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}{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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 50 "A modification of Newton's method for root-finding" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nana imo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 24.3.2007 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "load root-finding procedures" }} {PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 277 7 "roots.m " }{TEXT -1 38 " contains the code for the procedures " }{TEXT 0 9 "im pnewton" }{TEXT -1 5 " and " }{TEXT 0 14 "impnewton_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 foll ows, where the file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\\\Maple/procdrs/roots.m\";" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 64 "The iter ative formula for the improved \"leap-frog\" Newton method" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 110 " See: International Journal of Mathematical Education in Science and Te chnology, vol. 33, no. 4, pp 521 - 527. " }}{PARA 0 "" 0 "" {TEXT -1 182 "Leap-frogging Newton\222s method, by A. Bathi Kasturiarachi,\nDep t. of Mathematics & Computer Science, Kent State University - Stark Ca mpus, 6000 Frank Avenue, N.W. Canton, OH 44720, USA" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " } {XPPEDIT 18 0 "x = r" "6#/%\"xG%\"rG" }{TEXT -1 26 " is a root of an e quation " }{XPPEDIT 18 0 "f(x) = 0;" "6#/-%\"fG6#%\"xG\"\"!" }{TEXT -1 33 ", where f is differentiable near " }{XPPEDIT 18 0 "x = a;" "6#/ %\"xG%\"aG" }{TEXT -1 16 ". Suppose that " }{XPPEDIT 18 0 "x = a;" "6 #/%\"xG%\"aG" }{TEXT -1 34 " is an approximation for the root " } {TEXT 269 1 "r" }{TEXT -1 2 ". " }}{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 278 1 "x" }{TEXT -1 19 " axis is given by " } {XPPEDIT 18 0 "x = a-f(a)/`f '`(a);" "6#/%\"xG,&%\"aG\"\"\"*&-%\"fG6#F &F'-%$f~'G6#F&!\"\"F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 21 "This gives the usual " }{TEXT 266 20 "Newton approximation" }{TEXT -1 2 " " }{XPPEDIT 18 0 "b = a-f(a)/`f '`(a);" "6#/%\"bG,&%\"aG\"\"\" *&-%\"fG6#F&F'-%$f~'G6#F&!\"\"F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "We seek an improvement on this approximation by considering the " }{TEXT 266 11 "secant line" } {TEXT -1 19 " joining the points" }{XPPEDIT 18 0 "``(a,f(a));" "6#-%!G 6$%\"aG-%\"fG6#F&" }{TEXT -1 4 " and" }{XPPEDIT 18 0 "``(b,f(b));" "6# -%!G6$%\"bG-%\"fG6#F&" }{TEXT -1 14 " on the curve " }{XPPEDIT 18 0 "y = f(x);" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 30 "This secant line has equation " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y-f(a) = m*(x-a);" "6#/,&%\"yG\"\"\"-% \"fG6#%\"aG!\"\"*&%\"mGF&,&%\"xGF&F*F+F&" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 31 "where the slope of the line is " }{XPPEDIT 18 0 "m = (f(b)-f(a))/(b-a);" "6#/%\"mG*&,&-%\"fG6#%\"bG\"\"\"-F(6#%\"aG!\"\" F+,&F*F+F.F/F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "This s ecant line meets the " }{TEXT 280 1 "x" }{TEXT -1 12 " axis where " }} {PARA 258 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x = a-f(a)*``((b-a)/ (f(b)-f(a)));" "6#/%\"xG,&%\"aG\"\"\"*&-%\"fG6#F&F'-%!G6#*&,&%\"bGF'F& !\"\"F',&-F*6#F1F'-F*6#F&F2F2F'F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 7 "Since, " }{XPPEDIT 18 0 "b-a=-f(a)/`f '`(a)" "6#/,&%\"bG\" \"\"%\"aG!\"\",$*&-%\"fG6#F'F&-%$f~'G6#F'F(F(" }{TEXT -1 17 ", it foll ows that" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = a-f(a )^2/(`f '`(a)*(f(a)-f(b)));" "6#/%\"xG,&%\"aG\"\"\"*&-%\"fG6#F&\"\"#*& -%$f~'G6#F&F',&-F*6#F&F'-F*6#%\"bG!\"\"F'F7F7" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "We take t his value as the next approximation " }{TEXT 279 1 "c" }{TEXT -1 13 " \+ to the root " }{TEXT 270 1 "r" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "h = f(a)/`f '`(a);" "6#/%\"hG*&-%\"fG 6#%\"aG\"\"\"-%$f~'G6#F)!\"\"" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "b =a-h" "6#/%\"bG,&%\"aG\"\"\"%\"hG!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "c=a-k" "6#/%\"cG,&%\"aG\"\"\"%\"kG!\"\"" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "k = ``(f(a)/(f(a)-f(b)))*h;" "6#/%\"kG*&-%!G6#*&-%\"fG 6#%\"aG\"\"\",&-F+6#F-F.-F+6#%\"bG!\"\"F5F.%\"hGF." }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 506 400 400 {PLOTDATA 2 "65-%'CURVESG6%7S7$$!3A+++++++S!#=$ \"3;gT1QwJU7!#;7$$!3\"ommm1\"\\-LF*$\"3WX'Qi7v5;\"F-7$$!3sLLLtAf&p#F*$ \"3i6>ic%pW4\"F-7$$!3ymmm1628?F*$\"3!zNm9]7Q-\"F-7$$!3&pmmmuDgK\"F*$\" 3[C4kG#)Rp&*!#<7$$!3[OLLLzXAk!#>$\"3?1z/C/TV*)FB7$$!3]$*ommmC%H)!#@$\" 3gA^KJn,'R)FB7$$\"3)\\******fe7['FF$\"3;b&yPDH0'yFB7$$\"3#GLLLZ'*pK\"F *$\"3:E;GiU5QtFB7$$\"3n******R**o.?F*$\"3GUaf)fYq%oFB7$$\"3InmmYBv*p#F *$\"3-'pl\"o$=3P'FB7$$\"3ULLL$\\WGJ$F*$\"31R1i \"FB$\"3G<%f\\SQ1.#FB7$$\"3MLLL(Gs*o7FB$\"32%H%=.?C3=FB7$$\"3%)*****R \"yQI8FB$\"3)4.$)Hx\"4B;FB7$$\"3ULLLl#=nR\"FB$\"3X\"*[9o+RJ9FB7$$\"3s* ****>6W_Y\"FB$\"3$4v=DP(yT7FB7$$\"3-+++%G$GK:FB$\"3MNpfN=1k5FB7$$\"3u* *****Hr9(f\"FB$\"3[LFDiiR*)*)F*7$$\"3)******>Pn\"p;FB$\"3YUImxRfHsF*7$ $\"3Ommmi2)Qt\"FB$\"3YuzC%y)45dF*7$$\"3H+++?h(H!=FB$\"31Y\"3y+qz9%F*7$ $\"3GLLL$y'el=FB$\"3w6T5Hgy#y#F*7$$\"37+++wn.M>FB$\"3\"H3Eyc1:M\"F*7$$ \"3Ummm%)GW)*>FB$\"3S)>D-gVa6$!#?7$$\"3C+++Eewl?FB$!39I%)H&=dTH\"F*7$$ \"3rmmm1jeJ@FB$!3K)3,;+F)[DF*7$$\"3w******\\U\\+AFB$!3P<<7Y0o@QF*7$$\" 3oLLL*ygoE#FB$!3'pl'*e?63,&F*7$$\"3LLLLh,tMBFB$!3S72gI-*>>'F*7$$\"3(pm m1aP?S#FB$!3N)3\")[3>3L(F*7$$\"3++++Kg)QY#FB$!3g(eX_930N)F*7$$\"3\"pmm EQrZ`#FB$!3)y;T1$=o*[*F*7$$\"3?LLL&3s\")f#FB$!3[jq(4Nc$[5FB7$$\"3M+++e /xlEFB$!37m$[Q+$*=:\"FB7$$\"32+++E2ZIFFB$!3WJ!='H&)y[7FB7$$\"3#)****** *******z#FB$!3EyF)e.r1N\"FB-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fd[lFc[l- %*THICKNESSG6#\"\"\"-F$6%7$7$Fc[lFc[l7$Fc[l$\"3Q******)4c!*Q)FB-F][l6& F_[lFd[lFd[lFd[l-%*LINESTYLEG6#\"\"#-F$6%7$7$$Fh[lFd[lFc[l7$Fj\\l$\"3% )*****z#=G=FFBF`\\lFb\\l-F$6$7$F]\\lFi\\l-F][l6&F_[lFc[lFc[l$\"*++++\" !\")-F$6$7$F]\\l7$$\"3++++F$\\$z9FBFc[l-%&COLORG6&F_[lFc[l$\"\")Fb[lFc [l-F$6&7'F\\\\lF]\\lFi\\lFi]lF[]l-%'SYMBOLG6#%'CIRCLEG-F][l6&F_[l$\")! \\DP\"Fe]lFj^l$\")viobFe]l-%&STYLEG6#%&POINTG-F$6&Fc^l-Fe^l6#%&CROSSGF h^lF^_l-F$6&Fc^l-Fe^l6#%(DIAMONDGFh^lF^_l-F$6$7$7$F(Fc[l7$$\"3++++++++ NFBFc[lF`\\l-%%TEXTG6%7$$Fb[lFb[l$\"#7Fd[lQ)y~=~f(x)6\"-F][l6&F_[lFc]l Fc[lFc[l-Fd`l6%7$$\"#:!\"#$\"\"*Fd[lQ)(a,f(a))F[alF`\\l-Fd`l6%7$$\"$: \"Fcal$\"#NFb[lQ)(b,f(b))F[alF`\\l-Fd`l6%7$$Fe\\lFd[l$!\"%Fb[lQ\"rF[al F`\\l-Fd`l6%7$$\"$[\"FcalFcblQ\"cF[alF`\\l-Fd`l6%7$Fj\\lFcblQ\"bF[alF` \\l-Fd`l6%7$Fc[lFcblQ\"aF[alF`\\l-%+AXESLABELSG6%Q\"xF[alQ!F[al-%%FONT G6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$;Fcbl$\"#GFb[lF\\dl" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2 " "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9 " "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "C urve 16" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 259 "" 0 "" {TEXT -1 32 "If the initial approximation is " }{XPPEDIT 18 0 "a=x[0]" "6#/%\"aG&%\"xG6#\"\"!" }{TEXT -1 35 ", and the 1st new \+ approximation is " }{XPPEDIT 18 0 "c = x[1];" "6#/%\"cG&%\"xG6#\"\"\" " }{TEXT -1 7 ", then " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x[1] = x[0]-``(f(x[0])/(f(x[0])-f(x[0]-h)))*h;" "6#/&%\"xG6#\"\" \",&&F%6#\"\"!F'*&-%!G6#*&-%\"fG6#&F%6#F+F',&-F26#&F%6#F+F'-F26#,&&F%6 #F+F'%\"hG!\"\"FAFAF'F@F'FA" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "h = f(x[0])/`f '`(x[0]);" "6#/%\"hG*&-% \"fG6#&%\"xG6#\"\"!\"\"\"-%$f~'G6#&F*6#F,!\"\"" }{TEXT -1 1 "." }} {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 0 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x[n+1] = x[n]-``(f(x[ n])/(f(x[n])-f(x[n]-h)))*h;" "6#/&%\"xG6#,&%\"nG\"\"\"F)F),&&F%6#F(F)* &-%!G6#*&-%\"fG6#&F%6#F(F),&-F36#&F%6#F(F)-F36#,&&F%6#F(F)%\"hG!\"\"FB FBF)FAF)FB" }{TEXT -1 1 "," }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{TEXT 271 20 "____________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "with " }{XPPEDIT 18 0 "h = f(x[n])/`f '`(x[n]);" "6#/%\"hG*&-%\" fG6#&%\"xG6#%\"nG\"\"\"-%$f~'G6#&F*6#F,!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 133 "The iterative formula can be applied successiv ely until some member of the sequence generated gives the root to the \+ desired accuracy." }}{PARA 0 "" 0 "" {TEXT -1 140 "The sequence of app roximations usually converges more rapidly than a corresponding sequen ce of approximations obtained by Newton's method. " }}{PARA 0 "" 0 " " {TEXT -1 33 "This method has been called the \"" }{TEXT 266 9 "leap- frog" }{TEXT -1 49 "\" Newton method. It could also be called (the/a) \+ " }{TEXT 266 22 "improved Newton method" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 12 "It requires " }{TEXT 266 15 "two evaluations" } {TEXT -1 17 " of the function " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"x G" }{TEXT -1 5 " and " }{TEXT 266 14 "one evaluation" }{TEXT -1 19 " o f the derivative " }{XPPEDIT 18 0 "`f '`(x);" "6#-%$f~'G6#%\"xG" } {TEXT -1 16 " per iteration. " }}{PARA 0 "" 0 "" {TEXT -1 106 "The mor e rapid convergence therefore comes at the expense of one extra functi on evaluation per iteration. " }}{PARA 0 "" 0 "" {TEXT -1 31 "In a cas e where the difference " }{XPPEDIT 18 0 "f(x[n])-f(x[n]-h);" "6#,&-%\" fG6#&%\"xG6#%\"nG\"\"\"-F%6#,&&F(6#F*F+%\"hG!\"\"F2" }{TEXT -1 53 " in the values of the function at the starting value " }{XPPEDIT 18 0 "x= x[n]" "6#/%\"xG&F$6#%\"nG" }{TEXT -1 38 " and at the intermediate Newt on value " }{XPPEDIT 18 0 "x = x[n]-f(x[n])/`f '`(x[n]);" "6#/%\"xG,&& F$6#%\"nG\"\"\"*&-%\"fG6#&F$6#F(F)-%$f~'G6#&F$6#F(!\"\"F5" }{TEXT -1 116 " is zero, the method fails. In such an eventuality it may be desi rable to perform a simple Newton iteration instead." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Introductory example " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = 1/16;" "6#/-%\"fG6#% \"xG*&\"\"\"F)\"#;!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^2-x+31/8" " 6#,(*$%\"xG\"\"#\"\"\"F%!\"\"*&\"#JF'\"\")F(F'" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "g(x)" "6#- %\"gG6#%\"xG" }{TEXT -1 24 " has the two real zeros " }{XPPEDIT 18 0 " x=8" "6#/%\"xG\"\")" }{TEXT -1 3 " \261 " }{XPPEDIT 18 0 "sqrt(2)" "6# -%%sqrtG6#\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "f := x -> x^2/16-x+31/8:\n'f (x)'=f(x);\nplot(f(x),x=0..15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"fG6#%\"xG,(*&#\"\"\"\"#;F+*$)F'\"\"#F+F+F+F'!\"\"#\"#J\"\")F+" }} {PARA 13 "" 1 "" {GLPLOT2D 655 190 190 {PLOTDATA 2 "6%-%'CURVESG6$7S7$ $\"\"!F)$\"3+++++++vQ!#<7$$\"3')*****\\7t&pK!#=$\"3MEBE0SsaNF,7$$\"3$* ***\\(=7T9hF0$\"3)[)*pM9DpG$F,7$$\"3X****\\(=HPJ*F0$\"3'eIh30Vy*HF,7$$ \"3;++DJaU`7F,$\"3P::tvnw>FF,7$$\"3)***\\P%GZRd\"F,$\"3!\\Y**Rf%)eX#F, 7$$\"3%)**\\(=276(=F,$\"3Fs(>A?/FA#F,7$$\"3'***\\(o**3)y@F,$\"37l*GMa \"*G*>F,7$$\"3/+](ofHq\\#F,$\"3wdVg.xmnw\">AP)F07$$\"37++]7XM*Q%F,$\"3'4m$)el?!)*oF07$$\"3/+](o%QjtYF,$\" 3))e(4a'[WlcF07$$\"32++]i8o6]F,$\"3)ol=X0)H&F,$ \"3$*\\)GhTLQ9F07$$\"35+](=xZ&\\iF,$\"3q'fB%o(=0l'!#>7$$\"3;+]i:$4 wb'F,$\"3$[1O#)R!oI]!#?7$$\"3-++v=#R!zoF,$!3wu67$HVlk%F\\r7$$\"3q+]P4A @urF,$!3U\\6\")F,$!3`@*3**esAC\"F07$$\"3 w++voo6A%)F,$!34Pz>VejQ6F07$$\"3q*****\\xJLu)F,$!3KT4s,k D\"F07$$\"3*****\\i0XE.\"Fhu$\"3o^yMPEtK@F07$$\"3%**\\(o/Q*>1\"Fhu$\"3 M([9f5Z+/$F07$$\"3=++vQ(zS4\"Fhu$\"3SwUX'z!=bTF07$$\"3***\\(=-,FC6Fhu$ \"3@KKH)pV>K&F07$$\"33+v$4tFe:\"Fhu$\"3_9\"3:)eLjmF07$$\"3!****\\73\"o '=\"Fhu$\"395=Hi69&4)F07$$\"3-+voz;)*=7Fhu$\"3rM*4Y*Hg@(*F07$$\"31+++& *44]7Fhu$\"3yibi)*o8T6F,7$$\"35+]7jZ!>G\"Fhu$\"3(e,;XD^kK\"F,7$$\"34+v =(4bMJ\"Fhu$\"3-3RF,7$$\"3\"******\\;$R09Fhu$\"3]\"[Hk_Ic;#F,7$ $\"38+v=-*zqV\"Fhu$\"3l<-26Dp6CF,7$$\"33+D\"G:3uY\"Fhu$\"3A4PDl-'*eEF, 7$$\"#:F)$\"3++++++]PHF,-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG 6$Q\"x6\"Q!Ff[l-%%VIEWG6$;F(Fgz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 92 "We consider a single step of the improved \"leap-frog\" Newton method with the starting value " }{XPPEDIT 18 0 "x=1" "6#/%\"x G\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 18 "The derivative of " }{XPPEDIT 18 0 "f(x);" "6#-%\" fG6#%\"xG" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "`f '`(x) = x/8-1;" "6#/- %$f~'G6#%\"xG,&*&F'\"\"\"\"\")!\"\"F*F*F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 0 "`f '`(0) = -1;" "6#/-%$f~' G6#\"\"!,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 " The " }{TEXT 266 12 "tangent line" }{TEXT -1 13 " at the point" } {XPPEDIT 18 0 "``(0,f(0));" "6#-%!G6$\"\"!-%\"fG6#F&" }{TEXT -1 4 " is " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = f(0)+`f '`(0 )*x;" "6#/%\"yG,&-%\"fG6#\"\"!\"\"\"*&-%$f~'G6#F)F*%\"xGF*F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 258 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 31/8-x;" "6#/%\"yG,&*&\"#J\"\"\" \"\")!\"\"F(%\"xGF*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Th e " }{TEXT 285 1 "x" }{TEXT -1 35 "-intercept of this tangent line is \+ " }{XPPEDIT 18 0 "x = 31/8;" "6#/%\"xG*&\"#J\"\"\"\"\")!\"\"" }{TEXT -1 26 ". This is the preliminary " }{TEXT 266 15 "Newton estimate" } {TEXT -1 42 " for the zero bases on the starting value " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "The corresponding point on the cur ve " }{XPPEDIT 18 0 "y = f(x);" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 3 " \+ is" }{XPPEDIT 18 0 "``(31/8,f(31/8));" "6#-%!G6$*&\"#J\"\"\"\"\")!\"\" -%\"fG6#*&F'F(F)F*" }{TEXT -1 10 ", that is," }{XPPEDIT 18 0 "``(31/8, 961/1024)" "6#-%!G6$*&\"#J\"\"\"\"\")!\"\"*&\"$h*F(\"%C5F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "T he " }{TEXT 266 11 "secant line" }{TEXT -1 27 " passing through the po ints" }{XPPEDIT 18 0 "``(31/8,f(31/8));" "6#-%!G6$*&\"#J\"\"\"\"\")!\" \"-%\"fG6#*&F'F(F)F*" }{TEXT -1 4 " and" }{XPPEDIT 18 0 "``(31/8,f(31/ 8));" "6#-%!G6$*&\"#J\"\"\"\"\")!\"\"-%\"fG6#*&F'F(F)F*" }{TEXT -1 12 " has slope: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(f(0)-f(31/8))/(0-31/8) = ``(31/8-961/1 024)/``(-31/8);" "6#/*&,&-%\"fG6#\"\"!\"\"\"-F'6#*&\"#JF*\"\")!\"\"F0F *,&F)F**&F.F*F/F0F0F0*&-%!G6#,&*&F.F*F/F0F**&\"$h*F*\"%C5F0F0F*-F56#,$ *&F.F*F/F0F0F0" }{XPPEDIT 18 0 "``=31/128-1" "6#/%!G,&*&\"#J\"\"\"\"$G \"!\"\"F(F(F*" }{XPPEDIT 18 0 "``=-97/128" "6#/%!G,$*&\"#(*\"\"\"\"$G \"!\"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "This secant line therefore has the equation " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=31/8-97*x/128" "6#/%\"yG,&*&\"#J\"\"\"\"\")!\"\"F(*( \"#(*F(%\"xGF(\"$G\"F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "The secant line has the " }{TEXT 286 1 "x" }{TEXT -1 11 "-intercep t " }{XPPEDIT 18 0 "31/8" "6#*&\"#J\"\"\"\"\")!\"\"" }{TEXT -1 1 " " } {TEXT 287 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "128/97=496/97" "6#/*& \"$G\"\"\"\"\"#(*!\"\"*&\"$'\\F&F'F(" }{TEXT -1 1 " " }{TEXT 288 1 "~ " }{TEXT -1 26 " 5.113402062. This is the " }{TEXT 266 8 "improved" } {TEXT -1 2 " \"" }{TEXT 266 9 "leap-frog" }{TEXT -1 2 "\" " }{TEXT 266 15 "Newton estimate" }{TEXT -1 15 " for the zero. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 453 "f := x -> x^2/16-x+31/8:\n'f(x)'=f(x);\ng := x -> 31/8-x:\n'g(x)'=g(x);\nh : = x -> 31/8-97/128*x:\n'h(x)'=h(x);\na := 0: fa := f(a):\nb := 31/8: f b := f(b):\nc := 496/97:\np1 := plot([f(x),g(x),h(x)],x=0..10,y=-2..4. 2,color=[red,blue,green],thickness=[2$3]):\np2 := plot([[b,0],[b,fb]], color=black,linestyle=2):\np3 := plot([[[a,fa],[b,0],[b,fb],[c,0]]$3], style=point,color=black,\n symbol=[circle,diamo nd,cross]):\nplots[display]([p1,p2,p3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,(*&#\"\"\"\"#;F+*$)F'\"\"#F+F+F+F'!\"\"#\"#J\"\") F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,&#\"#J\"\")\"\"\" F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"hG6#%\"xG,&#\"#J\"\") \"\"\"*(\"#(*F,\"$G\"!\"\"F'F,F0" }}{PARA 13 "" 1 "" {GLPLOT2D 473 312 312 {PLOTDATA 2 "6+-%'CURVESG6%7S7$$\"\"!F)$\"3+++++++vQ!#<7$$\"3e mmm;arz@!#=$\"3EnV*G$z**fOF,7$$\"3[LL$e9ui2%F0$\"39gmB#fdxZ$F,7$$\"3nm mm\"z_\"4iF0$\"3KZsK22=yKF,7$$\"39ommT&phN)F0$\"3+2SrES-$3$F,7$$\"3KLL e*=)H\\5F,$\"3g&4$eyf^%*GF,7$$\"3smm\"z/3uC\"F,$\"3m(ye%HO%[s#F,7$$\"3 !****\\7LRDX\"F,$\"3[iF,7$$\"3D+++D.&4]#F,$\"3;]t+3<(\\w\"F,7$$ \"3;+++vB_-T;;c.>;F,7$$\"33+++v'Hi#HF,$\"3)H;G>eYR[\"F,7$$\"3 &om;z*ev:JF,$\"37*)f/G+*fO\"F,7$$\"3_LLL347TLF,$\"3g^IU\\@dJ7F,7$$\"3n LLLLY.KNF,$\"3%**HetlpE7\"F,7$$\"33++D\"o7Tv$F,$\"3@H,AXCs,5F,7$$\"3?L LL$Q*o]RF,$\"3Yie!Rzs!)**)F07$$\"3m++D\"=lj;%F,$\"3D5x?_CZNzF07$$\"3S+ +vV&RY2aF,$\"3]Wr?kNy]HF07$$\"3Znm;z Xu9cF,$\"3Y]U8R@!fI#F07$$\"34+++]y))GeF,$\"3)[^B*z*zgp\"F07$$\"3H++]i_ QQgF,$\"3iYd[Ox&\\:\"F07$$\"3b++D\"y%3TiF,$\"3MH(*=n@9Oo!#>7$$\"3+++]P ![hY'F,$\"3n')o!z^'Q/AF^u7$$\"3iKLL$Qx$omF,$!3JZjQUDQ<9F^u7$$\"3Y+++v. 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" }}{PARA 0 "" 0 "" {TEXT -1 12 "Derivative: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "f := x -> x^2/16-x+31/8:\n'f(x)'=f(x);\na := 0;\nDiff('f(x)',x)=d iff(f(x),x);\nEval(Diff('f(x)',x),x=a)=eval(rhs(%),x=a);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,(*&#\"\"\"\"#;F+*$)F'\"\"#F+F +F+F'!\"\"#\"#J\"\")F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6#%\"xGF*,&*&\"\" )!\"\"F*\"\"\"F/F/F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%EvalG6$-%% DiffG6$-%\"fG6#%\"xGF-/F-\"\"!!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 14 "T angent line: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "y=f(a)+D(f) (a)*(x-a);\neq := %:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,&#\"#J \"\")\"\"\"%\"xG!\"\"" }}}{PARA 0 "" 0 "" {TEXT 284 1 "x" }{TEXT -1 67 "-intercept of tangent line (preliminary Newton estimate for zero): " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "b := solve(eval(eq,y=0) ,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG#\"#J\"\")" }}}{PARA 0 " " 0 "" {TEXT -1 14 "Secant through" }{XPPEDIT 18 0 "``(0,f(0));" "6#-% !G6$\"\"!-%\"fG6#F&" }{TEXT -1 4 " and" }{XPPEDIT 18 0 "``(31/8,f(31/8 ));" "6#-%!G6$*&\"#J\"\"\"\"\")!\"\"-%\"fG6#*&F'F(F)F*" }{TEXT -1 15 " has equation: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "y=f(0)+(f (a)-f(b))/(a-b)*x;\neq2 := %:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"y G,&#\"#J\"\")\"\"\"*(\"#(*F)\"$G\"!\"\"%\"xGF)F-" }}}{PARA 0 "" 0 "" {TEXT 289 1 "x" }{TEXT -1 56 "-intercept of secant line (improved esti mate for zero): " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "c := sol ve(eval(eq2,y=0),x);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"cG#\"$'\\\"#(*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+i?S8^!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "This process can be automated by the following \+ procedure " }{TEXT 0 11 "next_approx" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 289 "f := x -> x^2/16-x+31/8:\n'f(x)'=f(x);\nDf := D(f):\nnext_approx := proc(x)\n \+ local fx,dfx,fxp,h,d;\n fx := evalf(f(x));\n dfx := evalf(Df(x)); \n h := fx/dfx;\n fxp := evalf(f(x-h));\n if fxp<>0 then\n \+ d := fx-fxp;\n if d<>0 then h := fx*h/d end if;\n end if;\n x -h;\nend proc:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,(*&# \"\"\"\"#;F+*$)F'\"\"#F+F+F+F'!\"\"#\"#J\"\")F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Four iterations are neede d to calculate the zero correct to 11 digits. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "Digits := 1 2:\nx0 := 0;\nx1 := next_approx(x0);\nx2 := next_approx(x1);\nx3 := ne xt_approx(x2);\nx4 := next_approx(x3);\nDigits := 10:\n" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#x0G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#x1G$\"-'=1-M6&!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x2G$\"-k%Q_v Y'!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x3G$\"-\\2Pg&e'!#6" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x4G$\"-hPky&e'!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f(x)=0;\n solve(%);\nevalf[12](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&#\"\" \"\"#;F'*$)%\"xG\"\"#F'F'F'F+!\"\"#\"#J\"\")F'\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$,&\"\")\"\"\"*$\"\"##F%F'F%,&F$F%F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"-PiN@9%*!#6$\"-jPky&e'F%" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 75 "A procedure for graphing successi ve stages of the leap-frog Newton method: " }{TEXT 0 14 "impnewton_ste p" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The \+ procedure " }{TEXT 0 14 "impnewton_step" }{TEXT -1 80 " enables the p rogress of the improved Newton method to be observed graphically." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "i mpnewton_step: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }{TEXT 272 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 273 2 " " }{TEXT -1 70 " impnewton_step( eqn, approxroot ) or newtonstep( 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 266 2 "OR" }{TEXT -1 34 " a function of the form x -> f(x)," }}{PARA 0 "" 0 "" {TEXT -1 96 " where f(x) evaluates t o a real or complex floating point number." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 15 "approxroot - \+ " }{TEXT 274 109 "an initial approximation for the root in the form of a real constant a, when the1st argument is a procedure, " }}{PARA 0 " " 0 "" {TEXT 276 115 " and in the form of an 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 14 "impnewton_step" }{TEXT -1 106 " performs a single iteration of the improved \"leap-frog\" Newton method, and returns the new approximation." }}{PARA 0 "" 0 "" {TEXT -1 47 "A picture can be drawn to illustrate the step. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 275 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 -1 264 "If c is a list of up to 5 colours, these colours will be applied in respective order to the curve, the tangent line, t he secant line, the ordinates of the initial and preliminary approxima tions, and the 3 points shown. A single colour is applied to the curve only." }}{PARA 0 "" 0 "" {TEXT -1 52 "The default is \"colour=[red,gr een,blue,navy,black]\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "thickness=t" }}{PARA 0 "" 0 "" {TEXT -1 175 "If t is is a list of 1 or 2 or 3 positive integers, then they will be applied in respective order to specify the thickness of the curve, the tangen t line and the secant line. " }}{PARA 0 "" 0 "" {TEXT -1 105 "A single thickness is applied to both the curve and the tangent line. The defa ult is \"thickness=[1,2,2]\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 15 "zoom=true/false" }}{PARA 0 "" 0 "" {TEXT -1 130 "This option allows a zoom feature (which causes an extra \"zoo med picture\" to be drawn in some situations) to be turned on or off. \+ " }}{PARA 0 "" 0 "" {TEXT -1 50 "See the next option \"max_ratio\" for more details. " }}{PARA 0 "" 0 "" {TEXT -1 28 "The default is \"zoom= false\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "max_ratio=n or maxratio=n" }}{PARA 0 "" 0 "" {TEXT -1 220 "Three x values are involved in each step of the method: the initial approxima tion x=a, the preliminary Newton approximation b=a-h, where h=f '(a)/f (a), and the improved approximation c=a-k, where k=f(a)/(f(a)-f(b))*h. " }}{PARA 0 "" 0 "" {TEXT -1 311 "If any one of the three x values \+ is an outlier, in the sense that its maximum distance from the other t wo values is greater than \"max_ratio\" times the distance between the other two values, then an additional \"zoomed picture\" is drawn to s how what is happening in the neighbourhood of the two closer x values. " }}{PARA 0 "" 0 "" {TEXT -1 31 "The default is \"max_ratio=25\". " }}{PARA 0 "" 0 "" {TEXT -1 79 "Decreasing \"max_ratio\" means that zoo med pictures are more likely to be drawn. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 16 "How to activate:" }{TEXT 256 1 "\n" }{TEXT -1 154 "To make the pro cedure active open the subsection, place the cursor anywhere after the prompt [ > and press [Enter].\nYou can then close up the subsection. " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "impnewton_step: implementatio n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "newton_ste p" {MPLTEXT 1 0 11663 "# to allow for different spellings\nimpnewtonst ep := proc() impnewton_step(args[1..nargs]) end:\n\nimpnewton_step := \+ proc(ff,approx)\n local x,x1,y1,x2,w2,h2,xL,xR,yT,yB,xrange,yrange,d f,m,pts,ord,\n f,fn,approxroot,lmr,sf,proctype,vars,Options,i,clr,co lr,thk,\n thik,drawpic,lft,rgt,mm,yy,xx2,h,xp,yp,mp,d,tgt,secnt,curv e,\n mindist,maxdist,mxr,zm;\n\n if nargs<2 then\n error \"at least 2 arguments are required; the basic syntax is: 'impnewton_step( f(x),x=a)'.\"\n end if;\n\n if type(ff,procedure) then\n if n ops([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,complexcon s) then\n approxroot := approx;\n else\n error \" the 2nd argument, %1, is invalid .. when the 1st argument is a procedu re, the 2nd argument should be a complex constant\",approx;\n end if;\n elif type(ff,algebraic) or type(ff,equation) then\n if t ype(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,na me) 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 should be an expression or an equation which depends only on a single variable\",ff;\n end if;\n \+ end if;\n if type(approx,name=complexcons) then\n proct ype := false;\n x := op(1,approx);\n if not member(x,v ars) then\n error \"the 1st argument, %1, is invalid .. it \+ should be an expression or an equation which depends only on the varia ble %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 should be an algebraic expression in a single v ariable, an equation in a single variable, or a procedure with a singl e argument\",ff;\n end if;\n\n # Get the options.\n # Set the de fault value to start with.\n drawpic := false;\n zm := false;\n \+ mxr := 25;\n clr := [COLOR(RGB,1,0,0),COLOR(RGB,0,1,0),COLOR(RGB,0,0 ,1),\n COLOR(RGB,.137,.137,.557),COLOR(RGB,0,0,.01)]; \n thk := [1,2,2];\n if nargs>2 then\n Options:=[args[3..narg s]];\n if not type(Options,list(equation)) then\n error \+ \"each optional argument must be an equation\"\n end if;\n i f hasoption(Options,'draw','drawpic','Options') then\n if draw pic<>true then drawpic := false end if;\n end if;\n if hasop tion(Options,'zoom','zm','Options') then\n if zm<>true then zm := false end if;\n end if;\n if hasoption(Options,'maxratio ','mxr','Options') then\n if not type(mxr,posint) then\n \+ error \"\\\"maxratio\\\" must be a positive integer\";\n \+ end if;\n elif hasoption(Options,'max_ratio','mxr','Options') th en\n if not type(mxr,posint) then\n error \"\\\"max _ratio\\\" must be a positive integer\";\n end if;\n end \+ if;\n if hasoption(Options,'color','colr','Options') or\n \+ hasoption(Options,'colour','colr','Options') then\n if type(c olr,list) then\n for i from 1 to min(nops(colr),5) 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,'thickness','thik','Option s') then\n if type(thik,list) then\n for i from 1 t o min(nops(thik),3) do\n thk[i] := thik[i];\n \+ end do;\n else\n thk := [thik$3];\n end if ;\n end if;\n if nops(Options)>0 then\n error \"%1 i s not a valid option for %2 .. the recognised options are \\\"draw\\\" , \\\"zoom\\\",\\\"max_ratio\\\", or (\\\"maxratio\\\"),\\\"colour\\\" , or (\\\"color\\\") and \\\"thickness\\\"\",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(ev alf(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 \"failed to evaluate derivative at %1\",x1 ;\n end if;\n if m=0 then\n error \"zero derivative obtained \"\n end if;\n y1 := traperror(evalf(fn(x1)));\n if y1=lasterror or not type(y1,numeric) then\n error \"failed to evaluate functi on at %1\",x1;\n end if;\n\n # This is where the Newton formula b= a-f(a)/f'(a) is applied.\n h := y1/m;\n xp := x1 - h;\n yp := tr aperror(evalf(fn(xp)));\n if yp=lasterror or not type(yp,numeric) th en\n error \"failed to evaluate function at %1\",xp;\n end if; \n d := y1-yp;\n if d<>0 then \n x2 := x1 - h*y1/d\n else \n x2 := xp\n end if;\n\n if drawpic then\n # recalulate for the picture\n Digits := max(Digits,15);\n mm := traperr or(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(x1)));\n if yy<>lasterror and type(yy,numeric) then\n \+ y1 := yy;\n end if;\n h := y1/m;\n xp := x1 - h; \n yy := traperror(evalf(fn(xxp)));\n if yy<>lasterror and t ype(yy,numeric) then\n yp := yy;\n end if;\n d := y1 - yp;\n if d=0 then\n xL := min(x1,xp);\n xR := \+ max(x1,xp);\n w2 := (xR-xL)/2;\n lft := xL-w2;\n \+ rgt := xR+w2;\n if lft<>rgt then\n xrange := lft ..rgt;\n yT := max(0,y1,yp);\n yB := min(0,y1,yp );\n h2 := (yT-yB)/2;\n yrange := yB-h2..yT+h2; \n curve := op(plot('fn'(x),x=xrange,yrange,color=op(1,clr ),thickness=thk[1],\n title=`standard Newton step `));\n tgt := CURVES([[x1,y1],[xp,0]],THICKNESS(thk[3]),LIN ESTYLE(1),op(3,clr));\n ord := CURVES([[x1,0],[x1,y1]],THIC KNESS(1),LINESTYLE(2),op(4,clr));\n pts := POINTS([x1,0],[x 1,y1],[xp,0],SYMBOL(CIRCLE),op(5,clr)),\n POINTS([x1,0],[ x1,y1],[xp,0],SYMBOL(CROSS),op(5,clr)),\n POINTS([x1,0],[ x1,y1],[xp,0],SYMBOL(DIAMOND),op(5,clr));\n print(PLOT(pts, tgt,ord,curve));\n else\n WARNING(\"the range for t he plot is empty\");\n end if;\n else # d<>0\n xx 2 := x1 - y1*h/d;\n if zm then \n mindist := min(ab s(x1-xp),abs(xx2-xp),abs(x1-xx2));\n maxdist := max(abs(x1- xp),abs(xx2-xp),abs(x1-xx2));\n end if;\n if not zm or maxdistrgt then \n xrange := lft..rgt;\n yT := max(0,y1,yp );\n yB := min(0,y1,yp);\n h2 := (yT-yB)/2 ;\n yrange := yB-h2..yT+h2;\n curve := op (plot('fn'(x),x=xrange,yrange,color=op(1,clr),thickness=thk[1]));\n \+ tgt := CURVES([[x1,y1],[xp,0]],THICKNESS(thk[3]),LINESTYLE (1),op(3,clr));\n secnt := CURVES([[x1,y1],[xx2,0],[xp,y p]],\n LINESTYLE(1),THICKNESS(thk[2 ]),op(2,clr));\n ord := CURVES([[x1,0],[x1,y1]],[[xp,0], [xp,yp]],\n THICKNESS(1),LINESTYLE(2),op(4,clr));\n \+ pts := POINTS([x1,0],[x1,y1],[xx2,0],[xp,0],[xp,yp],\n \+ SYMBOL(CIRCLE),op(5,clr)),\n POINTS([x1,0],[x1,y1],[xx2,0],[xp,0],[xp,yp],SYMBOL(CR OSS),op(5,clr)),\n POINTS([x1,0],[x1,y1],[xx2,0],[xp,0] ,[xp,yp],SYMBOL(DIAMOND),op(5,clr));\n print(PLOT(pts,tg t,secnt,ord,curve));\n else\n WARNING(\"the r ange for the plot is empty\");\n end if;\n else\n \+ if abs(xx2-xp)>max(abs(x1-xp),abs(x1-xx2)) then\n \+ xL := min(xx2,xp);\n xR := max(xx2,xp);\n \+ yT := max(0,yp);\n yB := min(0,yp);\n eli f abs(x1-xp)>max(abs(xx2-xp),abs(x1-xx2)) then\n xL := m in(x1,xp);\n xR := max(x1,xp);\n yT := max (0,y1,yp);\n yB := min(0,y1,yp);\n else\n \+ xL := min(x1,xx2);\n xR := max(x1,xx2);\n \+ yT := max(0,y1);\n yB := min(0,y1);\n \+ end if;\n w2 := (xR-xL)/2;\n lft := xL-w2;\n rgt := xR+w2;\n if lft<>rgt then\n \+ xrange := lft..rgt;\n yT := max(0,y1,yp);\n \+ yB := min(0,y1,yp);\n h2 := (yT-yB)/2;\n \+ yrange := yB-h2..yT+h2;\n curve := op(plot('fn'(x),x =xrange,yrange,color=op(1,clr),thickness=thk[1]));\n tgt := CURVES([[x1,y1],[xp,0]],THICKNESS(thk[3]),LINESTYLE(1),op(3,clr)); \n secnt := CURVES([[x1,y1],[xx2,0],[xp,yp]],LINESTYLE(1 ),\n THICKNESS(thk[2]),op(2,clr));\n \+ ord := CURVES([[x1,0],[x1,y1]],[[xp,0],[xp,yp]],\n \+ THICKNESS(1),LINESTYLE(2),op(4,clr));\n \+ pts := POINTS([x1,0],[x1,y1],[xx2,0],[xp,0],[xp,yp],\n \+ SYMBOL(CIRCLE),op(5,clr)),\n POINTS( [x1,0],[x1,y1],[xx2,0],[xp,0],[xp,yp],SYMBOL(CROSS),op(5,clr)),\n \+ POINTS([x1,0],[x1,y1],[xx2,0],[xp,0],[xp,yp],SYMBOL(DIAMOND ),op(5,clr));\n print(PLOT(pts,tgt,secnt,ord,curve,VIEW( xrange,yrange)));\n else\n WARNING(\"the rang e for the plot is empty\");\n end if;\n if abs(x x2-xp)rgt then\n xrange := lft..r gt;\n h2 := (yT-yB)/2;\n yrange := yB-h2.. yT+h2;\n curve := op(plot('fn'(x),x=xrange,yrange,colo r=op(1,clr),thickness=thk[1],\n title=`Zo om of previous picture`));\n tgt := CURVES([[x1,y1],[xp, 0]],THICKNESS(thk[3]),LINESTYLE(1),op(3,clr));\n secnt : = CURVES([[x1,y1],[xx2,0],[xp,yp]],\n \+ LINESTYLE(1),THICKNESS(thk[2]),op(2,clr));\n ord := C URVES([[x1,0],[x1,y1]],[[xp,0],[xp,yp]],\n THICKNESS(1 ),LINESTYLE(2),op(4,clr));\n pts := POINTS([x1,0],[x1,y1 ],[xx2,0],[xp,0],[xp,yp],\n \+ SYMBOL(CIRCLE),op(5,clr)),\n POINTS([x1,0],[x1,y1],[ xx2,0],[xp,0],[xp,yp],SYMBOL(CROSS),op(5,clr)),\n POINT S([x1,0],[x1,y1],[xx2,0],[xp,0],[xp,yp],SYMBOL(DIAMOND),op(5,clr));\n \+ print(PLOT(pts,tgt,secnt,ord,curve,VIEW(xrange,yrange))) ;\n else\n WARNING(\"the range for the plot i s empty\");\n end if;\n end if;\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 given in the next sectio n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 14 "impnewton_s tep" }{TEXT -1 10 ": examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 266 4 "Note" }{TEXT -1 53 ": The exam ples in this section require the procedure " }{TEXT 0 14 "impnewton_st ep" }{TEXT -1 3 " - " }{HYPERLNK 17 "impnewton_step" 1 "" "newton_step " }{TEXT -1 2 ". " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }} {PARA 0 "" 0 "" {TEXT -1 87 "We can graph a single step in the improve d \"leap-frog\" Newton iteration for computing " }{XPPEDIT 18 0 "8-sq rt(2);" "6#,&\"\")\"\"\"-%%sqrtG6#\"\"#!\"\"" }{TEXT -1 14 " as a zero of " }{XPPEDIT 18 0 "g(x) = 1/16;" "6#/-%\"gG6#%\"xG*&\"\"\"F)\"#;!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^2-x+31/8" "6#,(*$%\"xG\"\"#\"\" \"F%!\"\"*&\"#JF'\"\")F(F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "g := x -> x^2/16-x +31/8:\n'g(x)'=g(x);\nimpnewton_step(g(x),x=0,draw=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,(*&#\"\"\"\"#;F+*$)F'\"\"#F+F+F+ F'!\"\"#\"#J\"\")F+" }}{PARA 13 "" 1 "" {GLPLOT2D 486 394 394 {PLOTDATA 2 "6--%'POINTSG6)7$$\"\"!F(F(7$F'$\"++++vQ!\"*7$$\"0,&)41-M6 &!#9F(7$$\"0+++++](QF0F(7$F2$\"*ilZQ*F,-%'SYMBOLG6#%'CIRCLEG-%&COLORG6 &%$RGBGF(F($\"\"\"!\"#-F$6)F&F)F-F1F4-F86#%&CROSSGF;-F$6)F&F)F-F1F4-F8 6#%(DIAMONDGF;-%'CURVESG6&7$F)F1-%*THICKNESSG6#\"\"#-%*LINESTYLEG6#F@- F<6&F>F(F(F@-FM6&7%F)F-F4FTFP-F<6&F>F(F@F(-FM6'7$F&F)7$F1F4-FQFV-FUFR- F<6&F>$\"$P\"!\"$F`o$\"$d&Fbo-FM6#7S7$$!0]#\\I5qcDF0$\"04]//Y-%oF07$$! 0i8W!eyLBF0$\"0y=dc&>\\lF07$$!0iB\"e%G)R@F0$\"0&3^sv+,jF07$$!0#\\&o8.< #>F0$\"0I#*427v-'F07$$!0`m$R>8-<'\\&F07$$!0BrQD,5G\"F0$\"0YZr^h&e_F07$$!0Sx%3v@r5F0$\"0o\")GpOz,& F07$$!/N#*)e)eU&)F0$\"05niuo[x%F07$$!/2[%)*f*zjF0$\"0ID$HfVQXF07$$!/g' RkIa:%F0$\"0h%z;`L,VF07$$!/T+!*R1'>#F0$\"0`bYe?w4%F07$$\",si(y=(*F0$\" 0br-=GS(QF07$$\"/2F@(eXA#F0$\"0<88/Pcl$F07$$\"/f>(Rn*eVF0$\"0bhKjy4X$F 07$$\"/:%\\&\\@(H'F0$\"0\"Q7FG1qKF07$$\"/%=Cxy>g)F0$\"0()3qR[51$F07$$ \"0\"*=UgTa0\"F0$\"0&e-22=*)GF07$$\"0B-QZcDG\"F0$\"0_y/(HD&p#F07$$\"09 9>F0$\"0u&)*om\")*=#F07$$\"0bu'y!QL8#F0$\"0j&>Jw5E?F07$$\"0ub\"pJeMBF 0$\"0%3qT#f5)=F07$$\"052l=`;b#F0$\"091v=!GI=\"Gz7xFF0$\"0p_C a**)z:F07$$\"0K9NN/M(HF0$\"0Izq[mTX\"F07$$\"0`5,F)Q&=$F0$\"0L\\=Q!yB8F 07$$\"0e&f)R)Q/MF0$\"0=NW(y(\\>\"F07$$\"0Gl()HP'=OF0$\"0o&)\\>rZ2\"F07 $$\"0m_h7Mf#QF0$\"/1^*>m#R'*F07$$\"0\"[#)4>5cSF0$\"/Z0\"fd9Z)F07$$\"0( *z0!z\"HE%F0$\"/&Gh89'yuF07$$\"0X+Z6QP[%F0$\"/kvG5cxkF07$$\"0kI!\\S$Qo %F0$\"//)R)y4BcF07$$\"0t:vv\"f-\\F0$\"/4)y86iu%F07$$\"0L8IrD%3^F0$\"/l sg6vvRF07$$\"0tm[]\"eB`F0$\"/@ov*4qA$F07$$\"0q,$=c$R`&F0$\"/cF07$$\"0@V)HhDmfF0$\"/_(3Lr]L\"F07$$\"0I2lC eJ='F0$\".:FB728)F07$$\"0\"R8/VE)R'F0$\".BjuKZ`$F07$$\"0$=/JW#ff'F0$!- #4HPdy\"F07$$\"0>7CClC#oF0$!./l+*)F07$$\"0+rL8'*yW(F0$!/LGD$)[f5F07$$\"0^x94.,n( F0$!/?f=+)>=\"F0F\\o-%+AXESLABELSG6$Q\"x6\"Q!Fa_l-F<6&F>$F@F(F'F'-%%VI EWG6$;FioFi^l;$!0+++++v$>F0$\"0+++++D\"eF0" 1 2 0 1 10 1 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+h?S8^!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 ". . . or a number of successive steps . . ." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "g := x -> x^2/16-x+31/8:\nxin := 0;\nfor i from 1 to 4 do\n xout := impn ewton_step(g(x),x=xin,draw=is(i<=3),zoom=true);\n print(`approximate root =`,xout);\n xin := xout;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$xinG\"\"!" }}{PARA 13 "" 1 "" {GLPLOT2D 483 343 343 {PLOTDATA 2 "6--%'POINTSG6)7$$\"\"!F(F(7$F'$\"++++vQ!\"*7$$\"0,&)41-M6 &!#9F(7$$\"0+++++](QF0F(7$F2$\"*ilZQ*F,-%'SYMBOLG6#%'CIRCLEG-%&COLORG6 &%$RGBGF(F($\"\"\"!\"#-F$6)F&F)F-F1F4-F86#%&CROSSGF;-F$6)F&F)F-F1F4-F8 6#%(DIAMONDGF;-%'CURVESG6&7$F)F1-%*THICKNESSG6#\"\"#-%*LINESTYLEG6#F@- F<6&F>F(F(F@-FM6&7%F)F-F4FTFP-F<6&F>F(F@F(-FM6'7$F&F)7$F1F4-FQFV-FUFR- F<6&F>$\"$P\"!\"$F`o$\"$d&Fbo-FM6#7S7$$!0]#\\I5qcDF0$\"04]//Y-%oF07$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" } {TEXT -1 43 " has a single real zero which is near 4.3. 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A\\g%)R(F.7$$\"0sk(y6S6UF2$\"/,/^HVypF.7$$\"0&*e([wk;UF2$\"/+B7#odg'F. 7$$\"0Pjy&QSAUF2$\"/Kz1]&f>'F.7$$\"0mb,8&)yA%F2$\"/#)zf!*z/eF.7$$\"0Wp FE/OB%F2$\"/cV)[8dR&F.7$$\"0))G=@c)QUF2$\"/SP@U=>]F.7$$\"0:.%[6_WUF2$ \"/krM177YF.7$$\"0S-aU0/D%F2$\"/]JSlF)=%F.7$$\"09GeqFbD%F2$\"/2;q;Z=QF .7$$\"0RyV=p8F.7$$\"05pdIV\\H%F2$\".CM-C JY*F.7$$\"02\")\\El,I%F2$\".0-$G_AcF.7$$\"0$**pXU(eI%F2$\".oqd\">99F.7 $$\"05X;&fC6VF2$!-&\\uNYb#F27$$\"0_W@$4'oJ%F2$!-0/*oEr'F27$$\"0q2>g]BK %F2$!.<> F27$$\"0'zFXNHRVF2$!.(>2p9UBF27$$\"0W\\yA2\\M%F2$!.b5C7Fw#F27$$\"06=\" >c1]VF2$!.%3l+.]JF27$$\"0p2V!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3approximate~root~=G$\"+#p*y2V!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3approximate~root~=G$\"+\"p*y2V!\"*" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "evalf[14]( fsolve(g(x),x=4..5));\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"/:1\"p*y2V!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+\"p*y2V!\"*" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 " }{TEXT 293 36 ".. illustration of cubic convergence" }{TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 67 "In good situations the improved \"leap-frog\" Newton method exh ibits " }{TEXT 266 16 "cubic convegence" }{TEXT -1 106 ". Roughly spea king this means that the number of correct digits approximately triple s with each iteration." }}{PARA 0 "" 0 "" {TEXT -1 90 "Here is an exam ple where the number of correct digits exactly triples with each itera tion." }}{PARA 0 "" 0 "" {TEXT -1 13 "The equation " }{XPPEDIT 18 0 "1 /x=3" "6#/*&\"\"\"F%%\"xG!\"\"\"\"$" }{TEXT -1 18 " has the solution \+ " }{XPPEDIT 18 0 "x=1/3" "6#/%\"xG*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 23 " = 0.333333333 . . . . " }}{PARA 0 "" 0 "" {TEXT -1 76 "We can apply t he improved \"leap-frog\" Newton method with the starting value " } {XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 3 ".3." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "The preliminary Newton \+ correction for a general starting value " }{TEXT 294 1 "x" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "h=phi(x)/(phi*`'`(x))" "6#/%\"hG*&-%$phiG6#% \"xG\"\"\"*&F'F*-%\"'G6#F)F*!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "phi := x \+ -> 1/x-3:\n'phi(x)'=phi(x);\n'phi(x)'/Diff('phi(x)',x)=expand(phi(x)/d iff(phi(x),x));\nh := rhs(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$p hiG6#%\"xG,&*&\"\"\"F*F'!\"\"F*\"\"$F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%$phiG6#%\"xG\"\"\"-%%DiffG6$F%F(!\"\",&F(F-*&\"\"$F))F(\"\" #F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG,&%\"xG!\"\"*&\"\"$\"\" \")F&\"\"#F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The preliminary Newton estimate is " }{XPPEDIT 18 0 "chi= x-h" "6#/%$chiG,&%\"xG\"\"\"%\"hG!\"\"" }{XPPEDIT 18 0 "``=x-phi(x)/(p hi*`'`(x))" "6#/%!G,&%\"xG\"\"\"*&-%$phiG6#F&F'*&F*F'-%\"'G6#F&F'!\"\" F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "x-'phi(x)'/Diff('phi(x)',x)=expand(x-phi(x) /diff(phi(x),x));\nchi := rhs(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/,&%\"xG\"\"\"*&-%$phiG6#F%F&-%%DiffG6$F(F%!\"\"F.,&*&\"\"#F&F%F&F&*& \"\"$F&)F%F1F&F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$chiG,&*&\"\"#\" \"\"%\"xGF(F(*&\"\"$F()F)F'F(!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 66 "The factor needed to modify the prelimin ary Newton correction is " }{XPPEDIT 18 0 "r=phi(x)/(phi(x)-phi(chi)) " "6#/%\"rG*&-%$phiG6#%\"xG\"\"\",&-F'6#F)F*-F'6#%$chiG!\"\"F1" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "'phi'(x)/('phi'(x)-'phi'(chi))=normal(phi(x)/(ph i(x)-phi(rhs(%%))));\nr := rhs(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/*&-%$phiG6#%\"xG\"\"\",&F%F)-F&6#,&*&\"\"#F)F(F)F)*&\"\"$F))F(F/F)! \"\"F3F3,&F/F)*&F1F)F(F)F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG,& \"\"#\"\"\"*&\"\"$F'%\"xGF'!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 55 "The \"leap-frog\" Newton correction for a starting value " }{TEXT 295 1 "x" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "k = r*h;" "6#/%\"kG*&%\"rG\"\"\"%\"hGF'" }{XPPEDIT 18 0 "``=phi(x)*h/ (phi(x)-phi(chi))" "6#/%!G*(-%$phiG6#%\"xG\"\"\"%\"hGF*,&-F'6#F)F*-F'6 #%$chiG!\"\"F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "'r*h'=r*h;\nk := factor(valu e(rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"rG\"\"\"%\"hGF&*& ,&\"\"#F&*&\"\"$F&%\"xGF&!\"\"F&,&F-F.*&F,F&)F-F*F&F&F&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"kG,$*(,&\"\"\"!\"\"*&\"\"$F(%\"xGF(F(F(F,F(, &\"\"#F)*&F+F(F,F(F(F(F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 53 "The new estimate by the \"leap-frog\" Newton metho d is " }{XPPEDIT 18 0 "x-k" "6#,&%\"xG\"\"\"%\"kG!\"\"" }{XPPEDIT 18 0 "``=x-phi(x)*h/(phi(x)-phi(chi))" "6#/%!G,&%\"xG\"\"\"*(-%$phiG6#F&F '%\"hGF',&-F*6#F&F'-F*6#%$chiG!\"\"F3F3" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "'x-k' =x-k;\n``=convert(factor(rhs(%)),horner);\nunassign('h','chi','r','k') :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"xG\"\"\"%\"kG!\"\",&F%F&*(, &F&F(*&\"\"$F&F%F&F&F&F%F&,&\"\"#F(*&F-F&F%F&F&F&F&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%!G,$*(\"\"$\"\"\"%\"xGF(,&F(F(*&,&F'!\"\"*&F'F(F)F( F(F(F)F(F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "The improved \"leap-frog\" Newton iteration formula is:" }}{PARA 258 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x[n+1] = 3*x[n]*(1 -3*x[n]*(1-x[n]));" "6#/&%\"xG6#,&%\"nG\"\"\"F)F)*(\"\"$F)&F%6#F(F),&F )F)*(F+F)&F%6#F(F),&F)F)&F%6#F(!\"\"F)F5F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "D igits := 81:\nnext_approx := x -> 3*x*(1-3*x*(1-x));\nx0 := 0.3;\nnext _approx(%);\nnext_approx(%);\nnext_approx(%);\nnext_approx(%);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,next_approxGf*6#%\"xG6\" 6$%)operatorG%&arrowGF(,$*(\"\"$\"\"\"9$F/,&F/F/*(F.F/F0F/,&F/F/F0!\" \"F/F4F/F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$\"\"$!\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$L$!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*LLLL$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" " 0 "" {MPLTEXT 1 0 210 "phi := x -> 1/x-3:\n'ph i(x)'=phi(x);\nxin := 0.3;\nDigits :=81:\nfor i from 1 to 4 do\n xou t := impnewton_step(phi(x),x=xin,draw=is(i<=1));\n print(`approximat e root =`,xout);\n xin := xout;\nend do:\nDigits :=10:" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%$phiG6#%\"xG,&*&\"\"\"F*F'!\"\"F*\"\"$F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$xinG$\"\"$!\"\"" }}{PARA 13 "" 1 " " {GLPLOT2D 503 274 274 {PLOTDATA 2 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&*oSWj=\"*[@(\\arBJjn!y)y%=m*p9745$pB8)p7F.7$$\"\\p+++++++++++++++++++ +++++++++++++++++++]\\$F2$![p!G'RxT#p%G*\\\\YDyZMT*e7)o\"=F!>LKE%)*)G- ;7&e4np(Q\"F.F^o-%+AXESLABELSG6$Q\"x6\"Q!Fc_l-F>6&F@$FBF*$F*F*Fh_l-%%V IEWG6$;F[pF[_l;$!\\plmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm\"F2$\"\\p &********************************************************************* *********\\F2" 1 2 0 1 10 1 2 6 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 11 "" 1 "" {XPPMATH 20 "6$%3approximate~root~=G$\"\\p++++++ +++++++++++++++++++++++++++++++++L$!#\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3approximate~root~=G$\"\\p++++++++++++++++++++++++++++++++++++L LLL$!#\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3approximate~root~=G$\" \\p+++++++++++++++++++++++++++LLLLLLLLLLLLL$!#\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3approximate~root~=G$\"\\pLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLL$!#\")" }}}{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 73 "A procedure implement ing the \"leap-frog\" Newton method for root-finding: " }{TEXT 0 9 "im pnewton" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "impnewton: \+ usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }{TEXT 259 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 260 2 " \+ " }{TEXT -1 32 " impnewton( 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 variab le, say x," }}{PARA 0 "" 0 "" {TEXT -1 23 " " } {TEXT 266 2 "OR" }{TEXT -1 99 " a function of the form x -> f(x), wher e 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 262 68 "an initial approximation for the root \+ (which may be real or complex)" }}{PARA 0 "" 0 "" {TEXT 265 98 " \+ in the form of a constant a when the1st argume nt is a procedure, and" }}{PARA 0 "" 0 "" {TEXT 264 111 " \+ in the form of an equation x=a when the1st argument i s an expression or equation." }}{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 9 "impnewton" }{TEXT -1 28 " attempts to find a root of " }{XPPEDIT 18 0 "f(x) = 0;" "6#/-%\"fG6#%\"xG\"\"!" }{TEXT -1 74 " by the improve d \"leap-frog\" Newton method given an initial approximation." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 8 "Options: " }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 135 "maxiterations=n\nTh is option can be used to override the default value of Digits*5 for th e 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 "precision =fixed/variable" }}{PARA 0 "" 0 "" {TEXT -1 309 "If the computed value of the function exhibits a loss of significant digits as the successi ve approximations converge the root then the working precision is incr eased to compensate for this. This feature can be turned off via the o ption \"precision=fixed\". The default for this option is \"precision= variable\". " }}{PARA 0 "" 0 "" {TEXT -1 52 "The abreviated form \"prc sn=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 t he 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 t he computation to be monitored by printing the result of each Newton s tep as it occurs." }}{PARA 0 "" 0 "" {TEXT -1 96 "With the option \"in fo= 2\" the expressions for function and derivative being used are als o given." }}{PARA 0 "" 0 "" {TEXT -1 199 "The option \"info=3\" provid es additional information regarding the computations involved at each \+ step, together with information regarding any change in the working pr ecision used in the computation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 16 "How to a ctivate:" }{TEXT 256 1 "\n" }{TEXT -1 154 "To make the procedure activ e 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 25 "impnewton: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10096 "impn ewton := proc(ff,approx)\n local Options,x,df,dfx,fx,h,eps,saveDigit s,i,maxit,prntflg, \n f,fn,approxroot,xx,xp,fxp,d,lmr,sf,procty pe,complexround,vars,\n workingDigits,extraDigits,adjustDigits,ep s2,prsn,dfn,\n small,triedzero,f0;\n\n if nargs<2 then\n e rror \"at least 2 arguments are required; the basic syntax is: 'impnew ton(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 argumen t, %1, is invalid .. it should be a procedure with a single argument\" ,ff;\n end if;\n proctype := true;\n if type(approx,com plexcons) then\n approxroot := approx;\n else\n e rror \"the 2nd argument, %1, is invalid .. when the 1st argument is a \+ procedure, the 2nd argument should be a complex constant\",approx;\n \+ end if;\n elif type(ff,algebraic) 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 := inde ts(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 should be an expression or an equation which depends only on a single variable\",ff;\n end \+ if;\n end if;\n if type(approx,name=complexcons) then\n \+ proctype := false;\n x := op(1,approx);\n if not m ember(x,vars) then\n error \"the 1st argument, %1, is inval id .. it should be an expression or an 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 inval id .. it should have the form 'x=a', to provide a starting approximati on for a root\",approx;\n end if;\n else\n error \"the 1st argument, %1, is invalid .. it should be an algebraic expression in a single variable, an equation in a single variable, or a procedure wit h a single argument\",ff;\n end if;\n \n # Get the options \"max iterations\" and \"info\".\n # Set the default 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,li st(equation)) then\n error \"each optional 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','Optio ns') 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,'precision','prsn','Options') th en\n if not member(prsn,\{'fixed','variable'\}) 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','prsn','Options') then\n \+ if not member(prsn,\{'fixed','variable'\}) then\n erro r \"\\\"prcsn\\\" must be 'fixed' or 'variable'\"\n end if;\n \+ if prsn='fixed' then prsn := 0 else prsn := 1 end if;\n e nd if;\n if hasoption(Options,'info','prntflg','Options') then\n \+ if not member(prntflg,\{true,false,0,1,2,3\}) then\n \+ error \"\\\"info\\\" must be false <-> 0, true <-> 1,2 or 3\"\n \+ end if;\n if prntflg=false then prntflg := 0\n eli f prntflg=true then prntflg := 1 end if; \n end if;\n if nop s(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 comp lexround := 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 r e=0 then return Im(zz) end if;\n if not type(re,float) or not typ e(im,float) then\n return zz\n end if;\n eps := Floa t(1,-Digits);\n if abs(re)<=eps*abs(im) then return im*I\n e lif abs(im)<=eps*abs(re) then return re\n else return zz end if; \n end proc: # of complexround\n\n # Increase precision for the co mputation\n saveDigits := Digits;\n extraDigits := min(iquo(iquo(D igits,5)+1,2)+3,8);\n workingDigits := Digits + extraDigits;\n Dig its := workingDigits;\n\n if proctype then\n fn := ff;\n d fn := D(fn);\n else\n # Evaluate any real constants in f\n \+ fn := unapply(evalf(f),x);\n df := diff(f,x);\n dfn := unap ply(evalf(df),x);\n if prntflg>1 then\n print(`Attempting to calculate a zero of`);\n print(f); \n print(`by th e leap-frog Newton method, using the derivative`);\n print(df) ;\n print(``);\n end if;\n end if;\n if prntflg>2 the n\n print(`** working precision is `||Digits||` digits **`);\n \+ end if;\n\n xx := evalf(approxroot);\n\n eps := Float(1,-saveDigit s-min(iquo(Digits,10),2));\n eps2 := Float(1,-iquo(saveDigits,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 := trap error(evalf(fn(xx)));\n if fx=lasterror or not type(fx,complex(nu meric)) then\n error \"failed to evaluate function at %1\",eva lf[saveDigits](xx);\n end if;\n if prntflg>2 then\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(SFloatM antissa(Im(fx))));\n if adjustDigits>0 and (abs(h)<=eps2*abs(x x) or abs(fx)2 then\n \+ print(`** increasing working precision to `||Digits||` digits **`); \n end if;\n if not proctype then \n fn := unapply(evalf(f),x);\n dfn := una pply(evalf(df),x);\n end if;\n fx := traperror(e valf(fn(xx)));\n if fx=lasterror or not type(fx,complex(num eric)) then\n error \"failed to evaluate function at %1 \",evalf[saveDigits](xx);\n end if;\n if prntflg >2 then\n print(`value`=evalf[workingDigits](fx))\n \+ end if;\n end if;\n end if;\n dfx := traperro r(evalf(dfn(xx)));\n if dfx=lasterror or not type(dfx,complex(num eric)) then\n error \"failed to evaluate derivative at %1\",ev alf[saveDigits](xx);\n end if;\n if dfx=0 then\n err or \"zero derivative obtained\"\n end if;\n if prntflg>2 the n\n print(`derivative`=evalf[workingDigits](dfx))\n end i f;\n h := fx/dfx;\n if prntflg>2 then\n print(`preli minary Newton correction -> `,-`value`/`derivative`=evalf[workingDi gits](-h));\n end if;\n xp := xx - h;\n if prntflg>2 th en\n print(`preliminary Newton approximation -> `,evalf[wor kingDigits](xp))\n end if; \n fxp := traperror(evalf(fn(xp)) );\n if fxp=lasterror or not type(fxp,complex(numeric)) then\n \+ error \"failed to evaluate function at %1\",evalf[saveDigits](xp );\n end if;\n if prntflg>2 then\n print(`associated 'Newton' value`=evalf[workingDigits](fxp))\n end if;\n if p rsn=1 and fxp<>0 then\n adjustDigits := extraDigits-\n \+ max(length(SFloatMantissa(Re(fxp))),\n length(S FloatMantissa(Im(fxp))));\n if adjustDigits>0 and (abs(h)<=eps 2*abs(xx) or abs(fxp)2 then \n print(`** increasing working precision to `||Digits|| ` digits **`); \n end if;\n if not proct ype then\n fn := unapply(evalf(f),x);\n df n := unapply(evalf(df),x);\n end if;\n fxp := tr aperror(evalf(fn(xp)));\n if fxp=lasterror or not type(fxp, complex(numeric)) then\n error \"failed to evaluate func tion at %1\",evalf[saveDigits](xp);\n end if;\n \+ if prntflg>2 then\n print(`associated 'Newton' value`=ev alf[workingDigits](fxp))\n end if;\n end if;\n \+ end if;\n if fxp<>0 then\n d := fx - fxp;\n if d <>0 then\n h := fx*h/d;\n if prntflg>2 then\n \+ print(`modified correction -> `,-(`value times first co rrrection`)/(`value`-`'Newton' value`)=evalf[workingDigits](-h));\n \+ print(``);\n end if;\n xx := xx - h; \n else\n if prntflg>2 then\n print(` no difference between value and 'Newton' value`);\n prin t(` .. taking Newton approximation`);\n print(``);\n \+ end if;\n xx := xp;\n end if;\n else\n if prntflg>2 then\n print(`no difference between N ewton correction and modified correction`);\n print(``);\n \+ end if;\n xx := xp;\n end if;\n if prntflg>0 then\n print(`approximation `||i||` -> `,evalf[workingDigi ts](xx))\n end if; \n if prntflg>2 then print(``) end if;\n \+ if abs(h)<=eps*abs(xx) then\n Digits := saveDigits;\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 conve rging to 0`);\n print(``);\n end if;\n \+ return 0.0\n end if;\n triedzero := true;\n \+ end if;\n end do:\n Digits := saveDigits;\n print(`last iteratio n gives `,evalf(xx));\n error \"reached max, %1, iterations 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 in the secti ons which follow." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 9 "impn ewton" }{TEXT -1 10 ": examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }{TEXT 299 115 ".. a transcendental equation with two real solutions (differe nt levels of information via the option \"info=1/2/3\") " }}{PARA 0 " " 0 "" {TEXT 298 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 43 "Find the all the solutions of the equation " }{XPPEDIT 18 0 "x^2 = 3*ln(x)+2;" "6#/*$%\"xG\"\"#,&*&\"\"$\"\"\"-%#lnG6#F%F*F*F &F*" }{TEXT -1 68 " by a numerical method. Your answers should be corr ect to 10 digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 297 8 "Solution" }{TEXT -1 4 ": " }}{PARA 0 "" 0 "" {TEXT -1 34 "It is evident from the graphs of " }{XPPEDIT 18 0 "y = x^2;" "6#/ %\"yG*$%\"xG\"\"#" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "y = 3*ln(x)+2; " "6#/%\"yG,&*&\"\"$\"\"\"-%#lnG6#%\"xGF(F(\"\"#F(" }{TEXT -1 19 " tha t the equation " }{XPPEDIT 18 0 "x^2 = 3*ln(x)+2;" "6#/*$%\"xG\"\"#,&* &\"\"$\"\"\"-%#lnG6#F%F*F*F&F*" }{TEXT -1 25 " has two real solutions. 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Fi^m-%*AXESTICKSG6$Fez\"\"%-%%VIEWG6$;F(Fdz;$!\"\"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 21 "We use \+ the procedure " }{TEXT 0 9 "impnewton" }{TEXT -1 48 " which implements the \"leap-frog\" Newton method." }}{PARA 0 "" 0 "" {TEXT -1 83 "The \+ root with the lesser magnitude can be obtained with the starting appro ximation " }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 3 ".5." }} {PARA 0 "" 0 "" {TEXT -1 28 "( Incorporating the option \"" }{TEXT 277 9 "info=true" }{TEXT -1 6 "\" or \"" }{TEXT 277 6 "info=1" }{TEXT -1 9 "\" causes " }{TEXT 0 9 "impnewton" }{TEXT -1 44 " to show all th e successive approximations.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "impnewton(x^2=3*ln(x)+2,x=0. 5,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~-> ~~~G$\"/)*RhY\"3s&!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximati on~2~~->~~~G$\"/Sb~~~G$\"/K&GK.us&!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7 approximation~4~~->~~~G$\"/K&GK.us&!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+BLSFd!#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 277 6 "info=2" }{TEXT -1 7 "\" or \"" } {TEXT 277 6 "info=3" }{TEXT -1 3 "\". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "impnewton(x^2=3*ln(x)+2, x=0.5,info=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%BAttempting~to~calc ulate~a~zero~ofG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"#\" \"\"F(*&\"\"$F(-%#lnG6#F&F(!\"\"F'F." }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%Uby~the~leap-frog~Newton~method,~using~the~derivativeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"#\"\"\"%\"xGF&F&*&\"\"$F&F'!\"\"F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%7approximation~1~~->~~~G$\"/)*RhY\"3s&!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\"/Sb~~~G$\"/K&GK.us&!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\"/K&GK.us&!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+BLSFd!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "impnewton(x^ 2=3*ln(x)+2,x=0.5,info=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%BAttemp ting~to~calculate~a~zero~ofG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)% \"xG\"\"#\"\"\"F(*&\"\"$F(-%#lnG6#F&F(!\"\"F'F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Uby~the~leap-frog~Newton~method,~using~the~derivativeG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"#\"\"\"%\"xGF&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$\".)z;aT%H$!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG$!/++++++]!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Epreliminary~Newton~correction~~->~~~G/,$*&%&valueG\" \"\"%+derivativeG!\"\"F*$\"/gfL3$))e'!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Hpreliminary~Newton~approximation~~->~~~G$\"/'fL3$))ec!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%:associated~'Newton'~valueG$\"-SbV_I G!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;modified~correction~~->~~~G/ ,$*&%>value~times~first~corrrectionG\"\"\",&%&valueGF(%/'Newton'~value G!\"\"F,F,$\"/x*RhY\"3s!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\"/)*RhY\"3s &!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\",n;F))p#!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG$!/PQVa%)*4%!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%Epreliminary~Newton~correction~~->~~~G/,$*&%&valueG\"\"\"%+derivativ eG!\"\"F*$\"//&HV`Fe'!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Hprelimin ary~Newton~approximation~~->~~~G$\"/G$[T(RFd!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%:associated~'Newton'~valueG$\")k&yT#!#8" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%;modified~correction~~->~~~G/,$*&%>value~times~ first~corrrectionG\"\"\",&%&valueGF(%/'Newton'~valueG!\"\"F,F,$\"/S>ah l)e'!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\"/Sb~~~G/,$*&%&valueG\"\"\"%+derivativeG!\"\"F*$\"/J6c@@*H&!#B" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%Hpreliminary~Newton~approximation~~-> ~~~G$\"/K&GK.us&!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%:associated~' Newton'~valueG$\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%jnno~differ ence~between~Newton~correction~and~modified~correctionG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approxi mation~3~~->~~~G$\"/K&GK.us&!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG$!/+4N^\\#4%!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Epreliminary~Newton~correction~~->~~~G/,$*&%&valueG\" \"\"%+derivativeG!\"\"F*$\"\"!F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%H preliminary~Newton~approximation~~->~~~G$\"/K&GK.us&!#9" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%:associated~'Newton'~valueG$\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%jnno~difference~between~Newton~correction~a nd~modified~correctionG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\"/K&GK.us&! #9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+BLSFd!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "The root with the greater magnitude can be obta ined with the starting approximation " }{XPPEDIT 18 0 "x = 2;" "6#/%\" xG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "impnewton(x^2=3*ln(x)+2,x=2,info=1) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\"/QIBB BJ?!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\" /@%GJT7.#!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~ ~~G$\"/?%GJT7.#!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+88CJ?!\"*" } }}{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 302 54 ".. a transcendental equation with many real solutions " }} {PARA 0 "" 0 "" {TEXT 301 8 "Question" }{TEXT -1 4 ": " }}{PARA 0 " " 0 "" {TEXT -1 75 "Use a numerical method to find the positive real s olutions of the equation " }{XPPEDIT 18 0 "tan(x)=40/(x^2+4)" "6#/-%$t anG6#%\"xG*&\"#S\"\"\",&*$F'\"\"#F*\"\"%F*!\"\"" }{TEXT -1 82 " which \+ have a magnitude less than 10. Your answers should be correct to 10 di gits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 300 8 " Solution" }{TEXT -1 4 ": " }}{PARA 0 "" 0 "" {TEXT -1 34 "It is evid ent from the graphs of " }{XPPEDIT 18 0 "y = tan(x);" "6#/%\"yG-%$tan G6#%\"xG" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "y = 40/(x^2+4);" "6#/% \"yG*&\"#S\"\"\",&*$%\"xG\"\"#F'\"\"%F'!\"\"" }{TEXT -1 19 " that the \+ equation " }{XPPEDIT 18 0 "tan(x)=40/(x^2+4)" "6#/-%$tanG6#%\"xG*&\"#S \"\"\",&*$F'\"\"#F*\"\"%F*!\"\"" }{TEXT -1 65 " has four positive real solutions with a magnitude less than 10. 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They can all b e obtained correct to 10 digits by using the procedure " }{TEXT 0 9 "i mpnewton" }{TEXT -1 32 " with suitable starting values. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "impne wton(tan(x)=40/(x^2+4),x=4.2,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\"/s#e'4<9U!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\"/WS/w<9U!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$\"/WS/w<9U!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+/w<9U!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "impnewton(tan(x)=40/(x^2+4),x=6.9,info=true);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\"/:8D@kPp!# 8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\"/WqNP mPp!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$\" /WqNPmPp!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+OPmPp!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "impnewton(tan(x)=40/(x^2+4),x=9.8,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\"/o>95#[!)*!#8" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\"/d$*G5#[!)*!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$\"/d$*G5#[!) *!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+H5#[!)*!\"*" }}}{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 283 58 ".. de termining the points of intersection of two curves. " }}{PARA 0 "" 0 "" {TEXT 282 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 91 "Use a numerical method to find the coordinates of the points of in tersection of the curve " }{XPPEDIT 18 0 "y = x*exp(-x^2/5);" "6#/%\" yG*&%\"xG\"\"\"-%$expG6#,$*&F&\"\"#\"\"&!\"\"F/F'" }{TEXT -1 17 " with the circle " }{XPPEDIT 18 0 "x^2+y^2 = 5;" "6#/,&*$%\"xG\"\"#\"\"\"*$ %\"yGF'F(\"\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 303 1 "x" }{TEXT -1 5 " and " }{TEXT 304 1 "y" }{TEXT -1 58 " co ordinates of each point should be correct to 10 digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 281 8 "Solution" }{TEXT -1 4 ": " }}{PARA 0 "" 0 "" {TEXT -1 102 "It is evident from the follow ing picture that there are two points of intersection of the two curve s. 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!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$\"/6q c$*>a?!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G $\"/6qc$*>a?!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"+d$*>a?!\" *" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "The re is a slight difference in the 10 digit values for the corresponding " }{TEXT 306 1 "y" }{TEXT -1 59 " coordinate as computed by using eac h of the two functions " }{XPPEDIT 18 0 "f(x)=x*exp(-x^2/5)" "6#/-%\"f G6#%\"xG*&F'\"\"\"-%$expG6#,$*&F'\"\"#\"\"&!\"\"F1F)" }{TEXT -1 5 " an d " }{XPPEDIT 18 0 "g(x)=sqrt(5-x^2)" "6#/-%\"gG6#%\"xG-%%sqrtG6#,&\" \"&\"\"\"*$F'\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f(x1);\ng(x1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+A5EL))!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+<5EL))!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 111 "By recalculating with higher precision w e can be confident of obtaining both coordinates correct to 10 digits. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "Digits := 13:\nxx1 := impnewton(f(x)=g(x),x=2);\nf(x x1);\nyy1 := g(xx1);\nDigits := 10:\nx1 := evalf(xx1);\ny1 := evalf(yy 1);\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$xx1G$\".,nN*>a?!#7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\".sD-hK$))!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$yy1G$\".tD-hK$))!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"+d$*>a?!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G$\"+B 5EL))!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The point of intersection in the first quadrant is" }{XPPEDIT 18 0 "``(x[1],y[1])" "6#-%!G6$&%\"xG6#\"\"\"&%\"yG6#F)" }{TEXT -1 7 " whe re " }{XPPEDIT 18 0 "x[1]" "6#&%\"xG6#\"\"\"" }{TEXT -1 1 " " }{TEXT 307 1 "~" }{TEXT -1 17 " 2.054199357 and " }{XPPEDIT 18 0 "y[1]" "6#&% \"yG6#\"\"\"" }{TEXT -1 1 " " }{TEXT 308 1 "~" }{TEXT -1 15 " 0.883326 1023. " }}{PARA 0 "" 0 "" {TEXT -1 63 "By symmetry, the point of inter section in the third quadrant is" }{XPPEDIT 18 0 "``(x[2],y[2]);" "6#- %!G6$&%\"xG6#\"\"#&%\"yG6#F)" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "x[ 2];" "6#&%\"xG6#\"\"#" }{TEXT -1 1 " " }{TEXT 309 1 "~" }{TEXT -1 18 " -2.054199357 and " }{XPPEDIT 18 0 "y[2];" "6#&%\"yG6#\"\"#" }{TEXT -1 1 " " }{TEXT 310 1 "~" }{TEXT -1 16 " -0.8833261023. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 4 "Note" }{TEXT -1 10 ": Maple's " }{TEXT 0 6 "fsolve" }{TEXT -1 47 " can work with the o riginal pair of equations. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "fsolve(\{y=x*exp(-x^2/5),x^2+y^2=5 \},\{x=2,y=1\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"xG$\"+d$*>a? !\"*/%\"yG$\"+B5EL))!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "fsolve(\{y=x*exp(-x^2/5),x^2+y^2=5 \},\{x=-2,y=-1\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"xG$!+d$*>a ?!\"*/%\"yG$!+B5EL))!#5" }}}{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 292 104 ".. a transcendental equation \+ with one real solution (taking a starting value close to a critical va lue) " }}{PARA 0 "" 0 "" {TEXT -1 13 "The equation " }{XPPEDIT 18 0 "e xp(-x^2)=x^3-x" "6#/-%$expG6#,$*$%\"xG\"\"#!\"\",&*$F)\"\"$\"\"\"F)F+ " }{TEXT -1 33 " has exactly one real solution. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "g := x -> exp(-x^2)-x^3+x:\n 'g(x)'=g(x);\nimpnewton(g(x),x=1,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,(-%$expG6#,$*$)F'\"\"#\"\"\"!\"\"F0*$)F' \"\"$F0F1F'F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~-> ~~~G$\"/!Rzo9&=6!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation ~2~~->~~~G$\"/s]PP4?6!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approxim ation~3~~->~~~G$\"/SVSP4?6!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7app roximation~4~~->~~~G$\"/RVSP4?6!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"+SP4?6!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 172 "Taking a starting approximation close to a critical value leads to slow conver gence. The behaviour is rather different to that which occurs with the standard Newton method." }}{PARA 0 "" 0 "" {TEXT -1 90 "The critical \+ value associated with the local maximum point is approximately 0.35383 79575. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "g := x -> exp(-x^2)-x^3+x:\nDiff('g(x)',x)=D(g)(x);\n impnewton(D(g)(x),x=1,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%%DiffG6$-%\"gG6#%\"xGF*,(*(\"\"#\"\"\"F*F.-%$expG6#,$*$)F*F-F.!\"\"F .F5*&\"\"$F.F4F.F5F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximat ion~1~~->~~~G$\"/X\"3b!)ow$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7ap proximation~2~~->~~~G$\"/xGi\\RQN!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%7approximation~3~~->~~~G$\"/P[v&z$QN!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\"/P[v&z$QN!#9" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+v&z$QN!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "The starting value " }{XPPEDIT 18 0 "x= 0" "6#/%\"xG\"\"!" }{TEXT -1 57 ".356 is relatively close to the previ ous critical value. " }}{PARA 0 "" 0 "" {TEXT -1 174 "In the early ite rations the preliminary Newton approximation is large in magnitude, bu t the modified improved \"leap-frog\" Newton value is close to the ini tial approximation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "g := x -> exp(-x^2)-x^3+x:\n'g(x)'=g(x); \nimpnewton(g(x),x=0.356,info=true,maxiterations=60);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,(-%$expG6#,$*$)F'\"\"#\"\"\"!\"\"F0 *$)F'\"\"$F0F1F'F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~ 1~~->~~~G$\"/\\YVLYgN!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approxim ation~2~~->~~~G$\"/dT2n%4c$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7ap proximation~3~~->~~~G$\"/5\"eS^9c$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\"/)>$p)y>c$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~5~~->~~~G$\"/_\\e1`iN!#9" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%7approximation~6~~->~~~G$\"/Ywy%3Jc$!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~7~~->~~~G$\"/R`(>9Pc$! #9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~8~~->~~~G$\"/.wj )\\Vc$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~9~~->~~~G $\"/pTIx,lN!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~10~~ ->~~~G$\"/)e.G?dc$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximati on~11~~->~~~G$\"/I?c-YmN!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8appro ximation~12~~->~~~G$\"/\\'\\pSsc$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%8approximation~13~~->~~~G$\"/LNo\\1oN!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~14~~->~~~G$\"/p#)Ho$*oN!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~15~~->~~~G$\"/nwp/')pN!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~16~~->~~~G$\"/[V!eS3d$ !#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~17~~->~~~G$\"/ \\zKC)=d$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~18~~-> ~~~G$\"/Pyn>*Hd$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation ~19~~->~~~G$\"/lG0f~~~G$\"/)H\\(=WvN!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% 8approximation~21~~->~~~G$\"/\\#Rd)zwN!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~22~~->~~~G$\"/\\nefDyN!#9" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%8approximation~23~~->~~~G$\"/i\")ya#)zN!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~24~~->~~~G$\"/$*)\\O?: e$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~25~~->~~~G$\" /-=&)fN$e$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~26~~- >~~~G$\"/HJ*G]`e$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximatio n~27~~->~~~G$\"/~~~G$\"/7HkJ!**e$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%8approximation~29~~->~~~G$\"/%zxP;Df$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~30~~->~~~G$\"/fcW'*R&f$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~31~~->~~~G$\"/!QB2'f)f$!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~32~~->~~~G$\"/(faFe@g$ !#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~33~~->~~~G$\"/$ GG5^hg$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~34~~->~~ ~G$\"/n@Z`l5O!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~35 ~~->~~~G$\"/n:oGx:O!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximat ion~36~~->~~~G$\"/O<*yL;i$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8app roximation~37~~->~~~G$\"/]%G!oSGO!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%8approximation~38~~->~~~G$\"/*z.C9jj$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~39~~->~~~G$\"/fQNYlXO!#9" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%8approximation~40~~->~~~G$\"/Zf>u$ol$!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~41~~->~~~G$\"/\"\\q!zV qO!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~42~~->~~~G$\" /KR$[(G(o$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~43~~- >~~~G$\"/n;Avi3P!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation ~44~~->~~~G$\"/\\8CWQOP!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approx imation~45~~->~~~G$\"/m)*3$)otP!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %8approximation~46~~->~~~G$\"/9*)Qb\"f#Q!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~47~~->~~~G$\"/945#\\H!R!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~48~~->~~~G$\"/ln#yWX-%!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~49~~->~~~G$\"/.\\[D\\M U!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~50~~->~~~G$\"/ 3U\"*3\\UY!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~51~~- >~~~G$\"/!o'G3Obb!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximatio n~52~~->~~~G$\"/R!G5&z5x!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8appro ximation~53~~->~~~G$\"/5o$z`51\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%8approximation~54~~->~~~G$\"/2Y>^#*>6!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~55~~->~~~G$\"/.VSP4?6!#8" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%8approximation~56~~->~~~G$\"/RVSP4?6!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+SP4?6!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "We can see more details o f the computation in the first few iterations by means of the option \+ \"" }{TEXT 277 6 "info=3" }{TEXT -1 3 "\". " }}{PARA 0 "" 0 "" {TEXT -1 99 "There are large differences between the preliminary Newton corr ection and the modified correction. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "g := x -> exp(-x^2)-x^3+x: \n'g(x)'=g(x);\ntraperror(impnewton(g(x),x=0.356,info=3,maxiterations= 3)):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,(-%$expG6#,$*$) F'\"\"#\"\"\"!\"\"F0*$)F'\"\"$F0F1F'F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%BAttempting~to~calculate~a~zero~ofG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$expG6#,$*$)%\"xG\"\"#\"\"\"!\"\"F,*$)F*\"\"$F,F-F* F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Uby~the~leap-frog~Newton~method ,~using~the~derivativeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(\"\"#\" \"\"%\"xGF&-%$expG6#,$*$)F'F%F&!\"\"F&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$\"/7)4?[=>\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+deriva tiveG$!,1\\YfX(!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Epreliminary~Ne wton~correction~~->~~~G/,$*&%&valueG\"\"\"%+derivativeG!\"\"F*$\"/?LD. _)f\"!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Hpreliminary~Newton~appro ximation~~->~~~G$\"/?LD.3-;!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%:a ssociated~'Newton'~valueG$!/rtsu$=6%!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;modified~correction~~->~~~G/,$*&%>value~times~first~corrrectio nG\"\"\",&%&valueGF(%/'Newton'~valueG!\"\"F,F,$\"/)[!\\YVLY!#=" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%7approximation~1~~->~~~G$\"/\\YVLYgN!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"/+\" =&y%=>\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG$!,1Wkeh( !#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Epreliminary~Newton~correction ~~->~~~G/,$*&%&valueG\"\"\"%+derivativeG!\"\"F*$\"/]3JT&\\c\"!#6" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%Hpreliminary~Newton~approximation~~-> ~~~G$\"/&GWf9&o:!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%:associated~' Newton'~valueG$!/v'*3CweQ!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;mod ified~correction~~->~~~G/,$*&%>value~times~first~corrrectionG\"\"\",&% &valueGF(%/'Newton'~valueG!\"\"F,F,$\"/b03&RO$[!#=" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximatio n~2~~->~~~G$\"/dT2n%4c$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"/Xlzu%=>\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG$!,s3)p#y(!#8" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%Epreliminary~Newton~correction~~->~~~G/,$*&%&valueG \"\"\"%+derivativeG!\"\"F*$\"/T.YlSJ:!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Hpreliminary~Newton~approximation~~->~~~G$\"/:u#\\n\\`\"!#6" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%:associated~'Newton'~valueG$!/9()*4A kh$!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;modified~correction~~->~~ ~G/,$*&%>value~times~first~corrrectionG\"\"\",&%&valueGF(%/'Newton'~va lueG!\"\"F,F,$\"/`W`R)p/&!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$\"/5\"eS^9c $!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6last~iteration~gives~G$\"+19XhN!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 34 "Taking the starting approximation " } {XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 149 ".44 is not such an \+ extreme case. Because there are fewer iterations needed to obtain conv ergence in this case, it is reasonable to get the procedure " }{TEXT 0 9 "impnewton" }{TEXT -1 151 " to give more details of the computatio n. Initially there are large differences between the preliminary Newto n correction and the modified correction. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "g := x -> exp(-x^2 )-x^3+x:\n'g(x)'=g(x);\nimpnewton(g(x),x=0.44,info=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,(-%$expG6#,$*$)F'\"\"#\"\"\"!\"\"F0 *$)F'\"\"$F0F1F'F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%BAttempting~to~ calculate~a~zero~ofG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$expG6#,$* $)%\"xG\"\"#\"\"\"!\"\"F,*$)F*\"\"$F,F-F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Uby~the~leap-frog~Newton~method,~using~the~derivativeG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(\"\"#\"\"\"%\"xGF&-%$expG6#,$* $)F'F%F&!\"\"F&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$\"/LT*3fI!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%Epreliminary~Newton~correction~~->~~~ G/,$*&%&valueG\"\"\"%+derivativeG!\"\"F*$\"/_#)*\\XM&Q!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Hpreliminary~Newton~approximation~~->~~~G$\"/_ #)*\\XMH%!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%:associated~'Newton' ~valueG$!/8qGI0&[(!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;modified~co rrection~~->~~~G/,$*&%>value~times~first~corrrectionG\"\"\",&%&valueGF (%/'Newton'~valueG!\"\"F,F,$\"/vqr\"3Y(f!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~ 1~~->~~~G$\"/3<<3Y(*\\!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"/fWL\\$R:\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG$!.(eVMTy_!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Epreliminary~Newton~correction~~->~~~G/,$*&%&valu eG\"\"\"%+derivativeG!\"\"F*$\"/h()G(Rh=#!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Hpreliminary~Newton~approximation~~->~~~G$\"/Kf5e)eo#! #8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%:associated~'Newton'~valueG$!/ 7*R9J*o;!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;modified~correction~~ ->~~~G/,$*&%>value~times~first~corrrectionG\"\"\",&%&valueGF(%/'Newton '~valueG!\"\"F,F,$\"/,M)\\\"z89!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\"/4^ :BD6k!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"/UN\"\\b0/\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG$!//YbY?$3\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Epreliminary~Newton~correction~~->~~~G/,$*&%&valueG\" \"\"%+derivativeG!\"\"F*$\"/q+T'oig*!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Hpreliminary~Newton~approximation~~->~~~G$\"/=l&4_~~~G /,$*&%>value~times~first~corrrectionG\"\"\",&%&valueGF(%/'Newton'~valu eG!\"\"F,F,$\"/&fMm'[zG!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$\"//(*y*Q2H* !#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"/rv4uO*[&!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG$!/2)\\[PLP#!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%Epreliminary~Newton~correction~~->~~~G/,$*&%&valueG\"\"\"%+derivativ eG!\"\"F*$\"/5Pg;$HJ#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%Hprelimin ary~Newton~approximation~~->~~~G$\"/T$R1n.;\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%:associated~'Newton'~valueG$!.v\"zT\\=9!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;modified~correction~~->~~~G/,$*&%>value~ti mes~first~corrrectionG\"\"\",&%&valueGF(%/'Newton'~valueG!\"\"F,F,$\"/ p:HH)z$=!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%7approximation~4~~->~~~G$\"/F\"3>sG6\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&va lueG$\"-j'eK?W#!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG$! 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" }}{PARA 0 "" 0 "" {TEXT -1 96 "This means that we can obtain an accu rate estimate for the root with relatively few iterations. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "ev alf[65](impnewton(x^3-3*x^2-5,x=4,info=true));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\"doM[.8oA/>8YFn$zAF]OLg!e2ng \\pSo%\\LD,&HBX$!#s" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation ~2~~->~~~G$\"dow$RcNUbW<#\\p2c;CZ?Mn=kQlG$pY4lK8IQJ*fU$!#s" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$\"dof\"\\`**p;pcxP gX]4?T$yxnT;%y$\\*4o9A;Od())fU$!#s" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %7approximation~4~~->~~~G$\"doYs[?f^0(R2sa)HhYh5Nx+'H`V!=k2E@ihtv))fU$ !#s" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~5~~->~~~G$\"doz /GDS(pe8H__)HhYh5Nx+'H`V!=k2E@ihtv))fU$!#s" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~6~~->~~~G$\"doz/GDS(pe8H__)HhYh5Nx+'H`V !=k2E@ihtv))fU$!#s" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\\oS(pe8H__)H hYh5Nx+'H`V!=k2E@ihtv))fU$!#k" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 122 " \+ 3.4259887573616221260764180435329600773510614661298525 229135869740" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "The cubic equation has two complex roots (with nonzero imagina ry part) which form a complex conjugate pair." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "impnewton(x^ 3-3*x^2-5,x=I,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approxi mation~1~~->~~~G^$$!/9=t)fw.#!#9$\"/!zdEp7>\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G^$$!/Yvyj%*H@!#9$\"/uto^9*=\"! #8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G^$$!/\" 3oyV*H@!#9$\"/m53^9*=\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approx imation~4~~->~~~G^$$!/\"3oyV*H@!#9$\"/m53^9*=\"!#8" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#^$$!+(yV*H@!#5$\"+3^9*=\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "impnewton(x^ 3-3*x^2-5,x=-I);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!+(yV*H@!#5$!+3 ^9*=\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Maple's " }{TEXT 0 6 "fsolve" }{TEXT -1 79 " incorporates an alg orithm which calculates all complex zeros of a polynomial. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "fs olve(x^3-3*x^2-5,x,complex);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%^$$!+( yV*H@!#5$!+3^9*=\"!\"*^$F$$\"+3^9*=\"F)$\"+d())fU$F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The improved \"leap-fr og\" Newton method fails if the " }{TEXT 291 1 "x" }{TEXT -1 76 " coor dinate of a stationary point on the curve is used as a starting value. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "impnewton(x^3-3*x^2-5,x=2);" }}{PARA 8 "" 1 "" {TEXT -1 47 "Error, (in impnewton) zero derivative obtained\n" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "The method converg es slowly if a starting value close to a critical value is taken. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "impnewton(x^3-3*x^2-5,x=1e-2,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\"/.5zr$)o5!#:" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\"/VoB4CZ6!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$\"/hOf/EP7!# :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\"/m_YA bT8!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~5~~->~~~G$\" /mG)*[hj9!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~6~~->~ ~~G$\"/8r2'Q\"3;!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation ~7~~->~~~G$\"/*)Rux`\"y\"!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7appr oximation~8~~->~~~G$\"/TF.ay#*>!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %7approximation~9~~->~~~G$\"/<9p5waA!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~10~~->~~~G$\"/8%4X0le#!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~11~~->~~~G$\"/_#yZUr,$!#:" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%8approximation~12~~->~~~G$\"/X]M],$f$!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~13~~->~~~G$\"/g9SxM\"R %!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~14~~->~~~G$\"/ 0wL\"zza&!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~15~~-> ~~~G$\"/dA#HEvJ(!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation ~16~~->~~~G$\"/)[^5?9-\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8appro ximation~17~~->~~~G$\"/'*\\4X;P:!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%8approximation~18~~->~~~G$\"/xS@-9dD!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~19~~->~~~G$\"/j8(o?6)[!#9" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%8approximation~20~~->~~~G$\"/1@^%fB=\"!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~21~~->~~~G$\"/,Jd\\YvU !#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~22~~->~~~G$\"/5 K1PA$\\$!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~23~~->~ ~~G$\"/'Hi]egU$!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~ 24~~->~~~G$\"/;Od())fU$!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approx imation~25~~->~~~G$\"/;Od())fU$!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"+d())fU$!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exam ple 6" }{TEXT 296 106 " .. a pathological example for which the leap-f rog Newton method is superior to the standard Newton method" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "g(x) = (arctan((x-1)/41)^(1/3)+sinh(x)/179" "6#/-%\"gG6#%\"xG,&) -%'arctanG6#*&,&F'\"\"\"F/!\"\"F/\"#TF0*&F/F/\"\"$F0F/*&-%%sinhG6#F'F/ \"$z\"F0F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "To obtain real number cube roots of negative r eal numbers we must set up the cube root " }{XPPEDIT 18 0 "u^(1/3)" " 6#)%\"uG*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 26 " using the Maple function \+ " }{TEXT 277 9 "surd(u,3)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "g := x -> surd(ar ctan((x-1)/41),3)+sinh(x)/179:\n'g(x)'= eval(subs(\{surd=(x->x^(1/3)), 3=NULL\},g(x)));\nplot(g(x),x=-1..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,&*$)-%'arctanG6#,&*&\"#T!\"\"F'\"\"\"F2#F2F0F1#F2\" \"$F2F2*&#F2\"$z\"F2-%%sinhGF&F2F2" }}{PARA 13 "" 1 "" {GLPLOT2D 485 299 299 {PLOTDATA 2 "6%-%'CURVESG6$7in7$$!\"\"\"\"!$!3]8!*[L>_=P!#=7$$ !3PLLLLQ6G\"*F-$!3%*>%p^%\\XdOF-7$$!3immmT.\\p$)F-$!3D)z;KuILg$F-7$$!3 LLLL$))Qj^(F-$!3\\VZEmB?TNF-7$$!3ULLL$=Kvl'F-$!3;]Y9>B8xMF-7$$!3hnmmTs !G!eF-$!3Yu07Apj6MF-7$$!3iLLL3yO5]F-$!3'=1!yE]<\\LF-7$$!3i+++vE%)*=%F- $!3S24Sze^#G$F-7$$!3)RLL$3WDTLF-$!3(piI[J*=6KF-7$$!3'4++]d(Q&\\#F-$!3S U#=iuit8$F-7$$!3:mmmm&4`i\"F-$!3%R.Sa(\\>eIF-7$$!3GKLLLQW*e)!#>$!3y9ZQ 9UQ&)HF-7$$\"3HI#*******H,Q!#@$!3EYl0\"4p%**GF-7$$\"3Q(*******\\*3q)F] o$!38Y'[-h_$3GF-7$$\"3!********p=\\q\"F-$!3HCW>-S<:FF-7$$\"3_mmm\"fBIY 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1 to 11 do\n xout := impnewton_step(g(x) ,x=xin,draw=is(i<=8),\n color=[red,brown,coral,COLOR(RGB ,0,.55,0)$2]);\n print(`approximate root =`,xout);\n xin := xout; \nend do:\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#% \"xG,&*$)-%'arctanG6#,&*&\"#T!\"\"F'\"\"\"F2#F2F0F1#F2\"\"$F2F2*&#F2\" $z\"F2-%%sinhGF&F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$xinG$\"$,\" !\"#" }}{PARA 13 "" 1 "" {GLPLOT2D 476 476 476 {PLOTDATA 2 "6--%'POINT SG6)7$$\"$,\"!\"#\"\"!7$F'$\"0%ps*GdJ\"p!#;7$$\"0qb;;GF%**!#:F*7$$\"0C =P?R%p(*F2F*7$F4$!-V\\w@-F$6)F&F+F/F3F6-F;6#%(DIAMOND GF>-%'CURVESG6&7$F+F3-%*THICKNESSG6#\"\"#-%*LINESTYLEG6#\"\"\"-%'COLOU RG6&FA$\"*++++\"!\")$\")AR!)\\FjnFB-FP6&7%F+F/F6FWFS-Ffn6&FA$\")#)eqkF jn$\"))eqk\"FjnFdo-FP6'7$F&F+7$F3F6-FTFY-FXFUF>-FP6#7fn7$$\"0Oxb!)eTg* F2$!0RMt'ohg#*F.7$$\"0Q`[Pp&='*F2$!0-e9X))z8*F.7$$\"0bFT$z5J'*F2$!0OR* fbxG!*F.7$$\"0L)Rb)3_k*F2$!0a$[\\O)H!*)F.7$$\"0aGnC.%f'*F2$!0)*f!)zvHx 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "g := x -> surd(arctan((x-1) /41),3)+sinh(x)/179:\n'g(x)'= eval(subs(\{surd=(x->x^(1/3)),3=NULL\},g (x)));\nnewton(g(x),x=.99999,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,&*$)-%'arctanG6#,&*&\"#T!\"\"F'\"\"\"F2#F2F0F1#F2 \"\"$F2F2*&#F2\"$z\"F2-%%sinhGF&F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%7approximation~1~~->~~~G$\"/!pWw%))****!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\"/GXzR))****!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$\"/(ow(R))****!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\"/*ow(R))** **!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+yR))****!#5" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "g := x -> surd(arctan((x-1)/41),3)+sinh(x)/179:\n'g(x)'= eval(subs(\{surd= (x->x^(1/3)),3=NULL\},g(x)));\nnewton(g(x),x=1.001,info=false);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,&*$)-%'arctanG6#,&*&\"# T!\"\"F'\"\"\"F2#F2F0F1#F2\"\"$F2F2*&#F2\"$z\"F2-%%sinhGF&F2F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%6last~iteration~gives~G$!+W0%p'o!#5" }}{PARA 8 "" 1 "" {TEXT -1 67 "Error, (in newton) reached max, 50, ite rations without convergence\n" }}}{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 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 34 "Find the solution of the equation " }{XPPEDIT 18 0 "exp(-x^2) = ln(x);" "6#/-%$expG6#,$*$%\"xG\"\"#!\"\"-%#lnG6#F)" }{TEXT -1 44 " correct to 10 digits by a numerical method." }}{PARA 0 "" 0 "" {TEXT -1 31 "_______________________________" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 31 "_______________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }} {PARA 0 "" 0 "" {TEXT -1 79 "Use a numerical method to find all the po sitive real solutions of the equation " }{XPPEDIT 18 0 "sec(x) = 50/(x ^2+5);" "6#/-%$secG6#%\"xG*&\"#]\"\"\",&*$F'\"\"#F*\"\"&F*!\"\"" } {TEXT -1 46 ". Your answers should be correct to 10 digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "_______________ ________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 31 "_______________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 91 "Use a numerical \+ method to find the coordinates of the points of intersection of the cu rve " }{XPPEDIT 18 0 "y = arctan(x);" "6#/%\"yG-%'arctanG6#%\"xG" } {TEXT -1 17 " with the circle " }{XPPEDIT 18 0 "x^2+y^2 = 3;" "6#/,&*$ %\"xG\"\"#\"\"\"*$%\"yGF'F(\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 312 1 "x" }{TEXT -1 5 " and " }{TEXT 313 1 "y " }{TEXT -1 58 " coordinates of each point should be correct to 10 dig its." }}{PARA 0 "" 0 "" {TEXT -1 31 "_______________________________" }}{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 31 "_______________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "g(x)=tanh((2*x-1)/13)^(1/3)+x/7" "6# /-%\"gG6#%\"xG,&)-%%tanhG6#*&,&*&\"\"#\"\"\"F'F1F1F1!\"\"F1\"#8F2*&F1F 1\"\"$F2F1*&F'F1\"\"(F2F1" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 89 "To obtain real number cube roots of negative real numbers we must set up the cube root " }{XPPEDIT 18 0 "u^(1/3)" "6#)%\"uG*&\"\"\"F& \"\"$!\"\"" }{TEXT -1 26 " using the Maple function " }{TEXT 277 9 "su rd(u,3)" }{TEXT -1 31 ", that is, define the function " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 24 " in Maple via the code: " } {TEXT 267 38 "g := x -> surd(tanh((2*x-1)/13),3)+x/7" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 88 "Investigate the convergence of the \" leap-frog\" Newton method to the single real zero of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 33 " with different starting values . " }}{PARA 0 "" 0 "" {TEXT -1 31 "_______________________________" }} {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 31 "_______________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 25 "Code for drawin g pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 46 "1st step of improved \"leap-frog\" Newton method" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 718 "f := x -> exp(-x+2)+1-x:\nm := evalf(D(f)(0)):\ny0 : = evalf(f(0)):\nxp := -y0/m: yp := f(xp):\nx1 := y0*xp/(y0-yp):\np1 := plot(f(x),x=-.4..2.8,thickness=1):\np2 := plot([[[0,0],[0,y0]],[[xp,0 ],[xp,yp]]],linestyle=2,color=black):\np3 := plot([[0,y0],[xp,0]],colo r=blue):\np4 := plot([[0,y0],[x1,0]],color=COLOR(RGB,0,.8,0)):\np5 := \+ plot([[[0,0],[0,y0],[xp,0],[x1,0],[xp,yp]]$3],style=point,\n color= navy,symbol=[circle,cross,diamond]):\np6 := plot([[-.4,0],[3.5,0]],col or=black):\nt1:=plots[textplot]([-.1,12,`y = f(x)`],color=red):\nt2:=p lots[textplot]([[.15,9,`(a,f(a))`],[1.15,3.5,`(b,f(b))`],\n [2 ,-0.4,`r`],[1.48,-0.4,`c`],[1,-0.4,`b`],[0,-0.4,`a`]],color=black):\np lots[display]([p1,p2,p3,p4,p5,p6,t1,t2],axes=none);" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }