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" }{TEXT 257 0 "" }}{PARA 263 "" 0 "" {TEXT 256 7 "\251 2003 " }{TEXT -1 144 "Nathan Col lier, Autar Kaw, Jai Paul , Michael Keteltas, University of South Flor ida , kaw@eng.usf.edu , http://numericalmethods.eng.usf.edu/mws " }} {PARA 258 "" 0 "" {TEXT -1 176 "NOTE: This worksheet demonstrates the \+ use of Maple to illustrate how the Newton-Raphson method converges slo wly due to an inflection point occuring in the vicinity of the root." }}{SECT 0 {PARA 260 "" 0 "" {TEXT 258 12 "Introduction" }{TEXT -1 0 " " }}{PARA 259 "" 0 "" {TEXT 259 33 "Although, Newton-Raphson method [ " }{URLLINK 17 "text notes" 4 "http://numericalmethods.eng.usf.edu/mws /gen/03nle/mws_gen_nle_txt_newton.pdf" "" }{TEXT 270 2 "][" }{URLLINK 17 "PPT" 4 "http://numericalmethods.eng.usf.edu/mws/gen/03nle/mws_gen_ nle_ppt_newton.ppt" "" }{TEXT 271 400 "] converges faster than any oth er method it still has a few drawbacks.One of them is Divergence at in flection points. For a function f(x), the point where the concavity ch anges from up-to-down or down-to-up are called inflection points. Theo rem for a function f(x) that states : \"If f '(c) exists and f ''(c) c hanges sign at x = c , then the point (c, f(c)) is an inflection point of the graph of f. " }{TEXT -1 144 "The following simulation illustra tes the Newton-Raphson method converges slowly due to an inflection po int occuring in the vicinity of the root." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 262 17 "Sect ion I : Data." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Function in f(x)= 0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f(x):=(x-1)^3:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Initial guess" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 9 "x0:=-1.0:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Upper bound of range of 'x' that is desired" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "uxrange:=3.0:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Lower bound of range of 'x' that is desired" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "lxrange:=-3.0:" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 28 "Maximum number of iterations" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "nmax:=10:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{PARA 3 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 " " }{TEXT 263 29 "Section II: Before you start." }}{PARA 259 "" 0 "" {TEXT -1 235 "Because the method uses a line tangent to the function a t the initial guess, we must calculate the derivative of the function \+ to find the slope of the line at this point. Here we will define the d erivative of the function f(x) as g(x)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "g(x):=diff(f(x),x);" }{TEXT -1 0 "" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>-%\"gG6#%\"xG,$*&\"\"$\"\"\"),&F'F+F+!\"\"\"\"#F+F+ " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 226 "W e now plot the data. The following function determines the upper and l ower ranges on the Y-axis. This is done using the upper and lower rang es of the X-axis specified, and the value of the original functional a t these values." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 343 "yranger:=proc(uxrange,lxrange)\nlocal i,maxi, mini,tot;\nmaxi:=eval(f(x),x=lxrange);\nmini:=eval(f(x),x=lxrange);\nf or i from lxrange by (uxrange-lxrange)/10 to uxrange do\nif eval(f(x), x=i)maxi th en maxi:=eval(f(x),x=i) end if;\nend do;\ntot:=maxi-mini;\n-0.1*tot+mi ni..0.1*tot+maxi;\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "yrange:=yranger(uxrange,lxrange):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "xrange:=lxrange..uxrange:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "The following calls are needed \+ to use the plot function" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changec oords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(plottools):" }}{PARA 7 "" 1 "" {TEXT -1 43 "Warning, the name arrow has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "plot(f(x),x=xrange,y=yrange,title=\"Entered function on given int erval\",legend=[\"Function\"],thickness=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 549 549 549 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$!\"$\"\"!$!#kF*7 $$!3!******\\2<#pG!#<$!3;MAyuKa#z&!#;7$$!3#)***\\7bBav#F0$!3CW4c97N'H& F37$$!36++]K3XFEF0$!3]%*QPPZ9tZF37$$!3%)****\\F)H')\\#F0$!3u!eYqemCG%F 37$$!3#****\\i3@/P#F0$!3_L8g\\,rGQF37$$!3;++Dr^b^AF0$!3nT#ot!GuPMF37$$ !3$****\\7Sw%G@F0$!3%\\w=g\"R&>1$F37$$!3*****\\7;)=,?F0$!3Uwr41$4Kq#F3 7$$!3/++DO\"3V(=F0$!33y=V%*>luBF37$$!3#******\\V'zVn6Un\"=F37$$!3!******\\!)H%*\\\"F0$!3S:#)>G 6Vh:F37$$!3/+++vl[p8F0$!3_M]MI.MI8F37$$!3\"******\\>iUC\"F0$!3d7-R7/PI 6F37$$!3-++DhkaI6F0$!3!Hqz%Gm.r'*F07$$!3s******\\XF`**!#=$!3N(\\YSNgS% zF07$$!3u*******>#z2))Fgp$!3q3Q\\]w$Hl'F07$$!3S++]7RKvuFgp$!3BF=K(zNnL &F07$$!3s,+++P'eH'Fgp$!3HUs\"f\"4XFVF07$$!3q)***\\7*3=+&Fgp$!3W<3>j6Aw LF07$$!3[)***\\PFcpPFgp$!3Tmi144s5EF07$$!3;)****\\7VQ[#Fgp$!3G]*)z\\7c X>F07$$!32)***\\i6:.8Fgp$!3kN7Z_W5W9F07$$!3Wb+++v`hH!#?$!3thye]4\"*35F 07$$\"3]****\\(QIKH\"Fgp$!3hUOABPT+mFgp7$$\"38****\\7:xWCFgp$!3)z%)R! \\gj7VFgp7$$\"3E,++vuY)o$Fgp$!3?q7$$\"3s****\\(o 3lW(Fgp$!3;-7!p]d\\m\"F\\u7$$\"35*****\\A))oz)Fgp$!3T5*[uvx9u\"F`s7$$ \"3e******Hk-,5F0$\"3![PSs4/93\"!#E7$$\"36+++D-eI6F0$\"342*[1!*[lA#F`s 7$$\"3u***\\(=_(zC\"F0$\"3K$G))=+U[_\"F\\u7$$\"3M+++b*=jP\"F0$\"3?d8F` oFH`F\\u7$$\"3g***\\(3/3(\\\"F0$\"3XjY&pnI#G7Fgp7$$\"33++vB4JB;F0$\"3N tY&G$em@CFgp7$$\"3u*****\\KCnu\"F0$\"3%\\TW*eSrjTFgp7$$\"3s***\\(=n#f( =F0$\"39z$[H(o_?nFgp7$$\"3P+++!)RO+?F0$\"3*Ga#\\PB4,5F07$$\"30++]_!>w7 #F0$\"3.`E_3bzL9F07$$\"3O++v)Q?QD#F0$\"3))oQmgy3r>F07$$\"3G+++5jypBF0$ \"3-%)QQj*\\,d#F07$$\"3<++]Ujp-DF0$\"3#z1hw/LKR$F07$$\"3++++gEd@EF0$\" 3[ZEk!*=#RE%F07$$\"39++v3'>$[FF0$\"3ZM\"[^A^RM&F07$$\"37++D6EjpGF0$\"3 ;(p(*>f\\``'F07$$\"\"$F*$\"\")F*-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*Fd[l- %'LEGENDG6#Q)Function6\"-%*THICKNESSG6#Fjz-%+AXESLABELSG6$Q\"xFi[lQ\"y Fi[l-%&TITLEG6#QCEntered~function~on~given~intervalFi[l-%%VIEWG6$;$!#I Fc[l$\"#IFc[l;$!++++?r!\")$\"++++?:Fa]l" 1 2 0 1 10 3 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Function" }}}}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 264 25 "S ection III: Iteration 1." }}{PARA 259 "" 0 "" {TEXT -1 214 "The Newton Raphson Method works by taking a tangent line at the value of the fun ction at the initial guess, and seeing where that tangent line crosses the x-axis. This value will be the new estimate for the root.." }} {PARA 0 "" 0 "" {TEXT -1 34 "The first estimate of the root is," }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "x1:=x0-eval(f(x),x=x0)/eval( g(x),x=x0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$!+LLLLL!#5" }}} {PARA 0 "" 0 "" {TEXT -1 79 "How good is that answer? Find the absolut e relative approximate error to judge." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "epsilon:=abs((x1-x0)/x1)*100;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(epsilonG$\"+++++?!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 111 "While not necesary to the method, for graphing purposes we define the equation of the tangent line touching x0." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "tanline(x):=eval(g(x),x=x0)*x+eval(f(x),x=x0)-ev al(g(x),x=x0)*x0:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 280 "plot([f(x),[x0,t,t=yrange], [x1,t,t=yrange],tanline(x)],x=xrange,y=yrange,title=\"Entered function on given interval with current and next root\\n and tangent line of t he curve at the current root\",legend=[\"Function\", \"x0, Current roo t\", \"x1, New root\", \"Tangent line\"],thickness=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 549 549 549 {PLOTDATA 2 "6*-%'CURVESG6%7S7$$!\"$\"\"!$! #kF*7$$!3!******\\2<#pG!#<$!3;MAyuKa#z&!#;7$$!3#)***\\7bBav#F0$!3CW4c9 7N'H&F37$$!36++]K3XFEF0$!3]%*QPPZ9tZF37$$!3%)****\\F)H')\\#F0$!3u!eYqe mCG%F37$$!3#****\\i3@/P#F0$!3_L8g\\,rGQF37$$!3;++Dr^b^AF0$!3nT#ot!GuPM F37$$!3$****\\7Sw%G@F0$!3%\\w=g\"R&>1$F37$$!3*****\\7;)=,?F0$!3Uwr41$4 Kq#F37$$!3/++DO\"3V(=F0$!33y=V%*>luBF37$$!3#******\\V'zVn6Un\"=F37$$!3!******\\!)H%*\\\"F0$!3S :#)>G6Vh:F37$$!3/+++vl[p8F0$!3_M]MI.MI8F37$$!3\"******\\>iUC\"F0$!3d7- R7/PI6F37$$!3-++DhkaI6F0$!3!Hqz%Gm.r'*F07$$!3s******\\XF`**!#=$!3N(\\Y SNgS%zF07$$!3u*******>#z2))Fgp$!3q3Q\\]w$Hl'F07$$!3S++]7RKvuFgp$!3BF=K (zNnL&F07$$!3s,+++P'eH'Fgp$!3HUs\"f\"4XFVF07$$!3q)***\\7*3=+&Fgp$!3W<3 >j6AwLF07$$!3[)***\\PFcpPFgp$!3Tmi144s5EF07$$!3;)****\\7VQ[#Fgp$!3G]*) z\\7cX>F07$$!32)***\\i6:.8Fgp$!3kN7Z_W5W9F07$$!3Wb+++v`hH!#?$!3thye]4 \"*35F07$$\"3]****\\(QIKH\"Fgp$!3hUOABPT+mFgp7$$\"38****\\7:xWCFgp$!3) z%)R!\\gj7VFgp7$$\"3E,++vuY)o$Fgp$!3?q7$$\"3s*** *\\(o3lW(Fgp$!3;-7!p]d\\m\"F\\u7$$\"35*****\\A))oz)Fgp$!3T5*[uvx9u\"F` s7$$\"3e******Hk-,5F0$\"3![PSs4/93\"!#E7$$\"36+++D-eI6F0$\"342*[1!*[lA #F`s7$$\"3u***\\(=_(zC\"F0$\"3K$G))=+U[_\"F\\u7$$\"3M+++b*=jP\"F0$\"3? 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The second estimate of the root is, " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "x2:=x1-eval(f(x),x=x1)/e val(g(x),x=x1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x2G$\"+66666!#5 " }}}{PARA 0 "" 0 "" {TEXT -1 79 "How good is that answer? Find the ab solute relative approximate error to judge." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 29 "epsilon:=abs((x2-x1)/x2)*100;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(epsilonG$\"+++++S!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 111 "While not necesary to the method, for graphing purposes we define the equation of the tangent line touching x0." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "tanline(x):=eval(g(x),x=x1)*x+eval(f(x),x=x1)-ev al(g(x),x=x1)*x1:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 280 "plot([f(x),[x1,t,t=yrange], [x2,t,t=yrange],tanline(x)],x=xrange,y=yrange,title=\"Entered function on given interval with current and next root\\n and tangent line of t he curve at the current root\",legend=[\"Function\", \"x1, Current roo t\", \"x2, New root\", \"Tangent line\"],thickness=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 549 549 549 {PLOTDATA 2 "6*-%'CURVESG6%7S7$$!\"$\"\"!$! #kF*7$$!3!******\\2<#pG!#<$!3;MAyuKa#z&!#;7$$!3#)***\\7bBav#F0$!3CW4c9 7N'H&F37$$!36++]K3XFEF0$!3]%*QPPZ9tZF37$$!3%)****\\F)H')\\#F0$!3u!eYqe mCG%F37$$!3#****\\i3@/P#F0$!3_L8g\\,rGQF37$$!3;++Dr^b^AF0$!3nT#ot!GuPM F37$$!3$****\\7Sw%G@F0$!3%\\w=g\"R&>1$F37$$!3*****\\7;)=,?F0$!3Uwr41$4 Kq#F37$$!3/++DO\"3V(=F0$!33y=V%*>luBF37$$!3#******\\V'zVn6Un\"=F37$$!3!******\\!)H%*\\\"F0$!3S :#)>G6Vh:F37$$!3/+++vl[p8F0$!3_M]MI.MI8F37$$!3\"******\\>iUC\"F0$!3d7- R7/PI6F37$$!3-++DhkaI6F0$!3!Hqz%Gm.r'*F07$$!3s******\\XF`**!#=$!3N(\\Y SNgS%zF07$$!3u*******>#z2))Fgp$!3q3Q\\]w$Hl'F07$$!3S++]7RKvuFgp$!3BF=K (zNnL&F07$$!3s,+++P'eH'Fgp$!3HUs\"f\"4XFVF07$$!3q)***\\7*3=+&Fgp$!3W<3 >j6AwLF07$$!3[)***\\PFcpPFgp$!3Tmi144s5EF07$$!3;)****\\7VQ[#Fgp$!3G]*) z\\7cX>F07$$!32)***\\i6:.8Fgp$!3kN7Z_W5W9F07$$!3Wb+++v`hH!#?$!3thye]4 \"*35F07$$\"3]****\\(QIKH\"Fgp$!3hUOABPT+mFgp7$$\"38****\\7:xWCFgp$!3) z%)R!\\gj7VFgp7$$\"3E,++vuY)o$Fgp$!3?q7$$\"3s*** *\\(o3lW(Fgp$!3;-7!p]d\\m\"F\\u7$$\"35*****\\A))oz)Fgp$!3T5*[uvx9u\"F` s7$$\"3e******Hk-,5F0$\"3![PSs4/93\"!#E7$$\"36+++D-eI6F0$\"342*[1!*[lA #F`s7$$\"3u***\\(=_(zC\"F0$\"3K$G))=+U[_\"F\\u7$$\"3M+++b*=jP\"F0$\"3? d8F`oFH`F\\u7$$\"3g***\\(3/3(\\\"F0$\"3XjY&pnI#G7Fgp7$$\"33++vB4JB;F0$ \"3NtY&G$em@CFgp7$$\"3u*****\\KCnu\"F0$\"3%\\TW*eSrjTFgp7$$\"3s***\\(= n#f(=F0$\"39z$[H(o_?nFgp7$$\"3P+++!)RO+?F0$\"3*Ga#\\PB4,5F07$$\"30++]_ !>w7#F0$\"3.`E_3bzL9F07$$\"3O++v)Q?QD#F0$\"3))oQmgy3r>F07$$\"3G+++5jyp BF0$\"3-%)QQj*\\,d#F07$$\"3<++]Ujp-DF0$\"3#z1hw/LKR$F07$$\"3++++gEd@EF 0$\"3[ZEk!*=#RE%F07$$\"39++v3'>$[FF0$\"3ZM\"[^A^RM&F07$$\"37++D6EjpGF0 $\"3;(p(*>f\\``'F07$$\"\"$F*$\"\")F*-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*F d[l-%'LEGENDG6#Q)Function6\"-F$6%7S7$$!3z*****HLLLL$Fgp$!3H++++++?rF37 $F^\\l$!3q+++)es;$pF37$F^\\l$!3u++!Q\"*4yw'F37$F^\\l$!3)3++))>HNe'F37$ F^\\l$!3+++g^p-)R'F37$F^\\l$!35++?kjS8iF37$F^\\l$!3,++gY%RA/'F37$F^\\l $!3S++!y,1]'eF37$F^\\l$!3G++?_4r\"o&F37$F^\\l$!3s++?;P+*\\&F37$F^\\l$! 3@++Smo16`F37$F^\\l$!3Q++!o)>`X^F37$F^\\l$!3/++?>*y\"f\\F37$F^\\l$!3K+ ++o11sZF37$F^\\l$!3/++!3cPF37$F^\\l$!3>+++\\FqF_e8F37$F^\\l$!39+++wW'><\"F37$F^\\l$!3o+++&o:H+\"F37 $F^\\l$!3)y****z/25=)F07$F^\\l$!3/.++9@/UkF07$F^\\l$!3H+++)pAVi%F07$F^ \\l$!3y5++?(pr%GF07$F^\\l$!3=$)*****\\_l')*Fgp7$F^\\l$\"39o++?JT_!)Fgp 7$F^\\l$\"3f.++cVrPEF07$F^\\l$\"3%y++!)f8]X%F07$F^\\l$\"3T)****R'G#\\7 'F07$F^\\l$\"3x'****>LF)Q!)F07$F^\\l$\"3++++/jk](*F07$F^\\l$\"3]++gO-e d6F37$F^\\l$\"3L++?g4FK8F37$F^\\l$\"3$*************>:F3-F^[l6&F`[lFd[l Fa[lFd[l-Ff[l6#Q1x1,~Current~rootFi[l-F$6%7S7$$\"3$******46666\"FgpF` \\l7$F[flFc\\l7$F[flFf\\l7$F[flFi\\l7$F[flF\\]l7$F[flF_]l7$F[flFb]l7$F [flFe]l7$F[flFh]l7$F[flF[^l7$F[flF^^l7$F[flFa^l7$F[flFd^l7$F[flFg^l7$F [flFj^l7$F[flF]_l7$F[flF`_l7$F[flFc_l7$F[flFf_l7$F[flFi_l7$F[flF\\`l7$ F[flF_`l7$F[flFb`l7$F[flFe`l7$F[flFh`l7$F[flF[al7$F[flF^al7$F[flFaal7$ F[flFdal7$F[flFgal7$F[flFjal7$F[flF]bl7$F[flF`bl7$F[flFcbl7$F[flFfbl7$ F[flFibl7$F[flF\\cl7$F[flF_cl7$F[flFbcl7$F[flFecl7$F[flFhcl7$F[flF[dl7 $F[flF^dl7$F[flFadl7$F[flFddl7$F[flFgdl7$F[flFjdl7$F[flF]el7$F[flF`el- F^[l6&F`[lFa[lFa[lFd[l-Ff[l6#Q-x2,~New~rootFi[l-F$6%7S7$F($!3?++]e#f#f ;F37$F.$!3#H=(>l$3&*e\"F37$F5$!3A,/*e[=)G:F37$F:$!3-&f`#pjcg9F37$F?$!3 ?`OGL='=R\"F37$FD$!3IN-)z$Q[B8F37$FI$!33/8M$o)3g7F37$FN$!3sbLqsmW%>\"F 37$FS$!3*Hs*R6'fl7\"F37$FX$!3khEwk-*)e5F37$Fgn$!3YwWET-%G*)*F07$F\\o$! 3CQg'[4[(z#*F07$Fao$!3iJ8][:b*e)F07$Ffo$!3AKX!)))4_'*yF07$F[p$!3\\@n4H wlGsF07$F`p$!3HD(Ggrt@i'F07$Fep$!3c-4,$o05!fF07$F[q$!3!==yKw\"3!H&F07$ F`q$!3kedD5?VzXF07$Feq$!3k*HkQc'Q]RF07$Fjq$!3e5H$3,C-E$F07$F_r$!3%[V?6 0EIg#F07$Fdr$!3!*oPvC*3t\">F07$Fir$!3wj#H]a1wG\"F07$F^s$!3Snvf'euQ3'Fg p7$Fds$\"33Rz:.\"GIr*F\\u7$Fis$\"3cGbHua&G6(Fgp7$F^t$\"3s\"ps/1!fu8F07 $Fct$\"35c&Ro_[)f?F07$Fht$\"32giaY-CIFF07$F^u$\"3m\"[iiqy)yLF07$Fcu$\" 3#HRZf7\"3*4%F07$Fhu$\"3WO4LK]@YZF07$F^v$\"3x!)>O0'orV&F07$Fcv$\"3;!R@ %Q_FjgF07$Fhv$\"3_\"f)yk^xZnF07$F]w$\"3?xM<^i$=R(F07$Fbw$\"35vA@kc1l!) 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{TEXT -1 155 "Here we will \+ calculate the root and try to plot the absolute relative approximate e rror aginst the number of iterations to understand teh slow convergenc e." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Root Calculation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 193 "xr:=pro c(n)\n local p, q, i;\n p:=x0;\n q:=x0;\n for i from 1 to n do\n p:=q-eval(f(x),x=q)/eval(g(x),x=q);\n q:=p; \n end do;\n \n p; \nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "nrange:=1..nmax:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Absolute approximate error" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 100 "Ea:=proc(n)\n if n=1 then\n abs(xr(n) -x0);\n else\n abs(xr(n)-xr(n-1));\n end if;\nend proc:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Absolute relative approximate erro r" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "ea:=proc(n)\nabs(Ea(n) /xr(n))*100;\nend proc: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "plot(ea,nrange,title=\"Absolute relative approximate error 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45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 176 "As the above graph depict s, convergence will still happen, but very slowly. In this example, a fter 10 iterations, only 1 significant digit is correct in the value o f the root." }}}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 265 0 "" }{TEXT 266 23 "Section VI: Conclusion ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 252 "Map le helped us to apply our knowledge of numerical methods of finding ro ots of a nonlinear equation using the Newton-Raphson method to underst and how the method can have slow convergence due to an inflection poin t occuring in the vicinity of the root." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 4 "" 0 "" {TEXT 267 10 "References" }}{PARA 0 "" 0 "" {TEXT -1 4 "[1] " }{TEXT 269 174 "Nathan Collier, Autar Kaw, Jai Paul \+ , Michael Keteltas, Holistic Numerical Methods Institute, See http://n umericalmethods.eng.usf.edu/mws/gen/03nle/mws_gen_nle_txt_newton.pdf" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 0 "" } {TEXT 260 10 "Disclaimer" }{TEXT 268 1 ":" }{TEXT -1 248 " While every effort has been made to validate the solutions in this worksheet, Uni versity of South Florida and the contributors are not responsible for \+ any errors contained and are not liable for any damages resulting from the use of this material." }}}{MARK "0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }