znc1,0.^F*7$$\"3%GLL Lk;:p'F*$\"35[zV7rkXVF*7$$\"3[++v)R.g;(F*$\"3%>k5Gi#R]NF*7$$\"3:,+D'*p #yh(F*$\"3U5;%=Num#GF*7$$\"31NL$3_d#*3)F*$\"3**R(G&\\f70@F*7$$\"3yML32 zz**oMLZ\"F*7$$\"33++vd ZWG6F-$!3CEMM*[(*f)>F*7$$\"3A+](=lQI<\"F-$!3wP'p$)\\(4gCF*7$$\"3.++DoD bA7F-$!39#zoT%>WiHF*7$$\"3GLLLCI/n7F-$!3E7\"fTh[KR$F*7$$\"3;++]#3YXJ\" F-$!3F>FJ[$zF$QF*7$$\"3%omTN\"4fd8F-$!3G&y[NTvO@%F*7$$\"3K++]$G]YS\"F- $!3*4b*eEn37YF*7$$\"3EL$3K[H*[9F-$!3)\\O.*Hqeq\\F*7$$\"3E+]P0S@&\\\"F- $!3/:-$e6m\"H`F*7$$\"3PLL$eel/a\"F-$!3yuTwX;`kcF*7$$\"3)***\\(ozRye\"F -$!3!H6Ib\\#H+gF*7$$\"31nmm#zmMj\"F-$!3T)p=\"4Ab4jF*7$$\"3)pm;f)p7!o\" F-$!33L^rB==7mF*7$$\"3hL$3#43SEr]iO_rF*7$$\"3lLLeD`l<=F-$!329QyWi=IuF*7$$\"35nmm3LCh=F-$!3'>/*\\3 k;owF*7$$\"3F+]()*=F-$!3C)GfCv%\\6zF*7$$\"3W+]7C')>_>F-$!3-0(Q=u\" zM\")F*7$$\"\"#\"\"!$!3/]'3K^\"yk$)F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$ FgzFgzF^[l-%+AXESLABELSG6$Q\"x6\"Q!Ff[l-%%VIEWG6$;$!\"#!\"\"Fez%(DEFAU LTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "This zero can be calculat ed numerically by applying Newton's method to the function " } {XPPEDIT 18 0 "f(x) = 2*exp(-x)-arctan(x);" "6#/-%\"fG6#%\"xG,&*&\"\"# \"\"\"-%$expG6#,$F'!\"\"F+F+-%'arctanG6#F'F0" }{TEXT -1 39 " with ( fo r example) the starting value" }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x[0] = 1;" "6#/&%\"xG6#\"\"!\"\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 134 "The following Maple commands compute the first few \"Newton iterates\" which provide successively better appro ximations for the zero of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 267 4 "Note" }{TEXT -1 16 ": T he procedure " }{TEXT 268 38 "nextapprox := x -> evalf(x-f(x)/Df(x))" }{TEXT -1 8 ", where " }{TEXT 268 3 "Df " }{TEXT -1 48 "is the derivat ive f ' of f, is used to evaluate " }{XPPEDIT 18 0 "x-f(x)/`f '`(x)" " 6#,&%\"xG\"\"\"*&-%\"fG6#F$F%-%$f~'G6#F$!\"\"F-" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 260 "f := x -> 2*exp(-x)-arctan(x);\nDf := D(f); # Df is the derivativ e f' of f\nnextapprox := x -> evalf(x-f(x)/Df(x));\nDigits := 14:\nx0 \+ := 1;\nx1 := nextapprox(x0);\nx2 := nextapprox(x1);\nx3 := nextapprox( x2);\nx4 := nextapprox(x3);\nx5 := nextapprox(x4);\nDigits := 10:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,&*&\"\"#\"\"\"-%$expG6#,$9$!\"\"F/F/-%'arctanG6#F4F5F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DfGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,&*&\"\"#\"\"\"-%$expG6#,$9$!\"\"F/F5*&F/F/,&F/F/*$)F4F.F/F/F5F5F(F (F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+nextapproxGf*6#%\"xG6\"6$%)o peratorG%&arrowGF(-%&evalfG6#,&9$\"\"\"*&-%\"fG6#F0F1-%#DfGF5!\"\"F8F( F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"/_JK$4$)f*!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x2G$\"/LFhQ;1'*!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x3G$ \"/[`Yp;1'*!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x4G$\"/``Yp;1'*!# 9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x5G$\"/``Yp;1'*!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "The last two va lues both round to the same 10 digit value 0.3701538843 to give the ze ro of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 23 " correct to 10 digits. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 21 "evalf(x4);\nevalf(x5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Zp;1'*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Zp ;1'*!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "The result can be checked by computing " }{XPPEDIT 18 0 "f(x[4]) = ``;" "6#/-%\"fG6#&%\"xG6#\"\"%%!G" }{TEXT -1 17 "f(0.9606166947). " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(f(x4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"\"!#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 144 "This res ult is not exactly zero. However, if we change the last digit by 1 in \+ either direction the value is further away from zero in each case." }} {PARA 0 "" 0 "" {TEXT -1 97 "This shows that 0.3701538843 is the best \+ possible (most accurate) 10 digit value for the zero of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Digits := 14 :\nf(.9606166946);\nf(.9606166947);\nf(.9606166948);\nDigits := 10:" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%!)o!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!%tf!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!&F)=!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "The so lution of the equation " }{XPPEDIT 18 0 "2*exp(-x) = arctan(x);" "6#/* &\"\"#\"\"\"-%$expG6#,$%\"xG!\"\"F&-%'arctanG6#F+" }{TEXT -1 39 " corr ect to 10 digits is 0.9606166947. " }}{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 63 "A proce dure for graphing successive stages of Newton's method: " }{TEXT 0 11 "newton_step" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 11 "newton_step" }{TEXT -1 69 " enabl es the progress of Newton's method to be observed graphically." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "n ewton_step: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 297 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 298 2 " " }{TEXT -1 67 " newton_step( eqn, approxroot ) or ne wtonstep( eqn, approxroot ) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 4 " \+ " }}{PARA 0 "" 0 "" {TEXT 23 11 " eqn - " }{TEXT -1 66 " an \+ equation or expression involving a single variable, say x," }}{PARA 0 "" 0 "" {TEXT -1 26 " " }{TEXT 267 2 "OR" } {TEXT -1 34 " a function of the form x -> f(x)," }}{PARA 0 "" 0 "" {TEXT -1 96 " where f(x) evaluates to a real or complex floating point number." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 15 "approxroot - " } {TEXT 299 109 "an initial approximation for the root in the form of a \+ real constant a, when the1st argument is a procedure, " }}{PARA 0 "" 0 "" {TEXT 301 115 " and in the form of a n equation x=a when the1st argument is an expression or equation." }} {PARA 0 "" 0 "" {TEXT -1 10 " " }}{PARA 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 11 "newton_step" }{TEXT -1 90 " \+ performs a single Newton iteration and returns the new approximation b = a - f(a)/f '(a)." }}{PARA 0 "" 0 "" {TEXT -1 43 "A picture is drawn to illustrate the step. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 300 8 "Options:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "draw=true/false" }}{PARA 0 "" 0 "" {TEXT -1 79 "This option determines whether to draw the picture. The default is \" draw=true\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "color=c or colour=c" }}{PARA 0 "" 0 "" {TEXT -1 229 "If c is a \+ list of up to 4 colours, these colours will be applied in respective o rder to the curve, the tangent line, the ordinate of the initial appro ximation, and the 3 points shown. A single colour is applied to the cu rve only." }}{PARA 0 "" 0 "" {TEXT -1 46 "The default is \"colour=[red ,green,blue,navy]\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 11 "thickness=t" }}{PARA 0 "" 0 "" {TEXT -1 153 "If t is is a list of 1 or 2 positive integers, then they will be applied in resp ective order to specify the thickness of the curve and the tangent lin e. " }}{PARA 0 "" 0 "" {TEXT -1 103 "A single thickness is applied to \+ both the curve and the tangent line. The default is \"thickness=[1,2] \"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 16 "How to activate:" }{TEXT 256 1 "\n" } {TEXT -1 154 "To make the procedure 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 27 " newton_step: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "newton_step" {MPLTEXT 1 0 5328 "# to allow for differe nt spellings\nnewtonstep := proc() newton_step(args[1..nargs]) end:\n \nnewton_step := proc(ff,approx)\n local x,x1,y1,x2,w2,h2,xL,xR,yT,y B,xrange,yrange,df,m,graphs,\n pts,ord,f,fn,approxroot,lmr,sf,pro ctype,vars,Options,i,\n clr,colr,thk,thik,drawpic,lft,rgt,mm,yy,x x2;\n\n if nargs<2 then\n error \"at least 2 arguments are requ ired; the basic syntax is: 'newton_step(f(x),x=a)'.\"\n end if;\n\n \+ if type(ff,procedure) then\n if nops([op(1,eval(ff))])<>1 then \n error \"the 1st argument, %1, is invalid .. it should be a \+ procedure with a single argument\",ff;\n end if;\n proctype \+ := true;\n if type(approx,complexcons) then\n approxroot \+ := approx;\n else\n error \"the 2nd argument, %1, is inva lid .. when the 1st argument is a procedure, the 2nd argument should b e a complex constant\",approx;\n end if;\n elif type(ff,algebra ic) or type(ff,equation) then\n if type(ff,equation) then\n \+ lmr := lhs(ff)-rhs(ff);\n sf := traperror(simplify(lmr));\n if sf<>lasterror then\n f := sf;\n else\n \+ f := lmr;\n end if;\n else\n f := ff; \n end if;\n vars := indets(f,name) minus indets(f,complexco ns);\n if nops(vars)<>1 then \n if not has(indets(f),\{In t,Sum\}) then\n error \"the 1st argument, %1, is invalid .. it should be an expression or an equation which depends only on a sin gle variable\",ff;\n end if;\n end if;\n if type(app rox,name=complexcons) then\n proctype := false;\n x := op(1,approx);\n if not member(x,vars) then\n error \"the 1st argument, %1, is invalid .. it should be an expression or a n equation which depends only on the variable %2\",ff,x;\n end if;\n approxroot := op(2,approx);\n else\n error \"the 2nd argument, %1, is invalid .. it should have the form 'x=a', \+ to provide a starting approximation for a root\",approx;\n end if ;\n else\n error \"the 1st argument, %1, is invalid .. it shoul d be an algebraic expression in a single variable, an equation in a si ngle variable, or a procedure with a single argument\",ff;\n end if; \n\n # Get the options.\n # Set the default value to start with.\n drawpic := false;\n clr := [COLOR(RGB,1,0,0),COLOR(RGB,0,1,0),COL OR(RGB,0,0,1), COLOR(RGB,.137,.137,.557)];\n thk := [1,2];\n if na rgs>2 then\n Options:=[args[3..nargs]];\n if not type(Option s,list(equation)) then\n error \"each optional argument must b e an equation\"\n end if;\n if hasoption(Options,'draw','dra wpic','Options') then\n if drawpic<>true then drawpic := false end if;\n end if;\n if hasoption(Options,'color','colr','Op tions') or\n hasoption(Options,'colour','colr','Options') then \n if type(colr,list) then\n for i from 1 to min(no ps(colr),4) do\n clr[i] := `plot/color`(colr[i]);\n \+ end do;\n else\n clr[1] := `plot/color`(colr );\n end if;\n end if;\n if hasoption(Options,'thick ness','thik','Options') then\n if type(thik,list) then\n \+ for i from 1 to min(nops(thik),2) do\n thk[i] := t hik[i];\n end do;\n else\n thk := [thik, thik];\n end if;\n end if;\n if nops(Options)>0 then \n error \"%1 is not a valid option for %2 .. the recognised o ptions are \\\"draw\\\", \\\"colour\\\", or (\\\"color\\\") and \\\"th ickness\\\"\",op(1,Options),procname;\n end if;\n end if;\n\n \+ if proctype then\n fn := ff;\n else\n # Evaluate any real constants in f\n fn := unapply(evalf(f),x);\n end if;\n\n x1 := evalf(approxroot);\n df := D(fn);\n\n m := traperror(evalf(df( x1)));\n if m=lasterror or not type(m,numeric) then\n error \"f ailed to evaluate derivative at %1\",x1;\n end if;\n if m = 0 then \n error \"zero derivative obtained\"\n end if;\n y1 := trape rror(evalf(fn(x1)));\n if y1=lasterror or not type(y1,numeric) then \n error \"failed to evaluate function at %1\",x1;\n end if;\n \n # This is where the Newton formula b=a-f(a)/f'(a) is applied.\n \+ x2 := x1 - y1/m;\n\n if drawpic then\n # recalulate for the pi cture\n Digits := max(Digits,15);\n mm := traperror(evalf(df (x1)));\n if mm<>lasterror and type(mm,numeric) and mm<>0 then\n \+ m := mm \n end if;\n yy := traperror(evalf(fn(x 1)));\n if yy<>lasterror and type(yy,numeric) then\n y1 : = yy;\n end if;\n xx2 := x1 - y1/m;\n xL := min(x1,xx2) ;\n xR := max(x1,xx2);\n w2 := (xR-xL)/2;\n lft := xL-w 2;\n rgt := xR+w2;\n if lft<>rgt then\n xrange := lf t..rgt;\n yT := max(0,y1);\n yB := min(0,y1);\n \+ h2 := (yT-yB)/2;\n yrange := yB-h2..yT+h2;\n graphs \+ := plot(['fn'(x),m*(x-x1)+y1],x=xrange,yrange,\n color=[op( 1,clr),op(2,clr)],thickness=thk);\n ord := CURVES([[x1,0],[x1, y1]],LINESTYLE(2),op(3,clr));\n pts := POINTS([x1,0],[x1,y1],[ xx2,0],SYMBOL(CIRCLE),op(4,clr)),\n POINTS([x1,0],[x1,y1],[xx 2,0],SYMBOL(CROSS),op(4,clr)),\n POINTS([x1,0],[x1,y1],[xx2,0 ],SYMBOL(DIAMOND),op(4,clr));\n print(PLOT(pts,ord,op(graphs)) );\n else\n WARNING(\"the range for the plot is empty\"); \n end if;\n end if;\n x2;\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 -1 39 "Examples are gi ven 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 "newton_step" }{TEXT -1 10 ": examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 267 4 "Note" } {TEXT -1 53 ": The examples in this section require the procedure " } {TEXT 0 11 "newton_step" }{TEXT -1 3 " - " }{HYPERLNK 17 "newton_step " 1 "" "newton_step" }{TEXT -1 3 " . " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 66 "We can graph a s ingle step in the Newton iteration for computing " }{XPPEDIT 18 0 "sq rt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT -1 14 " as a root of " }{XPPEDIT 18 0 "x^2-2;" "6#,&*$%\"xG\"\"#\"\"\"F&!\"\"" }{TEXT -1 6 " . . ." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "f := x->x^2-2:\nnewton_step(f(x),x=2,draw=true);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6*-%'POINTSG6'7$$\"\"#\"\"!F)7$F 'F'7$$\"+++++:!\"*F)-%'SYMBOLG6#%'CIRCLEG-%&COLORG6&%$RGBG$\"$P\"!\"$F 7$\"$d&F9-F$6'F&F*F+-F06#%&CROSSGF3-F$6'F&F*F+-F06#%(DIAMONDGF3-%'CURV ESG6%7$F&F*-%*LINESTYLEG6#F(-F46&F6F)F)\"\"\"-FG6%7S7$$\"3+++++++]7!#< $!3+++++++vV!#=7$$\"3imm;arzr7FV$!3`#QFY7$$\"3OL$e9ui2H\"FV$!3 Ww'HEX:$RLFY7$$\"3smm\"z_\"478FV$!3_pIpB#eTy#FY7$$\"3pmmT&phNL\"FV$!3* \\Du^/Kh@#FY7$$\"3UL$e*=)H\\N\"FV$!33z`(p&=lT;FY7$$\"3sm;z/3uu8FV$!3%e g2k>x35\"FY7$$\"3++]7LRD&R\"FV$!3r.=#*4ikE`!#>7$$\"3fm;zR'okT\"FV$\"3- !GaK^2MQ'!#?7$$\"3.+]i5`hP9FV$\"3e7:MM\"yPn'Fgp7$$\"3YLL$3En$f9FV$\"3[ n=#*>!GvH\"FY7$$\"3cmmT!RE&y9FV$\"3ar`erGSg=FY7$$\"3)*****\\K]4+:FV$\" 3%>w6`1^G]#FY7$$\"3))****\\PAv@:FV$\"3oiDOB()HdJFY7$$\"30++]nHiU:FV$\" 3Gh]e)>coz$FY7$$\"3cm;z*ev:c\"FV$\"3g9>LEK=&Q%FY7$$\"3ELL$347Te\"FV$\" 3]YNCj66%4&FY7$$\"3>LLLjM?.;FV$\"3cT*R%[Mh-dFY7$$\"3#***\\7o7TD;FV$\"3 ,/sZ0zh>kFY7$$\"3CLLLQ*o]k\"FV$\"31.;p=\"=D1(FY7$$\"3-+]7=ljm;FV$\"3;O #ya$GxwxFY7$$\"3%***\\PaR<(o\"FV$\"3%fu8K_fbY)FY7$$\"3HLLe9Eg3*FY7$$\"3WL$eR\"3GGY2z\"FV$ \"3z8[rB>x17FV7$$\"3hmm\"zXu9\"=FV$\"3M\"[%o6(R9G\"FV7$$\"3'******\\y) )G$=FV$\"31w(y\")H\"[f8FV7$$\"3!****\\i_QQ&=FV$\"3kr(39GFV$\"31[))4:pEu;FV7$$\"3()****\\P+VQ>FV$\"35^AG55^dFV$\"3+D77EyuL=FV7$$\"3)*****\\#\\'Qz>FV$\"3m_9p')3(z \">FV7$$\"37L$e9S8&**>FV$\"37t9hUQ0)*>FV7$$\"3;+]i?=b??FV$\"3+.5$)f'HE 3#FV7$$\"3HLL$3s?6/#FV$\"3)f=:qzth;#FV7$$\"3'***\\7`Wli?FV$\"3s#R'*HRV XD#FV7$$\"3emmm'*RR$3#FV$\"3o1nMX0`SBFV7$$\"3`mmTvJg/@FV$\"3*f\"R(f_a$ HCFV7$$\"35L$e9tOc7#FV$\"36m[79:L=DFV7$$\"3v*****\\Qk\\9#FV$\"3uE%=H@s 3g#FV7$$\"3?LL3dg6n@FV$\"3o7%o[+#R'p#FV7$$\"3_mmmw(Gp=#FV$\"3yiF@uul#y #FV7$$\"3-+]7oK03AFV$\"3/$\\x[B*\\vGFV7$$\"3-+](=5s#GAFV$\"35IW**fl>lH FV7$$\"3+++++++]AFV$\"3++++++]iIFV-F46&F6FOF)F)-%*THICKNESSG6#FO-FG6%7 S7$FT$!\"\"F)7$Fen$!3fNLLLQ6G\"*FY7$Fjn$!3]lmmT.\\p$)FY7$F_o$!36JLL$)) Qj^(FY7$Fdo$!3IKLL$=Kvl'FY7$Fio$!3 $=FV7$Fay$\"3$*******pfa<>FV7$Ffy$\"3YKL$eg`!)*>FV7$F[z$\"3k++]#G2A3#F V7$F`z$\"3;LLL$)G[k@FV7$Fez$\"3#)****\\7yh]AFV7$Fjz$\"3Kmmm')fdLBFV7$F _[l$\"36mmm,FT=CFV7$Fd[l$\"3QKL$e#pa-DFV7$Fi[l$\"3+******Rv&)zDFV7$F^ \\l$\"3%GLL$GUYoEFV7$Fc\\l$\"33mmm1^rZFFV7$Fh\\l$\"34++]sI@KGFV7$F]]l$ \"34++]2%)38HFV7$Fb]l$\"\"$F)-F46&F6F)FOF)-Fi]lFL-%+AXESLABELSG6$Q\"x6 \"Q!Figl-%%VIEWG6$;$\"++++]7F.$\"++++]AF.;$!+++++5F.$\"+++++IF." 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2 " "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++:!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 44 ". . . or a number of successive steps . \+ . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "f := x->x^2-2:\nxin :=2;\nfor i from 1 to 3 do\n x out := newton_step(f(x),x=xin,draw=true);\n print(`approximate root \+ =`,xout);\n xin := xout;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%$xinG\"\"#" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6 *-%'POINTSG6'7$$\"\"#\"\"!F)7$F'F'7$$\"+++++:!\"*F)-%'SYMBOLG6#%'CIRCL EG-%&COLORG6&%$RGBG$\"$P\"!\"$F7$\"$d&F9-F$6'F&F*F+-F06#%&CROSSGF3-F$6 'F&F*F+-F06#%(DIAMONDGF3-%'CURVESG6%7$F&F*-%*LINESTYLEG6#F(-F46&F6F)F) \"\"\"-FG6%7S7$$\"3+++++++]7!#<$!3+++++++vV!#=7$$\"3imm;arzr7FV$!3`#QFY7$$\"3OL$e9ui2H\"FV$!3Ww'HEX:$RLFY7$$\"3smm\"z_\"478FV$!3_pIp B#eTy#FY7$$\"3pmmT&phNL\"FV$!3*\\Du^/Kh@#FY7$$\"3UL$e*=)H\\N\"FV$!33z` (p&=lT;FY7$$\"3sm;z/3uu8FV$!3%eg2k>x35\"FY7$$\"3++]7LRD&R\"FV$!3r.=#*4 ikE`!#>7$$\"3fm;zR'okT\"FV$\"3-!GaK^2MQ'!#?7$$\"3.+]i5`hP9FV$\"3e7:MM \"yPn'Fgp7$$\"3YLL$3En$f9FV$\"3[n=#*>!GvH\"FY7$$\"3cmmT!RE&y9FV$\"3ar` erGSg=FY7$$\"3)*****\\K]4+:FV$\"3%>w6`1^G]#FY7$$\"3))****\\PAv@:FV$\"3 oiDOB()HdJFY7$$\"30++]nHiU:FV$\"3Gh]e)>coz$FY7$$\"3cm;z*ev:c\"FV$\"3g9 >LEK=&Q%FY7$$\"3ELL$347Te\"FV$\"3]YNCj66%4&FY7$$\"3>LLLjM?.;FV$\"3cT*R %[Mh-dFY7$$\"3#***\\7o7TD;FV$\"3,/sZ0zh>kFY7$$\"3CLLLQ*o]k\"FV$\"31.;p =\"=D1(FY7$$\"3-+]7=ljm;FV$\"3;O#ya$GxwxFY7$$\"3%***\\PaR<(o\"FV$\"3%f u8K_fbY)FY7$$\"3HLLe9Eg3*FY7$$\"3WL$eR\"3GGY2z\"FV$\"3z8[rB>x17FV7$$\"3hmm\"zXu9\"=FV$\"3M \"[%o6(R9G\"FV7$$\"3'******\\y))G$=FV$\"31w(y\")H\"[f8FV7$$\"3!****\\i _QQ&=FV$\"3kr(39GFV$\"31[))4:pEu;FV7$$\" 3()****\\P+VQ>FV$\"35^AG55^dFV$\"3+D77EyuL=FV7$$\"3 )*****\\#\\'Qz>FV$\"3m_9p')3(z\">FV7$$\"37L$e9S8&**>FV$\"37t9hUQ0)*>FV 7$$\"3;+]i?=b??FV$\"3+.5$)f'HE3#FV7$$\"3HLL$3s?6/#FV$\"3)f=:qzth;#FV7$ $\"3'***\\7`Wli?FV$\"3s#R'*HRVXD#FV7$$\"3emmm'*RR$3#FV$\"3o1nMX0`SBFV7 $$\"3`mmTvJg/@FV$\"3*f\"R(f_a$HCFV7$$\"35L$e9tOc7#FV$\"36m[79:L=DFV7$$ \"3v*****\\Qk\\9#FV$\"3uE%=H@s3g#FV7$$\"3?LL3dg6n@FV$\"3o7%o[+#R'p#FV7 $$\"3_mmmw(Gp=#FV$\"3yiF@uul#y#FV7$$\"3-+]7oK03AFV$\"3/$\\x[B*\\vGFV7$ $\"3-+](=5s#GAFV$\"35IW**fl>lHFV7$$\"3+++++++]AFV$\"3++++++]iIFV-F46&F 6FOF)F)-%*THICKNESSG6#FO-FG6%7S7$FT$!\"\"F)7$Fen$!3fNLLLQ6G\"*FY7$Fjn$ !3]lmmT.\\p$)FY7$F_o$!36JLL$))Qj^(FY7$Fdo$!3IKLL$=Kvl'FY7$Fio$!3$=FV7$Fay$\"3$*******pfa<>FV7$Ffy$\"3YKL $eg`!)*>FV7$F[z$\"3k++]#G2A3#FV7$F`z$\"3;LLL$)G[k@FV7$Fez$\"3#)****\\7 yh]AFV7$Fjz$\"3Kmmm')fdLBFV7$F_[l$\"36mmm,FT=CFV7$Fd[l$\"3QKL$e#pa-DFV 7$Fi[l$\"3+******Rv&)zDFV7$F^\\l$\"3%GLL$GUYoEFV7$Fc\\l$\"33mmm1^rZFFV 7$Fh\\l$\"34++]sI@KGFV7$F]]l$\"34++]2%)38HFV7$Fb]l$\"\"$F)-F46&F6F)FOF )-Fi]lFL-%+AXESLABELSG6$Q\"x6\"Q!Figl-%%VIEWG6$;$\"++++]7F.$\"++++]AF. ;$!+++++5F.$\"+++++IF." 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}{PARA 11 "" 1 "" {XPPMATH 20 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Y&)F1$\"1=M%zlW]?\"Fjw7$$\"1LLe9tOc()F1$\"1$Hx6&*RgF\"Fjw7$$\"1+++]Qk \\*)F1$\"1&G]T,8AM\"Fjw7$$\"1LL$3dg6<*F1$\"1&p#z@-=>9Fjw7$$\"1mmmmxGp$ *F1$\"1:7Q\")*f!*[\"Fjw7$$\"1++D\"oK0e*F1$\"15OU9/nk:Fjw7$$\"1++v=5s#y *F1$\"1>F65%G\"Q;Fjw7$$\"\"\"F($\"1X!f%G=G= " 0 "" {MPLTEXT 1 0 245 "f := x -> exp(x)+x-2:\n'f(x)'=f(x) ;\nDf := D(f); # Df is the derivative f' of f\nnextapprox := x -> eval f(x-f(x)/Df(x));\nDigits := 13:\nx0 := 0.45;\nx1 := nextapprox(x0);\nx 2 := nextapprox(x1);\nx3 := nextapprox(x2);\nx4 := nextapprox(x3);\nDi gits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,(-%$expG F&\"\"\"F'F+\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DfGf*6#% \"xG6\"6$%)operatorG%&arrowGF(,&-%$expG6#9$\"\"\"F1F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+nextapproxGf*6#%\"xG6\"6$%)operatorG%&arr owGF(-%&evalfG6#,&9$\"\"\"*&-%\"fG6#F0F1-%#DfGF5!\"\"F8F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$\"#X!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\".&HM&*pGW!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#x2G$\".k2,W&GW!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x3G$\".D+ ,W&GW!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x4G$\".D+,W&GW!#8" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "There is \+ no change in the first 10 digits in the last two values given by " } {TEXT 0 10 "nextapprox" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 192 "By performing the calculation with higher than the required preci sion, and rounding the result, we can be reasonably confident that the value 0.4428544010 for the root is correct to 10 digits." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 4 "Note" }{TEXT -1 60 ": The same calculation can be performed using the procedure " } {TEXT 0 11 "newton_step" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 180 "eq := exp(x)=2-x;\nD igits := 13:\nx0 := 0.45;\nx1 := newton_step(eq,x=x0);\nx2 := newton_s tep(eq,x=x1);\nx3 := newton_step(eq,x=x2);\nx4 := newton_step(eq,x=x3) ;\nDigits := 10:\nevalf(x4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG /-%$expG6#%\"xG,&\"\"#\"\"\"F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#x0G$\"#X!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\".&HM&*pGW !#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x2G$\".k2,W&GW!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x3G$\".D+,W&GW!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x4G$\".D+,W&GW!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"+5SaGW!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "(b) Newton's method will still work with a starting app roximation which is further away from the root, such as " }{XPPEDIT 18 0 "x = 1" "6#/%\"xG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "x := 'x':\n f := x -> exp(x)+x-2:\nxin := 1:\nprint(x[0]=xin);\nfor i from 1 to 4 \+ do\n xout := newton_step(f(x),x=xin,draw=true,color=[magenta,blue,bl ack$2]);\n print(x[i]=xout);\n xin := xout;\nend do:" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"xG6#\"\"!\"\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6*-%'POINTSG6'7$$\"\"\"\"\"!F)7$F'$ \"+G=G=kU#oI!#=$!3')QV6;jtSLF R7$$\"3%3>e/U*ppKFR$!3=+niXEqiGFR7$$\"36A6@')e)\\W$FR$!3yZ\"p0-$>UCFR7 $$\"3]a%*e\\Q6UOFR$!3EZb&3q,T'>FR7$$\"33#\\p]][0%QFR$!3QWv!3x\">x9FR7$ $\"34K/FN+/QSFR$!3_<*pQ]e&o)*!#>7$$\"3OlAWk*R6A%FR$!3I.,STBOq_Fho7$$\" 3[$*y`N$H2T%FR$!3AKaZ8A+`X!#?7$$\"3,!RNV$H!og%FR$\"3!o\"Ra>!GKe%Fho7$$ \"3mf[hAxC-[FR$\"3/Fk'zRYim*Fho7$$\"3G([y0r'G.]FR$\"31N:$y<>f\\\"FR7$$ \"3O^Ygg>O!=&FR$\"3O$))R]9Rw'>FR7$$\"3u6_(>v1(z`FR$\"3QZ[,Jt)\\]#FR7$$ \"3K$z6A2q)zbFR$\"3)z_A-h*Q^IFR7$$\"31_5i?XwsdFR$\"3U\\.y'RrXe$FR7$$\" 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$GLN#FU7$Feo$!3'=*H\"*yq>^?FU7$Fjo$!3ck/AVF 3r'FU7$Fhx$\"3wP(*\\D5wvkFU7$F]y$\"31![E( *Q5#ynFU7$Fby$\"3ITP.r;ziqFU7$Fgy$\"3]P(o$36EgtFU7$F\\z$\"3/vl#e&=4^wF U7$Faz$\"3W&>Ccpkb&zFU7$Ffz$\"3'Q\\/yA3)[#)FU7$F[[l$\"3g$ogN4#p[&)FU7$ F`[l$\"3WN/]pF4Y))FU7$Fe[l$\"3!Gp\\Q5t$>\"*FU7$Fj[l$\"33?Po1KeK%*FU7$F _\\l$\"3&*HKYc?s7(*FU7$Fd\\l$\"3sV/&*\\39,5Fh\\l7$Fj\\l$\"3J8Eq`*G(H5F h\\l7$F_]l$\"3)4eDax]/1\"Fh\\l-F76&F9Fh]lFh]lFe]l-Fj]lFJ-%+AXESLABELSG 6$Q\"x6\"Q!Fjgl-%%VIEWG6$;$\"+Brw9WF)$\"+0w'*pWF);$!++g$[`$!#7$\"++3Xg 5!#6" 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" }}}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"xG6#\"\"$$\"+VscGW!#5" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6*-%'POINTSG6'7$$\"+VscGW!#5\"\"!7$ F'$\"%Tf!\"*7$$\"+5SaGWF)F*-%'SYMBOLG6#%'CIRCLEG-%'COLOURG6&%$RGBGF*F* F*-F$6'F&F+F/-F36#%&CROSSGF6-F$6'F&F+F/-F36#%(DIAMONDGF6-%'CURVESG6%7$ F&F+-%*LINESTYLEG6#\"\"#F6-FE6%7S7$$\"3z*****HRK&GW!#=$!3khLb*4T1(H!#B 7$$\"3>i$[eSL&GWFR$!3*eS!e,7k6FFU7$$\"372B7(GM&GWFR$!3%Q#)[Aw'G'[#FU7$ $\"3`Fq?y_`GWFR$!3$*=*\\;6_GB#FU7$$\"3h+8'eFO&GWFR$!3iFqv:vtx>FU7$$\"3 ?(*Qxos`GWFR$!3g.l=?a$Qs\"FU7$$\"34w4L*=Q&GWFR$!3)['zFicV)[\"FU7$$\"3U 5X^U\"R&GWFR$!3aulCLGpW7FU7$$\"3.&G)HG,aGWFR$!3)Hb*=mj8E**!#C7$$\"3^Q1 #46T&GWFR$!3#zZgbXGMT(Fhp7$$\"3\"*3&o;7U&GWFR$!3`W)3[O*zG[Fhp7$$\"3E$y N>,V&GWFR$!3!*HA!R_cAb#Fhp7$$\"3&\\(e;9SaGWFR$\"3q/zDKl;f5!#E7$$\"3z>7 ^?]aGWFR$\"3Kk&p)=L'Re#Fhp7$$\"3[I9J!*faGWFR$\"3+#oZb)Q)Q1&Fhp7$$\"3lI H)4(oaGWFR$\"3;(*Gcn%*)eJ(Fhp7$$\"3\\Cl==zaGWFR$\"3k[\"H$y?u$***Fhp7$$ \"3oI`I0)[&GWFR$\"3I;K@jMAE7FU7$$\"3(fRNs$)\\&GWFR$\"3E;'R64.,\\\"FU7$ $\"3Yd$o1v]&GWFR$\"35%3o$)G\"oBFU7 $$\"3yP@;2FbGWFR$\"3s;'=\"pa)RA#FU7$$\"3Y\"p()Gq`&GWFR$\"3kN,wLrgyCFU7 $$\"36#eusha&GWFR$\"3%[lt$>$HCr#FU7$$\"3]sVc.cbGWFR$\"3?K%Q'\\\"QY'HFU 7$$\"3NtA/GmbGWFR$\"3WDpEeAhEKFU7$$\"3#yH`)>vbGWFR$\"3]jC`.:maMFU7$$\" 3`NO.$[e&GWFR$\"3Kz*f#\\4'4q$FU7$$\"3_sJ4y%f&GWFR$\"3)e+JFR7a&RFU7$$\" 3;*zk:Xg&GWFR$\"3R([AJ]VV?%FU7$$\"3&Gl[MRh&GWFR$\"3O$Qej;(>XWFU7$$\"3` &3]#RCcGWFR$\"3s0:R6Ti7ZFU7$$\"3%o=\\*yLcGWFR$\"3aN8%*H\">H&\\FU7$$\"3 a_&yAQk&GWFR$\"3hk*HvO&[4_FU7$$\"3^\"3W9Hl&GWFR$\"3S%\\*\\DD(>W&FU7$$ \"3\"o9-aGm&GWFR$\"3#Req>TUhp&FU7$$\"39D#R1An&GWFR$\"3%)p#>0['HNfFU7$$ \"3)*[9B)>o&GWFR$\"3+4i029G&='FU7$$\"3sMl+a\"p&GWFR$\"3HNcSztoHkFU7$$ \"3!HZ9Y:q&GWFR$\"3in@f(zdbo'FU7$$\"34V)=$=6dGWFR$\"3!*pm(fV\"*>$pFU7$ $\"31_d%Q5s&GWFR$\"3u/,URa+%=(FU7$$\"3b+?@\"3t&GWFR$\"3\"yxVVlKRV(FU7$ $\"3dx+JzRdGWFR$\"3wD]S@)*ejwFU7$$\"3HC=j3]dGWFR$\"35FZ(y'G!o#zFU7$$\" 3Ul%p#HfdGWFR$\"3QoB)zwBA;)FU7$$\"3(*47'3\"pdGWFR$\"3ww\"3VcJKT)FU7$$ \"3gd(p.&ydGWFR$\"35)4Wu1yMl)FU7$$\"3y******f)y&GWFR$\"3$)ziWgdl6*)FU- F76&F9$\"*++++\"!\")$F*F*Fd]l-%*THICKNESSG6#\"\"\"-FE6%7S7$FP$!3#=\"HJ \"4y3(HFU7$FW$!3/?6b^w(=r#FU7$Ffn$!3#)Gup`F_'[#FU7$F[o$!3?&*e;-w3LAFU7 $F`o$!31t[xFU7$Feo$!3y_O!3&*pSs\"FU7$Fjo$!3Y_gXp(p')[\"FU7$F_p$! 3$fbjn^E\\C\"FU7$Fdp$!3y7uIg*o%G**Fhp7$Fjp$!3R$G!Qxpv:uFhp7$F_q$!3_y\" y?&Q7J[Fhp7$Fdq$!3+-UY*exXb#Fhp7$Fiq$\"3maJ1&o0VF)!#F7$F_r$\"3M>mQ;&\\ ;e#Fhp7$Fdr$\"3!y*R)[Ot:1&Fhp7$Fir$\"3HKGR-=e8tFhp7$F^s$\"3kp)*y_wV\"* **Fhp7$Fcs$\"3ea'R8G$*fA\"FU7$Fhs$\"372zD(>t)*[\"FU7$F]t$\"3JrZSN;XBFU7$Fgt$\"3.dBL*GcPA#FU7$F\\u$\"3e*\\L6\"QX\"pxD4_FU7$Fhx$\"3aNx3h\\uTaFU7$F]y$\"39I`)*p[\"fp&FU7$Fb y$\"3DxbIY*o]$fFU7$Fgy$\"3%)**HHlQ0&='FU7$F\\z$\"30WhK;)f%HkFU7$Faz$\" 3;Qr)p>I`o'FU7$Ffz$\"3!fUBXyj<$pFU7$F[[l$\"31!R44sxP=(FU7$F`[l$\"3tPZc a[qLuFU7$Fe[l$\"3M6/9M>OjwFU7$Fj[l$\"3\\]Y\"Q'[dEzFU7$F_\\l$\"3;^'Qik& *>;)FU7$Fd\\l$\"3ApAB-L+8%)FU7$Fi\\l$\"3Y%RUql\\Kl)FU7$F^]l$\"38saHvrU 6*)FU-F76&F9Fg]lFg]lFd]l-Fi]lFJ-%+AXESLABELSG6$Q\"x6\"Q!Fjgl-%%VIEWG6$ ;$\"+$RK&GWF)$\"+g)y&GWF);$!+++]qH!#:$\"+++]6*)Fghl" 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" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"x G6#\"\"%$\"+5SaGW!#5" }}}{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 35 "When computing the single z ero of " }{XPPEDIT 18 0 "x^3-1/100;" "6#,&*$%\"xG\"\"$\"\"\"*&F'F'\"$ +\"!\"\"F*" }{TEXT -1 53 " by Newton's method, the convergence is rat her slow." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 245 "f := x->x^3-1/100;\nxin := 1:\nprint(x[0]=xin);\nfor i from 1 to 9 do\n if i<=6 then\n xout := newton_step(f(x),x=x in,draw=true);\n else\n xout := newton_step(f(x),x=xin,draw=fal se);\n end if;\n print(x[i]=xout);\n xin := xout;\nend do:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 4 "Note" } {TEXT -1 91 ": The following loop provides an alternative way of contr olling when the picture is drawn. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "f := x->x^3-1/100;\nxin := \+ 1:\nprint(x[0]=xin);\nfor i from 1 to 9 do\n xout := newton_step(f(x ),x=xin,draw=is(i<=6));\n print(x[i]=xout);\n xin := xout;\nend do :" }}}{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 15 "The polynomial " }{XPPEDIT 18 0 "x^4-4* x^2+4 = (x^2-2)^2" "6#/,(*$%\"xG\"\"%\"\"\"*&F'F(*$F&\"\"#F(!\"\"F'F(* $,&*$F&F+F(F+F,F+" }{TEXT -1 19 " has the two zeros " }{XPPEDIT 18 0 " sqrt(2)" "6#-%%sqrtG6#\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "-sqrt (2)" "6#,$-%%sqrtG6#\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 68 "The convergence of Newton's method to either of these roo ts is slow." }}{PARA 0 "" 0 "" {TEXT -1 39 "The following loop draws s ome pictures." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "f := x->x^4-4*x^2+4:\nxin := 1.5;\nprint(x[0]=x in);\nfor i from 1 to 16 do\n xout := newton_step(f(x),x=xin,draw=is (i<=5 or irem(i,3)=0));\n print(x[i]=xout);\n xin := xout;\nend do :" }}}{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 59 "A procedure implementing Newton' s method for root-finding: " }{TEXT 0 6 "newton" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "newton: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 18 "Calling Sequenc e:\n" }}{PARA 0 "" 0 "" {TEXT 260 2 " " }{TEXT -1 29 " newton( eqn, approxroot ) " }{TEXT 261 1 "\n" }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT 23 11 " eqn - " }{TEXT -1 66 " an equation or \+ expression involving a single variable, say x," }}{PARA 0 "" 0 "" {TEXT -1 23 " " }{TEXT 267 2 "OR" }{TEXT -1 99 " a function of the form x -> f(x), where f(x) evaluates to a real or c omplex floating point number." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 15 "approxroot - " }{TEXT 262 68 "an initial approximation for the root (which may be real or co mplex)" }}{PARA 0 "" 0 "" {TEXT 265 98 " \+ in the form of a constant a when the1st argument is a procedure, and" }}{PARA 0 "" 0 "" {TEXT 264 111 " in the \+ form of an equation x=a when the1st argument is an expression or equat ion." }}{PARA 0 "" 0 "" {TEXT -1 10 " " }}{PARA 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 6 "newton" }{TEXT -1 28 " attempts to find a root of " }{XPPEDIT 18 0 "f(x) = 0;" "6#/-%\"fG6#% \"xG\"\"!" }{TEXT -1 51 " by Newton's method given an initial approxim ation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 8 "Options:" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 135 "maxiterati ons=n\nThis option can be used to override the default value of Digits *5 for the maximum number of iterations to be performed." }}{PARA 0 " " 0 "" {TEXT -1 50 "The abreviated form \"maxiter=n\" may also be used . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "pre cision=fixed/variable" }}{PARA 0 "" 0 "" {TEXT -1 309 "If the computed value of the function exhibits a loss of significant digits as the su ccessive approximations converge the root then the working precision i s increased to compensate for this. This feature can be turned off via the option \"precision=fixed\". The default for this option is \"prec ision=variable\". " }}{PARA 0 "" 0 "" {TEXT -1 52 "The abreviated form \"prcsn=fixed\" may also be used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 101 "info=true/false/0/1/2/3\n\"info=0\" is \+ the same as \"info=false\" and \"info=1\" is the same as \"info=true\" ." }}{PARA 0 "" 0 "" {TEXT -1 123 "This option allows the progress of \+ the computation to be monitored by printing the result of each Newton \+ step as it occurs." }}{PARA 0 "" 0 "" {TEXT -1 95 "With the option \"i nfo=2\" the expressions for function and derivative being used are als o given." }}{PARA 0 "" 0 "" {TEXT -1 302 "The option \"info=3\" provid es additional information regarding the value of the function, its der ivative and the correction term (the negative of the function value di vided by the derivative) at each step, together with information regar ding any change in the working precision used in the computation. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 267 16 "How to activate:" }{TEXT 256 1 "\n" } {TEXT -1 154 "To make the procedure 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 22 " newton: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7887 "newton := proc(ff,approx)\n loc al Options,x,df,dfx,fx,h,eps,saveDigits,i,maxit,prntflg, \n f,f n,approxroot,xx,lmr,sf,proctype,complexround,vars,\n workingDigit s,extraDigits,adjustDigits,eps2,prsn,dfn,\n small,triedzero,f0;\n \n if nargs<2 then\n error \"at least 2 arguments are required; the basic syntax is: 'newton(f(x),x=a)'.\"\n end if; \n\n if ty pe(ff,procedure) then\n if nops([op(1,eval(ff))])<>1 then\n \+ error \"the 1st argument, %1, is invalid .. it should be a procedur e with a single argument\",ff;\n end if;\n proctype := true; \n if type(approx,complexcons) then\n approxroot := appro x;\n else\n error \"the 2nd argument, %1, is invalid .. w hen the 1st argument is a procedure, the 2nd argument should be a comp lex constant\",approx;\n end if;\n elif type(ff,algebraic) or t ype(ff,equation) then\n if type(ff,equation) then\n lmr : = lhs(ff)-rhs(ff);\n sf := traperror(simplify(lmr));\n \+ if sf<>lasterror then\n f := sf;\n else\n \+ f := lmr;\n end if;\n else\n f := ff;\n e nd if;\n vars := indets(f,name) minus indets(f,complexcons);\n \+ if nops(vars)<>1 then \n if not has(indets(f),\{Int,Sum\}) \+ then\n error \"the 1st argument, %1, is invalid .. it shoul d be an expression or an equation which depends only on a single varia ble\",ff;\n end if;\n end if;\n if type(approx,name= complexcons) then\n proctype := false;\n x := op(1,app rox);\n if not member(x,vars) then\n error \"the 1s t argument, %1, is invalid .. it should be an expression or an equatio n which depends only on the variable %2\",ff,x;\n end if;\n \+ approxroot := op(2,approx);\n else\n error \"the 2n d argument, %1, is invalid .. it should have the form 'x=a', to provid e a starting approximation for a root\",approx;\n end if;\n els e\n error \"the 1st argument, %1, is invalid .. it should be an a lgebraic expression in a single variable, an equation in a single vari able, or a procedure with a single argument\",ff;\n end if;\n \n \+ # Get the options \"maxiterations\" and \"info\".\n # Set the defau lt values to start with.\n maxit := Digits*5;\n prntflg := 0;\n \+ prsn := 1;\n if nargs>2 then\n Options:=[args[3..nargs]];\n \+ if not type(Options,list(equation)) then\n error \"each opti onal argument must be an equation\"\n end if;\n if hasoption (Options,'maxiterations','maxit','Options') then\n if not type (maxit,posint) then\n error \"\\\"maxiterations\\\" must be a positive integer\"\n end if;\n elif hasoption(Options, 'maxiter','maxit','Options') then\n if not type(maxit,posint) \+ then\n error \"\\\"maxiter\\\" must be a positive integer\" \n end if;\n end if;\n if hasoption(Options,'precisi on','prsn','Options') then\n if not member(prsn,\{'fixed','var iable'\}) then\n error \"\\\"precision\\\" must be 'fixed' \+ or 'variable'\"\n end if;\n if prsn='fixed' then prsn \+ := 0 else prsn := 1 end if;\n elif hasoption(Options,'prcsn','prs n','Options') then\n if not member(prsn,\{'fixed','variable'\} ) then\n error \"\\\"prcsn\\\" must be 'fixed' or 'variable '\"\n end if;\n if prsn='fixed' then prsn := 0 else pr sn := 1 end if;\n end if;\n if hasoption(Options,'info','prn tflg','Options') then\n if not member(prntflg,\{true,false,0,1 ,2,3\}) then\n error \"\\\"info\\\" must be false <-> 0, tr ue <-> 1,2 or 3\"\n end if;\n if prntflg=false then pr ntflg := 0\n elif prntflg=true then prntflg := 1 end if; \n \+ end if;\n if nops(Options)>0 then\n error \"%1 is not \+ a valid option for %2 .. the recognised options are \\\"maxiterations \\\",(or \\\"maxiter\\\"),\\\"precision\\\",(or \\\"prcsn\\\") and \\ \"info\\\"\",op(1,Options),procname;\n end if;\n end if;\n\n \+ # local procedure\n complexround := proc(zz)\n local re,im,eps; \n re := Re(zz);\n im := Im(zz);\n if im=0 then return \+ Re(zz) end if;\n if re=0 then return Im(zz) end if;\n if not type(re,float) or not type(im,float) then\n return zz\n \+ end if;\n eps := Float(1,-Digits);\n if abs(re)<=eps*abs(im) then return im*I\n elif abs(im)<=eps*abs(re) then return re\n \+ else return zz end if;\n end proc: # of complexround\n\n # Incr ease precision for the computation\n saveDigits := Digits;\n extra Digits := min(iquo(iquo(Digits,5)+1,2)+3,8);\n workingDigits := Digi ts + extraDigits;\n Digits := workingDigits;\n\n if proctype then \n fn := ff;\n dfn := D(fn);\n else\n # Evaluate any \+ real constants in f\n fn := unapply(evalf(f),x);\n df := dif f(f,x);\n dfn := unapply(evalf(df),x);\n if prntflg>1 then\n print(`Attempting to calculate a zero of`);\n print(f ); \n print(`by Newton's method, using the derivative`);\n \+ print(df);\n print(``);\n end if;\n end if;\n if prntflg>2 then\n print(`** working precision is `||Digits||` dig its **`);\n end if;\n\n xx := evalf(approxroot);\n\n eps := Floa t(1,-saveDigits-min(iquo(Digits,10),2));\n eps2 := Float(1,-iquo(sav eDigits,2));\n small := abs(xx)*Float(1,-trunc(saveDigits*.75)-1);\n triedzero := false;\n h := xx;\n\n for i from 1 to maxit do\n \+ fx := traperror(evalf(fn(xx)));\n if fx=lasterror or not type (fx,complex(numeric)) then\n error \"failed to evaluate functi on at %1\",evalf[saveDigits](xx);\n end if;\n if prntflg>2 t hen\n print(`value`=evalf[workingDigits](fx))\n end if;\n if prsn=1 and fx<>0 then\n adjustDigits := extraDigits- \n max(length(SFloatMantissa(Re(fx))),\n \+ length(SFloatMantissa(Im(fx))));\n if adjustDigits>0 and (abs (h)<=eps2*abs(xx) or abs(fx)2 then\n print(`** increasing working precision to `||D igits||` digits **`); \n end if;\n if no t proctype then\n fn := unapply(evalf(f),x);\n \+ dfn := unapply(evalf(df),x);\n end if;\n fx := traperror(evalf(fn(xx)));\n if fx=lasterror or not type (fx,complex(numeric)) then\n error \"failed to evaluate \+ function at %1\",evalf[saveDigits](xx);\n end if;\n \+ if prntflg>2 then\n print(`value`=evalf[workingDigit s](fx))\n end if;\n end if;\n end if;\n d fx := traperror(evalf(dfn(xx)));\n if dfx=lasterror or not type(d fx,complex(numeric)) then\n error \"failed to evaluate derivat ive at %1\",evalf[saveDigits](xx);\n end if;\n if dfx=0 then \n error \"zero derivative obtained\"\n end if;\n if prntflg>2 then\n print(`derivative`=evalf[workingDigits](dfx) )\n end if;\n h := fx/dfx;\n if prntflg>2 then\n \+ print(`correction -> `,-`value`/`derivative`=evalf[workingDigits] (-h));\n print(``);\n end if;\n xx := xx - h;\n \+ if prntflg>0 then\n print(`approximation `||i||` -> `,eval f[workingDigits](xx))\n end if;\n if prntflg>2 then print(`` ) end if;\n if abs(h)<=eps*abs(xx) then\n Digits := saveD igits;\n return evalf(complexround(xx));\n end if;\n \+ if i>6 and not triedzero \n and abs(xx)0 then\n print(`The values appear to be converging to 0`);\n print(``);\n end \+ if;\n return 0.0\n end if;\n triedzero := t rue;\n end if;\n end do:\n Digits := saveDigits;\n print(`l ast iteration gives `,evalf(xx));\n error \"reached max, %1, iterati ons without convergence\",maxit;\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 -1 48 "Examples are given i n the sections which follow." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT 0 6 "newton" }{TEXT -1 10 ": examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Exampl e 1 " }{TEXT 316 37 ".. a cubic equation with 3 real roots" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 67 "We can use Newton's method to fin d the roots of the cubic equation " }{XPPEDIT 18 0 "x^3-3*x+1=0" "6#/, (*$%\"xG\"\"$\"\"\"*&F'F(F&F(!\"\"F(F(\"\"!" }{TEXT -1 63 " given as t he first example to illustrate the bisection method." }}{PARA 0 "" 0 " " {TEXT -1 80 "The convergence is much more rapid than can be obtained by the bisection method." }}{PARA 0 "" 0 "" {TEXT -1 131 "In fact, in typical good situation, we can expect the number of correct decimal d igits in the answer to double with each iteration." }}{PARA 0 "" 0 "" {TEXT -1 85 "We should take a starting approximation reasonably close \+ to the desired root however." }}{PARA 0 "" 0 "" {TEXT -1 28 "( Incorpo rating the option \"" }{TEXT 304 9 "info=true" }{TEXT -1 9 "\" causes \+ " }{TEXT 0 6 "newton" }{TEXT -1 44 " to show all the successive approx imations.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 24 "plot(x^3-3*x+1,x=-2..2);" }}{PARA 13 "" 1 "" {GLPLOT2D 260 185 185 {PLOTDATA 2 "6%-%'CURVESG6$7ao7$$!\"#\"\"!$!\"\" F*7$$!1nmm\"p0k&>!#:$!1o-8z:r*='!#;7$$!1LLL$Q6G\">F0$!15sYSy]-EF37$$!1 ++v3-)[(=F0$\"1FIJ(Q42T$!#<7$$!1nm;M!\\p$=F0$\"1#**)[$)zzAJF37$$!1++Dh 9H%z\"F0$\"1`N(Q%f&=1'F37$$!1LLL))Qj^VDP f\"F07$$!1nm;C2G!e\"F0$\"1k\"3&*RFWz\"F07$$!1++]_(e1a\"F0$\"1%pb+OB]'> F07$$!1LL$3yO5]\"F0$\"1;`kLs5@@F07$$!1++]nU)*=9F0$\"1W;\\xZ!)*R#F07$$! 1LL$3WDTL\"F0$\"1M@co)yxi#F07$$!1++]d(Q&\\7F0$\"1jW-\\Dl(z#F07$$!1nmmc 4`i6F0$\"14W$ffdk\"HF07$$!1++](p7U7\"F0$\"1a+>lrz^HF07$$!1LLLQW*e3\"F0 $\"1x'=(>FBxHF07$$!1nm;arvU5F0$\"1/_x4tV%*HF07$$!1,+++()>'***F3$\"1y%e lc*****HF07$$!1,+++Y0j&*F3$\"1^()p$ybV*HF07$$!1++++0\"*H\"*F3$\"1]#3T. Zz(HF07$$!1++++f\\7()F3$\"12ce[US_HF07$$!1++++83&H)F3$\"1?rUNLv`%HrBF07$$!1pmmmCC(>% F3$\"1%puGdI_=#F07$$!1*****\\FRXL$F3$\"1+F'G`%Gj>F07$$!1+++D=/8DF3$\"1 \")RRN$\"1opIC X(*f7F07$$!1qLLL$eV(>!#=$\"1sPInI#f+\"F07$$\"1Mmm;f`@')F>$\"1&3^')oZ*> uF37$$\"1)****\\nZ)H;F3$\"17$$\"1'*****\\5a`TF3$!1ctVm#eSu\"F37 $$\"1(****\\7RV'\\F3$!1jx/IFdpOF37$$\"1'*****\\@fkeF3$!1^['RSTnd&F37$$ \"1JLLL&4Nn'F3$!10ekYCV[qF37$$\"1*******\\,s`(F3$!1P()yk]wH$)F37$$\"1l mm\"zM)>$)F3$!17vqrPa+#*F37$$\"1KL$eCZwu)F3$!1.1!o^D\"\\&*F37$$\"1**** ***pfa<*F3$!1)p#3jdk,)*F37$$\"1km;zy*zd*F3$!1*H9))yDt%**F37$$\"1HLLeg` !)**F3$!1:1B?k))****F37$$\"1mm;W/8S5F0$!1o&o<9S5&**F37$$\"1++]#G2A3\"F 0$!1^vI#G.J'3#)>a6F37$$ \"1nm\"HB-7a\"F0$\"1&)yZKFw@PF>7$$\"1+++Sv&)z:F0$\"1ZqiZcsO?F37$$\"1nm ;%)3;C;F0$\"1TYHP.*)=TF37$$\"1LLLGUYo;F0$\"1GCLryK#R'F37$$\"1++]n'*33< F0$\"1\\<3?q-#f)F37$$\"1nmm1^rZYE&4\"F07$$\"1LLe*3k**y\"F0 $\"1bnZc:5l8F07$$\"1++]sI@K=F0$\"1#>Urv3Tl\"F07$$\"1+++S2ls=F0$\"17P( \\L)4\\>F07$$\"1++]2%)38>F0$\"1j^R>4YiAF07$$\"1++v.Uac>F0$\"1g+#*3i9?E F07$$\"\"#F*$\"\"$F*-%'COLOURG6&%$RGBG$\"#5F,F*F*-%+AXESLABELSG6$Q\"x6 \"%!G-%%VIEWG6$;F(F[al%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "newton(x^3-3*x+1,x=1.5,info= true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\" /LLLLLL:!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~ ~G$\"/\\FV14K:!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3 ~~->~~~G$\"/:C')))3K:!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approxim ation~4~~->~~~G$\"/!Qi)))3K:!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" +')))3K:!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "newton(x^3-3*x+1,x=0.35,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\"/X$HZMHZ$!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\"/]I`N'HZ$! #9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$\"/'QL bjHZ$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+`N'HZ$!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "newt on(x^3-3*x+1,x=-1.9,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7a pproximation~1~~->~~~G$!/+f'[$pz=!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%7approximation~2~~->~~~G$!/+27`Qz=!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$!/=dT_Qz=!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$!/=dT_Qz=!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+U_Qz=!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 256 4 "Note" }{TEXT -1 55 ": Newton's method can fail under certain circumstances." }}{PARA 0 "" 0 "" {TEXT -1 29 "For example, if you take the " }{TEXT 313 1 "x" }{TEXT -1 91 " coordinate of the minimum point on the curve as a starting value a zero derivati ve occurs." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "newton(x^3-3*x+1,x=1);" }}{PARA 8 "" 1 "" {TEXT -1 44 "Error, (in newton) zero derivative obtained\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "If you take a starting va lue close and to the right of the " }{TEXT 314 1 "x" }{TEXT -1 145 " c oordinate of the minimum point the next approximation is large and the n many iterations are wasted in returning to the neighbourhood of a ro ot." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "newton(x^3-3*x+1,x=1.00001,info=true);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\"/](QLenm\"!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\"/IeEA<66! \"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$\"/2) Q!\\\"yS(!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~ ~~G$\"/FCfLaQ\\!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~ 5~~->~~~G$\"/TZuBO#H$!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approxim ation~6~~->~~~G$\"/v5_%3\\>#!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7a pproximation~7~~->~~~G$\"/\"p]gsKY\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~8~~->~~~G$\"/Q!\\Db^v*!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~9~~->~~~G$\"//5nOW.l!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~10~~->~~~G$\"/E_a$RcL%!#6" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~11~~->~~~G$\"/-8&fT/*G !#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~12~~->~~~G$\"/V K)3%)p#>!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~13~~->~ ~~G$\"/.Um0p%G\"!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation ~14~~->~~~G$\"/m[>27l&)!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approx imation~15~~->~~~G$\"/t+#Ra3r&!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% 8approximation~16~~->~~~G$\"/VHjPR3Q!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~17~~->~~~G$\"/]K/zlSD!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~18~~->~~~G$\"/-tz#[jp\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~19~~->~~~G$\"/^GFjsM6!#7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~20~~->~~~G$\"/'[SIW9i( !#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~21~~->~~~G$\"/i fM>&!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~22~~->~~ ~G$\"/\">U?\"*Qc$!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximatio n~23~~->~~~G$\"/W7j%)[]D!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8appro ximation~24~~->~~~G$\"/E?UVk[>!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% 8approximation~25~~->~~~G$\"/$G\\)\\NW;!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~26~~->~~~G$\"/h#)y_(Ra\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~27~~->~~~G$\"/(*=:hCK:!#8" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~28~~->~~~G$\"/:M9*)3K :!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~29~~->~~~G$\"/ !Qi)))3K:!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~30~~-> ~~~G$\"/!Qi)))3K:!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+')))3K:!\" *" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "The default number of maximum iterations may be reached." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "newton(x^ 3-3*x+1,x=1.000000001);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6last~iter ation~gives~G$\"+CG6K:!\"*" }}{PARA 8 "" 1 "" {TEXT -1 67 "Error, (in \+ newton) reached max, 50, iterations without convergence\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "There is an optio n which allows you to change the maximum number of iterations if you w ish." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "newton(x^3-3*x+1,x=1.000000001,maxiterations=60);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+')))3K:!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 120 "You can usually avoid su ch problems simply by taking your starting approximation sufficiently \+ close to the desired root." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }{TEXT 328 7 ".. sin(" }{TEXT 334 1 "x" } {TEXT 335 8 ") = 1 - " }{TEXT 333 2 "x " }{TEXT 339 62 "(different lev els of information via the option \"info=1/2/3\") " }}{PARA 0 "" 0 "" {TEXT 303 8 "Question" }{TEXT -1 4 ": " }}{PARA 0 "" 0 "" {TEXT -1 41 "Find the single solution of the equation " }{XPPEDIT 18 0 "sin(x)= 1-x" "6#/-%$sinG6#%\"xG,&\"\"\"F)F'!\"\"" }{TEXT -1 70 " by Newton's m ethod. Your answer should be correct to about 10 digits." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 302 8 "Solution" }{TEXT -1 4 ": " }}{PARA 0 "" 0 "" {TEXT -1 33 "It is evident from the grap hs of " }{XPPEDIT 18 0 "y=sin(x)" "6#/%\"yG-%$sinG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=1-x" "6#/%\"yG,&\"\"\"F&%\"xG!\"\"" } {TEXT -1 19 " that the equation " }{XPPEDIT 18 0 "sin(x)=1-x" "6#/-%$s inG6#%\"xG,&\"\"\"F)F'!\"\"" }{TEXT -1 41 " has one solution, not too \+ far away from " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "plot([sin(x),1-x],x=-1..2,color=[red,blue],\n tickm arks=[3,4]);" }}{PARA 13 "" 1 "" {GLPLOT2D 278 205 205 {PLOTDATA 2 "6' -%'CURVESG6$7S7$$!\"\"\"\"!$!1l*y![)4ZT)!#;7$$!1*****\\P&3Y$*F-$!1*GE# )elO/)F-7$$!1++Dcx6x()F-$!1$*H5w\"*y#p(F-7$$!1,+]iTDP\")F-$!19345\"3&o sF-7$$!1****\\P\"\\J\\(F-$!1+'z6It8\"oF-7$$!1++DJa5_oF-$!1Ds)4Yg$GjF-7 $$!1,+DcexdiF-$!1Mia#*oFdeF-7$$!1++D1?QUcF-$!1'RVk`AxM&F-7$$!1++D13%f+ &F-$!1)Q\"o(3o%*z%F-7$$!1++D\"oS:P%F-$!1?&Q.#)GOB%F-7$$!1+++v@)*=PF-$! 1*yP\\RXQj$F-7$$!1++](G3U9$F-$!1O!)o`ul#4$F-7$$!1*****\\-\\r\\#F-$!1sa %p9x7Z#F-7$$!1+++vGVZ=F-$!1VTZa>%p$=F-7$$!1+++v4J@7F-$!1@I[eqF=7F-7$$! 1,+]iIKFl!#<$!135`.!*oAlFap7$$\"19+++DFOB!#=$\"1'\\0ZPqiL#Fgp7$$\"1,++ +!R5'fFap$\"1#HxDI4v&fFap7$$\"1++vV!QBE\"F-$\"1$QG+;))*e7F-7$$\"1***** *\\\"o?&=F-$\"1YM`b\")\\T=F-7$$\"1,+vVb4*\\#F-$\"1v:`VK;tCF-7$$\"1,+DJ '=_6$F-$\"1\"*eeRg2lIF-7$$\"1,+]P%y!ePF-$\"1Z9aU6CqOF-7$$\"1,+v=WU[VF- $\"1:4$R6vE@%F-7$$\"1++]7B>&)\\F-$\"1ZaLm^D\"y%F-7$$\"1++v$>:mk&F-$\"1 -Co%3*H^`F-7$$\"1++DcdQAiF-$\"1/GV;jbGeF-7$$\"1,+]PPBWoF-$\"1*z)3gNEAj F-7$$\"1******\\Nm'[(F-$\"1yuE4Mi1oF-7$$\"1****\\(yb^6)F-$\"1I>28LJ`sF -7$$\"1++vVVDB()F-$\"1\"\\S)3PEewF-7$$\"1++]7TW)R*F-$\"1'y#\\iKmu!)F-7 $$\"1+++:K^+5!#:$\"1TEE_;[<%)F-7$$\"1++]7,Hl5Fhu$\"1.Rh?XH\\()F-7$$\"1 +]P4w)R7\"Fhu$\"1nlFFhI=!*F-7$$\"1++]x%f\")=\"Fhu$\"1W#H%oM$oF*F-7$$\" 1+]P/-a[7Fhu$\"1,]p@IB&[*F-7$$\"1+](=Yb;J\"Fhu$\"1HOv$)[5m'*F-7$$\"1++ ]i@Ot8Fhu$\"1^o0S5t0)*F-7$$\"1+]PfL'zV\"Fhu$\"1D\"[\"fm!>\"**F-7$$\"1+ ++!*>=+:Fhu$\"1%='[N%y](**F-7$$\"1++DE&4Qc\"Fhu$\"1?O3Cfv****F-7$$\"1+ ]P%>5pi\"Fhu$\"1w\"H+IgU)**F-7$$\"1+++bJ*[o\"Fhu$\"1T-2^,)\\$**F-7$$\" 1++Dr\"[8v\"Fhu$\"1Dl4luWP)*F-7$$\"1+++Ijy5=Fhu$\"1l*y1^.Mr*F-7$$\"1+] P/)fT(=Fhu$\"1IGj:6PV&*F-7$$\"1+]i0j\"[$>Fhu$\"1#p#)QIJZM*F-7$$\"\"#F* $\"1Fhu7$F4$\"1+]iv$\"1++v8\\J\\ mP()F-7$Feq$\"1,++]=$z9)F-7$Fjq$\"1***\\iX/4](F-7$F_r$\"1*** \\(o8y%)oF-7$Fdr$\"1****\\i:#>C'F-7$Fir$\"1***\\7ev:l&F-7$F^s$\"1++](o 2[,&F-7$Fcs$\"1++D1[Q`VF-7$Fhs$\"1++vVUhxPF-7$F]t$\"1****\\iiwbJF-7$Fb t$\"1,++]kL8DF-7$Fgt$\"1,+]7U%[)=F-7$F\\u$\"1++Dccuw7F-7$Fau$\"1/++v)e b,'Fap7$Ffu$!1$*y*****\\@8&!#>7$F\\v$!10++]7,HlFap7$Fav$!1***\\P4w)R7F -7$Ffv$!1-++vZf\")=F-7$F[w$!1)**\\P/-a[#F-7$F`w$!1++v=Yb;JF-7$Few$!1** ***\\i@Ot$F-7$Fjw$!1***\\PfL'zVF-7$F_x$!1-+++*>=+&F-7$Fdx$!1++]i_4QcF- 7$Fix$!1-+vV>5piF-7$F^y$!1,++]:$*[oF-7$Fcy$!1,+]7<[8vF-7$Fhy$!1++++L'y 5)F-7$F]z$!1,+vV!)fT()F-7$Fbz$!1,+DcI;[$*F-7$FgzF(-F\\[l6&F^[lF*F*F_[l -%+AXESLABELSG6$Q\"x6\"%!G-%*AXESTICKSG6$\"\"$\"\"%-%%VIEWG6$;F(Fgz%(D EFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curv e 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We use th e procedure " }{TEXT 0 6 "newton" }{TEXT -1 33 " with the starting app roximation " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 3 ".5." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Note th at, if we give the procedure " }{TEXT 0 6 "newton" }{TEXT -1 14 " the \+ equation " }{XPPEDIT 18 0 "sin(x)=1-x" "6#/-%$sinG6#%\"xG,&\"\"\"F)F'! \"\"" }{TEXT -1 108 " as its first argument, it will automatically tra nspose all the terms to the left to construct the function " } {XPPEDIT 18 0 "f(x)=sin(x)-1+x" "6#/-%\"fG6#%\"xG,(-%$sinG6#F'\"\"\"F, !\"\"F'F," }{TEXT -1 68 " before applying Newton's method to find the \+ zero of this function. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "newton(sin(x)=1-x,x=0.5,info=1);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\"/=iH&z&4^ !#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\"/Hd $HM(4^!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G $\"/d)QHM(4^!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~ ->~~~G$\"/d)QHM(4^!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+%HM(4^!#5 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "More details concerning the computation can be seen by using the options \+ \"" }{TEXT 304 6 "info=2" }{TEXT -1 7 "\" or \"" }{TEXT 304 6 "info=3 " }{TEXT -1 3 "\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "newton(sin(x)=1-x,x=0.5,info=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%BAttempting~to~calculate~a~zero~ofG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$sinG6#%\"xG\"\"\"F(!\"\"F'F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%Iby~Newton's~method,~using~the~deriva tiveG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$cosG6#%\"xG\"\"\"F(F(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%7approximation~1~~->~~~G$\"/=iH&z&4^!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\"/Hd$HM(4^!#9" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$\"/d)QHM(4^!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\"/d)QHM(4^! #9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+%HM(4^!#5" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "newton(sin (x)=1-x,x=0.5,info=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%BAttempting ~to~calculate~a~zero~ofG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$sinG6 #%\"xG\"\"\"F(!\"\"F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Iby~Newton 's~method,~using~the~derivativeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,& -%$cosG6#%\"xG\"\"\"F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%E**~working~precision~is~14~digits~** G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$!.!eRhWd?!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG$\"//*=c#ex=!#8" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%2correction~~->~~~G/,$*&%&valueG\"\"\"%+derivat iveG!\"\"F*$\"/y@'H&z&4\"!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\"/=iH&z&4^ !#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$!+>!4w*G!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%+derivativeG$\"/-VckFs=!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2co rrection~~->~~~G/,$*&%&valueG\"\"\"%+derivativeG!\"\"F*$\"/-u5&Rwa\"!# =" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\"/Hd$HM(4^!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$!%ce!# 9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG$\"/i;)))oA(=!#8" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%2correction~~->~~~G/,$*&%&valueG\"\" \"%+derivativeG!\"\"F*$\"/KBnevFJ!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$\"/ d)QHM(4^!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%&valueG$\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%+derivativeG$\"/4:)))oA(=!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2co rrection~~->~~~G/,$*&%&valueG\"\"\"%+derivativeG!\"\"F*$!\"!\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%7approximation~4~~->~~~G$\"/d)QHM(4^!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+%HM(4^!#5" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Since t he procedure " }{TEXT 0 6 "newton" }{TEXT -1 166 " performs the calcul ation with higher precision than is required for the result, we can be reasonably confident that the rounded result given is correct to 10 d igits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 \+ " }{TEXT 317 60 ".. zooming in graphically to obtain a starting approx imation" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 305 8 "Question" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 37 "(a) Plot the graphs of \+ the functions " }{XPPEDIT 18 0 "f(x) = 4/(x+1);" "6#/-%\"fG6#%\"xG*&\" \"%\"\"\",&F'F*F*F*!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(x) = s in(2*x)+2;" "6#/-%\"gG6#%\"xG,&-%$sinG6#*&\"\"#\"\"\"F'F.F.F-F." } {TEXT -1 56 " in the same picture to show any points of intersection. " }}{PARA 0 "" 0 "" {TEXT -1 44 "(b) Find all the solutions of the equ ation " }{XPPEDIT 18 0 "4/(x+1) = sin(2*x)+2;" "6#/*&\"\"%\"\"\",&%\" xGF&F&F&!\"\",&-%$sinG6#*&\"\"#F&F(F&F&F/F&" }{TEXT -1 43 " correct t o 10 digits by Newton's method. " }}{PARA 0 "" 0 "" {TEXT -1 56 "(c) F ind the largest positive solution of the equation " }{XPPEDIT 18 0 "l n(x+1) = 2*cos(x^2)" "6#/-%#lnG6#,&%\"xG\"\"\"F)F)*&\"\"#F)-%$cosG6#*$ F(F+F)" }{TEXT -1 45 " correct to 10 digits by a numerical method." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 306 8 "Solution " }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plot([4/(x+1),sin(2*x)+2],x=-0.5..8,y=0.. 5);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVES G6$7Y7$$!3++++++++]!#=$\"\")\"\"!7$$!3qL$eRZ5o`%F*$\"3EO?QZ-t@t!#<7$$! 3%om;z%4itSF*$\"3+,*\\/!R[\\nF37$$!3****\\(=UJ/h$F*$\"3-\">;jc.-E'F37$ $!3oLL$e*=CZJF*$\"3HtQ.0e1PeF37$$!30+v$fV/7M#F*$\"3Rf**HDLvA_F37$$!3Tm ;/wp;N:F*$\"3Oca8XGVDZF37$$!3K(*\\i:b$pG'!#>$\"3M[Bb0$[$oUF37$$\"3[pm; H()zxFFP$\"3mKY_)4\"*=*QF37$$\"3Mnmm\"*>E!>\"F*$\"3Eu!=f]OXd$F37$$\"3w nmT5Tu-@F*$\"3))3h@)eN]I$F37$$\"3)HLe9hM!>RF*$\"39KcO@DwtGF37$$\"3Mm;H 2%oHg&F*$\"3))*QIIv9Oc#F37$$\"3#***\\i:VeYtF*$\"3N/@Jr%HfI#F37$$\"3Mo; 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aQIt@3&F3$\"3xk>wr\\&F3$\"3UnK^Y2++ 5F37$F^x$\"38')>ksT\"3+\"F37$$\"3dqn&)HCpSbF3$\"3Mc&F3$\"37#**o7(Hg35F37$$\"3;z>6.v9'e&F3$\"3q!*H*)4Xd:5F37$$\"3 Y$eR(R]()3cF3$\"3ttp_#ozX-\"F37$$\"3/#z%*H6IVl&F3$\"3u2'G`N7'[5F37$Fcx $\"3uu!)o/D]!3\"F37$$\"31x-'fF3$\"3%H\"*)H:&\\\")R\"F37$F]y$\"3qmI(Ho*z\\:F37$ $\"3%o\"z>,$3r8'F3$\"3a:`4rV)>r\"F37$Fby$\"3-sd6v0&H)=F37$$\"3=;HK*[Wg J'F3$\"3KJ(*Gp5nl?F37$Fgy$\"3!\\d$*yb(>YAF37$$\"3)Qe9;\"fq&\\'F3$\"3GD +FC$eBT#F37$F\\z$\"3=NVbnutlDF37$$\"3/oTN\")z)Rn'F3$\"37@\"fjx>Wq#F37$ Faz$\"3*pCFQEq-#GF37$$\"3%4v=Ug>N&oF3$\"3q&z>U\\7*3HF37$Ffz$\"3/KWF2%z &oHF37$$\"3'f(=<\"[$)R)pF3$\"3=![b4!)>d)HF37$$\"3\\hS;m'3(F3$\"3+:4uIkM**HF37$F[[l$\"3!>e7/V>q*HF37$$ \"3VnT&)y=M,sF3$\"3WtS&H,d\\'HF37$F`[l$\"3!**fJH4&y)*GF37$$\"3I,vVV0pz tF3$\"3qEW?m**e7GF37$Fe[l$\"39utV`JS.FF37$$\"39$ek.ptOb(F3$\"3]34$*z1: lDF37$Fj[l$\"3\")=3$z+E(3CF37$$\"3')****\\AGQHxF3$\"3klegYkiYAF37$F_\\ l$\"3s,UFl/Ex?F37$$\"3)*\\(oHVcw!zF3$\"3E!p&*[YcG*=F37$F+$\"3qM\\L$o'4 7 " 0 "" {MPLTEXT 1 0 77 "eq := 4/(x+1 )=sin(2*x)+2;\nnewton(eq,x=0.4);\nnewton(eq,x=2);\nnewton(eq,x=2.4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/,$*&\"\"%\"\"\",&%\"xGF)F)F)! \"\"F),&-%$sinG6#,$*&\"\"#F)F+F)F)F)F3F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+&R!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)[6cf#!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "(c) We start by plotting \+ the graphs of " }{XPPEDIT 18 0 "y=ln(x+1)" "6#/%\"yG-%#lnG6#,&%\"xG\" \"\"F*F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y= 2*cos(x^2)" "6#/%\"yG *&\"\"#\"\"\"-%$cosG6#*$%\"xGF&F'" }{TEXT -1 2 ". 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a>FL7$$\"3>]7.#Q?&=zFL$\"35ecef3T$)>FL7$$\"3I+DccB&R#zFL$\"3u\"yA(*ol \")*>FL7$$\"3S]P4JVQHzFL$\"37KO\"=*o7)*>FL7$$\"3^+]i0j\"[$zFL$\"3)=%)z oMkK)>FL7$$\"3s+voa-oXzFL$\"3k!f9%**z\\4>FL7$$\"3/++v.UaczFL$\"3WrpCF3 #)yIO\"FL7$$\"3!)*\\P40O\"*)zFL$\"3#>vgC0U**3\"FL7$Fbz$\"3:+5f3Y9PyF --Fgz6&FizF(FjzF(-%+AXESLABELSG6$Q\"x6\"Q\"yFe\\t-%%VIEWG6$;F(Fbz;$!\" $F)$\"\"$F)" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 " Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 80 "We can see from this graph that there are 11 positive s olutions of the equation " }{XPPEDIT 18 0 "ln(x+1)=2*cos(x^2)" "6#/-%# lnG6#,&%\"xG\"\"\"F)F)*&\"\"#F)-%$cosG6#*$F(F+F)" }{TEXT -1 17 " in th e interval " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "``< =6" "6#1%!G\"\"'" }{TEXT -1 75 ". However it is not clear whether ther e are more solutions in the interval " }{XPPEDIT 18 0 "6<=x" "6#1\"\"' %\"xG" }{XPPEDIT 18 0 "``<=7" "6#1%!G\"\"(" }{TEXT -1 39 ". We can che ck this with another plot. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot([ln(x+1),2*cos(x^2)],x=6..7); " }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$ 7S7$$\"\"'\"\"!$\"3C8`0\\,\"f%>!#<7$$\"3Rmm;arz@gF-$\"3-769!>>!\\>F-7$ $\"3#HLe9ui2/'F-$\"3)yJe#3lr^>F-7$$\"3]mm\"z_\"4igF-$\"3Q(yW!e7ua>F-7$ $\"3#pm;aphN3'F-$\"3i!eNb%oxd>F-7$$\"3UL$e*=)H\\5'F-$\"3Q[E!Q)))yg>F-7 $$\"3Fm;z/3uChF-$\"3#\\:=sMtN'>F-7$$\"3!)**\\7LRDXhF-$\"3;3es]$[k'>F-7 $$\"3#om\"zR'ok;'F-$\"3UpnU9ITp>F-7$$\"3e**\\i5`h(='F-$\"3+p@@\\%fB(>F -7$$\"3,LL$3En$4iF-$\"3#\\\"H*))=\"Qv>F-7$$\"3cmmT!RE&GiF-$\"3#p#Rk'>N !y>F-7$$\"3)*****\\K]4]iF-$\"3y4F-7$$\"3K++]PAvriF-$\"3SJRa&G (*R)>F-7$$\"3#)****\\nHi#H'F-$\"3%Hv/[Gjo)>F-7$$\"3bm;z*ev:J'F-$\"3A\" p&\\*ye%*)>F-7$$\"3/LL$347TL'F-$\"35YlD`j`#*>F-7$$\"3=LLLjM?`jF-$\"3%H ABO1O^*>F-7$$\"39+]7o7TvjF-$\"3#ebSem^\")*>F-7$$\"3+LLLQ*o]R'F-$\"3RY \"4%=M\"3+#F-7$$\"3C+]7=lj;kF-$\"3EnR'\\lDP+#F-7$$\"3&***\\PaR@V1s$[,#F-7$$\"3Om;zRQb@lF-$\"3F q!\\Uusx,#F-7$$\"3%)**\\(=>Y2a'F-$\"3!G'GcT6K??F-7$$\"3%om;zXu9c'F-$\" 3BA2H0i1B?F-7$$\"3u*****\\y))Ge'F-$\"3d&G_KB%*e-#F-7$$\"3n***\\i_QQg'F -$\"3am6.*=`'G?F-7$$\"3q**\\7y%3Ti'F-$\"3Fagv%R:8.#F-7$$\"3[***\\P![hY mF-$\"3wz&)eSIEM?F-7$$\"3ELLLQx$om'F-$\"3cXxqSU!p.#F-7$$\"3')****\\P+V )o'F-$\"3?#*GI1mrR?F-7$$\"3im;zpe*zq'F-$\"3S%*zr9#eA/#F-7$$\"3w****\\# \\'QHnF-$\"3ZrKP#\\H]/#F-7$$\"3cL$e9S8&\\nF-$\"3'GZ&[a+jZ?F-7$$\"3;+]i ?=bqnF-$\"3#fDV7=T.0#F-7$$\"3uLL$3s?6z'F-$\"3eJt8;Z)H0#F-7$$\"3&***\\7 `Wl7oF-$\"3+'4n[yWd0#F-7$$\"3emmm'*RRLoF-$\"3/OtqweRe?F-7$$\"3'pm;a<.Y &oF-$\"39nmq^(*4h?F-7$$\"35L$e9tOc(oF-$\"3[%[Ji.uP1#F-7$$\"3u*****\\Qk \\*oF-$\"3gmWhO^Am?F-7$$\"3mLL3dg6LL3x&)*3,'F-$\"3ILL%QjpB <&!#?7$F/$\"3'[cox'\\amEFf[l7$$\"35+D\"y%*z7.'F-$\"3*o\"=-G]u4\\Ff[l7$ F4$\"3Q)e5(4%QA4(Ff[l7$$\"3s*\\(oMrU^gF-$\"3h1'y\\.\"=Q%*Ff[l7$F9$\"3% z\"zm!))3J;\"F-7$$\"3qmmm6m#G2'F-$\"3#)z,%4.0YO\"F-7$F>$\"3r[@Ml'pKa\" F-7$FC$\"3!oShd76)==F-7$$\"3&)**\\(=JN[6'F-$\"3x&3d%3O%f!>F-7$FH$\"3)[ M)))y$e_'>F-7$$\"3**)*\\(o3p)HhF-$\"3Kv$*[q*)o%)>F-7$$\"3fK$e*ot*\\8'F -$\"3wxXm2+H'*>F-7$$\"3R******4:cPhF-$\"3QsQbFF8**>F-7$$\"3>m;/^c7ShF- $\"3#f**fV(e****>F-7$$\"3*HL$3#z*oUhF-$\"3;Vk+5h())*>F-7$FM$\"3A4ech?x &*>F-7$$\"3+n;z4wb]hF-$\"3MM5$*p'fI)>F-7$$\"3IL$ekGhe:'F-$\"3Ufhl!)\\* ='>F-7$$\"3i**\\7j\\;hhF-$\"3yg%o)>kMK>F-7$FR$\"3?/B`%f=X*=F-7$$\"3wK$ 3_(>/xhF-$\"3--n\\$y$*\\z\"F-7$FW$\"3M0ZD$o*ok;F-7$$\"3Gm\"HdG\"\\)>'F -$\"3aCi3H)H2]\"F-7$Ffn$\"3M.d6o'>#48F-7$$\"3B+]iDo%*=iF-$\"3Ct^`l!p.7 \"F-7$F[o$\"3e->$e#))H`\"*Ff[l7$$\"3sL$e9r5$RiF-$\"3%zP'p5v*oo'Ff[l7$F `o$\"3e\\qBMUB&4%Ff[l7$$\"3;+++NO#4E'F-$\"3?g*3#y#)o89Ff[l7$Feo$!3%oct NIX%)H\"Ff[l7$$\"3_++]-w=#G'F-$!35I4<\\\"=W*QFf[l7$Fjo$!3p())*e5VxFkFf [l7$$\"3kLeky#*4-jF-$!3mZG^Pn2P')Ff[l7$F_p$!3]8!fUP_E2\"F-7$$\"3N*\\7. %Q%GK'F-$!350P#*p!48I\"F-7$Fdp$!3)3$GwD%pR]\"F-7$$\"36LL3xxlVjF-$!3?hF clg7_;F-7$Fip$!3_EM4uPGwF-7$$\"3-%3xcoD.Q'F-$!3))\\gfN@c#)>F-7$$\"3-n\"HK5S_Q'F-$!3A VB=DP=&*>F-7$$\"3_3_+7tp(Q'F-$!3gxk$*zSb)*>F-7$$\"3+]7y?X:!R'F-$!3+TfF Zg&***>F-7$$\"3]\"Hd&HF-7$Fcq$!3SB.xS@%o*>F-7$$\"3 /]7GL3Y+kF-$!3Qyi*z#oM%)>F-7$$\"32n\"H#GF&eS'F-$!3yTP?\"\\(Qi>F-7$$\"3 @$3xJiW7T'F-$!39g.I\\]/J>F-7$Fhq$!3AjN6%pX/*=F-7$$\"3a++DO_!pU'F-$!3Q- 8qm^<)y\"F-7$F]r$!3u4gt*34Yl\"F-7$$\"3Gm\"zWG))yW'F-$!3'[J!)\\/`U[\"F- 7$Fbr$!3a)>,=b,`G\"F-7$$\"3/L3F9(ya!3\"F-7$Fgr$!3Wv[mz\\ ()z&)Ff[l7$$\"3%)*\\(=7O*))['F-$!32-uUH+P?gFf[l7$F\\s$!3_>WbG]eULFf[l7 $$\"3HmT5D,`5lF-$!3k6#Gkis\"**[!#>7$Fas$\"3S?KSNojxBFf[l7$$\"35LL$e,]6 `'F-$\"3xW[Zzj')R[Ff[l7$Ffs$\"3:$Q:sh'pHsFf[l7$$\"3MLe*[K56b'F-$\"3;%G 8dT)p&o*Ff[l7$F[t$\"3wi!Rin4n>\"F-7$$\"3GL$e9i\"=slF-$\"3vBb)>j*\\49F- 7$F`t$\"3EmhF(=RZf\"F-7$$\"3q**\\ibOO$f'F-$\"3+WZ8iEjXF-7$Fjt$\"33#H@4h+$*)>F-7$$\"3 7C\"G)o<#pi'F-$\"3%eM[g\\=c*>F-7$$\"3U\\7`f]tHmF-$\"3m8m3fU;**>F-7$$\" 3suVB]$[Dj'F-$\"3izLbK%H***>F-7$$\"3.+v$4kh`j'F-$\"3]9<$)4%4z*>F-7$$\" 3v\\PMA#))4k'F-$\"3AR9ybZ^&)>F-7$F_u$\"3m7MB27-i>F-7$$\"3Om;/riscmF-$ \"3-k<-tTF#*=F-7$Fdu$\"3!p,%[%eb\")y\"F-7$$\"3cmm\"z)QjxmF-$\"3t*\\/\\ +;5k\"F-7$Fiu$\"3OBM)3!fbf9F-7$$\"3CLek`H@)p'F-$\"3Us!*=Az\\o7F-7$F^v$ \"3%H@[3K!Rb5F-7$$\"3>Le9\"=\"p=nF-$\"35n,*QYGS,)Ff[l7$Fcv$\"3PT)>*=Iw /`Ff[l7$$\"3mm\"zp%*\\%RnF-$\"3abpA_)Q.l#Ff[l7$Fhv$!3s)yf0@^`n&F\\\\l7 $$\"3'omT5hK+w'F-$!3u>:fI\"3)))GFf[l7$F]w$!3KG\\*[wlom&Ff[l7$$\"3%p;H2 FO3y'F-$!3'=E#\\C]Tw#)Ff[l7$Fbw$!3?E=?HP!H2\"F-7$$\"3Sm\"zpe()=!oF-$!3 &)omLm#yvI\"F-7$Fgw$!3Sd6TFzf9:F-7$$\"3rLe*[ACI#oF-$!3=z*pjU`No\"F-7$F \\x$!3KL\\3:M3>=F-7$$\"3xm;/'e)*R%oF-$!3()o$\\doB,#>F-7$Fax$!3/7Dsn@!4 )>F-7$$\"37v=#*pBBdoF-$!3I3Y*3A`&*)>F-7$$\"3G$3FWch)foF-$!3a'f\"\\*f?c *>F-7$$\"3W\"HK*e2\\ioF-$!3%RemIV$4**>F-7$$\"3e*\\PM&*>^'oF-$!3o\"*Rt8 U'***>F-7$$\"3u2F%z9\\x'oF-$!3eje+:)G#)*>F-7$$\"3!f\"zWU$y.(oF-$!31U!f i]')Q*>F-7$$\"3%\\7`p`2I(oF-$!3w(z&zi*Rp)>F-7$Ffx$!3ePhYr_Rx>F-7$$\"3' o;H#e0I&)oF-$!3ycerx\"z,#>F-7$F[y$!34ze*eU/*G=F-7$$\"3rm;/@-/1pF-$!3M? >M0&RVo\"F-7$F`y$!3_&frdIr,]\"F-7$$\"3J+](oTAq#pF-$!3p(*3X!=a_I\"F-7$F ey$!3O>*H!Hc\\&3\"F-7$$\"3%R$eRA5\\ZpF-$!3EmY;]]$QG)Ff[l7$Fjy$!3y!zU5* *)eIbFf[l7$$\"3O*****\\oi\"opF-$!3Uv$fRqy)yFFf[l7$F_z$\"3OrWnwHz!>$F\\ \\l7$$\"3!)*\\P40O\"*)pF-$\"3]^ZV<(GZ0$Ff[l7$Fdz$\"3sTF([(3&=,'Ff[l-Fi z6&F[[lF_[lF\\[lF_[l-%+AXESLABELSG6$Q\"x6\"Q!Fhjm-%%VIEWG6$;F(Fdz%(DEF AULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve \+ 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "The two largest positive solutions lie in the interval " }{XPPEDIT 18 0 "6.1<=x" "6#1-%&FloatG6$\"#h!\"\"%\"xG" }{XPPEDIT 18 0 "``<=6.2" "6#1%!G-%&FloatG6$\"#i!\"\"" }{TEXT -1 138 ". A third plot w ill enable us to obtain a suitable starting value to use with Newton's method in order to calculate the largest solution. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plot([ln(x +1),2*cos(x^2)],x=6.1..6.2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$\"3k*************4'!#<$\"3yps/%y%4 g>F*7$$\"3$fm;arz@5'F*$\"3A2J5RF*7$$\"3?Le9ui2/hF*$\"3[w2UU(o1'>F* 7$$\"3@m;z_\"4i5'F*$\"3et@VI*o4'>F*7$$\"3gm;aphN3hF*$\"3C!>,t,r7'>F*7$ $\"3!H$e*=)H\\5hF*$\"3g2W#ydr:'>F*7$$\"3=m\"z/3uC6'F*$\"3m`\\Ab,&='>F* 7$$\"3X*\\7LRDX6'F*$\"3mY3BC&Q@'>F*7$$\"3om\"zR'ok;hF*$\"3w7OUomVi>F*7 $$\"3()*\\i5`h(=hF*$\"33rw-oPti>F*7$$\"3RLL3En$47'F*$\"3=JCUz#RI'>F*7$ $\"3mm;/RE&G7'F*$\"33z_t'H3L'>F*7$$\"3Q***\\K]4]7'F*$\"3K)HV*f5hj>F*7$ $\"37++vB_F*7$$\"3X***\\nHi#HhF*$\"3e>>Qkx?k>F*7$ $\"3#p;z*ev:JhF*$\"33_ntrNZk>F*7$$\"3[LL347TLhF*$\"3K>8Y]&*yk>F*7$$\"3 zKLLY.KNhF*$\"3w[_zZr0l>F*7$$\"3v*\\7o7Tv8'F*$\"3OM3/P$o`'>F*7$$\"3[LL $Q*o]RhF*$\"3-T#y9rVc'>F*7$$\"3H+D\"=lj;9'F*$\"3-.dm`d%f'>F*7$$\"3<+vV &RF*7$$\"33L$e9Ege9'F*$\"3_&H=Y?Ll'>F*7$$\"3#G$eR \"3Gy9'F*$\"3S)G?da3o'>F*7$$\"3Wm;/T1&*\\hF*$\"3=s7L`a5n>F*7$$\"3Gm\"z RQb@:'F*$\"3'z2$ejPTn>F*7$$\"3)**\\(=>Y2ahF*$\"3]Ks>r?on>F*7$$\"3]m;zX u9chF*$\"3-fF$*p<(z'>F*7$$\"3i****\\y))GehF*$\"3QQ<'*o4Fo>F*7$$\"3K+]i _QQghF*$\"3sntB!fj&o>F*7$$\"3q*\\7y%3TihF*$\"3i.m)[jY)o>F*7$$\"3%***\\ P![hY;'F*$\"3yu*zQ\"3;p>F*7$$\"3KLL$Qx$omhF*$\"3;q\"4N.V%p>F*7$$\"3;++ v.I%)ohF*$\"3WluIvUup>F*7$$\"3Im\"zpe*zqhF*$\"3')\\4!o;<+(>F*7$$\"3!)* **\\#\\'QH<'F*$\"3!yU'4CaJq>F*7$$\"3OLe9S8&\\<'F*$\"3/![h(zffq>F*7$$\" 3O+D1#=bq<'F*$\"3#e'\\:d\"*)3(>F*7$$\"3-LL3s?6zhF*$\"3?(*>d3dF*7$$ \"3W+DJXaE\"='F*$\"3s0fR8cZr>F*7$$\"3+nmm*RRL='F*$\"3u]CQsVwr>F*7$$\"3 qm;a<.Y&='F*$\"3YF*7$$\"3ALe9tOc(='F*$\"3#Hpc\\E_B(>F*7$$\"3' *****\\Qk\\*='F*$\"3/O.\"H8@E(>F*7$$\"3)QL3dg6<>'F*$\"3O\"zZq>HH(>F*7$ $\"3'ommw(Gp$>'F*$\"3A$G1Hl/K(>F*7$$\"3s*\\7oK0e>'F*$\"376-vi#)\\t>F*7 $$\"3[+v=5s#y>'F*$\"3!H^:T?zP(>F*7$$\"3;+++++++iF*$\"3k4?-E53u>F*-%'CO LOURG6&%$RGBG$\"#5!\"\"$\"\"!F_[lF^[l-F$6$7S7$F($\"3)y7(Q+5]l_w)=F*7$FL$\"3\\n'QMnKO! >F*7$FQ$\"3f'zlV)\\!*=>F*7$FV$\"303U!z?ZG$>F*7$Fen$\"3CTZ6jF%e%>F*7$Fj n$\"3)=on3U]h&>F*7$F_o$\"3OF5)*)Rpk'>F*7$Fdo$\"3sWr>mHXv>F*7$Fio$\"3!) )*ea-nz#)>F*7$F^p$\"3U,)fKFV$))>F*7$Fcp$\"3+!zu)GFa$*>F*7$Fhp$\"3g'oo0 )ov'*>F*7$F]q$\"3+zWrHz6**>F*7$Fbq$\"37G%=#4!p***>F*7$Fgq$\"3/w)edMi&* *>F*7$F\\r$\"3#f?+'3/(y*>F*7$Far$\"3Yli01![Z*>F*7$Ffr$\"3%Rai))eg1*>F* 7$F[s$\"39u66WP%\\)>F*7$F`s$\"3%*>1DJ3dx>F*7$Fes$\"3kN?X\"Rl*p>F*7$Fjs $\"3')o*zG)Q^g>F*7$F_t$\"37Bcns]S\\>F*7$Fdt$\"3dN'HS/@s$>F*7$Fit$\"3^y `!*)o*>C>F*7$F^u$\"3O'z4\"f)H$3>F*7$Fcu$\"3w7m&RM6G*=F*7$Fhu$\"3IH*fmo O\\(=F*7$F]v$\"3k$z/sd%ed=F*7$Fbv$\"3DU.zf'ot$=F*7$Fgv$\"3xPYd&>qr\"=F *7$F\\w$\"3i[\")zY)\\[z\"F*7$Faw$\"3qF8'*Qy%=x\"F*7$Ffw$\"3#4(ovp?`Yr4s\"F*7$F`x$\"3Ki4N2=l$p\"F*7$Fex$\"3)peW2r(Rl;F*7$Fj x$\"3)\\*4j>8VQ;F*7$F_y$\"38?!=uthjg\"F*7$Fdy$\"3%>(piMPkw:F*7$Fiy$\"3 w:Ef(*=!Ra\"F*7$F^z$\"37T-(R2i:^\"F*7$Fcz$\"3w$)H9)yQdZ\"F*-Fhz6&FjzF^ [lF[[lF^[l-%+AXESLABELSG6$Q\"x6\"Q!F\\el-%%VIEWG6$;$\"#hF][l$\"#iF][l% (DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cu rve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "The largest solution is approximately equal to 6.15. Thi s approximation can be refined by using Newton's method. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "newto n(ln(x+1)=2*cos(x^2),x=6.15,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\"/.m'R#ebh!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\"/Oltkhah!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$\"/Wr$)eeah!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\"/m$H&eeah!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~5~~->~~~G$\"/m$H&eeah! #8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+`eeah!\"*" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 4 " }{TEXT 318 28 ".. inv estigating convergence" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 271 8 "Question" }{TEXT -1 4 ": " }}{PARA 0 "" 0 "" {TEXT -1 46 "This ques tion is concerned with the function " }{XPPEDIT 18 0 "f(x) = x^2/10+x /4-1+cos(10*x)/12;" "6#/-%\"fG6#%\"xG,**&F'\"\"#\"#5!\"\"\"\"\"*&F'F- \"\"%F,F-F-F,*&-%$cosG6#*&F+F-F'F-F-\"#7F,F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 22 "(a) Plot the graph of " }{XPPEDIT 18 0 "f(x)" " 6#-%\"fG6#%\"xG" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "0< =x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "``<=4" "6#1%!G\"\"%" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 30 "(b) Find the positive zero of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 48 " correct to about 10 digits by Newton's method. " }}{PARA 0 "" 0 "" {TEXT -1 31 " Y ou may use the procedure " }{TEXT 0 6 "newton" }{TEXT -1 12 " for this . " }}{PARA 0 "" 0 "" {TEXT -1 97 "(c) Investigate the convergence of Newton's method to the positive zero with the starting value " } {XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 31 " You may use the procedure " }{TEXT 0 11 "newton_step " }{TEXT -1 10 " for this." }}{PARA 0 "" 0 "" {TEXT -1 54 "(d) What ha ppens if the starting value is taken to be " }{XPPEDIT 18 0 "x=2.65" " 6#/%\"xG-%&FloatG6$\"$l#!\"#" }{TEXT -1 5 "? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 8 "Solution" }{TEXT -1 4 ": \+ " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "f := x -> x^2/10+x/4-1+c os(10*x)/12:\n'f(x)'=f(x);\nplot(f(x),x=0..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,**&\"#5!\"\"F'\"\"#\"\"\"*&\"\"%F+F'F-F- F-F+*&#F-\"#7F--%$cosG6#,$*&F*F-F'F-F-F-F-" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7gq7$$\"\"!F)$!3Immmm mmm\"*!#=7$$\"3emmm;arz@!#>$!3QA;;1pTJ\"*F,7$$\"39LLLL3VfVF0$!3B8t#QD? 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This is because of the \"wiggly\" nature of the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 196 "The first few new approximations are ac tually futher away from the desired zero than the starting value and t he convergence to the desired zero happens by good luck rather than by good management." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 64 "f := x -> x^2/10+x/4-1+cos(10*x)/12:\nnewton (f(x),x=2,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximati on~1~~->~~~G$\"/'H6xFVS\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7appr oximation~2~~->~~~G$!-)yeRm7)!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7 approximation~3~~->~~~G$\"/x1`D(f1\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\"/!o;$z&Qe\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~5~~->~~~G$\"/t&)G(*=HA!#8" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%7approximation~6~~->~~~G$\"/lep$)4cA!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~7~~->~~~G$\"/Ltgfh`A!# 8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~8~~->~~~G$\"/ktIi f`A!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~9~~->~~~G$\" /uZHif`A!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~10~~->~ ~~G$\"/tZHif`A!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Hif`A!\"*" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The proc edure " }{TEXT 0 11 "newton_step" }{TEXT -1 21 " shows what happens. \+ " }}{PARA 0 "" 0 "" {TEXT -1 41 "(The output is not shown to save memo ry.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 193 "x := 'x':\nf := x -> x^2/10+x/4-1+cos(10*x)/12:\nxin := 2:\nprint(x[0]=xin);\nfor i from 1 to 10 do \n xout := newton_ step(f(x),x=xin,draw=is(i<=6));\n print(x[i]=xout);\n xin := xout; \nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "(d) With the starting value " }{XPPEDIT 18 0 "x = 2.5;" "6#/%\" xG-%&FloatG6$\"#D!\"\"" }{TEXT -1 58 " Newton's method does not conver ge to the positve zero of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 45 ". Convergence to the single negative zero of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " occurs instead. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "f := x -> x^2/10+x/4-1+cos(10*x)/12:\n'f(x)'=f(x);\nnewton(f(x),x= 2.5,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,**& \"#5!\"\"F'\"\"#\"\"\"*&\"\"%F+F'F-F-F-F+*&#F-\"#7F--%$cosG6#,$*&F*F-F 'F-F-F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G $\"/)edJ(Q8@!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~ ->~~~G$\"/L9'=K(zR!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximati on~3~~->~~~G$!.v,!)\\bO(!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7appro ximation~4~~->~~~G$\".7Wca>k&!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7 approximation~5~~->~~~G$\"/*\\T&=&pW\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~6~~->~~~G$!.RP83d*Q!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~7~~->~~~G$!/())\\yo!eK!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~8~~->~~~G$!/;?T\\dz7!# 8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~9~~->~~~G$\"/.\"f 0:od%!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~10~~->~~~G $!/4~~~G$!/bqeLEjH!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~ 12~~->~~~G$!/gBv$\\Rs$!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approxi mation~13~~->~~~G$!/9tvy3nU!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8ap proximation~14~~->~~~G$!/$R>3-lU%!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%8approximation~15~~->~~~G$!/%)R\\cM$f%!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~16~~->~~~G$!/1(fiLJ$R!#8" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%8approximation~17~~->~~~G$!/k\"y*HRPC!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~18~~->~~~G$!/7ov@*[h$! #8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~19~~->~~~G$!/f7% )[jqS!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~20~~->~~~G $!/7mgp:]]!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~21~~- >~~~G$!/ZB^\\Q(R%!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximatio n~22~~->~~~G$!/zdl#3n_%!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approx imation~23~~->~~~G$!/;:-9)p6%!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8 approximation~24~~->~~~G$!/nZ]kB2Y!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~25~~->~~~G$!/\"op!fU_K!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~26~~->~~~G$!/;)G^bG<\"!#8" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%8approximation~27~~->~~~G$!/~~~G$!/Yxd8[5P!# 8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~29~~->~~~G$!/Y7\\ R/8U!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~30~~->~~~G$ !/>q**[PHW!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~31~~- >~~~G$!/\\%ffYIg%!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximatio n~32~~->~~~G$!/(=0$=A$f$!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8appro ximation~33~~->~~~G$!/!*f$Q#*z2%!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%8approximation~34~~->~~~G$!/$)f$el<$\\!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~35~~->~~~G$!/M^&oLjv%!#8" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%8approximation~36~~->~~~G$!/C4JRWeZ!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~37~~->~~~G$!/@\"\\!zUe Z!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~38~~->~~~G$!/P *R!zUeZ!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~39~~->~~ ~G$!/P*R!zUeZ!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+/zUeZ!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 5 " }{TEXT 319 47 ".. a \"practical\" numerical root-finding problem" }{TEXT -1 1 " \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 272 8 "Question" }{TEXT -1 82 ": (See \"Animating Calculus\", b y Ed Packel and Stan Wagon, Springer-Verlag, 1997.)" }}{PARA 0 "" 0 " " {TEXT -1 102 "A straight and level stretch of railway line 1 kilomet re long is firmly fixed at the two ends A and B." }}{PARA 0 "" 0 "" {TEXT -1 226 "In the dead of night, a prankster cuts the track, and we lds an additional length of 10 cm into the gap. This causes the track, now 1000.1 meters long, to bow up from the ground into an arc of a ci rcle, as shown in the picture." }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "62-%'CURVESG6$7S7$$\"1E.96y1rq!#;$\"1E1f7y1rqF*7$$\"1 MYv$Hl[#oF*$\"13S\"er\")*3tF*7$$\"1'3&3Tn7/mF*$\"1zn%[cG!4vF*7$$\"1[:$ QP\"*)[jF*$\"1/vb3D.ExF*7$$\"1`<*o0oZ3'F*$\"1^@^$e;d$zF*7$$\"1=@QH(G] \"eF*$\"1(*3*f\\Va8)F*7$$\"1HJQPr3fbF*$\"1*=\\.ZLCJ)F*7$$\"1_^;t]S)G&F *$\"1,#G.D7s[)F*7$$\"1Yk!Qh$p-]F*$\"1(*[b?$)pe')F*7$$\"1E'Q#p)pBr%F*$ \"1Is\\Qj1?))F*7$$\"1pE5d]J3WF*$\"1c(z#\\()*e(*)F*7$$\"1j\"[7;Hi8%F*$ \"1'z`(pG[/\"*F*7$$\"15caT^XDQF*$\"1%zZ/[m$R#*F*7$$\"1H#*))Ry)*3NF*$\" 1S+IMR8k$*F*7$$\"1C;g\"om,?$F*$\"1,i!e7>TZ*F*7$$\"1.=l`!Rn\"HF*$\"1o_: j\"y^c*F*7$$\"1z!z*QEPwDF*$\"1wn?!4F*$\"1,!\\v26\"4)*F*7$$\"1xQ*pE+3k\"F*$\"15\"R$4.Z k)*F*7$$\"1kn^P9t08F*$\"1hRc*y'Q9**F*7$$\"1)[Ml@\"o_)*!#<$\"1`,%f'RM^* *F*7$$\"1KhRhU5)\\'F^r$\"12L1')\\')y**F*7$$\"1h*49%*z4T$F^r$\"1R?$[\"4 =%***F*7$$\"1*[%4?$\"0oIV*p******!#:7$$!1G:..D-&Q$F^r$\"11f?p\" pU***F*7$$!1%p+m\"z-'R'F^r$\"1r(R5XC&z**F*7$$!1Nc3h`QT'*F^r$\"1K.EGLT` **F*7$$!11q#3SQ$)H\"F*$\"1(*e]`wN:**F*7$$!1^iA^*oQi\"F*$\"1>EfS;Fn)*F* 7$$!1K%HpSmr$>F*$\"1'>)QG_d5)*F*7$$!1&=4Cc:FG#F*$\"18*\\O-vft*F*7$$!1F N%o)fy!f#F*$\"1.uPtDNF*$\"1%>vBQUyN*F*7$$!1(oX\\cr(>QF*$\"1 !*)\\Pl<:WF*$\"1Rh'y/hU* pL'F*$\"1E3'=n$zNxF*7$$!1J\\k%)*\\,f'F*$\"1VAvrzH@vF*7$$!1@u?$Qgc#oF*$ \"1i[!)3#R#3tF*7$$!1gkd " 0 "" {MPLTEXT 1 0 1 ";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot(t/sin(t)-1.0001,t=0..0.03,y); " }}{PARA 13 "" 1 "" {GLPLOT2D 265 217 217 {PLOTDATA 2 "6%-%'CURVESG6$ 7S7$$\"1+++]i9Rl!#>$!1&4*ogK(G***!#?7$$\"1+++XV)RQ*F*$!1p3XrMK&)**F-7$ $\"1+++WA)GA\"!#=$!1EFEzf2v**F-7$$\"1+++Qeui=F6$!1Lu'*3'p@%**F-7$$\"1+ ++j3&o]#F6$!13%[!p:E&*)*F-7$$\"1+++pX*y9$F6$!1cUV0e%[$)*F-7$$\"1+++WTA UPF6$!1s/$Hf&fm(*F-7$$\"1+++%*zhdVF6$!1k__A(=No*F-7$$\"1+++%>fS*\\F6$! 1^4$*p;K%e*F-7$$\"1+++>$f%GcF6$!1]\\ywa+s%*F-7$$\"1+++Dy,\"G'F6$!1P$Q] AxCM*F-7$$\"1+++8F;'R(F-7$$\"1+++j=_68F gp$!17KJtk7LrF-7$$\"1+++Wy!eP\"Fgp$!1MB**e\\=XoF-7$$\"1+++UC%[V\"Fgp$! 1vxVE&H'olF-7$$\"1+++J#>&)\\\"Fgp$!1vX$)Q@IdiF-7$$\"1+++>:mk:Fgp$!1hON Ttg>fF-7$$\"1+++w&QAi\"Fgp$!1:<#*o'oPh&F-7$$\"1+++uLU%o\"Fgp$!1ZCWf*R5 F&F-7$$\"1+++bjm[Fgp$!1Y4<)3k!GPF-7$$\"1+++ :K^+?Fgp$!1t))H<+gHLF-7$$\"1+++8,Hl?Fgp$!1(yMOY21*GF-7$$\"1+++4w)R7#Fg p$!1d'H^(>t![#F-7$$\"1+++y%f\")=#Fgp$!1T\"=;P%[>?F-7$$\"1+++/-a[AFgp$! 1B'>S%y%Hd\"F-7$$\"1+++ial6BFgp$!11C\\%4$>$4\"F-7$$\"1+++j@OtBFgp$!1;( *3J3.8h!#@7$$\"1+++fL'zV#Fgp$!1oiBFST?$*!#A7$$\"1+++!*>=+DFgp$\"1C1!3D J%*=%Fhw7$$\"1+++E&4Qc#Fgp$\"1\\;ZA)*Qg&*Fhw7$$\"1+++%>5pi#Fgp$\"1vdis 7--:F-7$$\"1+++bJ*[o#Fgp$\"1[_)oDHa,#F-7$$\"1+++r\"[8v#Fgp$\"1'3'[cAk< EF-7$$\"1+++Ijy5GFgp$\"1#frnxY(oJF-7$$\"1+++/)fT(GFgp$\"1/I\\$)zJpPF-7 $$\"1+++1j\"[$HFgp$\"15iiDsocVF-7$$\"1+++++++IFgp$\"1keY\\^d,]F--%'COL OURG6&%$RGBG$\"#5!\"\"\"\"!Fc[l-%+AXESLABELSG6$Q\"t6\"Q\"yFh[l-%%VIEWG 6$;Fc[l$\"\"$!\"#%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "t1 := newton(t/sin(t)=1.0001 ,t=0.025,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximatio n~1~~->~~~G$\"/NQ_h\"*\\C!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7appr oximation~2~~->~~~G$\"/d=pSS\\C!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %7approximation~3~~->~~~G$\"/To:SS\\C!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\"/,s:SS\\C!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~5~~->~~~G$\"/-s:SS\\C!#:" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#t1G$\"+;SS\\C!#6" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 44 "The maximum displacement in me tres is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "evalf(subs(t=t1,500*(1-cos(t))/sin(t)));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Ih\"Q7'!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "The maximum displacement \+ is about 6.12 metres." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 256 4 "Note" }{TEXT -1 110 ": You might be surprised at the large size of the displacement produced by adding such a short length of rail." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 6 \+ " }{TEXT 320 18 ".. \"crowded\" zeros" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 273 8 "Question" }{TEXT -1 4 ": " }}{PARA 0 "" 0 "" {TEXT -1 62 "Find, correct to 30 digits, the smallest zero of the function \+ " }{XPPEDIT 18 0 "f(x)=sin(1/x)" "6#/-%\"fG6#%\"xG-%$sinG6#*&\"\"\"F,F '!\"\"" }{TEXT -1 29 ", which is greater than 0.01." }}{PARA 0 "" 0 " " {TEXT 274 8 "Solution" }{TEXT -1 4 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 " sin(1/x)" "6#-%$sinG6#*&\"\"\"F'%\"xG!\"\"" }{TEXT -1 43 " has zeros c rowded ever closer together as " }{TEXT 315 1 "x" }{TEXT -1 72 " appro aches 1. We plot the graph over a small interval starting at 0.01." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot(sin(1/x),x=0.01..0.012);" }}{PARA 13 "" 1 "" {GLPLOT2D 272 194 194 {PLOTDATA 2 "6%-%'CURVESG6$7^s7$$\"1+++++++5!#<$!1)e(46klj]!#; 7$$\"1nmT:(z@+\"F*$!1/(H#3*G^!oF-7$$\"1LL$3VfV+\"F*$!1<#)zO!z0A)F-7$$ \"1+Dc*)fD15F*$!1b7!**eJ&Q\"*F-7$$\"1n;H[D:35F*$!1)[!H!p\"*Rt*F-7$$\"1 DJ&zw&o35F*$!1r7\"zr&pS)*F-7$$\"1$e9w)*=#45F*$!1b?e(\\z-#**F-7$$\"1UgF 2Av45F*$!1IF7i2hs**F-7$$\"1+v$pU&G55F*$!1%=!p,Ej(***F-7$$\"1e*)fY'=3, \"F*$!1gs/lIO&***F-7$$\"1W1.uNF-7$$ \"1L$e4;[\\-\"F*$!15!))fd@iv\"F-7$$\"1n;zt%**p-\"F*$\"1cAR,'*yL=F*7$$ \"1+]i'y]!H5F*$\"1e_*[GV%3@F-7$$\"1n;HdA+0\"F*$\"1\"*4!pQCDN)F-7$ $\"1+++FZ=_5F*$\"1>4WS$p:7(F-7$$\"1++]Z/Na5F*$\"1>%QVa&\\CcF-7$$\"1++] ?vVc5F*$\"1;#=D(H$e)RF-7$$\"1++]$fC&e5F*$\"1?`76p%[@#F-7$$\"1n\"Hd&)>/ 1\"F*$\"1N=Y5cQYaF*7$$\"1L$ez6:B1\"F*$!1[)H1Vu]8\"F-7$$\"1+D1o(oX1\"F* $!1P\"**[*RVzIF-7$$\"1nm;=C#o1\"F*$!1G`'y!yc%*[F-7$$\"1nmTb:to5F*$!1Ea IZoZziF-7$$\"1nmm#pS12\"F*$!12Jd5' )F-7$$\"1+]i`A3v5F*$!1/hq8*f.V*F-7$$\"1X'=)o&)*F-7$$\"1U5Si%Rv2\"F*$!1cx]`&4%>**F-7$$\"1]i:/ 4.y5F*$!1e&e?h?T'**F-7$$\"1e9\"fMA&y5F*$!1U'eAu$)4***F-7$$\"1nmm(y8!z5 F*$!04>er*******!#:7$$\"1e9;xHbz5F*$!1j\"4h%*)R*)**F-7$$\"1]ilm@4!3\"F *$!1?U9A8Wd**F-7$$\"1U5:c8j!3\"F*$!15MVz*eU!**F-7$$\"1LekX0<\"3\"F*$!1 u5PT(G+$)*F-7$$\"1'*F-7$$\"1+]i.tK$3\"F*$!1gmBJ $)oF$*F-7$$\"1++DZ5Q&3\"F*$!1&f0cvT$f&)F-7$$\"1+](3zMu3\"F*$!1E*zoFs[` (F-7$$\"1Le*olx&*3\"F*$!1^+b%*zQHiF-7$$\"1nm\"H_?<4\"F*$!1PAz-#fps%F-7 $$\"1nT&GM)o$4\"F*$!1?MG!\\Or@$F-7$$\"1n;zihl&4\"F*$!1PWng#pfi\"F-7$$ \"1+vVA(yx4\"F*$\"1Va;x`v98F*7$$\"1LL3#G,**4\"F*$\"1$pImGZ\"y=F-7$$\"1 L3-Dg5-6F*$\"14%yBa_Pi$F-7$$\"1L$ezw5V5\"F*$\"1!o(R2FjV_F-7$$\"1nm;.+B 16F*$\"1I,zLDI6lF-7$$\"1+]PQ#\\\"36F*$\"1r@\\SO1:wF-7$$\"1m\"z\\1A-6\" F*$\"1Ab*H(*4Tf)F-7$$\"1LLe\"*[H76F*$\"1/A7o_^F$*F-7$$\"1+v$zglL6\"F*$ \"1X3LX>*Rg*F-7$$\"1n;HCjV96F*$\"1s\")[+jR3)*F-7$$\"1](oCor\\6\"F*$\"1 R#zst/K))*F-7$$\"1LekSq]:6F*$\"1qP(eMC'R**F-7$$\"11 bNgx**F-7$$\"1+++dxd;6F*$\"1$*[u(=Dr***F-7$$\"1]7`+:5<6F*$\"1+S[2#y$)* **F-7$$\"1+D1W_i<6F*$\"1]0-\"QH?)**F-7$$\"1]Pf()*[\"=6F*$\"1B_\"\\(o:[ **F-7$$\"1+]7JFn=6F*$\"1x'*)\\Jpo*)*F-7$$\"1+v==-s>6F*$\"1t<)3WLEu*F-7 $$\"1++D0xw?6F*$\"1c;zxQy?&*F-7$$\"1+vV+ZzA6F*$\"1'Q\\c4+q!*)F-7$$\"1+ ]i&p@[7\"F*$\"1VW=0e/m!)F-7$$\"1+v=GB2F6F*$\"1.?y_vT&*oF-7$$\"1++vgHKH 6F*$\"1pDmlV%R^&F-7$$\"1L$3UDX88\"F*$\"1*=wDGn+8%F-7$$\"1nmmZvOL6F*$\" 1iF$)QAQ[EF-7$$\"1LLexn_N6F*$\"1O\\Z'>>4+\"F-7$$\"1++]2goP6F*$!16C+qY) Ho'F*7$$\"1n\"H2fU'R6F*$!1Hqq9Sog@F-7$$\"1L$eR<*fT6F*$!1*=o(RE8*f$F-7$ $\"1m\"HiBQP9\"F*$!1#[L&H%)es]F-7$$\"1++])Hxe9\"F*$!1aVzZ&4bS'F-7$$\"1 MeR*)**)y9\"F*$!1QM^]\\7,vF-7$$\"1n;H!o-*\\6F*$!1iBN%)4c=%)F-7$$\"1L$3 A_1?:\"F*$!1SGc%*R:m\"*F-7$$\"1+]7k.6a6F*$!1k()RY\")e\"o*F-7$$\"1d6F*$!1c'>&\\\"yv***F-7$$\"1f9m@*4x:\"F*$!1 g#QcQf')***F-7$$\"1nm;WTAe6F*$!1h]dE,.&)**F-7$$\"1]7yI3If6F*$!1$HnW$Q8 4**F-7$$\"1LeRmI*F-7$$\"1m\"z\\%[gk6F*$!12g*oPc$R')F-7$$\"1 LLL*zym;\"F*$!1.DVe&)=txF-7$$\"1L$3sr*zo6F*$!1JI=wV![q'F-7$$\"1LL3N1#4 <\"F*$!1A'4$y6EzaF-7$$\"1+vo!*R-t6F*$!19NaZX6RTF-7$$\"1n;HYt7v6F*$!1*z (\\h(ptq#F-7$$\"1Lek6,1x6F*$!1O[f]7AS8F-7$$\"1+++xG**y6F*$\"1x\"F*$\"1)3%Q$*er^yF-7$$\"1+ ++PDj$>\"F*$\"16`mi.()\\')F-7$$\"1+]P?Wl&>\"F*$\"1'fZ$z3tr#*F-7$$\"1++ +++++7F*$\"11r>t#4r'**F--%'COLOURG6&%$RGBG$\"#5!\"\"\"\"!Ffdm-%+AXESLA BELSG6$Q\"x6\"%!G-%%VIEWG6$;$\"\"\"!\"#$\"#7!\"$%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "From this graph, \+ we see that the first zero after 0.01 is between 0.0102 and 0.0103. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "evalf(newton(sin(1/x),x=0.01025,info=true),30);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\"E>$=%*[A'=`$QK*z\"e\"3 #o-\"!#P" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$ \"E2k'GW'>BYFE2HC%31o-\"!#P" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7appro ximation~3~~->~~~G$\"E&o.;g+X2)e(3%QY%31o-\"!#P" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\"EM-)QVM#GGf(3%QY%31o-\"!#P " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~5~~->~~~G$\"EM-)QV M#GGf(3%QY%31o-\"!#P" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?MWBGGf(3%Q Y%31o-\"!#J" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The required zero is 0.102680608446384087592828234434." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 4 "Note" } {TEXT -1 20 ": Maple's procedure " }{TEXT 0 6 "fsolve" }{TEXT -1 63 " \+ will look for the root of an equation inside a given interval." }} {PARA 0 "" 0 "" {TEXT -1 30 "We can find our zero by using " }{TEXT 0 6 "fsolve" }{TEXT -1 19 " with the interval " }{TEXT 304 16 "x=0.0102. .0.0103" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "evalf(fsolve(sin(1/x),x=0.0102..0.0 103),30);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?MWBGGf(3%QY%31o-\"!#J " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "If w e give " }{TEXT 0 6 "fsolve" }{TEXT -1 14 " the interval " }{TEXT 304 14 "x=0.01..0.0102" }{TEXT -1 28 ", we get no answer, because " } {XPPEDIT 18 0 "sin(1/x)" "6#-%$sinG6#*&\"\"\"F'%\"xG!\"\"" }{TEXT -1 30 " has no zero in this interval." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "evalf(fsolve(sin(1/x),x=0.01 ..0.0102),30);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'fsolveG6%-%$sinG6 #*&\"\"\"F*%\"xG!\"\"F+;$F*!\"#$\"$-\"!\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 7 " }{TEXT 321 40 ".. illustratio n of quadratic convergence" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 91 "In good situation s Newton's method converges rapidly. In fact it exhibits what is known as " }{TEXT 267 20 "quadratic convegence" }{TEXT -1 106 ". Roughly sp eaking this means that the number of correct digits approximately doub les with each iteration." }}{PARA 0 "" 0 "" {TEXT -1 90 "Here is an ex ample where the number of correct digits exactly doubles with each ite ration." }}{PARA 0 "" 0 "" {TEXT -1 13 "The equation " }{XPPEDIT 18 0 "1/x=3" "6#/*&\"\"\"F%%\"xG!\"\"\"\"$" }{TEXT -1 18 " has the solution " }{XPPEDIT 18 0 "x=1/3" "6#/%\"xG*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 20 " = 0.333333 . . . . " }}{PARA 0 "" 0 "" {TEXT -1 53 "We can apply New ton's method with the starting value " }{XPPEDIT 18 0 "x=0" "6#/%\"xG \"\"!" }{TEXT -1 3 ".3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "f := x -> 1/x-3:\n'f(x)'=f(x);\nx-' f(x)'/Diff('f(x)',x)=expand(x-f(x)/diff(f(x),x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&*&\"\"\"F*F'!\"\"F*\"\"$F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"xG\"\"\"*&-%\"fG6#F%F&-%%DiffG6$F(F%!\"\" F.,&*&\"\"#F&F%F&F&*&\"\"$F&)F%F1F&F." }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "The Newton iteration formula is" }} {PARA 258 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x[n+1]=x[n]*(2-3*x[n ])" "6#/&%\"xG6#,&%\"nG\"\"\"F)F)*&&F%6#F(F),&\"\"#F)*&\"\"$F)&F%6#F(F )!\"\"F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " } {XPPEDIT 18 0 "x[0]=0.3" "6#/&%\"xG6#\"\"!-%&FloatG6$\"\"$!\"\"" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x[1] = ``(.3)*`.`*(2-.9);" "6#/&%\"xG6#\"\"\"*(-%!G6#-%&FloatG6$\"\"$!\"\" F'%\".GF',&\"\"#F'-F-6$\"\"*F0F0F'" }{TEXT -1 3 " = " }{XPPEDIT 18 0 " ``(.3)*`.`*``(1.1) = .33;" "6#/*(-%!G6#-%&FloatG6$\"\"$!\"\"\"\"\"%\". GF--F&6#-F)6$\"#6F,F--F)6$\"#L!\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x[2] = ``(.33)*`.`*(2-.99);" "6#/&%\"x G6#\"\"#*(-%!G6#-%&FloatG6$\"#L!\"#\"\"\"%\".GF1,&F'F1-F-6$\"#**F0!\" \"F1" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "``(.33)*`.`*``(1.01) = .3333; " "6#/*(-%!G6#-%&FloatG6$\"#L!\"#\"\"\"%\".GF--F&6#-F)6$\"$,\"F,F--F)6 $\"%LL!\"%" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x[3] = ` `(.3333)*`.`*(2-.9999);" "6#/&%\"xG6#\"\"$*(-%!G6#-%&FloatG6$\"%LL!\"% \"\"\"%\".GF1,&\"\"#F1-F-6$\"%****F0!\"\"F1" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "``(.3333)*`.`*``(1.0001) = .33333333;" "6#/*(-%!G6#-%&F loatG6$\"%LL!\"%\"\"\"%\".GF--F&6#-F)6$\"&,+\"F,F--F)6$\")LLLL!\")" } {TEXT -1 7 " etc." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "evalf(newton(1/x=3,x=0.3,info=true),32);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\"G++++++++ ++++++++++L!#Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~- >~~~G$\"G+++++++++++++++++LL!#Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7a pproximation~3~~->~~~G$\"G+++++++++++++++LLLL!#Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\"G+++++++++++LLLLLLLL!#Q" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~5~~->~~~G$\"G+++LLLLLL LLLLLLLLLL!#Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~6~~-> ~~~G$\"GLLLLLLLLLLLLLLLLLLL!#Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7ap proximation~7~~->~~~G$\"GLLLLLLLLLLLLLLLLLLL!#Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"ALLLLLLLLLLLLLLLL!#K" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 10 "Example 8 " }{TEXT 322 46 ".. slow convergence and problems with accuracy" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 49 "Newton's method does not always converge rapidly." }}{PARA 0 "" 0 "" {TEXT -1 15 "The polynomial " }{XPPEDIT 18 0 "x^4-4*x^2+4=(x^2-2)^2 " "6#/,(*$%\"xG\"\"%\"\"\"*&F'F(*$F&\"\"#F(!\"\"F'F(*$,&*$F&F+F(F+F,F+ " }{TEXT -1 14 " has the zero " }{XPPEDIT 18 0 "x = sqrt(2)" "6#/%\"xG -%%sqrtG6#\"\"#" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 41 "Newt on's method, with the starting value " }{TEXT 289 1 "x" }{TEXT -1 45 " = 1.5, converges rather slowly to this zero." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "newton(x^4-4 *x^2+4,x=1.5,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approxim ation~1~~->~~~G$\"/LLLLLe9!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7app roximation~2~~->~~~G$\"/r&G92mV\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%7approximation~3~~->~~~G$\"/qT>w\\D9!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\"/Io@z()>9!#8" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%7approximation~5~~->~~~G$\"/SF\"R^qT\"!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~6~~->~~~G$\"/cg(*Qj:9! #8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~7~~->~~~G$\"/#\\ i3C\\T\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~8~~->~~ ~G$\"/]\\8*oXT\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation ~9~~->~~~G$\"/YBg7R99!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approxim ation~10~~->~~~G$\"/O%oT-VT\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8 approximation~11~~->~~~G$\"/g'3*zD99!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~12~~->~~~G$\"/3swdB99!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~13~~->~~~G$\"/#f)pYA99!#8" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%8approximation~14~~->~~~G$\"//!o6>UT\"!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~15~~->~~~G$\"/GvRj@99! #8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~16~~->~~~G$\"/zI ^\\@99!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~17~~->~~~ G$\"/>`cU@99!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~18~ ~->~~~G$\"/%\\$4R@99!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approxima tion~19~~->~~~G$\"/D$et8UT\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8a pproximation~20~~->~~~G$\"/[5\\O@99!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~21~~->~~~G$\"/$Qdg8UT\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~22~~->~~~G$\"/>1%e8UT\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~23~~->~~~G$\"/6AtN@99!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~24~~->~~~G$\"/tznN@99! #8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~25~~->~~~G$\"/Q3 lN@99!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~26~~->~~~G $\"/$GPc8UT\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~27 ~~->~~~G$\"/30jN@99!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximat ion~28~~->~~~G$\"/@riN@99!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8appr oximation~29~~->~~~G$\"/EaiN@99!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %8approximation~30~~->~~~G$\"/zXiN@99!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~31~~->~~~G$\"/bTiN@99!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~32~~->~~~G$\"/VRiN@99!#8" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%8approximation~33~~->~~~G$\"/PQiN@99!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+iN@99!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "More details concerning t he computation can be seen by using the option \"" }{TEXT 304 6 "info= 3" }{TEXT -1 3 "\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "newton(x^4-4*x^2+4,x=1.5,info=3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%BAttempting~to~calculate~a~zero~ofG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"%\"\"\"F(*&F'F()F&\"\" #F(!\"\"F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Iby~Newton's~method,~ using~the~derivativeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"%\"\" \")%\"xG\"\"$F&F&*&\"\")F&F(F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%E**~working~precision~is~14~d igits~**G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"$D'!\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG$\"%+:!\"$" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%2correction~~->~~~G/,$*&%&valueG\"\"\"%+derivat iveG!\"\"F*$!/nmmmmmT!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\"/LLLLLe9!# 8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"-'f=/ig\"!#8" }}{PARA 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" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\"/Io@z()>9!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%&valueG$\"+=%)3xD!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivati veG$\",de(\\<\"*!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2correction~~- >~~~G/,$*&%&valueG\"\"\"%+derivativeG!\"\"F*$!/Q+4/`EG!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approx imation~5~~->~~~G$\"/SF\"R^qT\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"*%Gbbk!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG$\",-C-Ub%!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2correction~~->~~~G/,$*&%&valueG\"\"\"%+derivativ eG!\"\"F*$!/)y$oO\\<9!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~6~~->~~~G$\"/cg(*Qj:9! #8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"*-.bh\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 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{XPPMATH 20 "6 #/%&valueG$\"(Tk_#!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivative G$\"+y;qVG!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2correction~~->~~~G/ ,$*&%&valueG\"\"\"%+derivativeG!\"\"F*$!/Ao(4RV)))!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approxima tion~10~~->~~~G$\"/O%oT-VT\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"'^;j!#8" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%+derivativeG$\"+:b!=U\"!#7" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%2correction~~->~~~G/,$*&%&valueG\"\"\"%+derivativeG! \"\"F*$!/ouixfUW!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~11~~->~~~G$\"/g'3*zD99!#8" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%&valueG$\"'=z:!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivative G$\"*&Q*)3r!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2correction~~->~~~G /,$*&%&valueG\"\"\"%+derivativeG!\"\"F*$!/HD~~~G$\"/3swdB99!#8" 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11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~17~~->~~~G$\"/>`cU@99!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%&valueG$\"$'Q!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%P**~increasing ~working~precision~to~16~digits~**G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%&valueG$\"%cQ!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG$ \"+Q\\l56!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2correction~~->~~~G/, $*&%&valueG\"\"\"%+derivativeG!\"\"F*$!/b/y`#=Z$!#?" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximatio n~18~~->~~~G$\"/%\\$4R@99!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"$j*!#:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%P**~increasing~working~precision~to~17~digits~**G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"%L'*!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG$\"+MEi^b!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2correction~~->~~~G/,$*&%&valueG\"\"\"%+derivativeG!\" \"F*$!/dfq$o^t\"!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~19~~->~~~G$\"/D$et8UT\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"%2C!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivative G$\"+4~~~G/ ,$*&%&valueG\"\"\"%+derivativeG!\"\"F*$!/IApYxs')!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximati on~20~~->~~~G$\"/[5\\O@99!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"$,'!#;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%P**~increasing~working~precision~to~18~digits~**G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"%=g!#<" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%+derivativeG$\",\"ytq(Q\"!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2correction~~->~~~G/,$*&%&valueG\"\"\"%+derivativeG!\" \"F*$!/FPM\"\\mL%!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~21~~->~~~G$\"/$Qdg8UT\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"%/:!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivative G$\"+4UVQp!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2correction~~->~~~G/ ,$*&%&valueG\"\"\"%+derivativeG!\"\"F*$!/%>C.Ow;#!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximati on~22~~->~~~G$\"/>1%e8UT\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"$x$!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%P**~increasing~working~precision~to~19~digits~**G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"%iP!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG$\",%=j@qM!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2correction~~->~~~G/,$*&%&valueG\"\"\"%+derivativeG!\" \"F*$!/iluA3%3\"!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~23~~->~~~G$\"/6AtN@99!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%&valueG$\"$T*!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%P**~increasing ~working~precision~to~20~digits~**G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%&valueG$\"%9%*!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG $\"-/>YoN~~~G/ ,$*&%&valueG\"\"\"%+derivativeG!\"\"F*$!/)\\nS'zBa!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approxima tion~24~~->~~~G$\"/tznN@99!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"%bB!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG$\",+zr(y')!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2correction~~->~~~G/,$*&%&valueG\"\"\"%+derivativeG!\" \"F*$!/s;+w^8F!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%8approximation~25~~->~~~G$\"/Q3lN@99!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%&valueG$\"$(e!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%P**~increasing ~working~precision~to~21~digits~**G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%&valueG$\"%ze!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG$ \"-:#fVrL%!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2correction~~->~~~G/ ,$*&%&valueG\"\"\"%+derivativeG!\"\"F*$!/v/H/]b8!#A" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximatio n~26~~->~~~G$\"/$GPc8UT\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"%p9!#?" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%+derivativeG$\"-K'*GMo@!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2correction~~->~~~G/,$*&%&valueG\"\"\"%+derivativeG!\" \"F*$!/+YX(eZx'!#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~27~~->~~~G$\"/30jN@99!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%&valueG$\"$p$!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%P**~increasing ~working~precision~to~22~digits~**G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%&valueG$\"%tO!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG$ \".(p%\\\"Q%3\"!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2correction~~-> ~~~G/,$*&%&valueG\"\"\"%+derivativeG!\"\"F*$!/B$RK%=(Q$!#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8app roximation~28~~->~~~G$\"/@riN@99!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"$@*!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%P**~increasing~working~precision~to~23~digi ts~**G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"%$>*!#A" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG$\".OG-?VU&!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2correction~~->~~~G/,$*&%&valueG\"\"\"%+der ivativeG!\"\"F*$!/*GujuZp\"!#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~29~~->~~~G$\"/EaiN @99!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"%)H#!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%+derivativeG$\".'[,1o7F!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2corr ection~~->~~~G/,$*&%&valueG\"\"\"%+derivativeG!\"\"F*$!/B(*p`Kr%)!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~30~~->~~~G$\"/zXiN@99!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"$v&! #A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%P**~increasing~working~precisio n~to~24~digits~**G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"%ed! #B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG$\"/b#>aosN\"!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2correction~~->~~~G/,$*&%&valueG\" \"\"%+derivativeG!\"\"F*$!/1JfRMUU!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~31~~->~~~G$ \"/bTiN@99!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"%R9!#B" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%+derivativeG$\".TM3N\\y'!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2c orrection~~->~~~G/,$*&%&valueG\"\"\"%+derivativeG!\"\"F*$!/:j=^(37#!#C " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~32~~->~~~G$\"/VRiN@99!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"$g$! #B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%P**~increasing~working~precisio n~to~25~digits~**G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"%%f$ !#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG$\"/7S$*[`\"R$!#B " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2correction~~->~~~G/,$*&%&valueG \"\"\"%+derivativeG!\"\"F*$!/Q3\">(pf5!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~ 33~~->~~~G$\"/PQiN@99!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+iN@99!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The main problem here is that " }{XPPEDIT 18 0 "x = \+ sqrt(2)" "6#/%\"xG-%%sqrtG6#\"\"#" }{TEXT -1 32 " is a zero of the pol ynomial of " }{TEXT 267 12 "multiplicity" }{TEXT -1 35 " 2, that is, t he associated factor " }{XPPEDIT 18 0 "x-sqrt(2);" "6#,&%\"xG\"\"\"-%% sqrtG6#\"\"#!\"\"" }{TEXT -1 43 " appears twice in the linear factoris ation " }{XPPEDIT 18 0 "x^4-4*x^2+4=(x-sqrt(2))*(x-sqrt(2))*(x+sqrt(2) )*(x+sqrt(2))" "6#/,(*$%\"xG\"\"%\"\"\"*&F'F(*$F&\"\"#F(!\"\"F'F(**,&F &F(-%%sqrtG6#F+F,F(,&F&F(-F06#F+F,F(,&F&F(-F06#F+F(F(,&F&F(-F06#F+F(F( " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 59 "This means that, as w ell as being a zero of the polynomial " }{XPPEDIT 18 0 "f(x) = (x^2-2) ^2;" "6#/-%\"fG6#%\"xG*$,&*$F'\"\"#\"\"\"F+!\"\"F+" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "x = sqrt(2);" "6#/%\"xG-%%sqrtG6#\"\"#" }{TEXT -1 38 " \+ is also a zero of the derivative f '(" }{TEXT 338 1 "x" }{TEXT -1 1 ") " }{XPPEDIT 18 0 "`` = 4*x*(x^2-2);" "6#/%!G*(\"\"%\"\"\"%\"xGF',&*$F( \"\"#F'F+!\"\"F'" }{TEXT -1 41 ". The zero occurs at a minimum point a nd " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 38 " does not c hange sign across the zero." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot(x^4-4*x^2+4,x=-1..2);" }} {PARA 13 "" 1 "" {GLPLOT2D 292 205 205 {PLOTDATA 2 "6%-%'CURVESG6$7_o7 $$!\"\"\"\"!$\"\"\"F*7$$!1*****\\P&3Y$*!#;$\"1.XOax,p7!#:7$$!1++Dcx6x( )F0$\"1F/ie.(>^\"F37$$!1,+]iTDP\")F0$\"10/1$F37$$!1++D\"oS:P%F0$\"1J27**e5sKF37$$!1+++v@)*=PF0$\"1*Rg*RF37$$\"19+++DFOB!#=$\"1:INn\"y***RF37$$\"1,++DJL(4$Fbp$ \"1bANJN;'*RF37$$\"1,+++!R5'fFbp$\"1txDK!*z&)RF37$$\"1++v=(4AH*Fbp$\"1 yhG!\\Ob'RF37$$\"1++vV!QBE\"F0$\"1j#3*HS^ORF37$$\"1******\\\"o?&=F0$\" 1!>D+MqR'QF37$$\"1,+vVb4*\\#F0$\"12Pue93aPF37$$\"1,+DJ'=_6$F0$\"1F%)H; WB@OF37$$\"1,+]P%y!ePF0$\"1I@!\\??]X$F37$$\"1,+v=WU[VF0$\"1<0`bCSzKF37 $$\"1++]7B>&)\\F0$\"1))eW-snnIF37$$\"1++v$>:mk&F0$\"1Umy(4!HEGF37$$\"1 ++DcdQAiF0$\"1p$[ni&=,EF37$$\"1,+]PPBWoF0$\"18tBj-pXBF37$$\"1******\\N m'[(F0$\"1\\*=`Zc@2#F37$$\"1****\\(yb^6)F0$\"1%)>5))yY*z\"F37$$\"1++vV VDB()F0$\"1c')G!yS_`\"F37$$\"1++]7TW)R*F0$\"1o%*y+@+Z7F37$$\"1+++:K^+5 F3$\"1X/?tmZz**F07$$\"1++]7,Hl5F3$\"1dP%[$f'\\[(F07$$\"1+]P4w)R7\"F3$ \"1\"3'fR&flU&F07$$\"1++]x%f\")=\"F3$\"1zqZY*)pgMF07$$\"1+]P/-a[7F3$\" 1F07$$\"1+](=Yb;J\"F3$\"1AQ8FlP:yFbp7$$\"1+v=7)3DM\"F3$\"1M/R ;kM2RFbp7$$\"1++]i@Ot8F3$\"1DjSyFy'H\"Fbp7$$\"1](=7$$\"1]i:gI\"=U\"F3$\"1'>fy)* \\]k%F\\z7$$\"1+]PfL'zV\"F3$\"1Z42L5_)e%Fbq7$$\"1+vouE2p9F3$\"1\\=())) z\">]#Fbp7$$\"1+++!*>=+:F3$\"1y(ez'*HtF'Fbp7$$\"1+]7ed*>`\"F3$\"1AoUhj ;/7F07$$\"1++DE&4Qc\"F3$\"1wwT)e/Z)>F07$$\"1+]P%>5pi\"F3$\"1$3po?yR=%F 07$$\"1+vou;!fl\"F3$\"1x#ovo$z0bF07$$\"1+++bJ*[o\"F3$\"11HzQR%p.(F07$$ \"1+]7j17=F37$$\"1+]P/)fT(=F3$\"1s&R>a!e(G#F37$$\"1+++b!)[/>F3$\" 1FoTZAPZEF37$$\"1+]i0j\"[$>F3$\"1)[=UaT)RIF37$$\"1](=#HA6^>F3$\"1tBgAp mkKF37$$\"1+D\"G:3u'>F3$\"1_)yM=*\\*\\$F37$$\"1]iSwSq$)>F3$\"1/qa:=aWP F37$$\"\"#F*$\"\"%F*-%'COLOURG6&%$RGBG$\"#5F)F*F*-%+AXESLABELSG6$Q\"x6 \"%!G-%%VIEWG6$;F(Fb`l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "You can see how this affects the convergence by using the procedure " }{TEXT 0 11 "newton_step" }{TEXT -1 23 " (see example 3 o f the " }{TEXT 0 11 "newton_step" }{TEXT -1 147 " examples). All the p ictures at the various scales look \"about the same\", with tangent li ne providing a rather poor approximation for the function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "There is another \+ potential problem in this example related to the accuracy of the compu ted zero." }}{PARA 0 "" 0 "" {TEXT -1 37 "The current version of the p rocedure " }{TEXT 0 6 "newton" }{TEXT -1 196 " working in the default \+ mode circumvents this problem by increasing the working precision for \+ the calculation as the computation procedes. This feature can be switc hed off by means of the option \"" }{TEXT 304 15 "precision=fixed" } {TEXT -1 3 "\". " }}{PARA 0 "" 0 "" {TEXT -1 5 "With " }{TEXT 268 6 "D igits" }{TEXT -1 61 " set to 10 the working precision remains fixed at 14 digits. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "newton(x^4-4*x^2+4,x=1.5,info=true,prcsn=fixed);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\"/LLLLLe9 !#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\"/r& G92mV\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~ G$\"/qT>w\\D9!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~ ~->~~~G$\"/Io@z()>9!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximat ion~5~~->~~~G$\"/SF\"R^qT\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7ap proximation~6~~->~~~G$\"/cg(*Qj:9!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%7approximation~7~~->~~~G$\"/#\\i3C\\T\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~8~~->~~~G$\"/]\\8*oXT\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~9~~->~~~G$\"/YBg7R99!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~10~~->~~~G$\"/O%oT-VT \"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~11~~->~~~G$\" /g'3*zD99!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~12~~-> ~~~G$\"/3swdB99!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~ 13~~->~~~G$\"/#f)pYA99!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approxi mation~14~~->~~~G$\"//!o6>UT\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% 8approximation~15~~->~~~G$\"/GvRj@99!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~16~~->~~~G$\"/zI^\\@99!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~17~~->~~~G$\"/?$QD9UT\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~18~~->~~~G$\"/cN5R@99!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~19~~->~~~G$\"/'4([P@99 !#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~20~~->~~~G$\"/ \"o^r8UT\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~21~~- >~~~G$\"/\"o^r8UT\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+:P@99!\" *" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "The resulting value is correct to 7 digits rather than 10 digits. " }} {PARA 0 "" 0 "" {TEXT -1 21 "A 10 digit value for " }{XPPEDIT 18 0 "sq rt(2)" "6#-%%sqrtG6#\"\"#" }{TEXT -1 9 " is . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf(sqrt(2 ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+iN@99!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 195 "The reason for the po or accuracy in the value calculated above using Newton's method with f ixed working precision is that rather inaccurate approximations for th e root will still give values for " }{XPPEDIT 18 0 "x^4-4*x^2" "6#,&*$ %\"xG\"\"%\"\"\"*&F&F'*$F%\"\"#F'!\"\"" }{TEXT -1 20 " which are equal to " }{XPPEDIT 18 0 "-4" "6#,$\"\"%!\"\"" }{TEXT -1 60 " correct to 1 0 digits, so that adding 4 effectively gives 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "To see why this happens l et " }{XPPEDIT 18 0 "X=sqrt(2)+d" "6#/%\"XG,&-%%sqrtG6#\"\"#\"\"\"%\"d GF*" }{TEXT -1 10 ", so that " }{TEXT 290 1 "X" }{TEXT -1 31 " deviate s from the solution by " }{TEXT 287 1 "d" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "X := \+ sqrt(2)+d;\nexpand(X^4-4*X^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" XG,&*$-%%sqrtG6#\"\"#\"\"\"F+%\"dGF+" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,*!\"%\"\"\"*$)%\"dG\"\"#F%\"\")*&-%%sqrtG6#F)F%)F(\"\"$F%\"\"%*$)F( F1F%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus " }{XPPEDIT 18 0 "X^4-4*X^2+4;" "6#,(*$%\"XG\"\"%\"\"\"*&F&F'*$F %\"\"#F'!\"\"F&F'" }{TEXT -1 15 " deviates from " }{XPPEDIT 18 0 "-4" "6#,$\"\"%!\"\"" }{TEXT -1 5 " by " }{XPPEDIT 18 0 "8*d^2+4*sqrt(2)*d ^3+d^4" "6#,(*&\"\")\"\"\"*$%\"dG\"\"#F&F&*(\"\"%F&-%%sqrtG6#F)F&F(\" \"$F&*$F(F+F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 59 "The main point here is that the error does not involve a \"" }{TEXT 288 1 "d " }{TEXT -1 7 "\" term." }}{PARA 0 "" 0 "" {TEXT -1 12 "In fact, if " }{TEXT 291 1 "d" }{TEXT -1 38 " is small, the error is approximately \+ " }{XPPEDIT 18 0 "8*`.`*d^2;" "6#*(\"\")\"\"\"%\".GF%%\"dG\"\"#" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 9 "Thus, if " }{XPPEDIT 18 0 "8*`.`*d^2 < 5*`.`*10^(-10);" "6#2*(\"\")\"\"\"%\".GF&%\"dG\"\"#*(\" \"&F&F'F&)\"#5,$F-!\"\"F&" }{TEXT -1 25 ", the value obtained for " } {XPPEDIT 18 0 "X^4-4*X^2" "6#,&*$%\"XG\"\"%\"\"\"*&F&F'*$F%\"\"#F'!\" \"" }{TEXT -1 52 " will register as -4.000000000 correct to 10 digits. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evalf(sqrt(5/8)*10^(-5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+]Tp0z!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Thus we can have an error of about 0" }{XPPEDIT 18 0 ".79*`.`*1 0^(-5);" "6#*(-%&FloatG6$\"#z!\"#\"\"\"%\".GF))\"#5,$\"\"&!\"\"F)" } {TEXT -1 72 " in the value of the root, and still effectively obtain t he value 0 for " }{XPPEDIT 18 0 "x^4-4*x^2+4" "6#,(*$%\"xG\"\"%\"\"\"* &F&F'*$F%\"\"#F'!\"\"F&F'" }{TEXT -1 38 " when working with 10 digit p recision." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "evalf(subs(x=sqrt(2.)+.79*10^(-5),x^4-4*x^2),15);\nev alf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!0s+&********R!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$!+++++S!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "The error in the root 1.4 1421398 calculated above is less than the maximum 0" }{XPPEDIT 18 0 ". 79*`.`*10^(-5);" "6#*(-%&FloatG6$\"#z!\"#\"\"\"%\".GF))\"#5,$\"\"&!\" \"F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 54 "The error problem here could be regarded as a type of " }{TEXT 267 17 "subtraction erro r" }{TEXT -1 9 " problem." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "sqrt(2.)-1.41421398;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$!$=%!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 104 "This error problem (but not the slow con vergence) can be avoided by using the alternative factored form " } {XPPEDIT 18 0 "(x^2-2)^2" "6#*$,&*$%\"xG\"\"#\"\"\"F'!\"\"F'" }{TEXT -1 18 "for the poynomial " }{XPPEDIT 18 0 "x^4-4*x^2+4" "6#,(*$%\"xG\" \"%\"\"\"*&F&F'*$F%\"\"#F'!\"\"F&F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "newton((x^ 2-2)^2,x=1.5,info=3,precision=fixed);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%BAttempting~to~calculate~a~zero~ofG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$),&*$)%\"xG\"\"#\"\"\"F*F)!\"\"F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Iby~Newton's~method,~using~the~derivativeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"%\"\"\",&*$)%\"xG\"\"#F&F&F+!\"\"F&F *F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%E**~working~precision~is~14~digits~**G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"$D'!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG$\"%+:!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%1r atio~correctionG$!/nmmmmmT!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\"/LLLLLe9 !#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"/S&f=/ig\"!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG$\"/!3[\")RHR(!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%1ratio~correctionG$!/thZ!>E<#!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\" /r&G92mV\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"/.q&=cb2%!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG$\"/*[UwC&oO!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%1ratio~correctionG$!/`&RM_46\"!#:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximatio n~3~~->~~~G$\"/vT>w\\D9!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"/$*y=NyE5!#;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%+derivativeG$\"/+)p?:r#=!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%1ratio~correctionG$!/-rLxp>c!#;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximatio n~4~~->~~~G$\"/Qo@z()>9!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"/FW<%)3xD!#<" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%+derivativeG$\"/O*pe(\\<\"*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%1ratio~correctionG$!/;*zSIl#G!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 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"6#/%&valueG$\"/o51 $\\/,\"!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG$\"/k6b8w( o&!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%1ratio~correctionG$!/sQGD`w ~~~G$\"/&f-E\"R99!#8" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG $\"/])y_Sk_#!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG$\"/C !o2-P%G!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%1ratio~correctionG$!/ \\_\">OV)))!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~10~~->~~~G$\"/w*oT-VT\"!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%&valueG$\"/T%\\1)\\;j!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+deri vativeG$\"/kmzj!=U\"!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%1ratio~co rrectionG$!/kqmmeUW!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~11~~->~~~G$\"/4.\"*zD9 9!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"/AT=T~~~G$\" 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Then " }{XPPEDIT 18 0 "f(x)" "6#-%\"f G6#%\"xG" }{TEXT -1 24 " has a single root at 0." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 116 "If we apply Newton's me thod with the starting approximation 4, we end up alternating forever \+ between the two values " }{XPPEDIT 18 0 "-4" "6#,$\"\"%!\"\"" }{TEXT -1 56 " and +4. This is illustrated by the following picture. " }} {PARA 258 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 385 267 267 {PLOTDATA 2 "6 ,-%'CURVESG6$7V7$$!\"'\"\"!$!3)y<$yU(*[\\C!#<7$$!3z******\\TVQdF-$!3qj 9F,H]&R#F-7$$!3l****\\-r%3^&F-$!39ia#3N>vM#F-7$$!3A+++l;!\\D&F-$!3)R(R #*GsN#H#F-7$$!3o*****\\lfs*\\F-$!3PNzuL^XNAF-7$$!3%)****\\s@%3u%F-$!31 KxV3vMx@F-7$$!3J++]U.6.XF-$!3np8oJL0A@F-7$$!3')****\\-G&pD%F-$!3IOW*fU QK1#F-7$$!3(*****\\AjP-SF-$!3(=D=C*Rf+?F-7$$!33++]sih[PF-$!3Jb^BjV8O>F -7$$!3%)******pGf([$F-$!31-e_/)4v'=F-7$$!3)******\\J$odKF-$!3K\"zLzM0 \\!=F-7$$!3y******4'f))*HF-$!3Zx7/u:sJF-$!3Ljkc0d!4T\"F-7$$!3'*******R%e:w\"F-$!33* >^`9PsK\"F-7$$!33++]#yk]\\\"F-$!3(f06tTGFA\"F-7$$!3M+++SFzkdP\"Go)F_r7$$!3I'*****\\ion\\F_r$!3'p_aI]\"=[qF_r7$$!3:'****\\K-jg #F_r$!3c(zL7r&>0^F_r7$$!3j)***\\PlwK8F_r$!3g$yVq^22l$F_r7$$!3*36+++vI# f!#?$!3Ab.sN;:'p(!#>7$$\"3)*)***\\7]hj7F_r$\"3[W+s%HOZb$F_r7$$\"3/**** *\\xgke#F_r$\"3]7-J#)es&3&F_r7$$\"3m)*******=+QPF_r$\"3vOX)35?R6'F_r7$ $\"3E)****\\-V&*)[F_r$\"3YAx)GuED*pF_r7$$\"3a-++]\\$pP(F_r$\"3M3-A;&3* )e)F_r7$$\"3y&******>am%**F_r$\"3EF9iK9Ht**F_r7$$\"3k*****\\JigC\"F-$ \"34oDX8;F;6F-7$$\"3%*****\\PF-7$$\"3v+++gzs+SF-$\"3Uj1s\")>=+?F-7$$\"3 5+++0\"Q_D%F-$\"3e37n[G#G1#F-7$$\"3q++]x2k2XF-$\"3y?bDG07B@F-7$$\"3d++ +?EdRZF-$\"3%z\\FC&f0x@F-7$$\"3M+++&o#R0]F-$\"3O\"4jN\\tsB#F-7$$\"3+++ +?`9V_F-$\"3g+v4X:z*G#F-7$$\"3G++]<#Rm\\&F-$\"3qCR=\\7\\WBF-7$$\"3F++] A_ERdF-$\"3MzH8gjn&R#F-7$$\"\"'F*$\"3)y<$yU(*[\\CF--%'COLOURG6&%$RGBG$ \"#5!\"\"$F*F*Fa\\l-F$6$7$7$F($!3++++++++]F_r7$Ff[l$\"3++++++++DF--%&C OLORG6&F]\\lFa\\l$\"\")F`\\lFa\\l-F$6$7$7$F($!3++++++++DF-7$Ff[l$\"3++ ++++++]F_rF[]l-F$6%7$7$$\"\"%F*Fa\\l7$F]^l$\"\"#F*-F[\\l6&F]\\lFa\\lFa \\l$\"*++++\"!\")-%*LINESTYLEG6#Fa^l-F$6%7$7$$!\"%F*$!\"#F*7$F^_lFa\\l Fb^lFg^l-F$6&7&F\\^lF_^lFb_lF]_l-%'SYMBOLG6#%'CIRCLEG-F[\\l6&F]\\lF*F* F*-%&STYLEG6#%&POINTG-F$6&Fe_l-Fg_l6#%(DIAMONDGFj_lF\\`l-F$6&Fe_l-Fg_l 6#%&CROSSGFj_lF\\`l-%+AXESLABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG-%%VI EWG6$;F(Ff[lFbal" 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" }}{TEXT -1 1 " " }}{PARA 258 "" 0 "" {TEXT -1 2 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "f := x -> piecewise(x < 0,- sqrt(-x),x >= 0, sqrt(x)):\n'f(x)'=f(x);\nnewton(f(x),x=4,info=true,ma xiter=10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWI SEG6$7$,$*$,$F'!\"\"#\"\"\"\"\"#F/2F'\"\"!7$*$F'F01F4F'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$!/++++++S!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\"/++++++S!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$!/++++++S!#8 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\"/+++++ +S!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~5~~->~~~G$!/+ +++++S!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~6~~->~~~G $\"/++++++S!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~7~~- >~~~G$!/++++++S!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~ 8~~->~~~G$\"/++++++S!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approxima tion~9~~->~~~G$!/++++++S!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8appro ximation~10~~->~~~G$\"/++++++S!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% 6last~iteration~gives~G$\"+++++S!\"*" }}{PARA 8 "" 1 "" {TEXT -1 67 "E rror, (in newton) reached max, 10, iterations without convergence\n" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "Similar behaviour occurs with any non-zero starting approximation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "ne wton(f(x),x=Pi,info=true,maxiter=10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$!/(*e`EfTJ!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\"/)*e`EfTJ!#8" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$!/(*e`EfTJ!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\"/)*e`EfTJ! #8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~5~~->~~~G$!/(*e` EfTJ!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~6~~->~~~G$ \"/)*e`EfTJ!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~7~~- >~~~G$!/(*e`EfTJ!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation ~8~~->~~~G$\"/)*e`EfTJ!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approxi mation~9~~->~~~G$!/(*e`EfTJ!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8ap proximation~10~~->~~~G$\"/)*e`EfTJ!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6last~iteration~gives~G$\"+aEfTJ!\"*" }}{PARA 8 "" 1 "" {TEXT -1 67 "Error, (in newton) reached max, 10, iterations without convergence \n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Example 11 " }{TEXT 361 53 ".. another pathological example giving no convergenc e" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 86 "Acknowledgement: This example is due to \+ Zdislav V. Kovarik, McMaster University, CA. " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)=x^3-5*x" "6#/-%\"fG6#%\"xG,&*$F '\"\"$\"\"\"*&\"\"&F+F'F+!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 116 "If we apply Newton's method with the starting approximat ion 1, we end up alternating forever between the two values " } {XPPEDIT 18 0 "1;" "6#\"\"\"" }{TEXT -1 56 " and -1. This is illustra ted by the following picture. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 504 339 339 {PLOTDATA 2 "6 ,-%'CURVESG6$7ao7$$!33+++++++E!#<$!3_++++++wXF*7$$!3]mm;*RFLa#F*$!3czl (=b7\\t$F*7$$!3OLLL)zam[#F*$!35zRV3o$G%HF*7$$!3?+]PrUMPCF*$!31')pv83o# H#F*7$$!3gmmTWP.)Q#F*$!3aOm8XO3y;F*7$$!34+]i**)yDL#F*$!3hLZ5s2\\G5F*7$ $!3eLL$[0CrF#F*$!3g``PgLP>U!#=7$$!3[LLL>;I@AF*$\"3!*QSO)3T?Y\"FK7$$!3Q LL$Q=za;#F*$\"3Gpoj`<9GnFK7$$!3P+]ii?#*4@F*$\"31ysjH-sc6F*7$$!3#pm;9% \\Oa?F*$\"31n-#3X^:g\"F*7$$!3=++Dyj&G+#F*$\"3gg[cvj&*z>F*7$$!3WLL3:yM^ >F*$\"3NDHEX#ykK#F*7$$!3?++vZ&zY%=F*$\"3yUm!y_Vi%HF*7$$!3ELL3tIOMV7F*$\"33ch#*fed%H%F*7$$!3;+++l$))o=\"F*$\"3ODeQ [uYiUF*7$$!33+++nWiK6F*$\"3szDX@%[,@%F*7$$!3++++p0Oy5F*$\"3Z#Q.\"Q)=y8 %F*7$$!3UML$3Lp!)z*FK$\"3U]hB?))ReRF*7$$!3hnmmw7m]?rOF* 7$$!3Znmm1*>Mj(FK$\"3c$3%[\\u\">P$F*7$$!3k++]dShykFK$\"3ma9D)y$QnHF*7$ $!3Mpmm1_TcaFK$\"3_N!f@mcdc#F*7$$!3*z***\\d5!\\L%FK$\"3I_oH6=*f3#F*7$$ !3?(***\\sV&pE$FK$\"3/wI07!4')f\"F*7$$!3OlmmTSm_@FK$\"31--WcmNm5F*7$$! 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#F*$\"3yb^J<&[,l*FK7$$\"3?++D%*p(=Q#F*$\"3q%*)Q!H$3Qg\"F*7$$\"3I+++ifW MCF*$\"3q[sFf-ebAF*7$$\"3Q++vH\\,([#F*$\"3O!o$*\\o " 0 "" {MPLTEXT 1 0 70 "f := x -> x^3-5*x:\n'f( x)'=f(x);\nnewton(f(x),x=1,info=true,maxiter=10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&*$)F'\"\"$\"\"\"F,*&\"\"&F,F'F,!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$!/++++++5!# 8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\"/++++ ++5!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$!/ ++++++5!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~ G$\"/++++++5!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~5~~ ->~~~G$!/++++++5!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation ~6~~->~~~G$\"/++++++5!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approxim ation~7~~->~~~G$!/++++++5!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7appr oximation~8~~->~~~G$\"/++++++5!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% 7approximation~9~~->~~~G$!/++++++5!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~10~~->~~~G$\"/++++++5!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6last~iteration~gives~G$\"+++++5!\"*" }}{PARA 8 "" 1 " " {TEXT -1 67 "Error, (in newton) reached max, 10, iterations without \+ convergence\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "Unlike the previous example, a slight change in the star ting approximation does lead to eventual convergence to one of the thr ee zeros of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "newton(f(x),x=1+10^(-9),info=true,maxiter=50);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&*$)F'\"\"$\"\"\"F,*&\"\"&F,F'F,!\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$!/++1+ ++5!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\" /++O+++5!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~ ~G$!/++;-++5!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~ ->~~~G$\"/4+'H,++\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximat ion~5~~->~~~G$!/L/wx++5!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approx imation~6~~->~~~G$\"//idm/+5!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7a pproximation~7~~->~~~G$!/7r%**z-+\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~8~~->~~~G$\"/L!Rt\"o,5!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~9~~->~~~G$!/!G$4W:55!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~10~~->~~~G$\"/J]b0Lj5!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~11~~->~~~G$!/&y%*4&Q&\\\"!#8 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~12~~->~~~G$!/'Q.[7 W\"R!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~13~~->~~~G$ !/1&yy-\"GH!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~14~~ ->~~~G$!/\"*>lm4BC!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximati on~15~~->~~~G$!/Au5uqbA!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approx imation~16~~->~~~G$!/=%G_@jB#!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8 approximation~17~~->~~~G$!/Ah?!ogB#!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~18~~->~~~G$!/)*\\xz1OA!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~19~~->~~~G$!/)*\\xz1OA!#8" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$!+xz1OA!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Example 12 " }{TEXT 337 57 ".. yet another pathologica l example giving no convergence" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 5 "Let " } {XPPEDIT 18 0 "f(x) = x^3-2^(1/3)*x+1;" "6#/-%\"fG6#%\"xG,(*$)%\"xG\" \"$\"\"\"F-*&)\"\"#*&F-F-F,!\"\"F-F+F-F2F-F-" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 119 "If we apply Newton's method with the sta rting approximation 0, we end up alternating forever between the two v alues: " }}{PARA 0 "" 0 "" {TEXT -1 7 "0 and " }{XPPEDIT 18 0 "2^(- 1/3)" "6#)\"\"#,$*&\"\"\"F'\"\"$!\"\"F)" }{TEXT -1 1 " " }{TEXT 362 1 "~" }{TEXT -1 19 " 0.79370052598410. " }}{PARA 0 "" 0 "" {TEXT -1 46 " This is illustrated by the following picture. 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a[l-F$6$7$7$$!3\"o*4%)f_+PzF7$\"\"#Fb[l7$$\"3]\\h(*)y]0>\"F*$!3+++++++ +]F7-%&COLORG6&F][lFa[l$\"\")F`[lFa[l-F$6$7$7$$!3S)\\?*HE]oRF7$!3+++++ +++DF77$F\\\\l$\"3++++++++vF7F`\\l-F$6%7$7$$\"3\"o*4%)f_+PzF7Fa[l7$Fd] l$\"3++++++++]F7-F[[l6&F][lFa[lFa[l$\"*++++\"!\")-%*LINESTYLEG6#Fj[l-F $6%7$7$Fa[lFa[l7$Fa[l$\"\"\"Fb[lFi]lF^^l-F$6&7&Fc]lFf]lFd^lFe^l-%'SYMB OLG6#%'CIRCLEG-F[[l6&F][lFb[lFb[lFb[l-%&STYLEG6#%&POINTG-F$6&Fj^l-F\\_ l6#%(DIAMONDGF__lFa_l-F$6&Fj^l-F\\_l6#%&CROSSGF__lFa_l-%*AXESTICKSG6$ \"\"$Fb`l-%+AXESLABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$F*F `[l$\"# " 0 "" {MPLTEXT 1 0 78 "f := x -> x^ 3-2^(1/3)*x+1:\n'f(x)'=f(x);\nnewton(f(x),x=0,info=true,maxiter=10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,(*$)F'\"\"$\"\"\"F,*& )\"\"##F,F+F,F'F,!\"\"F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approx imation~1~~->~~~G$\"/5%)f_+Pz!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7 approximation~2~~->~~~G$!#?!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7ap proximation~3~~->~~~G$\"/5%)f_+Pz!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%7approximation~4~~->~~~G$\"\"!F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%7approximation~5~~->~~~G$\"/5%)f_+Pz!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~6~~->~~~G$!#?!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~7~~->~~~G$\"/5%)f_+Pz!#9" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%7approximation~8~~->~~~G$\"\"!F%" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%7approximation~9~~->~~~G$\"/5%)f_+Pz!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~10~~->~~~G$!#?!#<" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%6last~iteration~gives~G$!#?!#<" }} {PARA 8 "" 1 "" {TEXT -1 67 "Error, (in newton) reached max, 10, itera tions without convergence\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Experimentation suggests that there is a well-d efined " }{TEXT 267 27 "interval of non-convergence" }{TEXT -1 25 " co ntaining zero given by" }}{PARA 262 "" 0 "" {TEXT -1 5 " -0." } {XPPEDIT 18 0 "137077432000014<=x" "6#1\"09++Ku2P\"%\"xG" }{XPPEDIT 18 0 "``<= 0" "6#1%!G\"\"!" }{TEXT -1 18 ".1360001251354250 " }}{PARA 0 "" 0 "" {TEXT -1 13 "approximately" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "newton(f(x),x=0.1360001251 354250,info=false,maxiter=1000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6 last~iteration~gives~G$\"+g_+Pz!#5" }}{PARA 8 "" 1 "" {TEXT -1 69 "Err or, (in newton) reached max, 1000, iterations without convergence\n" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "newton(f(x),x=0.1360001251354251,info=false,maxiter=200);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$!+s#)G/9!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "newton(f(x), x=-0.137077432000015,info=false,maxiter=200);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+s#)G/9!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "newton(f(x),x=-0.13707743200 0014,info=false,maxiter=1000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6la st~iteration~gives~G$!#E!#9" }}{PARA 8 "" 1 "" {TEXT -1 69 "Error, (in newton) reached max, 1000, iterations without convergence\n" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Example 13 " }{TEXT 360 48 ".. a pathological example with erratic behaviour" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 85 "Acknowledgement: This example is du e to Paul Garrett of the University of Minnesota. " }}{PARA 0 "" 0 "" {TEXT -1 92 "See: Pathological example for Newton's method (applet) Pa ul Garrett, garrett@math.umn.edu \n " }{URLLINK 17 "http://www.math.um n.edu/~garrett/qy/BadNewton.html" 4 "http://www.math.umn.edu/~garrett/ qy/BadNewton.html" "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 19 "The cubic function " }{XPPEDIT 18 0 "f(x)=5*x^3-15*x^2+ 10*x+2" "6#/-%\"fG6#%\"xG,**&\"\"&\"\"\"*$F'\"\"$F+F+*&\"#:F+*$F'\"\"# F+!\"\"*&\"#5F+F'F+F+F1F+" }{TEXT -1 61 " has one real zero, as can be seen from the following graph. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "f := x -> 5*x^3-15*x^2+10*x+ 2:\n'f(x)'=f(x);\nplot(f(x),x=-0.5..2.4,y=-4..7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,**&\"\"&\"\"\")F'\"\"$F+F+*&\"#:F+)F'\" \"#F+!\"\"*&\"#5F+F'F+F+F1F+" }}{PARA 13 "" 1 "" {GLPLOT2D 378 370 370 {PLOTDATA 2 "6%-%'CURVESG6$7U7$$!3++++++++]!#=$!3+++++++vt!#<7$$!3 QLLek7%Ro%F*$!3-Tp:G#\\')['F-7$$!3umm;HD)yO%F*$!3G_JQ*>.jk&F-7$$!3=+vV 8:)G4%F*$!3U%H$3klW[\\F-7$$!3gL$3x\\!)y\"QF*$!3esv!**QjDG%F-7$$!3smmT! pX$*>$F*$!3MI7.OrX)*GF-7$$!3emm\"H$3rwDF*$!3/cKr3i;e;F-7$$!3CL$3-DNq&> F*$!37a8QLR5!p&F*7$$!3sm;/hm^#Q\"F*$\"3_g[D\"4\"ovJF*7$$!3G/+v$Rfj(y!# >$\"3Uc#oYPlo6\"F-7$$!3:kmTgW4C:/z'377CF-7$$\"3Qnm;k0lr5F*$\"38oq7H*Qb!HF-7$$\"3ILL3A`EF;F*$\"3 +2\\2*Q6;D$F-7$$\"3+,+]Ufv_AF*$\"37Ql&G?#o[NF-7$$\"3[****\\()[\"3)GF*$ \"3mei#))\\\"\\bPF-7$$\"3?++]dg1'[$F*$\"3%QWv:R#*\\(QF-7$$\"3yL$eR5#pN SF*$\"3hK#Q\"eKJ@RF-7$$\"3Knm;M1D*o%F*$\"3[tvFX7X1RF-7$$\"3\\mmmO/!HC& F*$\"3;+69bSGSQF-7$$\"3+,]ivn#p)eF*$\"3_XTdwYi3PF-7$$\"3Clmm6#**pX'F*$ \"31&>%pDl7\\NF-7$$\"3%3+Dc-fC3(F*$\"37MI&>G*fMLF-7$$\"3#***\\(onW!ywF *$\"3-ho]BQQ)4$F-7$$\"3uom\"H#eZ*H)F*$\"36jzXeVnDGF-7$$\"33n;z/O9q))F* $\"3))4!*yikrdDF-7$$\"3%RL$3-fo&[*F*$\"3IKjs>oZcAF-7$$\"3MLeRNh]75F-$ \"3>DF..\"zu$>F-7$$\"31+vVcR;o5F-$\"3+)Qssxj2m\"F-7$$\"3VL$ez#fFG6F-$ \"3S#**\\&yS\"F-$\"3MIfQFCh#3\"F-7$$\"3!***\\7 E<8^7F-$\"3V_')zG))FY7$$\"3_m\"HU'))et;F-$\"3yb.D2(f;g\" F*7$$\"3?+D\")z-gMF-$\"3M1YF$*z Fp7F-7$$\"3UL$3(3#\\$y>F-$\"3q#RiPps/z\"F-7$$\"3/n\"H7_Y$R?F-$\"3qV\"> Y,#*pT#F-7$$\"3C++];nR&4#F-$\"3wp_'G(f\"[4$F-7$$\"3sm;alljf@F-$\"3c!z; #4R'*)*RF-7$$\"3@LLL_M4#*f`gF-7$$\"36+vV&4*)pL#F-$\"3]b*G.!4mksF-7$$\"3!**************R#F- $\"3'))***********>()F--%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F[\\lFj[l-%+A XESLABELSG6$Q\"x6\"Q\"yF`\\l-%%VIEWG6$;$!\"&Fi[l$\"#CFi[l;$!\"%F[\\l$ \"\"(F[\\l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C urve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "The real root is " }{XPPEDIT 18 0 "1-(675+75*sqrt(6))^(1/3)/15-5/((675+75*sqrt(6))^(1/3)) ;" "6#,(\"\"\"F$*&),&\"$v'F$*&\"#vF$-%%sqrtG6#\"\"'F$F$*&F$F$\"\"$!\" \"F$\"#:F1F1*&\"\"&F$),&F(F$*&F*F$-F,6#F.F$F$*&F$F$F0F1F1F1" }{TEXT -1 3 " " }{TEXT 356 1 "~" }{TEXT -1 2 " " }{XPPEDIT 18 0 "-0" "6#,$ \"\"!!\"\"" }{TEXT -1 13 ".1597048528. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "[solve(f(x),x)]:\nop(re move(has,%,Complex(1)));\nevalf(evalf(%,15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"#:!\"\",&\"$v'\"\"\"*&\"#vF)\"\"'#F)\"\"#F)#F)\" \"$F&*&\"\"&F)F'#F&F0F&F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+G&[q f\"!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The maximum point o ccurs where " }{XPPEDIT 18 0 "x = 1-1/sqrt(3);" "6#/%\"xG,&\"\"\"F&*&F &F&-%%sqrtG6#\"\"$!\"\"F," }{TEXT -1 1 " " }{TEXT 355 1 "~" }{TEXT -1 46 " 0.4226497308, and minimum point occurs where " }{XPPEDIT 18 0 "x= 1+1/sqrt(3)" "6#/%\"xG,&\"\"\"F&*&F&F&-%%sqrtG6#\"\"$!\"\"F&" }{TEXT -1 1 " " }{TEXT 354 1 "~" }{TEXT -1 14 " 1.577350269. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "solve(D (f)(x)=0);\nxmin,xmax := max(%),min(%);\nevalf(evalf(%,15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&\"\"\"F$*&\"\"$!\"\"F&#F$\"\"#F$,&F$F$*&F& F'F&F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%%xminG%%xmaxG6$,&\"\" \"F)*&\"\"$!\"\"F+#F)\"\"#F),&F)F)*&F+F,F+F-F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+p-Nx:!\"*$\"+3t\\EU!#5" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 60 "Taking a starting value for Newton's method in the inte rval " }{XPPEDIT 18 0 "1-1/sqrt(3)<=x" "6#1,&\"\"\"F%*&F%F%-%%sqrtG6# \"\"$!\"\"F+%\"xG" }{XPPEDIT 18 0 "`` " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 64 "typ ical behaviour for a starting approximation in the interval " } {XPPEDIT 18 0 "1-1/sqrt(3)<=x" "6#1,&\"\"\"F%*&F%F%-%%sqrtG6#\"\"$!\" \"F+%\"xG" }{XPPEDIT 18 0 "``~~~G$\".: " 0 "" {MPLTEXT 1 0 61 "f := \+ x -> 5*x^3-15*x^2+10*x+2:\nnewton(f(x),x=1.51,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\"/PQ\"o*48;!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\"/Y,A]8x9!# 8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$\"/5$o_ fkd\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$ \"/gxG&*Guk!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~5~~- >~~~G$\"/TX+v2'o%!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximatio n~6~~->~~~G$\"/t~~~G$\"/ZUbl!Ru#!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %7approximation~8~~->~~~G$\"/b(e>zkD#!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~9~~->~~~G$\"/_Fb1za>!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~10~~->~~~G$\"/OzSP'Gx\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~11~~->~~~G$\"/XXX*e2m\"!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~12~~->~~~G$\"/07)p^7d \"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~13~~->~~~G$\" /YF%e\\GH#!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~14~~- >~~~G$\"/=:/5'p(>!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximatio n~15~~->~~~G$\"/e\"**Gvhy\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8ap proximation~16~~->~~~G$\"/\"*RAiTp;!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~17~~->~~~G$\"/d=cmm!e\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~18~~->~~~G$\".$o/F4%o#!#8" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%8approximation~19~~->~~~G$!.*)HG&4M&*!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~20~~->~~~G$!/0'*o35_Y! #9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~21~~->~~~G$!/!eT A>()H#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~22~~->~~~ G$!/]D?'>uk\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~23 ~~->~~~G$!/B*)3pL(f\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approxim ation~24~~->~~~G$!/A=P&[qf\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8a pproximation~25~~->~~~G$!/'[w_[qf\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~26~~->~~~G$!/'[w_[qf\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+G&[qf\"!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 56 "Individual steps can be illustrated using the procedure " }{TEXT 0 11 "newton_step" }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 55 "behaviour f or a starting approximation in the \"window\" " }{XPPEDIT 18 0 "1.5773 50269 < x" "6#2-%&FloatG6$\"+p-Nx:!\"*%\"xG" }{XPPEDIT 18 0 "``< 1.581 106866" "6#2%!G-%&FloatG6$\"+mo5\"e\"!\"*" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "Consider the start ing value for Newton's method given by the mid-point of the \"window\" . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "(1.577350269+1.581106866)/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+o&G#z:!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 56 "Successive iterates can be obtained using the procedure " }{TEXT 0 6 "newton" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "f := x -> \+ 5*x^3-15*x^2+10*x+2:\nnewton(f(x),x=1.579228568,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$!.D'owF'Q(!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$!/'Q\"Rx;BN! #9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$!/%*fl .h;>!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$! /NY;l93;!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~5~~->~~ ~G$!/iy?E1(f\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~6 ~~->~~~G$!/9nF&[qf\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approxima tion~7~~->~~~G$!/'[w_[qf\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7app roximation~8~~->~~~G$!/'[w_[qf\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$!+G&[qf\"!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "The first step of Newton's method is illustrated by the f ollowing picture. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 518 "f := x -> 5*x^3-15*x^2+10*x+2:\nx0 := 1.579 228568: x1 := x0-f(x0)/D(f)(x0):\np1 := plot(f(x), x=1.3..1.8,color=r ed,adaptive=false,numpoints=16):\np2 := plot(f(x), x=-.2..-0.1,color=r ed,adaptive=false,numpoints=5):\np3 := plot(f(x0)+D(f)(x0)*(x-x0), x=- 1..2,y=-.05..0.4,\n color=green,thicknes s=2):\np4 := plot([[x0,0],[xx,f(x0)]],color=blue,linestyle=2):\np5 := \+ plot([[[x1,0],[x0,0],[x0,f(x0)]]$3],\n style=point,symbol=[circle ,diamond,cross],color=black):\nplots[display]([p1,p2,p3,p4,p5]);" }} {PARA 13 "" 0 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 667 271 271 {PLOTDATA 2 "6+-%'CURVESG6$727$$\"3/+++++++8!#<$\"35!)**********\\ j!#=7$$\"3!ommmWv[L\"F*$\"3vc+X'\\+R8&F-7$$\"3ULLL'Q?_O\"F*$\"3YPAZQPu uTF-7$$\"3!ommYWY$*R\"F*$\"3E\\Ky_!=q@$F-7$$\"3rmmm7()pL9F*$\"3G(pYJ3$ )QR#F-7$$\"3WLLL5x)yY\"F*$\"3!3R>'[y3Fb7F-7$$\"3)******HH1C`\"F*$\"3)ybNwEbQD*!#>7$$\"3kmmmB)\\jc\"F*$\" 3kd7k^A/awFQ7$$\"34+++(\\%=+;F*$\"39u$o$=0T2!)FQ7$$\"3`LLL0_,q\" F*$\"3Csev\\cd`@F-7$$\"3!*******zN![t\"F*$\"3#48\")RNxr4$F-7$$\"3/+++[ n>oKXFas7$$!3b++]iTDP\")F-$!3agm\"FQ7$$!3#*****\\(G3U9$F -$\"3[@W,#=@BQ\"FQ7$$!3Y*****\\-\\r\\#F-$\"3!zf)*z(><$f\"FQ7$$!3?+++vG VZ=F-$\"3Z7#4wa))[!=FQ7$$!3_*****\\(4J@7F-$\"3y-ab#p;*3?FQ7$$!3;,+]iIK FlFQ$\"3eW@T%z$>%>#FQ7$$\"3(R,++]siL#Fas$\"3nsZ!*)=1XT#FQ7$$\"3K,+++!R 5'fFQ$\"3M!RkaoR6g#FQ7$$\"3!)***\\P/QBE\"F-$\"3bz,<5)Q#=GFQ7$$\"39**** **\\\"o?&=F-$\"3y9ec@$3/,$FQ7$$\"3k++vVb4*\\#F-$\"3mHfzK([7A$FQ7$$\"3w ++DJ'=_6$F-$\"3!*))4qj'=?U$FQ7$$\"3#4++vVy!ePF-$\"3S%f^W-,:j$FQ7$$\"3' 4+](=WU[VF-$\"3r&)\\F-$\"3T\"[qWZo8.%FQ7$$ \"3w***\\P>:mk&F-$\"3s%*op+)**oC%FQ7$$\"3d***\\iv&QAiF-$\"3r+-x`/_MWFQ 7$$\"3j++]PPBWoF-$\"36C^7**e:PYFQ7$$\"3%*)*****\\Nm'[(F-$\"3oa$p?9)\\Y [FQ7$$\"36****\\(yb^6)F-$\"3QG.V&o)H^]FQ7$$\"3')***\\PMaKs)F-$\"3c4)z< u`%\\_FQ7$$\"3a****\\7TW)R*F-$\"3khOMZ6ZpaFQ7$$\"3z*****\\@80+\"F*$\"3 o9A\">dmrm&FQ7$$\"31++]7,Hl5F*$\"3'Hh#)[j[#yeFQ7$$\"3()**\\P4w)R7\"F*$ \"3)R)p*oN?&pgFQ7$$\"3;++]x%f\")=\"F*$\"39x4aq3jyiFQ7$$\"3!)**\\P/-a[7 F*$\"3oyl^nvQvkFQ7$$\"3/+](=Yb;J\"F*$\"3=P8s3]0\"o'FQ7$$\"3')****\\i@O t8F*$\"3SiY3DD8#)oFQ7$$\"3')**\\PfL'zV\"F*$\"3#QtA)\\?k#4(FQ7$$\"3>+++ !*>=+:F*$\"3K!*fzTxQ&H(FQ7$$\"3-++DE&4Qc\"F*$\"3Gxx/&\\CF](FQ7$$\"3=+] P%>5pi\"F*$\"3Oq0I&RW$3xFQ7$$\"39+++bJ*[o\"F*$\"39chxGxG(*yFQ7$$\"33++ Dr\"[8v\"F*$\"3!=U$=6\"QQ6)FQ7$$\"3++++Ijy5=F*$\"3YIVsaL_2$)FQ7$$\"31+ ]P/)fT(=F*$\"3,!='p@A.9&)FQ7$$\"31+]i0j\"[$>F*$\"39JfKdvo6()FQ7$$\"\"# F\\q$\"3?******3_4C*)FQ-Fep6&FgpF[qFhpF[q-%*THICKNESSG6#F`bl-F$6%7$7$$ \"3++++o&G#z:F*F[q7$F\\cl$\"3m*********oHb(FQ-Fep6&FgpF[qF[qFhp-%*LINE STYLEGFgbl-F$6&7%7$$!3e******fxF'Q(F-F[qF[clF^cl-%'SYMBOLG6#%'CIRCLEG- Fep6&FgpF\\qF\\qF\\q-%&STYLEG6#%&POINTG-F$6&Fgcl-F\\dl6#%(DIAMONDGF_dl Fadl-F$6&Fgcl-F\\dl6#%&CROSSGF_dlFadl-%+AXESLABELSG6%Q\"x6\"Q!Fcel-%%F ONTG6#%(DEFAULTG-%%VIEWG6$;F]sF_bl;$!\"&!\"#$\"\"%F^s" 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" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Further steps can be illustrated using the procedure " }{TEXT 0 11 "n ewton_step" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 87 "Since the \+ picture for the first step is not as good as the one above, it is not \+ drawn. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "f := x -> 5*x^3-15*x^2+10*x+2:\nxin := 1.579228568: \nprint(x[0]=xin);\nfor i from 1 to 6 do\n xout := newton_step(f(x), x=xin,draw=is(i>1 and i<=3));\n print(x[i]=xout);\n xin := xout;\n end do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\"\"!$\"+o&G#z:!\" *" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\"\"\"$!*wxiQ(!\"*" }} {PARA 13 "" 1 "" {GLPLOT2D 492 208 208 {PLOTDATA 2 "6*-%'POINTSG6'7$$! *wxiQ(!\"*\"\"!7$F'$!03hs?r%e:!#87$$!0(px%ynJ_$!#:F*-%'SYMBOLG6#%'CIRC LEG-%&COLORG6&%$RGBG$\"$P\"!\"$F;$\"$d&F=-F$6'F&F+F/-F46#%&CROSSGF7-F$ 6'F&F+F/-F46#%(DIAMONDGF7-%'CURVESG6%7$F&F+-%*LINESTYLEG6#\"\"#-F86&F: F*F*\"\"\"-FK6%7S7$$!0_6wuKyJ*F2$!0+\"*>&)4'QCF.7$$!02E-9B%\\\"*F2$!0n f)o%zNN#F.7$$!0F@8%3*G+*F2$!0&\\p'3@4G#F.7$$!04oe%**4Q))F2$!0vYH.r1?#F .7$$!0F.7$$!0R?;\"*ob>)F2$!0DE)fWI->F.7$$!0`HJD \"F.7$$!0*=?=FKg\">\"F.7$$!0(*eWBVaE'F2$!0UYXVe$Q6F.7$$!0w *=pt!))4'F2$!0PTJXO73\"F.7$$!0+/&>18SfF2$!0XVQa*3G5F.7$$!0/\"\\x!oXx&F 2$!/'y%*y'>R(*F.7$$!01\"*3)*HDi&F2$!/%*Q(GxJD*F.7$$!0duSNO&eaF2$!/*>Wd \"3T()F.7$$!0R\"QlI>)G&F2$!/pNs@OA#)F.7$$!0;>=t3*R^F2$!0bV\"\\bl\"y(!# 97$$!0))*=*ed(z\\F2$!0qnBA&*oJ(Fiv7$$!0D`!)y0V\"[F2$!0(oP^c&)[oFiv7$$! 0#3DzMW_YF2$!0^n=3WFS'Fiv7$$!0hzEPLe\\%F2$!0>d$)er?)fFiv7$$!0]C1bW>K%F 2$!0c7-pxu_&Fiv7$$!0>!**4xplTF2$!01lE'34I^Fiv7$$!0p#Rd3())*RF2$!0?t(f^ CFiv7$$!0wuOwqCb#F2$!0,yU\"\\)Gh\"Fiv 7$$!0$eGh59.CF2$!0:(Q[ezQ8Fiv7$$!0%)*\\qB*>B#F2$!0\"[]Zu&[.\"Fiv7$$!0] #=T\\\"*y?F2$!/FL=AA@xFiv7$$!04MQZ-d\">F2$!/%zc/AM,&Fiv7$$!04SrJ([fF.7$Ffp$!09eeme\\)=F.7$F[q$!0;PokL)==F.7$F`q$!0\"3f\"p?Hv\"F.7$Feq$ !0)yQ:67&o\"F.7$Fjq$!0Ek<@.ai\"F.7$F_r$!02Jk*\\zO7F.7$F]t$!0E$Q1:dn6F.7$Fbt$!0@S.o*H16F.7$Fgt$!0(*3hxu!R5F.7$F\\u$ !/\"e(*p21v*F.7$Fau$!/2'3q)o#3*F.7$Ffu$!/r7e0Lp%)F.7$F[v$!/?QR4u2yF.7$ F`v$!/2)=EN07(F.7$Fev$!/4X*p?B_'F.7$F[w$!/KS$eKi(eF.7$F`w$!/$e(R.w3_F. 7$Few$!/W+p*odb%F.7$Fjw$!/$4;CmR#RF.7$F_x$!/KDKqXAKF.7$Fdx$!/Uw1)>@f#F .7$Fix$!/zwv-5>>F.7$F^y$!/YJ]XC48F.7$Fcy$!.#)Qe6^U'F.7$Fhy$!-OL'*p;:F. 7$F]z$\".s,9%)eS'F.7$Fbz$\"/9\"))=4F.7$F\\[l$\" /'oK&GM*f#F.7$Fa[l$\"/\"*\\T2UgKF.7$Ff[l$\"/T/rX-;RF.7$F[\\l$\"0A;?Sc% =XFiv7$F`\\l$\"0^2ZE6*3_Fiv7$Fe\\l$\"0vq[Kik#eFiv7$Fj\\l$\"0o()*e0!\\[ 'Fiv7$F_]l$\"09jQ'*4^6(Fiv7$Fd]l$\"0Y0j.cBz(Fiv-F86&F:F*FTF*-F[^lFP-%+ AXESLABELSG6$Q\"x6\"Q!Fjgl-%%VIEWG6$;FYFd]l;Fa^l$\"0S0j.cBz(Fiv" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2 " "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\"\"#$!+&ynJ_$!#5" }}{PARA 13 "" 1 "" {GLPLOT2D 412 332 332 {PLOTDATA 2 "6*-%'POINTSG6'7$$!+&ynJ_$!#5\"\"!7$ F'$!0)y.$zMPg$!#97$$!0'\\#*y.h;>!#:F*-%'SYMBOLG6#%'CIRCLEG-%&COLORG6&% $RGBG$\"$P\"!\"$F;$\"$d&F=-F$6'F&F+F/-F46#%&CROSSGF7-F$6'F&F+F/-F46#%( DIAMONDGF7-%'CURVESG6%7$F&F+-%*LINESTYLEG6#\"\"#-F86&F:F*F*\"\"\"-FK6% 7S7$$!0_P!)[YkK%F2$!0>P%3D3RbF.7$$!0)=7H(4kD%F2$!00#*ynI&f`F.7$$!0!Rn> 6Z&>%F2$!0_bqv5]?&F.7$$!0Xz*z#Rp7%F2$!0%*\\B#f7L]F.7$$!0'ewmJ&z0%F2$!0 U5(**>6i[F.7$$!0F1D$\\H*)RF2$!04g[p.Rp%F.7$$!0*=[g*Rc#RF2$!0C))\\i?(RX F.7$$!0`!*R#*G(fQF2$!0v'QHn&=Q%F.7$$!0Wkdjj:z$F2$!0J%p$f$\\?UF.7$$!0?# [aphBPF2$!0;97?h:1%F.7$$!0lu5WDPl$F2$!0uZ)[F1+RF.7$$!0aA!e\\;#f$F2$!0n \\7]s%fPF.7$$!0imYViG_$F2$!0d(4o)\\Ig$F.7$$!0M\"\\[`F`MF2$!0BUiRXzW$F. 7$$!0'zy'H:iQ$F2$!0e/D\"GK+LF.7$$!0_t`M=`K$F2$!0\"4PBA$y;$F.7$$!0]i+)f !HD$F2$!0U]..$F2$!0)*)e(3A?)GF.7$$!0/x<)p??JF2$!0v -yeZCt#F.7$$!0Ge/jWq0$F2$!0(yk)H@/v9s#F2$!0<7c-@K$>F.7$$!0vRC'Q#F2$!0#*G-I1$38F.7$$!0k&e#H96K #F2$!0!\\@tex\">\"F.7$$!0)GN%)))z[AF2$!0>b;JDU1\"F.7$$!0.)pt-#Q=#F2$!/ o>9*[D^*F.7$$!0CF2$!/8HK!Rd6'F.7$$!0w1f(Q<=>F2$!/L!H(3r`]F.7$$!07e'>]d]=F2$ !/k8%fo&fRF.7$$!04HF.7$F[v$ !0#)G%G\\U0=F.7$F`v$!0Od3:F.7$Fjv$!02k26#ze8F.7 $F_w$!0o/'***[W?\"F.7$Fdw$!094nPaM0\"F.7$Fiw$!/')e]Jft!*F.7$F^x$!/OAwl X^uF.7$Fcx$!/zr;Z*Q*fF.7$Fhx$!/fM@jjPWF.7$F]y$!/9m4\\VFIF.7$Fby$!/ho$G 7d[\"F.7$Fgy$!-q*=Wr]$F.7$F\\z$\"/<$4Am7[\"F.7$Faz$\"/vjZMwjHF.7$Ffz$ \"/kI`1!e^%F.7$F[[l$\"/HOYpf5gF.7$F`[l$\"/bAa?CRvF.7$Fe[l$\"/*)z3#H_0* F.7$Fj[l$\"0n^fRE[/\"F.7$F_\\l$\"0-t()*Q[/7F.7$Fd\\l$\"0L(esMGZ8F.7$Fi \\l$\"0f%H-w`*\\\"F.7$F^]l$\"0j8;Bk_k\"F.7$Fc]l$\"0$*=lRn=!=F.-F86&F:F *FTF*-Fj]lFP-%+AXESLABELSG6$Q\"x6\"Q!Figl-%%VIEWG6$;FYFc]l;$!0#ob*=-cS &F.$\"0%*=lRn=!=F." 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" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\"\"$$!+!Q5m\">!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\"\"%$!+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Example 14 " }{TEXT 325 25 ". . finding a complex root" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 131 "We can use Newton's method to find a complex root, but the starti ng value must be a complex number with a non-zero imaginary part. " }} {PARA 0 "" 0 "" {TEXT -1 26 "For example, the equation " }{XPPEDIT 18 0 "x^3+1 = 0;" "6#/,&*$%\"xG\"\"$\"\"\"F(F(\"\"!" }{TEXT -1 19 " has t he solutions " }{XPPEDIT 18 0 "x = -1" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "x = 1/2+sqrt(3)/2;" "6#/%\"xG,&*&\"\"\"F'\" \"#!\"\"F'*&-%%sqrtG6#\"\"$F'F(F)F'" }{TEXT -1 1 " " }{TEXT 340 1 "i" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x = 1/2-sqrt(3)/2*i;" "6#/%\"xG,&* &\"\"\"F'\"\"#!\"\"F'*(-%%sqrtG6#\"\"$F'F(F)%\"iGF'F)" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 86 "The starting approximation should be \+ close to the required root in the complex plane. " }}{PARA 258 "" 0 " " {TEXT -1 1 " " }{GLPLOT2D 367 343 343 {PLOTDATA 2 "64-%'CURVESG6&7#7 $$!\"\"\"\"!$F*F*-%'SYMBOLG6#%'CIRCLEG-%&COLORG6&%$RGBGF+F+$\"\"\"F*-% &STYLEG6#%&POINTG-F$6&F&-F-6#%(DIAMONDGF0F6-F$6&F&-F-6#%&CROSSGF0F6-F$ 6&7#7$$\"3++++++++]!#=$\"3a+++SSDg')FJF,-F16&F3F+$\"\"'F)F+F6-F$6&FFF< FMF6-F$6&FFFAFMF6-F$6&7#7$FH$!3a+++SSDg')FJF,-F16&F3F4F+F+F6-F$6&FWF " 0 "" {MPLTEXT 1 0 38 "newton(x^3+1=0,x=0.6+1.5*I,i nfo=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~ G^$$\"/*\\Q%\\#[#\\!#9$\"/43m&y!)3\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G^$$\"/w\"ydl`#[!#9$\"/h5&oK&4!*F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G^$$\"/;\"H/$Q &)\\!#9$\"/,t)GZEm)F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximati on~4~~->~~~G^$$\"/rb+V++]!#9$\"/$)e4DBg')F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~5~~->~~~G^$$\"/<'*********\\!#9$\"/P(y. a-m)F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~6~~->~~~G^$$ \"/++++++]!#9$\"/V%y.a-m)F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"++ +++]!#5$\"+QSDg')F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Example 15 " }{TEXT 326 29 ".. a complex solution of sin(" }{TEXT 329 1 "x" }{TEXT 330 4 ") = " }{TEXT 331 1 "x" }{TEXT 332 1 " " }} {PARA 0 "" 0 "" {TEXT -1 13 "The equation " }{XPPEDIT 18 0 "sin(x) = x ;" "6#/-%$sinG6#%\"xGF'" }{TEXT -1 5 " has " }{XPPEDIT 18 0 "x = 0;" " 6#/%\"xG\"\"!" }{TEXT -1 76 " as its only real solution, but it has ot her solutions in the complex plane." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "z := newton(sin(x)=x,x=7+3* I,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~-> ~~~G^$$\"/d\"fX2ZP(!#8$\"/btqFT$o#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%7approximation~2~~->~~~G^$$\"/\"z%\\!p%4v!#8$\"/h/&z5`w#F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G^$$\"/]_^Kk( \\(!#8$\"/+F~~~G^$$\"/R(*zin(\\(!#8$\"/VM&Gy'oFF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~5~~->~~~G^$$\"/kxxin(\\(!#8$\"/t)HGy'oF F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~6~~->~~~G^$$\"/k xxin(\\(!#8$\"/t)HGy'oFF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG^$$ \"+yin(\\(!\"*$\"+$Gy'oFF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "We can check that " }{XPPEDIT 18 0 "sin(z);" "6 #-%$sinG6#%\"zG" }{TEXT -1 30 " is approximately equal to z. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sin(z);\nz := 'z':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"+yin(\\ (!\"*$\"+\"Gy'oFF&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 " Example 16 " }{TEXT 327 44 ".. solving an equation involving an integr al" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 25 "We can use the proc edure " }{TEXT 0 6 "newton" }{TEXT -1 48 " to solve certain equations \+ involving integrals." }}{PARA 0 "" 0 "" {TEXT -1 44 "The following gra ph shows that the equation " }{XPPEDIT 18 0 "Int(sin(t^2),t=0..x)=1-x/ 2" "6#/-%$IntG6$-%$sinG6#*$%\"tG\"\"#/F+;\"\"!%\"xG,&\"\"\"F2*&F0F2F,! \"\"F4" }{TEXT -1 21 " has a solution for " }{TEXT 293 1 "x" }{TEXT -1 10 " near 1.1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot([Int(sin(t^2),t=0..x),1-x/2],x=0..2);" } }{PARA 13 "" 1 "" {GLPLOT2D 290 175 175 {PLOTDATA 2 "6&-%'CURVESG6$7U7 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+D\"3QDf%Ff\\l7$$\"1LL$e\"*[H7\"F]dl$\"1ML$3Ub_Q%Ff\\l7$$\"1+++qvxl6F] dl$\"1+++]@6rTFf\\l7$$\"1++]_qn27F]dl$\"1,+]PZhhRFf\\l7$$\"1++Dcp@[7F] dl$\"1,+v=_\"*ePFf\\l7$$\"1++]2'HKH\"F]dl$\"1,+]i>&Q`$Ff\\l7$$\"1nmmwa nL8F]dl$\"1nmm;EiJLFf\\l7$$\"1+++v+'oP\"F]dl$\"1+++D'*p:JFf\\l7$$\"1LL eR<*fT\"F]dl$\"1ML3-8/?HFf\\l7$$\"1+++&)Hxe9F]dl$\"1+++v]81FFf\\l7$$\" 1mm\"H!o-*\\\"F]dl$\"1omT&)f'[]#Ff\\l7$$\"1++DTO5T:F]dl$\"1,+v$z\"[%H# Ff\\l7$$\"1nmmT9C#e\"F]dl$\"1nmm\"z#z)3#Ff\\l7$$\"1++D1*3`i\"F]dl$\"1+ +voaXt=Ff\\l7$$\"1LLL$*zym;F]dl$\"1LLLL+1m;Ff\\l7$$\"1LL$3N1#4F]dl$\"1)***\\(=tY>%Fc\\l7$$ \"1++v.Uac>F]dl$\"1)***\\7)*ys@Fc\\l7$F_[lF(-Fd[l6&Ff[lF(Fg[lF(-%+AXES LABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(F_[l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "newton( Int(sin(t^2),t=0..x)=1-x/2,x=1.1,info=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%BAttempting~to~calculate~a~zero~ofG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,(-%$IntG6$-%$sinG6#*$)%\"tG\"\"#\"\"\"/F,;\"\"!%\"xG F.F.!\"\"*&F-F3F2F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Iby~Newton's ~method,~using~the~derivativeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-% $sinG6#*$)%\"xG\"\"#\"\"\"F+#F+F*F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\"/L \")4JEN6!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~ ~G$\"/B:IB&\\8\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation ~3~~->~~~G$\"/XC4B&\\8\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7appro ximation~4~~->~~~G$\"/XC4B&\\8\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"+4B&\\8\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 13 "The equation " }{XPPEDIT 18 0 "2*ln(x+1) = 1-x^2;" "6#/*&\"\"#\"\" \"-%#lnG6#,&%\"xGF&F&F&F&,&F&F&*$F+F%!\"\"" }{TEXT -1 39 " has exactly one real number solution. " }}{PARA 0 "" 0 "" {TEXT -1 102 "(a) Const ruct a Maple plot to illustrate this solution as the point of intersec tion of the two graphs " }{XPPEDIT 18 0 "y=2*ln(x+1)" "6#/%\"yG*&\"\"# \"\"\"-%#lnG6#,&%\"xGF'F'F'F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = 1-x^2;" "6#/%\"yG,&\"\"\"F&*$%\"xG\"\"#!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 24 "(b) Set up the function " }{XPPEDIT 18 0 "f(x) = 2*ln(x+1)-1+x^2;" "6#/-%\"fG6#%\"xG,(*&\"\"#\"\"\"-%#lnG6#,&F' F+F+F+F+F+F+!\"\"*$F'F*F+" }{TEXT -1 70 " in Maple and plot a graph to illustrate the solution of the equation " }{XPPEDIT 18 0 "2*ln(x+1) = 1-x^2;" "6#/*&\"\"#\"\"\"-%#lnG6#,&%\"xGF&F&F&F&,&F&F&*$F+F%!\"\"" } {TEXT -1 27 " as a zero of the function " }{XPPEDIT 18 0 "f(x)" "6#-% \"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 35 "(c) Set up a procedure to evaluate " }{XPPEDIT 18 0 "x-f(x)/`f '`(x)" "6#,&%\"xG \"\"\"*&-%\"fG6#F$F%-%$f~'G6#F$!\"\"F-" }{TEXT -1 43 " and use it to c alculate \"Newton iterates\" " }{XPPEDIT 18 0 "x[0],x[1], ` . . . `" " 6%&%\"xG6#\"\"!&F$6#\"\"\"%(~.~.~.~G" }{TEXT -1 7 ", with " }{XPPEDIT 18 0 "x[0]=0" "6#/&%\"xG6#\"\"!F'" }{TEXT -1 10 ".5, where " } {XPPEDIT 18 0 "x[1] = x[0]-f(x[0])/`f '`(x[0]);" "6#/&%\"xG6#\"\"\",&& F%6#\"\"!F'*&-%\"fG6#&F%6#F+F'-%$f~'G6#&F%6#F+!\"\"F7" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "x[2] = x[1]-f(x[1])/`f '`(x[1]),` . . . `;" "6$/&%\" xG6#\"\"#,&&F%6#\"\"\"F+*&-%\"fG6#&F%6#F+F+-%$f~'G6#&F%6#F+!\"\"F7%(~. ~.~.~G" }{TEXT -1 43 ", etc., until you can estimate the zero of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 23 " correct to 10 di gits. " }}{PARA 0 "" 0 "" {TEXT -1 67 "(d) Check that the result obtai ned in (c) is correct to 10 digits. " }}{PARA 0 "" 0 "" {TEXT -1 36 "_ ___________________________________" }}{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 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 36 "__ __________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }} {PARA 0 "" 0 "" {TEXT -1 43 "Find the positive solution of the equatio n " }{XPPEDIT 18 0 "sin(Pi*x)=x" "6#/-%$sinG6#*&%#PiG\"\"\"%\"xGF)F*" }{TEXT -1 41 " correct to 10 digits by Newton's method." }}{PARA 0 "" 0 "" {TEXT -1 112 "Indicate how you have chosen your starting approxim ation by means of a graph. You may use the special procedure " }{TEXT 0 6 "newton" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 "_________ ___________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 36 "____________________________________" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 35 "Find the solution of the equation " }{XPPEDIT 18 0 "ln(x) = 1/x;" "6#/-%#lnG6#%\"xG*&\"\" \"F)F'!\"\"" }{TEXT -1 43 " correct to 10 digits by Newton's method. \+ " }}{PARA 0 "" 0 "" {TEXT -1 77 "Indicate how you have chosen your sta rting approximation by means of a graph." }}{PARA 0 "" 0 "" {TEXT -1 36 "____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 36 "____________________ ________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 31 "(a) Find the two solutions of " }{XPPEDIT 18 0 "exp(x-1)-5*x+2*x^2 = 0" "6#/,(-%$expG6#,&%\"xG\"\"\"F*!\"\"F**&\"\"&F*F)F*F+*&\"\"#F**$F)F/ F*F*\"\"!" }{TEXT -1 163 " correct to 10 digits by using Newton's meth od.\n Include any graphs which you have used to obtain initial ap proximations for the roots as part of your answer." }}{PARA 0 "" 0 "" {TEXT -1 33 "_________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 33 "____________ _____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 " (b) Find the value of " }{TEXT 294 1 "x" }{TEXT -1 34 " which gives the minimum value of " }{XPPEDIT 18 0 "exp(x-1)-5*x+2 *x^2" "6#,(-%$expG6#,&%\"xG\"\"\"F)!\"\"F)*&\"\"&F)F(F)F**&\"\"#F)*$F( F.F)F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 33 "______________ ___________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 33 "_________________________________" }}{PARA 0 "" 0 "" {TEXT -1 60 "(c) What happens if you try to use Newton's method to solve " } {XPPEDIT 18 0 "exp(x)-4+x^2 = 0;" "6#/,(-%$expG6#%\"xG\"\"\"\"\"%!\"\" *$F(\"\"#F)\"\"!" }{TEXT -1 19 " with the value of " }{TEXT 295 1 "x" }{TEXT -1 78 " from part (b) as the starting value, and what happens i f you take a value of " }{TEXT 296 1 "x" }{TEXT -1 51 " close to the v alue from (b) as the starting value?" }}{PARA 0 "" 0 "" {TEXT -1 33 "_ ________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 33 "_________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5" }}{PARA 0 "" 0 "" {TEXT -1 18 "Find the smallest " } {TEXT 268 8 "positive" }{TEXT -1 27 " solution of the equation " } {XPPEDIT 18 0 "tan(x) = x;" "6#/-%$tanG6#%\"xGF'" }{TEXT -1 42 " corre ct to 10 digits by Newton's method. " }}{PARA 0 "" 0 "" {TEXT -1 77 "I ndicate how you have chosen your starting approximation by means of a \+ graph." }}{PARA 0 "" 0 "" {TEXT -1 36 "_______________________________ _____" }}{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 36 "____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q 6" }}{PARA 0 "" 0 "" {TEXT -1 32 "Find, correct to 10 digits, the " } {TEXT 267 16 "largest positive" }{TEXT -1 26 " solution of the equatio n " }{XPPEDIT 18 0 "sqrt(x)=3*cos(x^2)" "6#/-%%sqrtG6#%\"xG*&\"\"$\"\" \"-%$cosG6#*$F'\"\"#F*" }{TEXT -1 21 " by Newton's method. " }}{PARA 0 "" 0 "" {TEXT -1 96 "Indicate how you have chosen your starting appr oximation by means of a suitable graph or graphs." }}{PARA 0 "" 0 "" {TEXT -1 36 "____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 36 "____________________________________" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q7 " }} {PARA 0 "" 0 "" {TEXT -1 21 "(Example by S. Smale)" }}{PARA 0 "" 0 "" {TEXT -1 33 "Let f be the function defined by " }{XPPEDIT 18 0 "f(x) = x^3-2*x+2;" "6#/-%\"fG6#%\"xG,(*$F'\"\"$\"\"\"*&\"\"#F+F'F+!\"\"F-F+ " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 33 "(a) Calculate the si ngle zero of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 42 " \+ correct to 10 digits by Newton's method. " }}{PARA 0 "" 0 "" {TEXT -1 87 "(b) Check what happens when Newton's method is applied with the st arting approximation " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 103 "(c) Check what happens when \+ Newton's method is applied with starting approximations which are clos e to " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 "____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 36 "__ __________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q8" }} {PARA 0 "" 0 "" {TEXT -1 45 "This question is concerned with the funct ion " }{XPPEDIT 18 0 "f(x)=(sqrt(20*x^2-38*x+22)-2-arctan(x-1))/(x-1) " "6#/-%\"fG6#%\"xG*&,(-%%sqrtG6#,(*&\"#?\"\"\"*$F'\"\"#F0F0*&\"#QF0F' F0!\"\"\"#AF0F0F2F5-%'arctanG6#,&F'F0F0F5F5F0,&F'F0F0F5F5" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 22 "(a) Plot the graph of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "-5<=x" "6#1,$\"\"&!\"\"%\"xG" }{XPPEDIT 18 0 "``<=5" "6#1%!G\"\" &" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 28 "(b) Find the single zero of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 42 " corr ect to 10 digits by Newton's method. " }}{PARA 0 "" 0 "" {TEXT -1 31 " You may use the procedure " }{TEXT 0 6 "newton" }{TEXT -1 10 " fo r this." }}{PARA 0 "" 0 "" {TEXT -1 73 "(c) Investigate the convergenc e of Newton's method to the single zero of " }{XPPEDIT 18 0 "f(x)" "6# -%\"fG6#%\"xG" }{TEXT -1 26 " with the starting value " }{XPPEDIT 18 0 "x = 1.97;" "6#/%\"xG-%&FloatG6$\"$(>!\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 31 " You may use the procedure " }{TEXT 0 11 "n ewton_step" }{TEXT -1 10 " for this." }}{PARA 0 "" 0 "" {TEXT -1 54 "( d) What happens if the starting value is taken to be " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 3 "? " }}{PARA 0 "" 0 "" {TEXT -1 36 "____________________________________" }}{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 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 36 "____________________________________" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 25 "Code for drawing pic tures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 27 "1st step of Newton's method" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 483 "p1 := plot( x^2-2,x=1..2.5,thickness=1):\np2 := plot([[2,0],[2,2]],linestyle=2,col or=black):\np3 := plot([[[1.5,0],[2,2]]],color=blue):\np4 := plot([[[1 .5,0],[2,0],[2,2]]$3],style=point,\n color=navy,symbol=[circle,cros s,diamond]):\np5 := plot([[1,0],[4,0]],color=black):\nt1:=plots[textpl ot]([2.45,3.4,`y = f(x)`],color=red):\nt2:=plots[textplot]([[2.12,2,`( a,f(a))`],\n [1.414,-0.2,`r`],[1.5,-0.2,`b`],[2,-0.2,`a`]],col or=black):\nplots[display]([p1,p2,p3,p4,p5,t1,t2],axes=none);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 19 "Bowing rail example" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 392 "rr := evalf(sqrt(2)/2):\np1 := plot([[cos(t),sin(t),t=Pi/4..3*Pi/ 4],[[0,0],[0,1]],\n[[0,0],[-rr,rr],[rr,rr],[0,0]]],scaling=constrained ,color=black,axes=none):\nt1 := plots[textplot]([[0.05,0.15,t],[0.05,0 .87,h],[-0.75,0.72,A],\n[0.75,0.72,B],[0,-0.03,C],[0,1.05,M],[-0.055,0 .67,X],[-0.67,0.37,`AB = 1 km`],\n[0,1.15,`arc AB = 1 km + 10 cm = 100 0.1 meters`]],color=black):\nplots[display]([p1,t1]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 0 "" 0 "" {TEXT -1 21 "Pathological examples" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 382 "f : = x -> piecewise(x<0,-sqrt(-x),x>=0,sqrt(x)):\np1 := plot(f(x),x=-6..6 ):\np2 := plot([[[-6,-.5],[6,2.5]],[[-6,-2.5],[6,.5]]],color=COLOR(RGB ,0,.8,0)):\np3 := plot([[[4,0],[4,2]],[[-4,-2],[-4,0]]],color=blue,lin estyle=2):\np4 := plot([[[4,0],[4,2],[-4,0],[-4,-2]]$3],style=point,co lor=black,\n symbol=[circle,diamond,cross]):\nplots[displa y]([p1,p2,p3,p4],labels=[`x`,`y`]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 375 "f := x -> x^3-5*x:\np1 := plot(f(x),x=-2.6..2.6):\np2 := plot([[[-1.4,.8],[1.4,-4.8]],[[-1.4,4. 8],[1.4,-.8]]],\n color=COLOR(RGB,0,.8,0)):\np3 := plot([[[1, 0],[1,-4]],[[-1,0],[-1,4]]],color=blue,linestyle=2):\np4 := plot([[[1, 0],[1,-4],[-1,0],[-1,4]]$3],style=point,color=black,\n sym bol=[circle,diamond,cross]):\nplots[display]([p1,p2,p3,p4],labels=[`x` ,`y`]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 441 "aa := 2^(1/3):\nf := x -> x^3-aa*x+1:\np1 := plot(f( x),x=-1.7..1.7,-.7..2.3):\np2 := plot([[[-1/aa,2],[3/(2*aa),-.5]],[[-1 /(2*aa),-.25],[3/(2*aa),.75]]],\n color=COLOR(RGB,0,.8,0)):\n p3 := plot([[[1/aa,0],[1/aa,.5]],[[0,0],[0,1]]],color=blue,linestyle=2 ):\np4 := plot([[[1/aa,0],[1/aa,.5],[0,0],[0,1]]$3],style=point,color= black,\n symbol=[circle,diamond,cross]):\nplots[display]([ p1,p2,p3,p4],labels=[`x`,`y`],tickmarks=[3,3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 522 334 334 {PLOTDATA 2 "6--%'CURVESG6$7S7$$!3%************* *p\"!#<$!3%[ry^@M6x\"F*7$$!3ALL$en*)ei\"F*$!3ytAtdZd\\7F*7$$!3um;/zmSh :F*$!33=v!H.gWR)!#=7$$!3KLL30)))))[\"F*$!3WW<)**z7nC%F77$$!3QLLeN-*eT \"F*$!3%z=#3i$=*ea!#>7$$!3om;a:'QKM\"F*$\"3#>\")*)GMEyo#F77$$!3WL$3PE \")eF\"F*$\"3Ct@(3-q`I&F77$$!30+]PFm817F*$\"3c^r?]*3*\\wF77$$!3OL$3Zi1 S8\"F*$\"36Kn!zb\"f/(*F77$$!33+](Q%z5i5F*$\"3\\g#p!fl.S6F*7$$!3+mmmJJ^ \"))*F7$\"338Wv*=>,G\"F*7$$!3ZLL$es-,B*F7$\"3+/)p)\\Lcw8F*7$$!3k)**** \\*)on\\)F7$\"3KD<^u35d9F*7$$!3c*****\\#RUgxF7$\"3U3&*\\#)pQ5:F*7$$!35 *****\\5>30(F7$\"32t%\\qwAy`\"F*7$$!38ML3Z*HkS'F7$\"3cj#G&)oCUa\"F*7$$ !3\"fmm;\"*)=ScF7$\"3u^J:F*7$$!35nmmYA3\"*\\F7$\"3K@vu(f/X]\"F*7$ $!32++v$)o,OUF7$\"3E;W#)yUpd9F*7$$!31nmm'4cwc$F7$\"37#y6\"on3/9F*7$$!3 r***\\PQeV$GF7$\"3*pvHJrOVL\"F*7$$!3))***\\7b&3O@F7$\"3;3]b_KQf7F*7$$! 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