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-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 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 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" -1 260 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 37 "The bisection method for root fin ding" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Ca nada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 24.3.2007" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restar t;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "load root-finding procedures including: " }{TEXT 0 6 "bisect" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 339 7 "roots.m" }{TEXT -1 37 " contains the code for the procedure " } {TEXT 0 6 "bisect" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 122 "It can be read into a Maple session by a command similar to the one that follows, where the file path gives its locati on. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\\\Maple/p rocdrs/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 23 "Graphs, roots and zeros" }}{PARA 0 "" 0 "" {TEXT -1 13 "The equation " }{XPPEDIT 18 0 "x^2=x+2" "6#/*$%\"xG\"\"#,&F%\"\"\"F&F (" }{TEXT -1 27 " clearly has the solutions " }{XPPEDIT 18 0 "x = -1" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x = 2" "6# /%\"xG\"\"#" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve(x^2=x+2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$!\"\"\"\"#" } }}{PARA 0 "" 0 "" {TEXT -1 52 "We can interpret these solutions graphi cally as the " }{TEXT 330 1 "x" }{TEXT -1 57 " coordinates of the poin ts of intersection of the graphs " }{XPPEDIT 18 0 "y=x^2" "6#/%\"yG*$% \"xG\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=x+2" "6#/%\"yG,&%\"xG \"\"\"\"\"#F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot([x^2,x+2],x=-1.5..2.5); " }}{PARA 13 "" 1 "" {GLPLOT2D 271 213 213 {PLOTDATA 2 "6&-%'CURVESG6$ 7S7$$!1+++++++:!#:$\"1++++++]AF*7$$!1LLL$Q6GT\"F*$\"1Zi([+Og*>F*7$$!1n m;M!\\pL\"F*$\"1$=f*>FV(y\"F*7$$!1LLL))Qj^7F*$\"1>WU!R(em:F*7$$!1LLL=K vl6F*$\"1CX0m0)*e8F*7$$!1nm;C2G!3\"F*$\"1eg+Vk+n6F*7$$!1LL$3yO5+\"F*$ \"1j6eOY2-5F*7$$!1,++vE%)*=*!#;$\"1<^7R3KX%)FN7$$!1MLL3WDT$)FN$\"1GSX5 DldpFN7$$!1,++vvQ&\\(FN$\"1Rk%**[$3=cFN7$$!1mmmm&4`i'FN$\"1%[;aos%*Q%F N7$$!1LLL$QW*eeFN$\"1@$*pGHsKMFN7$$!1,+++()>'*\\FN$\"1q\"))\\9+i\\#FN7 $$!1++++0\"*HTFN$\"1D5!Q2;cq\"FN7$$!1++++83&H$FN$\"1p4Oxgv&3\"FN7$$!1L LL3k(p`#FN$\"1,*RkH\\iV'!#<7$$!1nmmmj^N;FN$\"1OXj&y8\\n#Fcp7$$!1!ommm9 '=()Fcp$\"1M:eqTU,w!#=7$$\"1+*****\\s]k\"F^q$\"1@fDINE1F!#@7$$\"19LLL` dF!)Fcp$\"1[TBtl>WkF^q7$$\"1,++D2Yl;FN$\"1Hv^E%fPx#Fcp7$$\"1+++v\"ep[# FN$\"1K$*>k4'\\='Fcp7$$\"1MLL$e/TM$FN$\"1,rUYNI=6FN7$$\"1LLLeDBJTFN$\" 1QL5X#3nq\"FN7$$\"1mmm;kD!)\\FN$\"1]\\d(R&H![#FN7$$\"1jmm\"f`@'eFN$\"1 O!HKZ%[OMFN7$$\"1)****\\nZ)HmFN$\"1)Gq$>!)[&R%FN7$$\"1lmm;$y*euFN$\"1Z .&GvNOc&FN7$$\"1*******R^bJ)FN$\"1%>/'3&R[\"pFN7$$\"1'*****\\5a`\"*FN$ \"1.NSv8ty$)FN7$$\"1(****\\7RV'**FN$\"1q0!)>a!)G**FN7$$\"1+++:#fk3\"F* $\"1;%eei$R!=\"F*7$$\"1LLL`4Nn6F*$\"1U#[#[#3FO\"F*7$$\"1+++],s`7F*$\"1 Ag^9U\"=d\"F*7$$\"1mm;zM)>L\"F*$\"1PHx))*zTx\"F*7$$\"1+++qfa<9F*$\"1TK 1xlV4?F*7$$\"1LL$eg`!)\\\"F*$\"1^-&fgkTC#F*7$$\"1++]#G2Ae\"F*$\"1Mgz%) )zL]#F*7$$\"1LLL$)G[k;F*$\"1Y'4*oK]qFF*7$$\"1++]7yh]F*$\"1**f\"RH2.o$F*7$$ \"1LL$e#pa-?F*$\"1LX;!>%>5SF*7$$\"1+++Sv&)z?F*$\"1^[p'Q2eK%F*7$$\"1LLL GUYo@F*$\"1y7c4rB-ZF*7$$\"1nmm1^rZAF*$\"1Wvt+KA_]F*7$$\"1++]sI@KBF*$\" 1\"*)Rb\"y@RaF*7$$\"1++]2%)38CF*$\"1()3Tic*H#eF*7$$\"1+++++++DF*$\"1++ ++++]iF*-%'COLOURG6&%$RGBG$\"#5!\"\"\"\"!Fb[l-F$6$7S7$F($\"1+++++++]FN 7$F.$\"1nmmmh)=(eFN7$F3$\"1MLLe'40j'FN7$F8$\"1nmm;6m$[(FN7$F=$\"1omm;y YU$)FN7$FB$\"1KLLeF>(>*FN7$FG$\"1mmm\">K'*)**FN7$FL$\"1++]Kd,\"3\"F*7$ FR$\"1nm;fX(e;\"F*7$FW$\"1++]U7Y]7F*7$Ffn$\"1LLLV!puL\"F*7$F[o$\"1nmmh b599F*7$F`o$\"1+++I,Q+:F*7$Feo$\"1+++]*3qe\"F*7$Fjo$\"1+++q=\\q;F*7$F_ p$\"1nm;fBIYF*7$F`q$\"1++]s ]k,?F*7$Ffq$\"1LLL`dF!3#F*7$F[r$\"1++]sgam@F*7$F`r$\"1++]L$F*7$Ffv$\"1+++qfa " 0 "" {MPLTEXT 1 0 26 "plot(x^2-x-2,x=-1.5..2.5);" }}{PARA 13 "" 1 "" {GLPLOT2D 235 200 200 {PLOTDATA 2 "6%-%'CURVESG6$7S 7$$!1+++++++:!#:$\"1++++++]'*\\F<$!1$=,b6g2D\"F*7$$!1 ++++0\"*HTF<$!1)*)>EMZkT\"F*7$$!1++++83&H$F<$!1.REii\">c\"F*7$$!1LLL3k (p`#F<$!1oA?m)R>o\"F*7$$!1nmmmj^N;F<$!1))pZD#*p4=F*7$$!1!ommm9'=()!#<$ !1_F;HC@0>F*7$$\"1+*****\\s]k\"FL$!1qkBmBk,?F*7$$\"19LLL`dF!)F\\q$!1*4 gnbJQ2#F*7$$\"1,++D2Yl;F<$!1D[By%3)Q@F*7$$\"1+++v\"ep[#F<$!12!ey?Yo=#F *7$$\"1MLL$e/TM$F<$!1B1p.,eAAF*7$$\"1LLLeDBJTF<$!1+IKJCXUAF*7$$\"1mmm; kD!)\\F<$!1s\"4>5'**\\AF*7$$\"1jmm\"f`@'eF<$!1jP%=\"pcUAF*7$$\"1)**** \\nZ)HmF<$!1rHclfVBAF*7$$\"1lmm;$y*euF<$!1K;QcU`*=#F*7$$\"1*******R^bJ )F<$!1!eR\"*=r+9#F*7$$\"1'*****\\5a`\"*F<$!1\\'fu'4[x?F*7$$\"1(****\\7 RV'**F<$!1V*>0P`N+#F*7$$\"1+++:#fk3\"F*$!1%eT\"*elg!>F*7$$\"1LLL`4Nn6F *$!1#4&30Fk/=F*7$$\"1+++],s`7F*$!1yR[Nf!>o\"F*7$$\"1mm;zM)>L\"F*$!1HPR !\\.yb\"F*7$$\"1+++qfa<9F*$!1fn$HR4\"39F*7$$\"1LL$eg`!)\\\"F*$!1#3$))* **))QD\"F*7$$\"1++]#G2Ae\"F*$!1mRq(RF)y5F*7$$\"1LLL$)G[k;F*$!1poBWhzR* )F<7$$\"1++]7yh]F*$!1zm]xS0\"Q#F<7$$\"1LL$e#pa-?F*$\"1****>JVEZwFL7$$\"1+++Sv&) z?F*$\"1:&[pY)\\fCF<7$$\"1LLLGUYo@F*$\"1_%zA\")GxL&F<7$$\"1nmm1^rZAF*$ \"1r(32%43X!)F<7$$\"1++]sI@KBF*$\"1\"*)RIu/q5\"F*7$$\"1++]2%)38CF*$\"1 ()3\"\\D2*49F*7$$\"1+++++++DF*F+-%'COLOURG6&%$RGBG$\"#5!\"\"\"\"!F_[l- %+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;$F*F^[l$\"#DF^[l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Now consider th e equation " }{XPPEDIT 18 0 "cos(x)=x" "6#/-%$cosG6#%\"xGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 "Maple's " }{TEXT 0 5 "solve" } {TEXT -1 37 " procedure does not give a solution. " }{TEXT 0 6 "RootOf " }{TEXT -1 54 " rephrases the question, but does not give a solution. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve(cos(x)=x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RootOfG 6#,&%#_ZG\"\"\"-%$cosG6#F'!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 23 "On plotting the graphs " }{XPPEDIT 18 0 " y=cos(x)" "6#/%\"yG-%$cosG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=x" "6#/%\"yG%\"xG" }{TEXT -1 38 " we see that there is a solution n ear " }{XPPEDIT 18 0 "x= 0" "6#/%\"xG\"\"!" }{TEXT -1 4 ".73." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot([cos(x),x],x=0..Pi/2,y=0..1.2);" }}{PARA 13 "" 1 "" {GLPLOT2D 297 219 219 {PLOTDATA 2 "6&-%'CURVESG6$7S7$\"\"!$\"\"\"F(7$$ \"1GK5j*))QU$!#<$\"1EIyk!RT***!#;7$$\"1XXYUk*HS'F.$\"1j$H4#y]z**F17$$ \"1N68yVJ`(*F.$\"1PWDATZ_**F17$$\"16YeRSe78F1$\"1>nIl(zR\"**F17$$\"1,, hQPB[;F1$\"1`p97NZk)*F17$$\"1B*ed(RUf>F1$\"1G6B)GY'3)*F17$$\"1IX[TMk\" G#F1$\"10*pfUK3u*F17$$\"1hw(QF$)[h#F1$\"1%3c\"zG1g'*F17$$\"1G?!>Saq%HF 1$\"1$Hg3Tx)o&*F17$$\"12n4OKt)G$F1$\"1X#yiMoSY*F17$$\"1Y]BRTo*e$F1$\"1 o4Josfi$*F17$$\"1t79wN[GRF1$\"15G)z'RAQ#*F17$$\"1)F1$\"1AKL\"Gew#oF17$$\"1w)\\AI@S\\)F1$\"1H 5]d>K/mF17$$\"1o()=V,i>))F1$\"1p$f4=xjN'F17$$\"1`\"[dg&*f:*F1$\"18&*yStc &F17$$\"1m\"[l:+d,\"!#:$\"1zV,H.Dq_F17$$\"1\"3XyEmu/\"F`u$\"1cj1lEn(* \\F17$$\"1K_*>P$Q\"3\"F`u$\"1F8Py$y5q%F17$$\"1M\"G%4t676F`u$\"1g[y,0kF WF17$$\"1p*Q4iU([*Q$Q F17$$\"1k&>q'*z.@\"F`u$\"18u?D(Qm_$F17$$\"1Q$[)>&*oU7F`u$\"1OldzS^AKF1 7$$\"1^lOFY^w7F`u$\"1O^_O]_+HF17$$\"1\")f6EA448F`u$\"1FKPO+F(e#F17$$\" 1;T7EvSU8F`u$\"1:%3md%3kAF17$$\"1`wiepWv8F`u$\"1+Nc*p#4T>F17$$\"1$obdw 1eS\"F`u$\"14+*zL?Ck\"F17$$\"1%)o#3`-1W\"F`u$\"1C?\")3IE)H\"F17$$\"1#* )4zFC " 0 "" {MPLTEXT 1 0 32 "fsolve(cos(x)=x,x=0.73);\nco s(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+K8&3R(!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+K8&3R(!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 73 "The second command above simply checks th at cos(.7390851332)=.7390851332." }}{PARA 0 "" 0 "" {TEXT -1 9 "Actual ly " }{TEXT 0 6 "fsolve" }{TEXT -1 49 " can manage here without having a starting value." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 34 "evalf(fsolve(cos(x)=x,x=0.73),15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0h^@L^3R(!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Note that a " }{TEXT 275 8 "sol ution" }{TEXT -1 4 " or " }{TEXT 275 4 "root" }{TEXT -1 1 " " } {XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 31 ".7390851332 of the e quation of " }{XPPEDIT 18 0 "cos(x)=x" "6#/-%$cosG6#%\"xGF'" }{TEXT -1 12 " is a value " }{TEXT 286 1 "x" }{TEXT -1 20 " which the functio n " }{XPPEDIT 18 0 "f(x)=cos(x)-x" "6#/-%\"fG6#%\"xG,&-%$cosG6#F'\"\" \"F'!\"\"" }{TEXT -1 34 " gives the value 0. It provides a " }{TEXT 275 4 "zero" }{TEXT -1 21 " of the the function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "How does " }{TEXT 0 6 "fso lve" }{TEXT -1 6 " work?" }}{PARA 0 "" 0 "" {TEXT -1 112 "There are a \+ number of techniques for refining an approximate root of an equation i nto a more accurate solution. " }}{PARA 0 "" 0 "" {TEXT -1 15 "The fir st such " }{TEXT 275 19 "root-finding method" }{TEXT -1 29 " we shall \+ investigate is the " }{TEXT 275 16 "bisection method" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "The bisection method \+ idea" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 62 "Suppose that we want to find solutions for the cubic eq uation " }{XPPEDIT 18 0 "x^3+x-1 = 0;" "6#/,(*$%\"xG\"\"$\"\"\"F&F(F(! \"\"\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 62 "A solution o f this equation is a zero or root of the function " }{XPPEDIT 18 0 "f( x) = x^3+x-1;" "6#/-%\"fG6#%\"xG,(*$F'\"\"$\"\"\"F'F+F+!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 26 "By plotting the graph of " } {XPPEDIT 18 0 "y = x^3+x-1;" "6#/%\"yG,(*$%\"xG\"\"$\"\"\"F'F)F)!\"\" " }{TEXT -1 41 " we can see that there is a solution for " }{TEXT 287 1 "x" }{TEXT -1 9 " between " }{TEXT 298 1 "x" }{TEXT -1 11 " = 0.6 an d " }{TEXT 299 1 "x" }{TEXT -1 37 " = 0.8 because the graph crosses th e " }{TEXT 288 1 "x" }{TEXT -1 10 " axis for " }{TEXT 289 1 "x" } {TEXT -1 22 " between these values." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "f := x -> x^3+x-1;\nplot(f( x),x=-1..1.3,y);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$!\"\"\"\"!$!\"$F*7$$!1nm;aam)\\*!#;$!1t())yE!)o! G!#:7$$!1M$ek%pXi!*F0$!1#*fw*HG0l#F37$$!1mm\"z&[*=d)F0$!1!\\gS()Hq[#F3 7$$!1mmT053y!)F0$!1za0\\k%\\L#F37$$!1M$eR;9me(F0$!1!)ek,?K&>#F37$$!1n; z*[h48(F0$!1\"[NK!*4d2#F37$$!1+]7Q&f\"fmF0$!1uINEC@h>F37$$!1n;z%G@7<'F 0$!1Y^*zuY@&=F37$$!1+]iby%[o&F0$!1?u^G[?_O\"F37$$!1n; zM9wLGF0$!1Rv^D=818F37$$!1ML$3\">U:BF0$!1P*R:ebRC\"F37$$!1MLLV.Kw=F0$! 1GG^mxB%>\"F37$$!1,]7L3al8F0$!1Yor4/5R6F37$$!1[LL$=WT8*!#<$!1N'Rr_.@4 \"F37$$!1-+DJ3gtTFfq$!1N7.3(3=/\"F37$$\"1K**\\i]4+b!#=$!1I(QbQ(*\\%**F 07$$\"1om;a8gyaFfq$!1v&)Rea\\]%*F07$$\"1m;/@(e/+\"F0$!1pL,6v_*)*)F07$$ \"1LLeRuk)[\"F0$!1$yZK2j$y%)F07$$\"1L$3_JQd*>F0$!1PK-)*>xCzF07$$\"1)* \\78C;PCF0$!1E5Utf2=uF07$$\"1ML3KD\"R\"HF0$!1'Q\\&p/nQoF07$$\"1*****\\ 0UkS$F0$!1P^k/'z#)>'F07$$\"1)**\\P5'G))QF0$!1(R%)*3H&Q_&F07$$\"1)*\\(o *\\\\aVF0$!1xO6=>#)>[F07$$\"1)**\\i[S@([F0$!1/!=L%GKrRF07$$\"1lmm\")zE P`F0$!1\"=wq$\\LUJF07$$\"1++]i3*Q$eF0$!1T,/Nie!=#F07$$\"1I$3_+0RG'F0$! 1t*\\lQSZB\"F07$$\"1++]F$*)ex'F0$!1+Y\\ta=J6Ffq7$$\"1l;aL#3)QsF0$\"1*G !*epo>.\"F07$$\"1)*\\Pu=pAxF0$\"1V'f\\(H]GBF07$$\"1lm;zlx&>)F0$\"1n+)= [T4q$F07$$\"1)*\\(=U_5p)F0$\"1vC]Wgyb_F07$$\"1LLLB>1o\"*F0$\"1)\\CX'f7 uoF07$$\"1JLeMI(el*F0$\"1Aw3A;he')F07$$\"1mTN#[kR,\"F3$\"1)[()GnXk0\"F 37$$\"1++]&3=%e5F3$\"1;o%>s4TC\"F37$$\"1n;HJpO46F3$\"1!QbPDfYZ\"F37$$ \"1LLL'=O\\:\"F3$\"1kJO)ozap\"F37$$\"1+vo;D_.7F3$\"1_5t,XyY>F37$$\"1+D JMe-]7F3$\"1a^:N>F.AF37$$\"1+++++++8F3$\"1,+++++(\\#F3-%'COLOURG6&%$RG BG$\"#5F)F*F*-%+AXESLABELSG6$Q\"x6\"Q\"yFe[l-%%VIEWG6$;F($\"#8F)%(DEFA ULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Wit hout drawing the graph, we can detect that there is a zero of the func tion " }{XPPEDIT 18 0 "f(x) = x^3+x-1;" "6#/-%\"fG6#%\"xG,(*$F'\"\"$\" \"\"F'F+F+!\"\"" }{TEXT -1 9 " between " }{TEXT 300 1 "x" }{TEXT -1 11 " = 0.6 and " }{TEXT 301 1 "x" }{TEXT -1 25 " = 0.8 by observing th at " }{TEXT 275 40 "the values of the function at these two " }{TEXT 290 1 "x" }{TEXT 275 27 " values have opposite signs" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "f := x -> x^3+x-1;\n'f(0.6)'=f(0.6);\n'f(0.8)'=f(0.8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(* $)9$\"\"$\"\"\"F1F/F1F1!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"fG6#$\"\"'!\"\"$!$%=!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\" fG6#$\"\")!\"\"$\"$7$!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "We could make a guess that the root is " }{TEXT 275 15 "mid-way between" }{TEXT -1 119 " 0.6 and 0.8 and try the numbe r 0.7. It is not very likely that we will hit the actual root exactly \+ by performing this " }{TEXT 275 9 "bisection" }{TEXT -1 11 " but maybe " }{TEXT 304 1 "x" }{TEXT -1 67 " = 0.7 will be closer to the actual \+ root than either of the values " }{TEXT 302 1 "x" }{TEXT -1 11 " = 0.6 and " }{TEXT 303 1 "x" }{TEXT -1 7 " = 0.8." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "'f(0.7)'=f(0.7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#$\"\"(!\"\"$\"#V!\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "What do you think?" }}{PARA 0 " " 0 "" {TEXT -1 83 "f(0.7) is closer to 0 than either of f(0.6) and f( 0.8), so things are looking good." }}{PARA 0 "" 0 "" {TEXT -1 50 "Noti ce that f(0.7) = 0.043 and f(0.8) = 0.312 are " }{TEXT 275 13 "both po sitive" }{TEXT -1 27 ", while f(0.6) = -0.184 is " }{TEXT 275 8 "negat ive" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 "This means that t he root is between " }{TEXT 291 1 "x" }{TEXT -1 12 " = 0.6 and " } {TEXT 292 1 "x" }{TEXT -1 8 " = 0.7." }}{PARA 0 "" 0 "" {TEXT -1 23 " We can confirm this by " }{TEXT 275 10 "zooming in" }{TEXT -1 27 " and plotting the graph of " }{XPPEDIT 18 0 "f(x) = x^3+x-1;" "6#/-%\"fG6# %\"xG,(*$F'\"\"$\"\"\"F'F+F+!\"\"" }{TEXT -1 5 " for " }{TEXT 293 1 "x " }{TEXT -1 21 " between 0.6 and 0.8." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "f := x -> x^3+x-1;\nplot (f(x),x=0.6..0.8,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"x G6\"6$%)operatorG%&arrowGF(,(*$)9$\"\"$\"\"\"F1F/F1F1!\"\"F(F(F(" }} {PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6%-%'CURVESG6$7S7$ $\"3w**************f!#=$!3_++++++S=F*7$$\"3ALLL3VfVgF*$!3-BKqE4)*[(48F*7$$\"3C++Dmy]!H'F*$!3A.Y%pR2.A\"F*7$$\"3uKL ezs$HL'F*$!3=dH#\\!*or7\"F*7$$\"3%)***\\7iI_P'F*$!3;we2Z$[O.\"F*7$$\"3 \"pmm;_M(=kF*$!3;!yCvVqtO*!#>7$$\"3MLL$3y_qX'F*$!3]nf(>TRx])Fin7$$\"3' ******\\1!>+lF*$!3^R6o0->LvFin7$$\"3V+++vW]VlF*$!3#R3C)[rLZlFin7$$\"3M +++NfC&e'F*$!3O02P\"3;.f&Fin7$$\"36LLez6:BmF*$!3^S:$Q6'G:ZFin7$$\"3_mm m\"=C#omF*$!3[hK&\\$oNnOFin7$$\"30nmmEpS1nF*$!3Y1!3x6mKx#Fin7$$\"3i*** \\i`A3v'F*$!3\"pDJ]sUes\"Fin7$$\"3Ymmmwy8!z'F*$!3UrX@2HI?z!#?7$$\"3\\+ +DOIFLoF*$\"3)ou7oygvR#Fgq7$$\"3M++v3zMuoF*$\"3#>C[=q]$H7Fin7$$\"3Znm; H_?uGX)Fin7$$\"3,++]_qn2sF*$\"3RI\"Q\\R\"4@&*F in7$$\"3%)***\\i&p@[sF*$\"3!=#ecgs=c5F*7$$\"3E++]2'HKH(F*$\"3%=#eZ;ees 6F*7$$\"3vmmmwanLtF*$\"3'*o)ot>JzF\"F*7$$\"3=+++v+'oP(F*$\"33%\\mNb/7R \"F*7$$\"38MLeR<*fT(F*$\"3Gv0g$yfX\\\"F*7$$\"3k+++&)HxeuF*$\"3UNz[@TL3 ;F*7$$\"3Amm\"H!o-*\\(F*$\"3V\\'HfuMhr\"F*7$$\"3))***\\7k.6a(F*$\"3cGr !*pofH=F*7$$\"3CnmmT9C#e(F*$\"34%**>$f7IT>F*7$$\"38++D1*3`i(F*$\"3ak:S %Qq!f?F*7$$\"3#QLLL*zymwF*$\"3kT3PK#)Ht@F*7$$\"3;ML$3N1#4xF*$\"3S-lqt7 $4H#F*7$$\"3'om;HYt7v(F*$\"3#4Nx4@7%3CF*7$$\"3#3+++xG**y(F*$\"32OCL&[! 4 " 0 "" {MPLTEXT 1 0 18 "'f(0.65)'=f( 0.65);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#$\"#l!\"#$!&v`(!\"' " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Since this value of f(0.6 5) is " }{TEXT 275 8 "negative" }{TEXT -1 18 ", while f(0.7) is " } {TEXT 275 8 "positive" }{TEXT -1 74 ", this narrows down the interval \+ containing the root to the interval from " }{TEXT 305 1 "x" }{TEXT -1 11 " = 0.65 to " }{TEXT 306 1 "x" }{TEXT -1 7 " = 0.7." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "We can continue this process until we have found the root to the desired accuracy." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 " A single step of the bisection method" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 61 "A single step for t he bisection method for finding a zero of " }{XPPEDIT 18 0 "f(x)" "6#- %\"fG6#%\"xG" }{TEXT -1 15 " is as follows." }}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " is " }{TEXT 275 10 "continuous" }{TEXT -1 25 " throughout the i nterval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 37 ". cons isting of all the real numbers " }{TEXT 307 1 "x" }{TEXT -1 6 " with \+ " }{XPPEDIT 18 0 "a <= x" "6#1%\"aG%\"xG" }{XPPEDIT 18 0 "``<= b" "6#1 %!G%\"bG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 45 "Roughly speak ing, this means that there are \"" }{TEXT 275 22 "no breaks in the gra ph" }{TEXT -1 5 "\" of " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"x G" }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG% \"bG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 9 "Then, if " }{XPPEDIT 18 0 "f(a)" "6#-%\"fG6#%\"aG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(b)" "6#-%\"fG6#%\"bG" }{TEXT -1 6 " have " }{TEXT 275 14 "opposite signs" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 11 " must have " }{TEXT 275 17 "at least one zero" }{TEXT -1 19 " somewhere between " }{TEXT 308 1 "a" }{TEXT -1 5 " and " }{TEXT 309 1 "b" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 4 "Set " }{XPPEDIT 18 0 "m = (a+b)/2;" "6#/%\"mG*&,&% \"aG\"\"\"%\"bGF(F(\"\"#!\"\"" }{TEXT -1 23 " , that is, the number " }{TEXT 275 23 "mid-way between a and b" }{TEXT -1 16 ", and calculate \+ " }{XPPEDIT 18 0 "f(m)" "6#-%\"fG6#%\"mG" }{TEXT -1 28 " taking note o f the sign of " }{XPPEDIT 18 0 "f(m)" "6#-%\"fG6#%\"mG" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "The number " }{XPPEDIT 18 0 "m = (a+b )/2;" "6#/%\"mG*&,&%\"aG\"\"\"%\"bGF(F(\"\"#!\"\"" }{TEXT -1 1 " " } {TEXT 275 7 "bisects" }{TEXT -1 14 " the interval " }{XPPEDIT 18 0 "[a ,b]" "6#7$%\"aG%\"bG" }{TEXT -1 29 " into the left-hand interval " } {XPPEDIT 18 0 "[a, (a+b)/2];" "6#7$%\"aG*&,&F$\"\"\"%\"bGF'F'\"\"#!\" \"" }{TEXT -1 29 " and the right-hand interval " }{XPPEDIT 18 0 "[(a+b )/2,b]" "6#7$*&,&%\"aG\"\"\"%\"bGF'F'\"\"#!\"\"F(" }{TEXT -1 1 "." }} {PARA 259 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 343 301 301 {PLOTDATA 2 "6 1-%'CURVESG6$7S7$$\"\"!F)$!\"\"F)7$$\"3emmm;arz@!#>$!3)=l\"e'*[#>y*!#= 7$$\"3[LL$e9ui2%F/$!3[#zf7F&p\"f*F27$$\"3nmmm\"z_\"4iF/$!3QFWCq3pw$*F2 7$$\"3[mmmT&phN)F/$!3[w+(GI[&e\"*F27$$\"3CLLe*=)H\\5F2$!3q6ZGd([\"R*)F 27$$\"3gmm\"z/3uC\"F2$!3'zGy![>=L()F27$$\"3%)***\\7LRDX\"F2$!3sE3xpR\" o^)F27$$\"3]mm\"zR'ok;F2$!3Q4t=K@=*G)F27$$\"3w***\\i5`h(=F2$!3#\\#o^Ls !y0)F27$$\"3WLLL3En$4#F2$!3D>m&e)>b9yF27$$\"3qmm;/RE&G#F2$!3%*yzri(*Q& 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#sF^]lQ)y~=~f(x)F`]l-Fgz6&Fiz$\"*++++\"!\")F(F(Fa]l-%*AXESSTYLEG6#%%NO NEG-%+AXESLABELSG6%Q\"xF`]lQ!F`]l-Fb]l6#%(DEFAULTG-%%VIEWG6$;F(FdzF`al " 1 2 0 1 10 0 2 9 1 1 2 1.000000 46.000000 45.000000 0 0 "Curve 1" "C urve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "C urve 9" "Curve 10" "Curve 11" "Curve 12" }}{TEXT -1 1 " " }}{PARA 15 " " 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "f(m)" "6#-%\"fG6#%\"mG" } {TEXT -1 9 " has the " }{TEXT 275 13 "opposite sign" }{TEXT -1 4 " to \+ " }{XPPEDIT 18 0 "f(a)" "6#-%\"fG6#%\"aG" }{TEXT -1 9 " but the " } {TEXT 275 9 "same sign" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "f(b)" "6#-% \"fG6#%\"bG" }{TEXT -1 21 ", the root is in the " }{TEXT 275 9 "left-h and" }{TEXT -1 10 " interval " }{XPPEDIT 18 0 "[a,m]" "6#7$%\"aG%\"mG " }{TEXT -1 35 ".\nIn this case make the assignment " }{TEXT 267 6 "b \+ := m" }{TEXT -1 56 " and repeat the bisection process in the \"new\" i nterval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 3 "If " } {XPPEDIT 18 0 "f(m)" "6#-%\"fG6#%\"mG" }{TEXT -1 8 " has the" }{TEXT 258 1 " " }{TEXT 275 9 "same sign" }{TEXT -1 4 " as " }{XPPEDIT 18 0 " f(a)" "6#-%\"fG6#%\"aG" }{TEXT -1 5 " but " }{TEXT 275 13 "opposite si gn" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "f(b)" "6#-%\"fG6#%\"bG" }{TEXT -1 21 ", the root is in the " }{TEXT 275 10 "right-hand" }{TEXT -1 10 " interval " }{XPPEDIT 18 0 "[m,b]" "6#7$%\"mG%\"bG" }{TEXT -1 35 ".\n In this case make the assignment " }{TEXT 267 6 "a := m" }{TEXT -1 56 " and repeat the bisection process in the \"new\" interval " } {XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "Automating th e bisection method" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "f(x) = x^3 +x-1;" "6#/-%\"fG6#%\"xG,(*$F'\"\"$\"\"\"F'F+F+!\"\"" }{TEXT -1 20 " h as a zero between " }{TEXT 310 1 "x" }{TEXT -1 11 " = 0.6 and " } {TEXT 311 1 "x" }{TEXT -1 25 " = 0.8 so start off with " }{TEXT 312 1 "a" }{TEXT -1 11 " = 0.6 and " }{TEXT 313 1 "b" }{TEXT -1 7 " = 0.8." }}{PARA 0 "" 0 "" {TEXT -1 31 "We can use the Maple procedure " } {TEXT 0 4 "sign" }{TEXT -1 67 " to determine the sign of the value of \+ the function f at these two " }{TEXT 314 1 "x" }{TEXT -1 8 " values." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sign(y) = `+`*1" "6#/ -%%signG6#%\"yG*&%\"+G\"\"\"F*F*" }{TEXT -1 6 " when " }{TEXT 315 1 "y " }{TEXT -1 26 " is positive or zero, and " }{XPPEDIT 18 0 "sign(y) = \+ -1" "6#/-%%signG6#%\"yG,$\"\"\"!\"\"" }{TEXT -1 7 " when " }{TEXT 316 1 "y" }{TEXT -1 13 " is negative." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "f := x -> x^3+x-1;\na := 0.6;\nb := 0.8;\nsignleft := sign(f(a));\nsignright := sign(f(b));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arro wGF(,(*$)9$\"\"$\"\"\"F1F/F1F1!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG$\"\"'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"bG$\"\")!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)signleftG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*signrightG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Calculate the value of f at the mid-poi nt " }{XPPEDIT 18 0 "m = (a+b)/2;" "6#/%\"mG*&,&%\"aG\"\"\"%\"bGF(F(\" \"#!\"\"" }{TEXT -1 13 " and rename " }{TEXT 317 1 "a" }{TEXT -1 5 " \+ and " }{TEXT 318 1 "b" }{TEXT -1 30 " depending on how the sign of " } {XPPEDIT 18 0 "f(m)" "6#-%\"fG6#%\"mG" }{TEXT -1 27 " compares with th e sign of " }{XPPEDIT 18 0 "f(a)" "6#-%\"fG6#%\"aG" }{TEXT -1 47 " in \+ the way suggested in the previous section. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "f(a)" "6#-% \"fG6#%\"aG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f(m)" "6#-%\"fG6#%\"m G" }{TEXT -1 45 " have opposite signs, which is the case when " } {XPPEDIT 18 0 "f(m)" "6#-%\"fG6#%\"mG" }{TEXT -1 22 " has the same sig n as " }{XPPEDIT 18 0 "f(a)" "6#-%\"fG6#%\"aG" }{TEXT -1 6 ", set " } {TEXT 267 6 "b := m" }{TEXT -1 16 ", otherwise set " }{TEXT 267 6 "a : = m" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "m := evalf((a + b)/2);\nif sign(f(m))=signl eft then a := m else b := m end if;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"mG$\"+++++q!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG$\"+++++q !#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 " Execute these two lines of Maple code repeatedly until you have calcul ated the root to the desired accuracy." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 390 "m := evalf((a + b)/2); \nif sign(f(m))=signleft then a := m else b := m end if;\nm := evalf(( a + b)/2);\nif sign(f(m))=signleft then a := m else b := m end if;\nm \+ := evalf((a + b)/2);\nif sign(f(m))=signleft then a := m else b := m e nd if;\nm := evalf((a + b)/2);\nif sign(f(m))=signleft then a := m els e b := m end if;\nm := evalf((a + b)/2);\nif sign(f(m))=signleft then \+ a := m else b := m end if;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG $\"+++++l!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG$\"+++++l!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG$\"++++]n!#5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"aG$\"++++]n!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"mG$\"++++vo!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG$\"++++vo !#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG$\"+++]7o!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG$\"+++]7o!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG$\"+++vVo!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"bG$\"+++vVo!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 162 "In order to try and make the progress of the bisection p rocess clearer, the following lines of code include extra print statem ents and other output is suppressed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "The lines of code to be repeated are en closed in a " }{TEXT 0 3 "for" }{TEXT -1 79 " loop and you can easily \+ change the number of times that the loop is executed. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 570 "f := x -> x^3+x-1; \na := 0.6;\nb := 0.8;\nsignleft := sign(f(a));\nsignrigh t := sign(f(b));\nprint(`starting interval --- `,[a,b]);\nfor i from 1 to 20 do\n m := evalf((a + b)/2);\n s := sign(f(m));\n if s=sig nleft then\n a := m;\n print(`mid-point`,a,`becomes left end point of new interval`);\n print(` ----> \+ `);\n else\n b := m;\n print(`mid-point`,b,`becomes ri ght end point of new interval`);\n print(` \+ <----`);\n end if;\n print(`step `||i||` --- `,[a,b]);\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#% \"xG6\"6$%)operatorG%&arrowGF(,(*$)9$\"\"$\"\"\"F1F/F1F1!\"\"F(F(F(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG$\"\"'!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"bG$\"\")!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%)signleftG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*signrightG\"\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7starting~interval~---~G7$$\"\" '!\"\"$\"\")F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"+++++ q!#5%Hbecomes~right~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<--- -G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~1~---~G7$$\"\"'!\"\"$\"++ +++q!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"+++++l!#5%Gb ecomes~left~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@~~~---->~~~~~~~~~~~~~~~~~~~~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~2~---~G7$$\"+++++l!#5$\"+++++qF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"++++]n!#5%Gbecomes~left~end~point~of~new ~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@~~~---->~~~~~~~~~~~~~~ ~~~~~~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~3~---~G7$$\"++++] n!#5$\"+++++qF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"++++ vo!#5%Hbecomes~right~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<--- -G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~4~---~G7$$\"++++]n!#5$\"+ +++voF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"+++]7o!#5%Gb ecomes~left~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@~~~---->~~~~~~~~~~~~~~~~~~~~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~5~---~G7$$\"+++]7o!#5$\"++++voF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"+++vVo!#5%Hbecomes~right~end~point~of~ne w~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %,step~6~---~G7$$\"+++]7o!#5$\"+++vVoF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"++]7Go!#5%Hbecomes~right~end~point~of~new~interva lG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~7~ ---~G7$$\"+++]7o!#5$\"++]7GoF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mi d-pointG$\"++DJ?o!#5%Gbecomes~left~end~point~of~new~intervalG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%@~~~---->~~~~~~~~~~~~~~~~~~~~~~~G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~8~---~G7$$\"++DJ?o!#5$\"++]7GoF '" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"+](=U#o!#5%Hbecome s~right~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %W~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~9~---~G7$$\"++DJ?o!#5$\"+](=U#oF'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"+DcEAo!#5%Gbecomes~left ~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@~~~-- -->~~~~~~~~~~~~~~~~~~~~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~ 10~---~G7$$\"+DcEAo!#5$\"+](=U#oF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6% %*mid-pointG$\"+(=UK#o!#5%Gbecomes~left~end~point~of~new~intervalG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%@~~~---->~~~~~~~~~~~~~~~~~~~~~~~G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~11~---~G7$$\"+(=UK#o!#5$\"+](=U #oF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"+p/tBo!#5%Hbeco mes~right~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%W~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~12~---~G7$$\"+(=UK#o!#5$\"+p/tBoF'" } }{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"+Gj[Bo!#5%Hbecomes~rig ht~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~13~---~G7$$\"+(=UK#o!#5$\"+Gj[BoF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"+eUOBo!#5%Hbecomes~right~end~ point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~14~---~G7$$\"+(=UK#o!#5$\"+eUOBoF'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6%%*mid-pointG$\"+BKIBo!#5%Hbecomes~right~end~poin t~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~15~---~G7$$\"+(=UK#o!#5$\"+BKIBoF'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6%%*mid-pointG$\"+1FFBo!#5%Gbecomes~left~end~point ~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@~~~---->~~~~~~~ ~~~~~~~~~~~~~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~16~---~G7$ $\"+1FFBo!#5$\"+BKIBoF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-point G$\"+lzGBo!#5%Hbecomes~right~end~point~of~new~intervalG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%W~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~17~---~G7$$\"+1FFB o!#5$\"+lzGBoF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"+N.G Bo!#5%Hbecomes~right~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<--- -G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~18~---~G7$$\"+1FFBo!#5$\" +N.GBoF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"+@lFBo!#5%G becomes~left~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@~~~---->~~~~~~~~~~~~~~~~~~~~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~19~---~G7$$\"+@lFBo!#5$\"+N.GBoF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"+G%yK#o!#5%Hbecomes~right~end~point~of~n ew~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%W~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%-step~20~---~G7$$\"+@lFBo!#5$\"+G%yK#oF'" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 162 "Looking at the way the digits of the mid-point values are changing towards the end of this output w e can see that the root is approximately 0.682328 to 6 digits." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f(0.682328);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$q%!\"*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 52 "Working towards a procedure for the bisection method" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 53 "Introduction and version 1 . . using exact arithmetic" }}{PARA 15 "" 0 "" {TEXT -1 25 "If a continuous function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 29 " changes sign in an interv al " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 50 ", it must h ave at least one zero in the interval. " }}{PARA 15 "" 0 "" {TEXT -1 25 "In terms of the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG " }{TEXT -1 11 " the points" }{XPPEDIT 18 0 "``(a, f(a))" "6#-%!G6$%\" aG-%\"fG6#F&" }{TEXT -1 4 " and" }{XPPEDIT 18 0 "``(b, f(b))" "6#-%!G6 $%\"bG-%\"fG6#F&" }{TEXT -1 17 " on the graph of " }{XPPEDIT 18 0 "f(x )" "6#-%\"fG6#%\"xG" }{TEXT -1 30 " are on opposite sides of the " } {TEXT 319 1 "x" }{TEXT -1 64 " axis, so if the graph is a continuous c urve, it must cross the " }{TEXT 320 1 "x" }{TEXT -1 57 " axis in orde r to get from the first point to the second." }}{PARA 15 "" 0 "" {TEXT -1 108 "First we illustrate the method with a loop which finds t he square root of 2 as the solution of the equation " }{XPPEDIT 18 0 " f(x) = 0;" "6#/-%\"fG6#%\"xG\"\"!" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "f(x) = x^2-2;" "6#/-%\"fG6#%\"xG,&*$F'\"\"#\"\"\"F*!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "f := x -> x^2-2:\nplot(f(x),x=1..2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"\"\"\"\"! $!\"\"F*7$$\"3hmm;arz@5!#<$!3M9!ptv0$f&*!#=7$$\"3OL$e9ui2/\"F0$!3EV8#* f\"H\"o\"*F37$$\"3smm\"z_\"4i5F0$!3E/kFjeh>()F37$$\"3qmmT&phN3\"F0$!3O (edA_S*e#)F37$$\"3UL$e*=)H\\5\"F0$!3;Yqw^4I\"z(F37$$\"3sm;z/3uC6F0$!3U RfO?7e\\tF37$$\"3-+]7LRDX6F0$!3Kzrh'GMR)oF37$$\"3em;zR'ok;\"F0$!3G\"zD Q74NR'F37$$\"3-+]i5`h(=\"F0$!3oY3pR()p&*eF37$$\"3YLL$3En$47F0$!3y*zWUG 3VP&F37$$\"3cmmT!RE&G7F0$!3Dhz\\!3Hs!\\F37$$\"3)*****\\K]4]7F0$!3=S#)= (4CEP%F37$$\"3))****\\PAvr7F0$!3'*Qu8kCYEQF37$$\"3/++]nHi#H\"F0$!3'y$ \\\"*Q'e7H$F37$$\"3bm;z*ev:J\"F0$!3Q?kiAZp(z#F37$$\"3ELL$347TL\"F0$!3q =J#4H\\9?#F37$$\"3=LLLjM?`8F0$!3SCnAoQS)o\"F37$$\"3#***\\7o7Tv8F0$!3q' zZ^VQC3\"F37$$\"3ALLLQ*o]R\"F0$!3+U1vHdEy`!#>7$$\"3-+]7=lj;9F0$\"3'\\ \\B`[C!fo!#?7$$\"3&***\\PaR&[#F37$$\"3em;zRQb@:F0$\"3_Xw%Q(3E^JF37$$\"3%)**\\(=>Y2a\"F0$ \"3pQJxx#))*QPF37$$\"3imm\"zXu9c\"F0$\"3Cz9EF[-#Q%F37$$\"3'******\\y)) Ge\"F0$\"3#=w(yc!p`0&F37$$\"3!****\\i_QQg\"F0$\"3.=x$G=!)Hs&F37$$\"3#* **\\7y%3Ti\"F0$\"3!R.vr[$GxjF37$$\"3#****\\P![hY;F0$\"3q2lG>JS8rF37$$ \"3ELLLQx$om\"F0$\"3K9=Kf/[$y(F37$$\"3')****\\P+V)o\"F0$\"3h6DK:*fz])F 37$$\"3im;zpe*zq\"F0$\"3!p\"RD7*)\\s\"*F37$$\"3)*****\\#\\'QH&)fOM637F07$$\"3&***\\7`Wl7=F0$ \"3w#*QVmhr&G\"F07$$\"3emmm'*RRL=F0$\"3%HP8qaL8O\"F07$$\"3_mmTvJga=F0$ \"3&Hel#QHbR9F07$$\"3KL$e9tOc(=F0$\"3,+dR[J,=:F07$$\"3'******\\Qk\\*=F 0$\"3?G%=/-!*3f\"F07$$\"3@LL3dg6<>F0$\"3IYnKwRLv;F07$$\"3_mmmw(Gp$>F0$ \"3IH%ze3$p^F0$\"3-$*\\\"3gsR$=F07$$\"3-+](=5s#y>F0$ \"3KIp040c8>F07$$\"\"#F*Fgz-%'COLOURG6&%$RGBG$\"#5F,$F*F*F_[l-%+AXESLA BELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F(Fgz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 34 "The graph shows that the function " }{XPPEDIT 18 0 "f(x ) = x^2-2;" "6#/-%\"fG6#%\"xG,&*$F'\"\"#\"\"\"F*!\"\"" }{TEXT -1 36 " \+ has a zero or root in the interval " }{XPPEDIT 18 0 "[1,2]" "6#7$\"\" \"\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "The sign of the function at the left end of the inte rval " }{XPPEDIT 18 0 "[1,2]" "6#7$\"\"\"\"\"#" }{TEXT -1 6 " is " } {XPPEDIT 18 0 "[``-``];" "6#7#,&%!G\"\"\"F%!\"\"" }{TEXT -1 50 " and t he sign at the right end of the interval is " }{XPPEDIT 18 0 "[``+``]; " "6#7#,&%!G\"\"\"F%F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 55 "The sign of the value of the function at the mid-point " }{XPPEDIT 18 0 "3/2" "6#*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 10 " must be " } {XPPEDIT 18 0 "[``+``];" "6#7#,&%!G\"\"\"F%F&" }{TEXT -1 4 " or " } {XPPEDIT 18 0 "[``-``];" "6#7#,&%!G\"\"\"F%!\"\"" }{TEXT -1 9 ", unles s " }{XPPEDIT 18 0 "f(3/2)" "6#-%\"fG6#*&\"\"$\"\"\"\"\"#!\"\"" } {TEXT -1 38 " is exactly 0 (which is not the case.)" }}{PARA 0 "" 0 " " {TEXT -1 6 "Since " }{XPPEDIT 18 0 "f(3/2) = 1/4" "6#/-%\"fG6#*&\"\" $\"\"\"\"\"#!\"\"*&F)F)\"\"%F+" }{TEXT -1 14 ", its sign is " } {XPPEDIT 18 0 "[``+``];" "6#7#,&%!G\"\"\"F%F&" }{TEXT -1 54 ", and the re is no sign change in the right-hand half " }{XPPEDIT 18 0 "[3/2,2] " "6#7$*&\"\"$\"\"\"\"\"#!\"\"F'" }{TEXT -1 26 " of the original inter val." }}{PARA 0 "" 0 "" {TEXT -1 46 "The sign change occurs in the lef t-hand half " }{XPPEDIT 18 0 "[1,3/2]" "6#7$\"\"\"*&\"\"$F$\"\"#!\"\" " }{TEXT -1 26 " of the original interval." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "In general, if a continuous fun ction changes sign just once in the interval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 49 ", bisecting the interval into the two i ntervals, " }{XPPEDIT 18 0 "[a, (a+b)/2];" "6#7$%\"aG*&,&F$\"\"\"%\"bG F'F'\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "[(a+b)/2, b];" "6# 7$*&,&%\"aG\"\"\"%\"bGF'F'\"\"#!\"\"F(" }{TEXT -1 75 ", means that the sign change must occur in just one of these two intervals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 327 "Once we have p icked the interval in which the sign change occurs, by comparing the s ign of the function at the mid-point with the sign at the end points, \+ we can repeat the bisection process on the new interval. We can then c ontinue bisecting until we have \"pinned down\" the zero or root of th e function to the desired accuracy." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 53 "This is achieved in the following loop b y means of a " }{TEXT 275 16 "renaming process" }{TEXT -1 6 ". Let " } {XPPEDIT 18 0 "m=(a+b)/2" "6#/%\"mG*&,&%\"aG\"\"\"%\"bGF(F(\"\"#!\"\" " }{TEXT -1 34 " be the mid-point of the interval " }{XPPEDIT 18 0 "[a ,b]" "6#7$%\"aG%\"bG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 17 " In the case that " }{TEXT 275 58 "the sign of the function at m is the same as the sign at a" }{TEXT -1 54 " so that the change occurs in th e right-hand interval " }{XPPEDIT 18 0 "[m,b]" "6#7$%\"mG%\"bG" } {TEXT -1 53 ", we rename m as a ,that is, we make the assignment " } {TEXT 267 6 "a := m" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 28 "Similarly, in the case that " }{TEXT 275 58 "the sign of the function at m is the same as the sign at b" } {TEXT -1 53 " so that the change occurs in the left-hand interval " } {XPPEDIT 18 0 "[a,m]" "6#7$%\"aG%\"mG" }{TEXT -1 53 ", we rename m as \+ b, that is, we make the assignment " }{TEXT 267 6 "b := m" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 53 " We then repeat the above process on the new interval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 164 "The loop terminates when the subdi vision has become fine enough to obtain the root to roughly the defaul t Maple precision of 10 digits.\nMaple's calculated value of " } {XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT -1 25 " is given f or comparison." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 712 "f := x -> x^2-2:\n\na := 1:\nb := 2:\neps := 10 ^(-10):\nsignleft := sign(f(a));\nsignright := sign(f(b));\nprint(`sta rting interval --- `,[a,b]);\n\nwhile abs(a-b) > eps do\n midpoint : = (a+b)/2;\n s := sign(f(midpoint));\n if s=signleft then\n a : = midpoint;\n print(`mid-point`,a,`becomes left end point of new \+ interval`);\n print(` ----> `);\n elif s=signrigh t then\n b := midpoint;\n print(`mid-point`,b,`becomes right e nd point of new interval`);\n print(` \+ <----`);\n end if;\n print(`step `||i||` --- `,[a,b]);\nend do :\nmidpoint;\nprint(`evaluating this number as a decimal --> `,evalf(m idpoint));\nprint(`value using \"sqrt\" --> `,evalf(sqrt(2)));" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "Version 2 . . using flo ating point arithmetic" }}{PARA 0 "" 0 "" {TEXT -1 195 "The computatio ns can be performed in approximate arithmetic, but the loop may not te rminate if the tolerance \"eps\" is too small compared to the spacing \+ of distinct floating point numbers between " }{TEXT 321 1 "a" }{TEXT -1 5 " and " }{TEXT 322 1 "b" }{TEXT -1 31 ".\nTry the following with \+ eps = " }{XPPEDIT 18 0 "10^(-10);" "6#)\"#5,$F$!\"\"" }{TEXT -1 3 " , \+ " }{XPPEDIT 18 0 "10^(-9);" "6#)\"#5,$\"\"*!\"\"" }{TEXT -1 4 "and " } {XPPEDIT 18 0 "10^(-8);" "6#)\"#5,$\"\")!\"\"" }{TEXT -1 143 ".\nA loo p counter \"i\" is included with a maximum set arbitrarily at 50, whic h is large enough so that the loop will always eventually terminate." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 628 "f := x -> x^2-2:\n\na := evalf(1):\nb := evalf(2):\neps := 10^( -9):\nsignleft := sign(f(a));\nsignright := sign(f(b));\n\nprint(`star ting interval --- `,[a,b]);\n\nfor i from 1 to 50 while abs(a-b) > eps do\n midpoint := (a+b)/2;\n s := sign(f(midpoint));\n if s=signle ft then\n a := midpoint;\n print(`mid-point`,a,`becomes left end point of new interval`);\n print(` ----> `);\n elif s=s ignright then\n b := midpoint;\n print(`mid-point`,b,`becomes \+ right end point of new interval`);\n print(` \+ <----`);\n end if;\n print(`step `||i||` --- \+ `,[a,b]);\nend do:\nmidpoint;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)si gnleftG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*signrightG\"\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%7starting~interval~---~G7$$\"\"\"\" \"!$\"\"#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"+++++:! \"*%Hbecomes~right~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%M~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<----G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~1~---~G7$$\"\"\"\"\"!$\"+++++:! \"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"++++]7!\"*%Gbeco mes~left~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%,~---->~~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~2~---~G7$$\" ++++]7!\"*$\"+++++:F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$ \"++++v8!\"*%Gbecomes~left~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,~---->~~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %,step~3~---~G7$$\"++++v8!\"*$\"+++++:F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"+++]P9!\"*%Hbecomes~right~end~point~of~new~interv alG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%M~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~4~---~G7$$ \"++++v8!\"*$\"+++]P9F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-point G$\"+++D19!\"*%Gbecomes~left~end~point~of~new~intervalG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%,~---->~~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%,step~5~---~G7$$\"+++D19!\"*$\"+++]P9F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"++](=U\"!\"*%Hbecomes~right~end~point~of ~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%M~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~ 6~---~G7$$\"+++D19!\"*$\"++](=U\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 %%*mid-pointG$\"++D199!\"*%Gbecomes~left~end~point~of~new~intervalG" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%,~---->~~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~7~---~G7$$\"++D199!\"*$\"++](=U\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"+](ozT\"!\"*%Hbecomes~right~end~ point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%M~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%,step~8~---~G7$$\"++D199!\"*$\"+](ozT\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"+Dc,;9!\"*%Hbecomes~right~end~point~of~n ew~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%M~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~9~ ---~G7$$\"++D199!\"*$\"+Dc,;9F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*m id-pointG$\"+i!R]T\"!\"*%Hbecomes~right~end~point~of~new~intervalG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%M~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~10~---~G7$$\"++D19 9!\"*$\"+i!R]T\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"+ \"y]XT\"!\"*%Hbecomes~right~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%M~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<----G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~11~---~G7$$\"++D199!\"*$\"+\"y ]XT\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"+SmI99!\"*%H becomes~right~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%M~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~12~---~G7$$\"++D199!\"*$\"+SmI99F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"+qX=99!\"*%Gbecomes~lef t~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,~--- ->~~~~~G" 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"6#%M~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~21~---~G7$$\"+!=8UT\"!\"*$\"+dO@99F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"+=M@99!\"*%Gbecomes~lef t~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,~--- ->~~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~22~---~G7$$\"+=M@99 !\"*$\"+dO@99F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"+QN@ 99!\"*%Gbecomes~left~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,~---->~~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-ste p~23~---~G7$$\"+QN@99!\"*$\"+dO@99F'" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6%%*mid-pointG$\"+)f8UT\"!\"*%Hbecomes~right~end~point~of~new~interval G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%M~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~24~---~G7$$\" +QN@99!\"*$\"+)f8UT\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-point G$\"+oN@99!\"*%Hbecomes~right~end~point~of~new~intervalG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%M~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<----G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~25~---~G7$$\"+QN@99!\"*$\"+ oN@99F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"+`N@99!\"*%G becomes~left~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,~---->~~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~26~---~ G7$$\"+`N@99!\"*$\"+oN@99F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-p ointG$\"+gN@99!\"*%Gbecomes~left~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,~---->~~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~27~---~G7$$\"+gN@99!\"*$\"+oN@99F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"+kN@99!\"*%Hbecomes~right~end~point~of~n ew~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%M~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~28 ~---~G7$$\"+gN@99!\"*$\"+kN@99F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%* mid-pointG$\"+iN@99!\"*%Gbecomes~left~end~point~of~new~intervalG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%,~---->~~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~29~---~G7$$\"+iN@99!\"*$\"+kN@99F'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6%%*mid-pointG$\"+jN@99!\"*%Hbecomes~right~end~poi nt~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%M~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%- step~30~---~G7$$\"+iN@99!\"*$\"+jN@99F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+jN@99!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "Version 3 . . allowing a change of precision" }}{PARA 0 "" 0 "" {TEXT -1 86 "The following loop can be used with different settings fo r the global variable Digits." }}{PARA 0 "" 0 "" {TEXT -1 66 "The valu e of \"eps\" and the maximum number of iterations allowed -\"" }{TEXT 339 5 "maxit" }{TEXT -1 29 "\" - are chosen automatically." }}{PARA 0 "" 0 "" {TEXT 266 25 "Try changing the value of" }{TEXT 257 1 " " } {TEXT 0 6 "Digits" }{TEXT 266 18 " in the first line" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 742 "Digits := 8;\nf := x -> x^2-2:\na := evalf(1):\nb := evalf(2): \neps := 10^(-Digits+1):\nmaxit := Digits*5:\n\nsignleft := sign(f(a)) ;\nsignright := sign(f(b));\n\nprint(`starting interval --- `,[a,b]); \n\nfor i from 1 to maxit while abs(a-b)>eps do\n midpoint := (a+b)/ 2;\n s := sign(f(midpoint));\n if s=signleft then\n a := midpoi nt;\n print(`mid-point`,a,`becomes left end point of new interval `);\n print(` ----> `);\n elif s=signright then\n b := m idpoint;\n print(`mid-point`,b,`becomes right end point of new in terval`);\n print(` <--- -`);\n end if;\n print(`step `||i||` --- `,[a,b]);\nend do:\nmidpo int;\nprint(``);print(`value using \"sqrt\" --> `,evalf(sqrt(2)));\nDi gits := 10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"\")" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%)signleftG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*signrightG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %7starting~interval~---~G7$$\"\"\"\"\"!$\"\"#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\")+++:!\"(%Hbecomes~right~end~point~of~ne w~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%M~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~1~- --~G7$$\"\"\"\"\"!$\")+++:!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mi d-pointG$\")++]7!\"(%Gbecomes~left~end~point~of~new~intervalG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%-~~---->~~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~2~---~G7$$\")++]7!\"($\")+++:F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\")++v8!\"(%Gbecomes~left~end~point~o f~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-~~---->~~~~~G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~3~---~G7$$\")++v8!\"($\")+++:F' " }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\")+]P9!\"(%Hbecomes~ right~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%M ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~4~---~G7$$\")++v8!\"($\")+]P9F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\")+D19!\"(%Gbecomes~left~end~point~o f~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-~~---->~~~~~G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~5~---~G7$$\")+D19!\"($\")+]P9F' " }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\")](=U\"!\"(%Hbecome s~right~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %M~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~6~---~G7$$\")+D19!\"($\")](=U\"F'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6%%*mid-pointG$\")D199!\"(%Gbecomes~left~end~point ~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-~~---->~~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~7~---~G7$$\")D199!\"($\")](=U \"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"))ozT\"!\"(%Hbe comes~right~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%M~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~8~---~G7$$\")D199!\"($\"))ozT\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\")c,;9!\"(%Hbecomes~right~end~ point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%M~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%,step~9~---~G7$$\")D199!\"($\")c,;9F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\")!R]T\"!\"(%Hbecomes~right~end~point~of~new~inter valG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%M~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~10~---~G7$ $\")D199!\"($\")!R]T\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-poin tG$\")3b99!\"(%Hbecomes~right~end~point~of~new~intervalG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%M~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<----G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~11~---~G7$$\")D199!\"($\")3 b99F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\")mI99!\"(%Hbec omes~right~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%M~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<----G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%-step~12~---~G7$$\")D199!\"($\")mI99F'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6%%*mid-pointG$\")Y=99!\"(%Gbecomes~left~end~point ~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-~~---->~~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~13~---~G7$$\")Y=99!\"($\")mI9 9F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\")cC99!\"(%Hbecom es~right~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%M~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~14~---~G7$$\")Y=99!\"($\")cC99F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\")^@99!\"(%Hbecomes~right~end~point~ of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%M~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-ste p~15~---~G7$$\")Y=99!\"($\")^@99F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6% %*mid-pointG$\"))*>99!\"(%Gbecomes~left~end~point~of~new~intervalG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%-~~---->~~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~16~---~G7$$\"))*>99!\"($\")^@99F'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6%%*mid-pointG$\")u?99!\"(%Gbecomes~left~end~point ~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-~~---->~~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~17~---~G7$$\")u?99!\"($\")^@9 9F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\")7@99!\"(%Gbecom es~left~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %-~~---->~~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~18~---~G7$$ \")7@99!\"($\")^@99F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$ \")K@99!\"(%Gbecomes~left~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-~~---->~~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% -step~19~---~G7$$\")K@99!\"($\")^@99F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\")U@99!\"(%Hbecomes~right~end~point~of~new~interva lG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%M~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~20~---~G7$$ \")K@99!\"($\")U@99F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$ \")P@99!\"(%Hbecomes~right~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%M~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<----G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~21~---~G7$$\")K@99!\"($\")P@99 F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\")M@99!\"(%Gbecome s~left~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% -~~---->~~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~22~---~G7$$\" )M@99!\"($\")P@99F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\" )O@99!\"(%Hbecomes~right~end~point~of~new~intervalG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%M~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<----G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~23~---~G7$$\")M@99!\"($\")O@99F '" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\")N@99!\"(%Gbecomes ~left~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%- ~~---->~~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~24~---~G7$$\") N@99!\"($\")O@99F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\")N@99!\"(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%8value~using~\"sqrt\"~-->~G$\")O@99!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 27 "Version 4 . . final touches" }}{PARA 0 "" 0 "" {TEXT -1 102 "If we want to ensure the accuracy of the result, we can perfor m the calculations within the loop to a " }{TEXT 275 16 "higher precis ion" }{TEXT -1 27 " than the current value of " }{TEXT 0 6 "Digits" } {TEXT -1 22 " and round the answer." }}{PARA 0 "" 0 "" {TEXT -1 101 "I n order that the routine can cope with other functions, the error cont rol can be framed in terms of " }{TEXT 275 14 "relative error" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 46 "These ideas are incorporated i n the procedure " }{TEXT 0 6 "bisect" }{TEXT -1 27 " given in the next section." }}{PARA 0 "" 0 "" {TEXT 0 6 "bisect" }{TEXT -1 99 " also in cludes a number \"book keeping\" commands which provide some flexibilt y in input parameters. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 970 "Digits := 12;\nf := x -> x^2-2:\n \n# Take eps less than the machine epsilon 10^(1-Digits).\neps := Floa t(1,-Digits):\nmaxit := Digits*5:\n\n# Increase precision for the comp utation.\nsaveDigits := Digits:\nDigits := Digits + min(trunc(Digits/3 ),5):\n\nmaxit := Digits*5:\n\nsignleft := sign(f(a));\nsignright := s ign(f(b));\n\nprint(`starting interval --- `,[a,b]);\n\nfor i from 1 t o maxit while abs(a-b) > eps*abs(a) do\n midpoint := (a+b)/2;\n s : = sign(f(midpoint));\n if s=signleft then\n a := midpoint;\n \+ print(`mid-point`,a,`becomes left end point of new interval`);\n \+ print(` ----> `);\n elif s=signright then\n b := midpoint; \n print(`mid-point`,b,`becomes right end point of new interval`) ;\n print(` <----`);\n \+ end if;\n print(`step `||i||` --- `,[a,b]);\n\n print(abs(a-b)>ep s*abs(a));\nend do:\nDigits := saveDigits;\nevalf(midpoint);\nprint(`` );print(`value using \"sqrt\" --> `,evalf(sqrt(2)));\nDigits := 10;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)signleftG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% *signrightG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7starting~interv al~---~G7$$\"-EiN@99!#6$\"-QiN@99F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 %%*mid-pointG$\"1++KiN@99!#:%Gbecomes~left~end~point~of~new~intervalG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-~~---->~~~~~G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%,step~1~---~G7$$\"1++KiN@99!#:$\"-QiN@99!#6" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#2$\"1++KiN@99!#F$\"&++'!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"1++NiN@99!#:%Gbecomes~left~en d~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-~~---->~ ~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~2~---~G7$$\"1++NiN@99! #:$\"-QiN@99!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#2$\"1++NiN@99!#F$\" &++$!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"1+]OiN@99!#: %Gbecomes~left~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-~~---->~~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,st ep~3~---~G7$$\"1+]OiN@99!#:$\"-QiN@99!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#2$\"1+]OiN@99!#F$\"&+]\"!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%% *mid-pointG$\"1+DPiN@99!#:%Gbecomes~left~end~point~of~new~intervalG" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%-~~---->~~~~~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~4~---~G7$$\"1+DPiN@99!#:$\"-QiN@99!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#2$\"1+DPiN@99!#F$\"%+v!#:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6%%*mid-pointG$\"1]iPiN@99!#:%Hbecomes~right~end~point~ of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%M~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,ste p~5~---~G7$$\"1+DPiN@99!#:$\"1]iPiN@99F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#2$\"1+DPiN@99!#F$\"%]P!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*m id-pointG$\"1vVPiN@99!#:%Hbecomes~right~end~point~of~new~intervalG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%M~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~<----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~6~---~G7$$\"1+DPiN @99!#:$\"1vVPiN@99F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#2$\"1+DPiN@99! #F$\"%v=!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"1QMPiN@9 9!#:%Hbecomes~right~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%M~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<----G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~7~---~G7$$\"1+DPiN@99!#:$\"1QMP iN@99F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#2$\"1+DPiN@99!#F$\"$Q*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"-PiN@99!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%8value~using~\"sqrt\"~-->~G$\"-PiN@9 9!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#5" }}}{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 47 "A procedure implementing the bisection method: " } {TEXT 0 6 "bisect" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "bis ect: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 260 3 " " }{TEXT -1 45 "bisect( eqn, rng ) or bisection( eqn, rng ) " }{TEXT 261 1 "\n" }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 11 "P arameters:" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT 23 10 " eqn - " }{TEXT -1 63 " an equation or expression inv olving a single variable, say x," }}{PARA 0 "" 0 "" {TEXT -1 22 " \+ " }{TEXT 275 2 "OR" }{TEXT -1 73 " a function of the \+ form x -> f(x), where f(x) evaluates to a real number," }}{PARA 0 "" 0 "" {TEXT -1 22 " " }{TEXT 275 2 "OR" }{TEXT -1 71 " a numerical procedure which evaluates to a real floating point nu mber." }}{PARA 0 "" 0 "" {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 8 "rng - " }{TEXT 264 89 "a range x=a..b (or simply a..b when the1st argument is a procedure) where a and b are two" }} {PARA 0 "" 0 "" {TEXT 265 161 " distinct initial a pproximations for the root, and x is the variable appearing in the 1st argument.\n The numbers a and b " }{TEXT 275 21 " must bracket the root" }{TEXT 263 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The procedure " }{TEXT 0 6 "bisect" }{TEXT -1 30 " attempts to locate a root of " }{XPPEDIT 18 0 "f(x) = 0;" "6#/-%\"fG6#%\"xG\"\"!" }{TEXT -1 48 " in the interva l (a,b) by the bisection method.\n" }}{PARA 0 "" 0 "" {TEXT 262 8 "Opt ions:" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 148 "maxiterations= n or maxiter=n\nThis option can be used to override the default value \+ of Digits*5 for the maximum number of iterations to be performed." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "info=tru e/false/0/1/2/3\n\"info=0\" is the same as \"info=false\" and \"info=1 \" is the same as \"info=true\"." }}{PARA 0 "" 0 "" {TEXT -1 119 "This option allows the progress of the procedure to be monitored by printi ng the result of each bisection as it occurs." }}{PARA 0 "" 0 "" {TEXT -1 80 "Three formats given by \"info=1\", \"info=2\" and \"info =3\" are available for this. " }}{PARA 0 "" 0 "" {TEXT -1 151 "\"info= 3\" is a modified form of \"info=1\" which can be used for high settin gs of \"Digits\" if the printout for a single step extends beyond a si ngle line." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 275 15 "How to activate" }{TEXT -1 1 " :" }}{PARA 0 "" 0 "" {TEXT -1 154 "To make the procedure active open t he subsection, place the cursor anywhere after the prompt [ > and pre ss [Enter].\nYou can then close up the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "bisect: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12052 "# to allow for \+ different names for the procedure\nbisection := proc() bisect(args[1.. nargs]) end:\n\nbisect := proc(ff,rng)\n local Options,a,b,c,fa,fb,m id,fmid,eps,saveDigits,i,maxit,\n prntflg,x,f,fn,rg,lmr,sf,procty pe,vars,signleft,signright,\n s,sgn,new_fmid,prev_fmid,h,k,tiny,d iverg,triedzero,f0,\n digtinc,startDigits,extraDigits,val,len,pri ntg;\n\n if nargs<2 then\n error \"at least 2 arguments are req uired; the basic syntax is: 'bisect(f(x),x=a..b)'.\"\n end if;\n\n \+ sgn := proc(u)\n if u > 0 then 1 elif u < 0 then -1 else 0 end i f;\n end proc: # of sgn \n \n if type(ff,procedure) then\n \+ if nops([op(1,eval(ff))])<>1 then\n error \"the 1st argum ent, %1, is invalid .. it should be a procedure with a single argument \",ff;\n end if;\n proctype := true;\n if type(rng,real cons..realcons) then\n rg := rng\n else\n error \+ \"the 2nd argument, %1, is invalid .. when the 1st argument is a proce dure, the 2nd argument should be a range of real constants\",rng;\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 := indet s(f,name) minus indets(f,realcons);\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 equat ion which depends only on a single variable\",ff;\n end if;\n \+ end if;\n if type(rng,name=realcons..realcons) then\n \+ proctype := false;\n x := op(1,rng);\n if not member (x,vars) then\n error \"the 1st argument, %1, is invalid .. it should be an expression or an equation which depends only on the v ariable %2\",ff,x;\n end if;\n rg := op(2,rng);\n \+ else\n error \"the 2nd argument, %1, is invalid .. it should \+ have the form 'x=a..b', to provide two real number starting values\",r ng;\n end if;\n else\n error \"the 1st argument, %1, is in valid .. it should be an algebraic expression in a single variable, an equation in a single variable, or a procedure with a single real argu ment\",ff;\n end if;\n \n # Get the options \"maxiterations\" an d \"info\".\n # Set the default values to start with.\n maxit := D igits*5;\n prntflg := 0;\n if nargs>2 then\n Options:=[args[3 ..nargs]];\n if not type(Options,list(equation)) then\n e rror \"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 \"\\\"maxiterat ions\\\" must be a positive integer\"\n end if;\n elif ha soption(Options,'maxiter','maxit','Options') then\n if not typ e(maxit,posint) then\n error \"\\\"maxiter\\\" must be a po sitive integer\"\n end if;\n end if;\n if hasoption( Options,'info','prntflg','Options') then\n if not member(prntf lg,\{true,false,0,1,2,3\}) then\n error \"\\\"info\\\" must be false <-> 0, true <-> 1, 2 or 3\"\n end if;\n if p rntflg=false then prntflg := 0\n elif prntflg=true then prntfl g := 1 end if; \n end if;\n if nops(Options)>0 then\n \+ if type(op(1,Options),'name'='range') \n \+ or type(op(1,Options),'range') then\n error \"%1 is not \+ a valid argument for %2 .. only one range is allowed as an argument\", op(1,Options),procname;\n else\n error \"%1 is not \+ a valid option for %2 .. the recognised options are \\\"maxiterations \\\",(or \\\"maxiter\\\") and \\\"info\\\"\",op(1,Options),procname;\n end if;\n end if;\n end if;\n\n # Increase precision for the computation\n saveDigits := Digits;\n extraDigits := iquo (Digits,3);\n Digits := Digits + min(iquo(iquo(Digits,5)+1,2)+3,8); \n startDigits := Digits;\n\n if proctype then\n fn := ff;\n \+ else\n # Evaluate any real constants in f\n fn := unapply( evalf(f),x);\n end if;\n\n a := evalf(min(op(rg)));\n b := evalf (max(op(rg)));\n if a=b then\n error \"distinct starting values are required\"\n end if;\n\n fa := traperror(evalf(fn(a)));\n i f fa=lasterror or not type(fa,numeric) then\n error \"evaluation \+ failed at %1\",evalf[saveDigits](a);\n end if;\n signleft := sgn(f a);\n fb := traperror(evalf(fn(b)));\n if fb=lasterror or not type (fb,numeric) then\n error \"evaluation failed at %1\",evalf[saveD igits](b);\n end if;\n signright := sgn(fb);\n\n # Indicate an e rror if the function has the same sign at a and b.\n if signleft*sig nright=1 then\n error \"the same sign occurs at both ends %1, and % 2 of the interval; try a different starting interval\",a,b;\n elif s ignleft*signright<>-1 then\n error \"could not determine signs at \+ end points %1, and %2 of the interval\",a,b;\n end if;\n\n eps := \+ Float(1,-saveDigits-min(iquo(Digits,10),2));\n tiny := abs(b-a)*Floa t(5,-saveDigits);\n triedzero := false;\n diverg := 0;\n digtinc := 0;\n\n # procedure to print a float with d+7 characters\n prin tg := proc(x::float,d::posint)\n local lg,wdth,dcpl,fmt,f,e;\n \+ if x<>0 then lg := ilog10(x) else lg := 0 end if;\n wdth := co nvert(d+7,string);\n if lg>-7 and lg12 and abs(mid)0 then break end if ;\n end do; \n end if;\n if s=0 then\n if no t proctype then\n val := `convert/real_rat`(mid,Digits);\n \+ if eval(f,x=val)=0 then return evalf[saveDigits](mid) end i f;\n else\n WARNING(\"the accuracy of the result is uncertain.\");\n return evalf[saveDigits](mid);\n \+ end if; \n end if;\n # check for divergence of the func tion values\n h := abs(new_fmid-fmid);\n k := abs(fmid-prev_ fmid); \n if i>6 then\n if h>k then diverg := diverg +1 end if;\n if diverg>5 then\n error \"there may b e a discontinuity near %1; try a different (narrower) starting interva l\",evalf[iquo(saveDigits,2)](mid);\n end if;\n end if;\n if s=signleft then\n a := mid;\n if irem(prntflg ,2)=1 then\n printf(\" step %d\",i);\n len := 5 -length(i);\n printf(cat(\" \"$len));\n if signl eft=1 then\n printf(\"[+]-> \");\n printg( a,startDigits);\n if prntflg=1 then\n p rintf(\" \");\n else\n printf(\"\\n\" );printf(cat(\" \"$30));\n end if; \n prin tg(b,startDigits); \n printf(\" [-]\\n\"); \n else\n printf(\"[-]-> \");\n \+ printg(a,startDigits);\n if prntflg=1 then\n \+ printf(\" \");\n else\n printf (\"\\n\");printf(cat(\" \"$30));\n end if; \n \+ printg(b,startDigits); \n printf(\" [+ ]\\n\");\n end if;\n if prntflg=3 then printf(\" \\n\") end if;\n elif prntflg=2 then\n print(`mid-p oint`,evalf[startDigits](a),`becomes left end point of new interval`); \n if signleft=1 then\n print(` \+ [+]---->[+] [-]`);\n else\n \+ print(` [-]---->[-] [+]`);\n \+ end if;\n end if;\n elif s=signright then\n b := \+ mid;\n if irem(prntflg,2)=1 then\n printf(\" step \+ %d\",i);\n len := 5-length(i);\n printf(cat(\" \+ \"$len)); \n if signleft=1 then\n printf(\"[+ ] \");\n printg(a,startDigits);\n if prn tflg=1 then\n printf(\" \");\n else\n printf(\"\\n\");printf(cat(\" \"$30));\n \+ end if; \n printg(b,startDigits); \n \+ printf(\" <-[-]\\n\");\n else\n pri ntf(\"[-] \");\n printg(a,startDigits);\n \+ if prntflg=1 then\n printf(\" \");\n \+ else\n printf(\"\\n\");printf(cat(\" \"$30));\n \+ end if;\n printg(b,startDigits); \n printf(\" <-[+]\\n\");\n end if;\n \+ if prntflg=3 then printf(\"\\n\") end if;\n elif prntflg=2 then\n print(`mid-point`,evalf[startDigits](b),`becomes ri ght end point of new interval`); \n if signleft=1 then\n \+ print(` [+] [-]<----[-]`); \n else\n print(` [-] \+ [+]<----[+]`);\n end if;\n end if;\n e lse \n error \"could not determine sign at %1\",mid;\n en d if;\n if prntflg=2 then\n print(`step `||i||` --- `,[ev alf[startDigits](a),evalf[startDigits](b)]);\n end if;\n if \+ abs(b-a)<=eps*abs(a) then\n ## perform one last bisection\n \+ mid := a+(b-a)/2;\n return evalf[saveDigits](mid);\n \+ end if;\n end do;\n if member(prntflg,\{0,1,3\}) then\n pri ntf(\" last iteration gives \");\n printg(evalf[saveDigits](mid) ,saveDigits);\n else\n print(`last iteration gives `,evalf[save Digits](mid));\n end if;\n error \"reached max, %1, iterations wit hout 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 39 "Examples are given in the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 6 "bisect " }{TEXT -1 10 ": examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }{TEXT 333 29 " \+ .. solving a cubic equation " }}{PARA 0 "" 0 "" {TEXT 268 8 "Question " }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 42 "Find all the roots o f the cubic equation " }{XPPEDIT 18 0 "x^3-3*x+1 = 0;" "6#/,(*$%\"xG \"\"$\"\"\"*&F'F(F&F(!\"\"F(F(\"\"!" }{TEXT -1 21 " correct to10 digit s." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 270 8 "S olution" }{TEXT 269 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 40 "First we plo t the graph of the equation " }{XPPEDIT 18 0 "y = x^3-3*x+1;" "6#/%\"y G,(*$%\"xG\"\"$\"\"\"*&F(F)F'F)!\"\"F)F)" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot( x^3-3*x+1,x=-2..2);" }}{PARA 13 "" 1 "" {GLPLOT2D 285 185 185 {PLOTDATA 2 "6%-%'CURVESG6$7ao7$$!\"#\"\"!$!\"\"F*7$$!1nmm\"p0k&>!#:$! 1k-8z:r*='!#;7$$!1LLL$Q6G\">F0$!13sYSy]-EF37$$!1++v3-)[(=F0$\"1#)HJ(Q4 2T$!#<7$$!1nm;M!\\p$=F0$\"1#**)[$)zzAJF37$$!1++Dh9H%z\"F0$\"1\\N(Q%f&= 1'F37$$!1LLL))Qj^VDPf\"F07$$!1nm;C2G!e\" F0$\"1j\"3&*RFWz\"F07$$!1++]_(e1a\"F0$\"1&pb+OB]'>F07$$!1LL$3yO5]\"F0$ \"1;`kLs5@@F07$$!1++]nU)*=9F0$\"1W;\\xZ!)*R#F07$$!1LL$3WDTL\"F0$\"1M@c o)yxi#F07$$!1++]d(Q&\\7F0$\"1jW-\\Dl(z#F07$$!1nmmc4`i6F0$\"14W$ffdk\"H F07$$!1++](p7U7\"F0$\"1a+>lrz^HF07$$!1LLLQW*e3\"F0$\"1x'=(>FBxHF07$$!1 nm;arvU5F0$\"1/_x4tV%*HF07$$!1,+++()>'***F3$\"1y%elc*****HF07$$!1,+++Y 0j&*F3$\"1^()p$ybV*HF07$$!1++++0\"*H\"*F3$\"1]#3T.Zz(HF07$$!1++++f\\7( )F3$\"13ce[US_HF07$$!1++++83&H)F3$\"1?rUNLv`%HrBF07$$!1pmmmCC(>%F3$\"1%puGdI_=#F07$$ !1*****\\FRXL$F3$\"1+F'G`%Gj>F07$$!1+++D=/8DF3$\"1\")RRN$\"1opICX(*f7F07$$!1qLLL$eV( >!#=$\"1sPInI#f+\"F07$$\"1Mmm;f`@')F>$\"1&3^')oZ*>uF37$$\"1)****\\nZ)H ;F3$\"17$$\"1'*****\\5a`TF3$!1ctVm#eSu\"F37$$\"1(****\\7RV'\\F3 $!1jx/IFdpOF37$$\"1'*****\\@fkeF3$!1^['RSTnd&F37$$\"1JLLL&4Nn'F3$!10ek YCV[qF37$$\"1*******\\,s`(F3$!1P()yk]wH$)F37$$\"1lmm\"zM)>$)F3$!17vqrP a+#*F37$$\"1KL$eCZwu)F3$!1.1!o^D\"\\&*F37$$\"1*******pfa<*F3$!1)p#3jdk ,)*F37$$\"1km;zy*zd*F3$!1,V\"))yDt%**F37$$\"1HLLeg`!)**F3$!1:1B?k))*** *F37$$\"1mm;W/8S5F0$!1q&o<9S5&**F37$$\"1++]#G2A3\"F0$!1^vI#G.J'3#)>a6F37$$\"1nm\"HB-7a\"F0$ \"1&)yZKFw@PF>7$$\"1+++Sv&)z:F0$\"1ZqiZcsO?F37$$\"1nm;%)3;C;F0$\"1QYHP .*)=TF37$$\"1LLLGUYo;F0$\"1LCLryK#R'F37$$\"1++]n'*33YE&4\"F07$$\"1LLe*3k**y\"F0$\"1bnZc:5l8F07$$ \"1++]sI@K=F0$\"1#>Urv3Tl\"F07$$\"1+++S2ls=F0$\"17P(\\L)4\\>F07$$\"1++ ]2%)38>F0$\"1j^R>4YiAF07$$\"1++v.Uac>F0$\"1g+#*3i9?EF07$$\"\"#F*$\"\"$ F*-%'COLOURG6&%$RGBG$\"#5F,F*F*-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F( F[al%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 196 "It is clear from this graph that the equation has a solution b etween 1 and 2. In fact by clicking on the graph and using the cursor, it is possible to make a guess of about 1.53 for this solution." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The proce dure " }{TEXT 0 6 "bisect" }{TEXT -1 64 " can be used to determine thi s solution to any desired accuracy." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 141 "Any two starting values which \"straddl e\" or \"bracket\" this solution only will do, but we may as well take numbers which are reasonably close." }}{PARA 0 "" 0 "" {TEXT -1 28 "( Incorporating the option \"" }{TEXT 339 9 "info=true" }{TEXT -1 9 "\" causes " }{TEXT 0 6 "bisect" }{TEXT -1 35 " to show all the bisection steps.) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "x1 := bisect(x^3-3*x+1,x=1.5..1.6,info=true);" }} {PARA 6 "" 1 "" {TEXT -1 71 " start [-] 1.5000000000000 \+ 1.6000000000000 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 1 [-] 1.5000000000000 1.5500000000000 <-[+]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 2 [-]-> 1.5250000000000 \+ 1.5500000000000 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 3 [-] 1.5250000000000 1.5375000000000 <-[+]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 4 [-]-> 1.5312500000000 \+ 1.5375000000000 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 5 [-] 1.5312500000000 1.5343750000000 <-[+]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 6 [-] 1.5312500000000 \+ 1.5328125000000 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 7 [-]-> 1.5320312500000 1.5328125000000 [+]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 8 [-] 1.5320312500000 \+ 1.5324218750000 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 9 [-] 1.5320312500000 1.5322265625000 <-[+]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 10 [-] 1.5320312500000 \+ 1.5321289062500 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 11 [-]-> 1.5320800781250 1.5321289062500 [+]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 12 [-] 1.5320800781250 \+ 1.5321044921875 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 13 [-] 1.5320800781250 1.5320922851562 <-[+]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 14 [-]-> 1.5320861816406 \+ 1.5320922851562 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 15 [-] 1.5320861816406 1.5320892333984 <-[+]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 16 [-]-> 1.5320877075195 \+ 1.5320892333984 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 17 [-]-> 1.5320884704590 1.5320892333984 [+]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 18 [-]-> 1.5320888519287 \+ 1.5320892333984 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 19 [-] 1.5320888519287 1.5320890426636 <-[+]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 20 [-] 1.5320888519287 \+ 1.5320889472962 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 21 [-] 1.5320888519287 1.5320888996124 <-[+]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 22 [-]-> 1.5320888757706 \+ 1.5320888996124 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 23 [-] 1.5320888757706 1.5320888876915 <-[+]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 24 [-]-> 1.5320888817310 \+ 1.5320888876915 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 25 [-]-> 1.5320888847112 1.5320888876915 [+]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 26 [-]-> 1.5320888862014 \+ 1.5320888876915 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 27 [-] 1.5320888862014 1.5320888869464 <-[+]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 28 [-] 1.5320888862014 \+ 1.5320888865739 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 29 [-] 1.5320888862014 1.5320888863876 <-[+]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 30 [-] 1.5320888862014 \+ 1.5320888862945 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 31 [-] 1.5320888862014 1.5320888862480 <-[+]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 32 [-]-> 1.5320888862247 \+ 1.5320888862480 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 33 [-]-> 1.5320888862364 1.5320888862480 [+]" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"+')))3K:!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "We can check this so lution by using " }{TEXT 0 4 "subs" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "subs(x=x1,x^ 3-3*x+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!\"\"!\"*" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 144 "The result is not exactly zero, but you can check that changing the last digit by 1 in \+ each direction produces a result with a larger magnitude." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "subs( x=1.532088885,x^3-3*x+1);\nsubs(x=1.532088886,x^3-3*x+1);\nsubs(x=1.53 2088887,x^3-3*x+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!\"&!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$!\"\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"$!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "Thus the value 1.532088886 is the best we can do \+ with 10 digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "There is 2nd solution between 0.3 and 0.4 and a 3rd betw een -2 and 1.8." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 71 "x2 := bisect(x^3-3*x+1,x=0.3..0.4);\nx3 := bis ect(x^3-3*x+1,x=-2..-1.8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x2G$ \"+`N'HZ$!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x3G$!+U_Qz=!\"*" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 275 4 "Note" } {TEXT -1 68 ": You can work with a function rather than the equation i f you wish." }}{PARA 0 "" 0 "" {TEXT -1 141 "In the following sequence of commands the 3 solutions are put into a list. Then a second list o f points is constructed with the solutions as " }{TEXT 278 1 "x" } {TEXT -1 26 " coordinates and with the " }{TEXT 279 1 "y" }{TEXT -1 156 " coordinates all zero. These points are plotted to indicate how t he solutions correspond to the intersection points of the graph of the function f with the " }{TEXT 277 1 "x" }{TEXT -1 6 " axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 307 "f := x -> x^3-3*x+1;\np1 := plot(f,-2..2):\nsols:=[bisect(f,-2..-1.8),b isect(f,0.3..0.4),bisect(f,1.5..1.6)];\npts := map(_X->[_X,0],sols);\n p1 := plot(f,-2..2,color=red):\np2 := plot([pts$3],style=[point$3],col or=[black$3],\n symbol=[circle,diamond,c ross]):\nplots[display]([p1,p2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(*$)9$\"\"$\"\"\"F1*&F0F1F/F1 !\"\"F1F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solsG7%$!+U_Qz=! \"*$\"+`N'HZ$!#5$\"+')))3K:F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pt sG7%7$$!+U_Qz=!\"*\"\"!7$$\"+`N'HZ$!#5F*7$$\"+')))3K:F)F*" }}{PARA 13 "" 1 "" {GLPLOT2D 309 214 214 {PLOTDATA 2 "6(-%'CURVESG6$7ao7$$!\"#\" \"!$!\"\"F*7$$!3ymmm\"p0k&>!#<$!39o-8z:r*='!#=7$$!3MLLL$Q6G\">F0$!3t4s YSy]-EF37$$!31++v3-)[(=F0$\"3vEIJ(Q42T$!#>7$$!3bmm;M!\\p$=F0$\"3&=**)[ $)zzAJF37$$!3#)***\\7Y\"H%z\"F0$\"33`N(Q%f&=1'F37$$!3MLLL))Qj^VDPf\"F07$$!3wmm;C2G!e\"F0$\"3mj\"3&*RFW z\"F07$$!3<++]_(e1a\"F0$\"37%pb+OB]'>F07$$!3OLL$3yO5]\"F0$\"3%eJXOB267 #F07$$!3&*****\\nU)*=9F0$\"3)[k\"\\xZ!)*R#F07$$!3SLL$3WDTL\"F0$\"3RM@c o)yxi#F07$$!35++]d(Q&\\7F0$\"3eiW-\\Dl(z#F07$$!3gmmmc4`i6F0$\"3z3W$ffd k\"HF07$$!3(*****\\(p7U7\"F0$\"3k`+>lrz^HF07$$!3KLLLQW*e3\"F0$\"3Ux'=( >FBxHF07$$!3qmm;arvU5F0$\"3s._x4tV%*HF07$$!3w++++()>'***F3$\"3wx%elc** ***HF07$$!3_++++Y0j&*F3$\"3)3v)p$ybV*HF07$$!3E++++0\"*H\"*F3$\"3!*\\#3 T.Zz(HF07$$!3=++++f\\7()F3$\"3o2ce[US_HF07$$!35++++83&H)F3$\"37?rUNLv< HF07$$!3\\LLL3k(p`(F3$\"3@\"=0)Rt%H$GF07$$!3Anmmmj^NmF3$\"3Ea'R/h#\\)p #F07$$!3)zmmmYh=(eF3$\"3U;:'[&R5fDF07$$!3+,++v#\\N)\\F3$\"3q/<$>`%HrBF 07$$!3commmCC(>%F3$\"3o$puGdI_=#F07$$!39*****\\FRXL$F3$\"3W+F'G`%Gj>F0 7$$!3t*****\\#=/8DF3$\"3A\")RRN$\"3+opICX(*f7F07$$!3IqLLL$eV(>!#?$\"3,sPInI#f+\" F07$$\"3)Qjmm\"f`@')F>$\"37&3^')oZ*>uF37$$\"3%z****\\nZ)H;F3$\"3O7$$\"3'e*****\\5a`TF3$!31ctVm#eSu\"F37$$\"3'o****\\7RV'\\F3$!3? jx/IFdpOF37$$\"3Y'*****\\@fkeF3$!33^['RSTnd&F37$$\"3_ILLL&4Nn'F3$!3/2e kYCV[qF37$$\"3A*******\\,s`(F3$!3oO()yk]wH$)F37$$\"3%[mm;zM)>$)F3$!3D7 vqrPa+#*F37$$\"33KL$eCZwu)F3$!3%Gg+o^D\"\\&*F37$$\"3M*******pfa<*F3$!3 3)p#3jdk,)*F37$$\"3Ckm;zy*zd*F3$!3_)H9))yDt%**F37$$\"39HLLeg`!)**F3$!3 ![hI-U'))****F37$$\"3Lmm;W/8S5F0$!3)yco<9S5&**F37$$\"3w****\\#G2A3\"F0 $!3;^vI#G.J'3#)>a6F37$$\"3emm\"HB-7a\"F0$\"3e%)yZKFw@PF>7$$\"3!*******Rv&)z:F0 $\"3yYqiZcsO?F37$$\"3gmm;%)3;C;F0$\"3gSYHP.*)=TF37$$\"3ILLLGUYo;F0$\"3 =PCLryK#R'F37$$\"3\"*****\\n'*33YE&4\"F07$$\"3ILLe*3k**y\"F0$\"3eanZc:5l8F07$$\"34++]sI@K=F0$ \"3k!>Urv3Tl\"F07$$\"33+++S2ls=F0$\"357P(\\L)4\\>F07$$\"34++]2%)38>F0$ \"3Ok^R>4YiAF07$$\"3/++v.Uac>F0$\"3,g+#*3i9?EF07$$\"\"#F*$\"\"$F*-%'CO LOURG6&%$RGBG$\"*++++\"!\")$F*F*Ffal-F$6&7%7$$!3!******>C&Qz=F0Ffal7$$ \"3E+++`N'HZ$F3Ffal7$$\"3$******f)))3K:F0Ffal-F`al6&FbalF*F*F*-%&STYLE G6#%&POINTG-%'SYMBOLG6#%'CIRCLEG-F$6&FialFcblFebl-Fjbl6#%(DIAMONDG-F$6 &FialFcblFebl-Fjbl6#%&CROSSG-%+AXESLABELSG6%Q!6\"Fjcl-%%FONTG6#%(DEFAU LTG-%%VIEWG6$;F(F[alF_dl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }{TEXT 334 70 " .. plot ting the graph of two functions to obtain a starting interval " }} {PARA 0 "" 0 "" {TEXT 282 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 40 "Find all the solutions of the equation " }{XPPEDIT 18 0 "exp(x) = 4-x^2;" "6#/-%$expG6#%\"xG,&\"\"%\"\"\"*$F'\"\"#!\"\"" }{TEXT -1 22 " correct to 10 digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 270 8 "Solution" }{TEXT 281 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 48 "This time it is best to graph the two equations " }{XPPEDIT 18 0 "y = exp(x);" "6#/%\"yG-%$expG6#%\"xG" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "y = 4-x^2;" "6#/%\"yG,&\"\"%\"\"\"*$%\"xG\"\"# !\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 30 "The solutions of the equation " }{XPPEDIT 18 0 "exp(x) = 4-x^2;" "6#/-%$expG6#%\"xG,& \"\"%\"\"\"*$F'\"\"#!\"\"" }{TEXT -1 18 " are given by the " }{TEXT 284 1 "x" }{TEXT -1 63 " coordinates of the points of intersection of \+ these two graphs." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plot([exp(x),4-x^2],x=-2.2..2,y=0..5);" }} {PARA 13 "" 1 "" {GLPLOT2D 293 209 209 {PLOTDATA 2 "6&-%'CURVESG6$7S7$ $!1+++++++A!#:$\"1RLi$eJ!36!#;7$$!1++]_>X3@F*$\"1:k4RzD97F-7$$!1+](e[' zG?F*$\"1/qd/o$\\J\"F-7$$!1++v#e:#R>F*$\"1Hjdyr;Q9F-7$$!1++Dz3/\\=F*$ \"1R'y3[!)Qd\"F-7$$!1+]PgZHf')3w;F*$\"1oe BIQ/r=F-7$$!1+](3[L**e\"F*$\"1bkQr)\\pF-7$$!1**** \\(=!pQFF-$\"1;'*G5nJ/wF-7$$!1'**\\P\"e?7>F-$\"1Cq&yy.&f#)F-7$$!1**** \\i2t?5F-$\"1VpmScjH!*F-7$$!1A,+vG(QZ*!#=$\"1RctG'3d!**F-7$$\"1&)**\\( e+M6(!#<$\"1,)*p,^st5F*7$$\"1++]Ks#>e\"F-$\"1vUTF>Rr6F*7$$\"1,++q*G8[# F-$\"1c9pX-j\"G\"F*7$$\"1****\\-\"=7O$F-$\"16$H!*[4&*R\"F*7$$\"1,+D\"3 cD@%F-$\"1!oTJutQ_\"F*7$$\"1,+]d<#y:&F-$\"16lM$4[\\n\"F*7$$\"1)*****4] =2gF-$\"1O?,l%GM#=F*7$$\"1.++v:19pF-$\"1jIG$)4_'*>F*7$$\"1,+DJl#et(F-$ \"1`kSCy^n@F*7$$\"10++&oKUj)F-$\"1`yu*Gk7P#F*7$$\"1(**\\7'Gcz%*F-$\"1A U)[gI/e#F*7$$\"1+]iYwJO5F*$\"1*\\6d*z\")=GF*7$$\"1++]FqqA6F*$\"1&>nw3i J2$F*7$$\"1+]7.([J@\"F*$\"18ZR@/1kLF*7$$\"1+++'ya-I\"F*$\"1wz9s;BqOF*7 $$\"1++vOLL*Q\"F*$\"1Q6*>Ru@,%F*7$$\"1+]7sUnx9F*$\"1jSBc2u#Q%F*7$$\"1+ ++'F*7$$\"1,](y#Gu3>F *$\"16t'yL/Yu'F*7$$\"\"#\"\"!$\"1]1$*)4c!*Q(F*-%'COLOURG6&%$RGBG$\"#5! \"\"FhzFhz-F$6$7S7$F($!12++++++%)F-7$F/$!1L1,gjpbWF-7$F4$!1$\\()4\"=:g 6F-7$F9$\"1sd>O#HWR#F-7$F>$\"1kQio#y/\"eF-7$FC$\"102)[!*F-7$FH$\"1 \"F*7$FM$\"11FRE:6s9F*7$FR$\"1$pG'pT]ZY#F*7$F`o$\"1a+kayTyEF*7$Feo$\"1O%QZ 2!GzGF*7$Fjo$\"1`f%=(4>dIF*7$F_p$\"1+I\"z>Pa?$F*7$Fdp$\"1z$)Q_DAlLF*7$ Fip$\"1M>N7Ec'[$F*7$F^q$\"132+u6`6OF*7$Fcq$\"14!G&*>Kwq$F*7$Fhq$\"1%*) yC-'Q(z$F*7$F]r$\"1_%Qe1*fnQF*7$Fbr$\"17*o0w&*\\#RF*7$Fgr$\"12'e#*oMM' RF*7$F\\s$\"1i[5(3\"e*)RF*7$Fas$\"1[FtX-\"***RF*7$Fgs$\"1w\"3K&*R\\*RF *7$F]t$\"1\\2Ji](\\(RF*7$Fbt$\"1(QEa1I%QRF*7$Fgt$\"1jUnG@-()QF*7$F\\u$ \"1OKi7PaAQF*7$Fau$\"1'fyruoRt$F*7$Ffu$\"18jb#GP\"ROF*7$F[v$\"1&3r`_d> _$F*7$F`v$\"1'3%yy)p:S$F*7$Fev$\"1wFTf-]aKF*7$Fjv$\"1)4'fz)y85$F*7$F_w $\"1@OHNd/EHF*7$Fdw$\"19@SI*G&RFF*7$Fiw$\"1Jh5C-FGDF*7$F^x$\"1$4%[\"\\ P$4BF*7$Fcx$\"16^RzGvp?F*7$Fhx$\"1^)*\\X(yk\"=F*7$F]y$\"1D*=uP&)*p:F*7 $Fby$\"1*='R')yEr7F*7$Fgy$\"1b&oo)*\\U*)*F-7$F\\z$\"12_Bg,nOnF-7$Faz$ \"1`uNq\"3qc$F-7$FfzFhz-F\\[l6&F^[lFhzF_[lFhz-%+AXESLABELSG6$Q\"x6\"Q \"yF\\el-%%VIEWG6$;$!#AFa[lFfz;Fhz$\"\"&Fhz" 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 71 "It is clear that there exactly 2 real number solutions of the equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 6 "bisect" }{TEXT -1 47 " will accept an equation as its first argumen t." }}{PARA 0 "" 0 "" {TEXT -1 36 "The solutions are given as follows. " }}{PARA 0 "" 0 "" {TEXT -1 26 "( You can use the option \"" }{TEXT 339 9 "info=true" }{TEXT -1 4 "\", \"" }{TEXT 339 6 "info=1" }{TEXT -1 6 "\" or \"" }{TEXT 339 6 "info=2" }{TEXT -1 34 "\" to see all the \+ bisection steps.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "bisect(exp(x)=4-x^2,x=-2..-1.9,info=1);" }} {PARA 6 "" 1 "" {TEXT -1 71 " start [+] -2.0000000000000 \+ -1.9000000000000 [-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 1 [+] -2.0000000000000 -1.9500000000000 <-[-]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 2 [+]-> -1.9750000000000 \+ -1.9500000000000 [-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 3 [+] -1.9750000000000 -1.9625000000000 <-[-]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 4 [+]-> -1.9687500000000 \+ -1.9625000000000 [-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 5 [+]-> -1.9656250000000 -1.9625000000000 [-]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 6 [+] -1.9656250000000 \+ -1.9640625000000 <-[-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 7 [+]-> -1.9648437500000 -1.9640625000000 [-]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 8 [+] -1.9648437500000 \+ -1.9644531250000 <-[-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 9 [+]-> -1.9646484375000 -1.9644531250000 [-]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 10 [+] -1.9646484375000 \+ -1.9645507812500 <-[-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 11 [+] -1.9646484375000 -1.9645996093750 <-[-]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 12 [+] -1.9646484375000 \+ -1.9646240234375 <-[-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 13 [+]-> -1.9646362304688 -1.9646240234375 [-]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 14 [+] -1.9646362304688 \+ -1.9646301269532 <-[-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 15 [+] -1.9646362304688 -1.9646331787110 <-[-]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 16 [+] -1.9646362304688 \+ -1.9646347045899 <-[-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 17 [+] -1.9646362304688 -1.9646354675294 <-[-]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 18 [+]-> -1.9646358489991 \+ -1.9646354675294 [-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 19 [+]-> -1.9646356582642 -1.9646354675294 [-]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 20 [+] -1.9646356582642 \+ -1.9646355628968 <-[-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 21 [+]-> -1.9646356105805 -1.9646355628968 [-]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 22 [+] -1.9646356105805 \+ -1.9646355867386 <-[-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 23 [+]-> -1.9646355986596 -1.9646355867386 [-]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 24 [+] -1.9646355986596 \+ -1.9646355926991 <-[-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 25 [+] -1.9646355986596 -1.9646355956794 <-[-]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 26 [+] -1.9646355986596 \+ -1.9646355971695 <-[-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 27 [+]-> -1.9646355979146 -1.9646355971695 [-]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 28 [+]-> -1.9646355975420 \+ -1.9646355971695 [-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 29 [+] -1.9646355975420 -1.9646355973558 <-[-]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 30 [+] -1.9646355975420 \+ -1.9646355974489 <-[-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 31 [+]-> -1.9646355974954 -1.9646355974489 [-]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 32 [+] -1.9646355974954 \+ -1.9646355974722 <-[-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 33 [+] -1.9646355974954 -1.9646355974838 <-[-]" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$!+(fNY'>!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "bisect(exp(x )=4-x^2,x=1..1.1,info=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%5startin g~interval~-~G7$$\"\"\"\"\"!$\"#6!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\".+++++0\"!#7%Gbecomes~left~end~point~of~new~inter valG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%P~~~~~~~~~~~~~~~~~~~~~[-]---- >[-]~~~~~~~~~~~~[+]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~1~---~G 7$$\".+++++0\"!#7$\"#6!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-p ointG$\".++++]2\"!#7%Hbecomes~right~end~point~of~new~intervalG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%P~~~~~~~~~~~~~~~~~~~~~[-]~~~~~~~~~~~~ [+]<----[+]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~2~---~G7$$\".++ +++0\"!#7$\".++++]2\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-point G$\".++++D1\"!#7%Hbecomes~right~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%P~~~~~~~~~~~~~~~~~~~~~[-]~~~~~~~~~~~~[+]<-- --[+]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~3~---~G7$$\".+++++0\" !#7$\".++++D1\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\".+ ++]i0\"!#7%Gbecomes~left~end~point~of~new~intervalG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%P~~~~~~~~~~~~~~~~~~~~~[-]---->[-]~~~~~~~~~~~~[+]G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~4~---~G7$$\".+++]i0\"!#7$\".++ ++D1\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\".+++v$f5!#7 %Hbecomes~right~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%P~~~~~~~~~~~~~~~~~~~~~[-]~~~~~~~~~~~~[+]<----[+]G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~5~---~G7$$\".+++]i0\"!#7$\".+++ v$f5F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\".++]7y0\"!#7% Gbecomes~left~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%P~~~~~~~~~~~~~~~~~~~~~[-]---->[-]~~~~~~~~~~~~[+]G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%,step~6~---~G7$$\".++]7y0\"!#7$\".+++v$f5F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\".++v$fe5!#7%Hbecomes~ri ght~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%P~~ ~~~~~~~~~~~~~~~~~~~[-]~~~~~~~~~~~~[+]<----[+]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~7~---~G7$$\".++]7y0\"!#7$\".++v$fe5F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\".+]7.#e5!#7%Hbecomes~right~en d~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%P~~~~~~~~ ~~~~~~~~~~~~~[-]~~~~~~~~~~~~[+]<----[+]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%,step~8~---~G7$$\".++]7y0\"!#7$\".+]7.#e5F'" }}{PARA 11 "" 1 " " {XPPMATH 20 "6%%*mid-pointG$\".+D\"y+e5!#7%Hbecomes~right~end~point~ of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%P~~~~~~~~~~~~~~~~ ~~~~~[-]~~~~~~~~~~~~[+]<----[+]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%, step~9~---~G7$$\".++]7y0\"!#7$\".+D\"y+e5F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\".]i:5z0\"!#7%Gbecomes~left~end~point~of~ new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%P~~~~~~~~~~~~~~~~~~~ ~~[-]---->[-]~~~~~~~~~~~~[+]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-ste p~10~---~G7$$\".]i:5z0\"!#7$\".+D\"y+e5F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\".vV)*ez0\"!#7%Gbecomes~left~end~point~of ~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%P~~~~~~~~~~~~~~~~~~ ~~~[-]---->[-]~~~~~~~~~~~~[+]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-st ep~11~---~G7$$\".vV)*ez0\"!#7$\".+D\"y+e5F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\".Q%)R$)z0\"!#7%Gbecomes~left~end~point~o f~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%P~~~~~~~~~~~~~~~~~ ~~~~[-]---->[-]~~~~~~~~~~~~[+]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-s tep~12~---~G7$$\".Q%)R$)z0\"!#7$\".+D\"y+e5F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\".pag&*z0\"!#7%Gbecomes~left~end~point~of 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"6#%P~~~~~~~~~~~~~~~~~~~~~[-]---->[-]~~~~~~~~~~~~[+]G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%-step~20~---~G7$$\".))>Q1!e5!#7$\".ctZ1!e5F'" } }{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\".s'Hk+e5!#7%Hbecomes~r ight~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%P~ ~~~~~~~~~~~~~~~~~~~~[-]~~~~~~~~~~~~[+]<----[+]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~21~---~G7$$\".))>Q1!e5!#7$\".s'Hk+e5F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\".IeS1!e5!#7%Hbecomes~right~en d~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%P~~~~~~~~ ~~~~~~~~~~~~~[-]~~~~~~~~~~~~[+]<----[+]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~22~---~G7$$\".))>Q1!e5!#7$\".IeS1!e5F'" }}{PARA 11 "" 1 " " {XPPMATH 20 "6%%*mid-pointG$\".4RR1!e5!#7%Gbecomes~left~end~point~of ~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%P~~~~~~~~~~~~~~~~~~ ~~~[-]---->[-]~~~~~~~~~~~~[+]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-st ep~23~---~G7$$\".4RR1!e5!#7$\".IeS1!e5F'" }}{PARA 11 "" 1 "" {XPPMATH 20 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"" {XPPMATH 20 "6$%-step~27~---~G7$$\".:1S1!e5!#7$\".g8S 1!e5F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\".))4S1!e5!#7% Gbecomes~left~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%P~~~~~~~~~~~~~~~~~~~~~[-]---->[-]~~~~~~~~~~~~[+]G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%-step~28~---~G7$$\".))4S1!e5!#7$\".g8S1!e5F'" } }{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\".u6S1!e5!#7%Hbecomes~r ight~end~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%P~ ~~~~~~~~~~~~~~~~~~~~[-]~~~~~~~~~~~~[+]<----[+]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~29~---~G7$$\".))4S1!e5!#7$\".u6S1!e5F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\".\"3,k+e5!#7%Gbecomes~left~en d~point~of~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%P~~~~~~~~ ~~~~~~~~~~~~~[-]---->[-]~~~~~~~~~~~~[+]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~30~---~G7$$\".\"3,k+e5!#7$\".u6S1!e5F'" }}{PARA 11 "" 1 " " {XPPMATH 20 "6%%*mid-pointG$\".G6S1!e5!#7%Hbecomes~right~end~point~o f~new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%P~~~~~~~~~~~~~~~~~ ~~~~[-]~~~~~~~~~~~~[+]<----[+]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-s tep~31~---~G7$$\".\"3,k+e5!#7$\".G6S1!e5F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\"./6S1!e5!#7%Hbecomes~right~end~point~of~ new~intervalG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%P~~~~~~~~~~~~~~~~~~~ ~~[-]~~~~~~~~~~~~[+]<----[+]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-ste p~32~---~G7$$\".\"3,k+e5!#7$\"./6S1!e5F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*mid-pointG$\".#4,k+e5!#7%Hbecomes~right~end~point~of~new~inter valG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%P~~~~~~~~~~~~~~~~~~~~~[-]~~~~ ~~~~~~~~[+]<----[+]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~33~---~ G7$$\".\"3,k+e5!#7$\".#4,k+e5F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*m id-pointG$\".'3,k+e5!#7%Gbecomes~left~end~point~of~new~intervalG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%P~~~~~~~~~~~~~~~~~~~~~[-]---->[-]~~~~ ~~~~~~~~[+]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-step~34~---~G7$$\".' 3,k+e5!#7$\".#4,k+e5F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+,k+e5!\" *" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }{TEXT 335 35 " .. an equation with two solutions " }}{PARA 0 "" 0 "" {TEXT 274 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 40 "Find all the solutions of the equation " }{XPPEDIT 18 0 "ln(x) = 2*x ^2-3*x-1;" "6#/-%#lnG6#%\"xG,(*&\"\"#\"\"\"*$F'F*F+F+*&\"\"$F+F'F+!\" \"F+F/" }{TEXT -1 22 " correct to 10 digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 270 8 "Solution" }{TEXT 326 1 ":" }} {PARA 0 "" 0 "" {TEXT -1 30 "The solutions of the equation " } {XPPEDIT 18 0 "ln(x) = 2*x^2-3*x-1;" "6#/-%#lnG6#%\"xG,(*&\"\"#\"\"\"* $F'F*F+F+*&\"\"$F+F'F+!\"\"F+F/" }{TEXT -1 18 " are given by the " } {TEXT 283 1 "x" }{TEXT -1 96 " coordinates of the points of intersecti on of the two graphs of the equations the two equations " }{XPPEDIT 18 0 "y = ln(x);" "6#/%\"yG-%#lnG6#%\"xG" }{TEXT -1 6 " and " } {XPPEDIT 18 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" }}{PARA 0 "" 0 "" {TEXT -1 36 "The solutions are given as follows . " }}{PARA 0 "" 0 "" {TEXT -1 26 "( You can use the option \"" } {TEXT 339 9 "info=true" }{TEXT -1 4 "\", \"" }{TEXT 339 6 "info=1" } {TEXT -1 6 "\" or \"" }{TEXT 339 6 "info=2" }{TEXT -1 34 "\" to see al l the bisection steps.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "bisect(ln(x)=2*x^2-3*x-1,x=0.1..0.3 ,info=true);" }}{PARA 6 "" 1 "" {TEXT -1 71 " start [-] 0. 10000000000000 0.30000000000000 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 1 [-]-> 0.20000000000000 0.30000000 000000 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 2 [-] \+ 0.20000000000000 0.25000000000000 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 3 [-] 0.20000000000000 0.22500000 000000 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 4 [-]-> \+ 0.21250000000000 0.22500000000000 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 5 [-] 0.21250000000000 0.21875000 000000 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 6 [-] \+ 0.21250000000000 0.21562500000000 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 7 [-] 0.21250000000000 0.21406250 000000 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 8 [-] \+ 0.21250000000000 0.21328125000000 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 9 [-] 0.21250000000000 0.21289062 500000 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 10 [-]-> \+ 0.21269531250000 0.21289062500000 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 11 [-] 0.21269531250000 0.21279296 875000 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 12 [-] \+ 0.21269531250000 0.21274414062500 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 13 [-]-> 0.21271972656250 0.21274414 062500 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 14 [-]-> \+ 0.21273193359375 0.21274414062500 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 15 [-] 0.21273193359375 0.21273803 710938 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 16 [-]-> \+ 0.21273498535157 0.21273803710938 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 17 [-]-> 0.21273651123047 0.21273803 710938 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 18 [-] \+ 0.21273651123047 0.21273727416993 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 19 [-]-> 0.21273689270020 0.21273727 416993 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 20 [-]-> \+ 0.21273708343506 0.21273727416993 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 21 [-]-> 0.21273717880249 0.21273727 416993 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 22 [-]-> \+ 0.21273722648620 0.21273727416993 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 23 [-] 0.21273722648620 0.21273725 032806 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 24 [-]-> \+ 0.21273723840713 0.21273725032806 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 25 [-]-> 0.21273724436759 0.21273725 032806 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 26 [-] \+ 0.21273724436759 0.21273724734783 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 27 [-] 0.21273724436759 0.21273724 585772 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 28 [-]-> \+ 0.21273724511266 0.21273724585772 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 29 [-] 0.21273724511266 0.21273724 548519 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 30 [-]-> \+ 0.21273724529893 0.21273724548519 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 31 [-] 0.21273724529893 0.21273724 539206 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 32 [-] \+ 0.21273724529893 0.21273724534549 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 33 [-] 0.21273724529893 0.21273724 532220 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 34 [-]-> \+ 0.21273724531056 0.21273724532220 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 35 [-] 0.21273724531056 0.21273724 531638 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 36 [-] \+ 0.21273724531056 0.21273724531347 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 37 [-]-> 0.21273724531202 0.21273724 531347 [+]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+`CPF@!#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "bisect(ln(x)=2*x^2-3*x-1,x=1.8..2.2,info=true);" }}{PARA 6 "" 1 " " {TEXT -1 71 " start [+] 1.8000000000000 2.20000 00000000 [-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 1 [+] \+ 1.8000000000000 2.0000000000000 <-[-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 2 [+]-> 1.9000000000000 2.0000000 000000 [-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 3 [+] \+ 1.9000000000000 1.9500000000000 <-[-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 4 [+]-> 1.9250000000000 1.9500000 000000 [-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 5 [+] \+ 1.9250000000000 1.9375000000000 <-[-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 6 [+] 1.9250000000000 1.9312500 000000 <-[-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 7 [+]-> \+ 1.9281250000000 1.9312500000000 [-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 8 [+] 1.9281250000000 1.9296875 000000 <-[-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 9 [+]-> \+ 1.9289062500000 1.9296875000000 [-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 10 [+]-> 1.9292968750000 1.9296875 000000 [-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 11 [+] \+ 1.9292968750000 1.9294921875000 <-[-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 12 [+]-> 1.9293945312500 1.9294921 875000 [-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 13 [+]-> \+ 1.9294433593750 1.9294921875000 [-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 14 [+] 1.9294433593750 1.9294677 734375 <-[-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 15 [+]-> \+ 1.9294555664062 1.9294677734375 [-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 16 [+] 1.9294555664062 1.9294616 699218 <-[-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 17 [+] \+ 1.9294555664062 1.9294586181640 <-[-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 18 [+]-> 1.9294570922851 1.9294586 181640 [-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 19 [+] \+ 1.9294570922851 1.9294578552246 <-[-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 20 [+] 1.9294570922851 1.9294574 737548 <-[-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 21 [+] \+ 1.9294570922851 1.9294572830200 <-[-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 22 [+]-> 1.9294571876526 1.9294572 830200 [-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 23 [+] \+ 1.9294571876526 1.9294572353363 <-[-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 24 [+]-> 1.9294572114944 1.9294572 353363 [-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 25 [+]-> \+ 1.9294572234154 1.9294572353363 [-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 26 [+]-> 1.9294572293758 1.9294572 353363 [-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 27 [+]-> \+ 1.9294572323560 1.9294572353363 [-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 28 [+]-> 1.9294572338462 1.9294572 353363 [-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 29 [+]-> \+ 1.9294572345912 1.9294572353363 [-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 30 [+] 1.9294572345912 1.9294572 349638 <-[-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 31 [+]-> \+ 1.9294572347775 1.9294572349638 [-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 32 [+]-> 1.9294572348706 1.9294572 349638 [-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 33 [+] \+ 1.9294572348706 1.9294572349172 <-[-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 34 [+]-> 1.9294572348939 1.9294572 349172 [-]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 35 [+] \+ 1.9294572348939 1.9294572349056 <-[-]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+NsXH>!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 4" }{TEXT 336 95 " .. an example where it is di fficult to display all solutions graphically in a single picture " }} {PARA 0 "" 0 "" {TEXT 271 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 22 "Find all the solutions" }{TEXT 275 1 " " }{TEXT -1 17 "of the equation " }{XPPEDIT 18 0 "x^3-10*x^2-5*x+1 = arctan(x);" "6#/,**$%\"xG\"\"$\"\"\"*&\"#5F(*$F&\"\"#F(!\"\"*&\"\"&F(F&F(F-F(F(-%' arctanG6#F&" }{TEXT -1 23 " correct to 10 digits. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 270 8 "Solution" }{TEXT 325 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 30 "The solutions of the equation " } {XPPEDIT 18 0 "x^3-10*x^2-5*x+1 = arctan(x)" "6#/,**$%\"xG\"\"$\"\"\"* &\"#5F(*$F&\"\"#F(!\"\"*&\"\"&F(F&F(F-F(F(-%'arctanG6#F&" }{TEXT -1 18 " are given by the " }{TEXT 280 1 "x" }{TEXT -1 62 " coordinates of the points of intersection of the two graphs " }{XPPEDIT 18 0 "y=x^3 -10*x^2-5*x+1" "6#/%\"yG,**$%\"xG\"\"$\"\"\"*&\"#5F)*$F'\"\"#F)!\"\"*& \"\"&F)F'F)F.F)F)" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "y = arctan(x); " "6#/%\"yG-%'arctanG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot([x^3-10 *x^2-5*x+1,arctan(x)],x=-1...1,y=-2..2);" }}{PARA 13 "" 1 "" {GLPLOT2D 428 329 329 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$!\"\"\"\"!$!\"&F *7$$!3ommm;p0k&*!#=$!3(zXY*Gc#*RU!#<7$$!3wKL$37$$!33LLL$yaE\"eF0$\"3F)zwQ>*R7LF07$$!3hm mm\">s%HaF0$\"3%e/]TVHw1'F07$$!3Q+++]$*4)*\\F0$\"33wwA$eC4w)F07$$!39++ +]_&\\c%F0$\"3=vK1_4;F37$$!3EMLLL7i)4# F0$\"3K^inPnk*f\"F37$$!3c****\\P'psm\"F0$\"3G*RK\"e8-^:F37$$!3')****\\ 74_c7F0$\"3Ok\"F37$$!3]^omm;zr)*!#@$\"3)QBW'\\h#\\+\"F37$$\"3%pJ L$ezw5VFfn$\"3G%z$)[()*efwF07$$\"3s*)***\\PQ#\\\")Ffn$\"3)p+&zn=pm_F07 $$\"3GKLLe\"*[H7F0$\"3Yp9$>6'\\fBF07$$\"3I*******pvxl\"F0$!3P*3pmV%R:* *Ffn7$$\"3#z****\\_qn2#F0$!3(oUj!GyD2YF07$$\"3U)***\\i&p@[#F0$!3r)R#)) oH3>%)F07$$\"3B)****\\2'HKHF0$!3Y*fql7r2I\"F37$$\"3ElmmmZvOLF0$!3KwOq= %>Yu\"F37$$\"3i******\\2goPF0$!3k9*\\La75D#F37$$\"3UKL$eR<*fTF0$!3`(= \"RkHYQFF37$$\"3m******\\)Hxe%F0$!3&\\jYfFK?I$F37$$\"3ckm;H!o-*\\F0$!3 145%H**R6'QF37$$\"3y)***\\7k.6aF0$!3yH93s#=]Z%F37$$\"3#emmmT9C#eF0$!31 !GsF37$$\"3Nmm;HYt7vF0$!3iB<$e& yXwzF37$$\"3Y*******p(G**yF0$!3-*yT'=Gh'p)F37$$\"3]mmmT6KU$)F0$!3]Sl)) *e9+b*F37$$\"3fKLLLbdQ()F0$!32X8%fnDQ.\"!#;7$$\"3[++]i`1h\"*F0$!3AX>?/ ,U?6F\\z7$$\"3W++]P?Wl&*F0$!3%>.H)Qts07F\\z7$$\"\"\"F*$!#8F*-%'COLOURG 6&%$RGBG$\"#5F)$F*F*Fb[l-F$6$7S7$F($!3!G[uRj\")R&yF07$F.$!3%o9iE$**=Jw F07$F5$!3\\NrMHwGHuF07$F:$!3#3*p!Q1DH>(F07$F?$!3)f^V^t'oWpF07$FD$!3qB! *fl)**po'F07$FI$!3F0*[6'zKQkF07$FN$!3e%z*>i:oqhF07$FS$!3'ohq[WnF)eF07$ FX$!3'[&)R:eMVe&F07$Fhn$!3cqW!H/-`E&F07$F]o$!3'fE7;(G7u\\F07$Fbo$!3BR> Taa&\\j%F07$Fgo$!3)>0JifCCG%F07$F\\p$!3&)))pq8MgJRF07$Fap$!33l=\")yc!R g$F07$Ffp$!3^zcHl8[.KF07$F[q$!3k\"oO)[cobGF07$F`q$!3k*e,mhV?W#F07$Feq$ !3c))fMu8go?F07$Fjq$!3IHNtUM2_;F07$F_r$!3i8j4?+(*\\7F07$Fdr$!3o#p'R#)f jg#)Ffn7$Fir$!3$yDE)z\"36M%Ffn7$F^s$!3O8q!*f%)yr)*F`s7$Fds$\"3[rlOQ253 VFfn7$Fis$\"3Bwk?'>q78)Ffn7$F^t$\"3xh#yKe\\LA\"F07$Fct$\"3lG+ie[$Gk\"F 07$Fht$\"3+(*zVbKmZ?F07$F]u$\"3yv!4`,)*HV#F07$Fbu$\"3q.A)f4RB&GF07$Fgu $\"3z**R$y^%e?KF07$F\\v$\"3=QjmGU+/OF07$Fav$\"3d\"H)Hh*e@%RF07$Ffv$\"3 q/I?iaD,VF07$F[w$\"35Se(H[(oGYF07$F`w$\"3uHA5_M()f\\F07$Few$\"3;j')=cQ fs_F07$Fjw$\"3C[sN$>9#)e&F07$F_x$\"3YdwGUf'3)eF07$Fdx$\"3?)Q`'o.yohF07 $Fix$\"3#pCb*pi:VkF07$F^y$\"33&)Qr#3(p&o'F07$Fcy$\"37,#=#)p%o_pF07$Fhy $\"3A(ohs$e#==(F07$F^z$\"3)HkVJ5GkT(F07$Fcz$\"3,!Qi@H8>j(F07$Fhz$\"3!G [uRj\")R&yF0-F][l6&F_[lFb[lF`[lFb[l-%+AXESLABELSG6$Q\"x6\"Q\"yF_el-%%V IEWG6$;F(Fhz;$!\"#F*$\"\"#F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "It is not easy to identify all \+ the intersection points by using a single graph. " }}{PARA 0 "" 0 "" {TEXT -1 22 "The graph above shows " }{TEXT 275 24 "two intersection \+ points" }{TEXT -1 22 ", but there must be a " }{TEXT 275 5 "third" } {TEXT -1 24 " because eventually, as " }{TEXT 276 1 "x" }{TEXT -1 16 " increases, the " }{XPPEDIT 18 0 "x^3" "6#*$%\"xG\"\"$" }{TEXT -1 9 " \+ term in " }{XPPEDIT 18 0 "x^3-10*x^2-5*x+1" "6#,**$%\"xG\"\"$\"\"\"*& \"#5F'*$F%\"\"#F'!\"\"*&\"\"&F'F%F'F,F'F'" }{TEXT -1 26 " will predomi nate so that " }{XPPEDIT 18 0 "x^3-10*x^2-5*x+1" "6#,**$%\"xG\"\"$\"\" \"*&\"#5F'*$F%\"\"#F'!\"\"*&\"\"&F'F%F'F,F'F'" }{TEXT -1 10 " tends to " }{XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT -1 4 " as " } {XPPEDIT 18 0 "x -> infinity" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"% )infinityGF*F*F*" }{TEXT -1 33 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 26 "( You can use the option \+ \"" }{TEXT 339 9 "info=true" }{TEXT -1 34 "\" to see all the bisection steps.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "x1 := bisect(x^3-10*x^2-5*x+1=arctan(x),x=-0.8..-0.6 );\nx2 := bisect(x^3-10*x^2-5*x+1=arctan(x),x=0..0.2);\nx3 := bisect(x ^3-10*x^2-5*x+1=arctan(x),x=10..11);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#x1G$!+/@iio!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x2G$\"+wsvi8 !#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x3G$\"+!pO\"[5!\")" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }{TEXT 337 49 " .. constructing a polynomial with a given zero " }}{PARA 0 "" 0 "" {TEXT 272 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 128 "Find the fifth root of 7 correct to 30 decimal digits by using th e bisection method to solve an appropriate polynomial equation." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 270 8 "Solutio n" }{TEXT 273 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 34 "The equation we ne ed to solve is " }{XPPEDIT 18 0 "x^5-7 = 0;" "6#/,&*$%\"xG\"\"&\"\"\" \"\"(!\"\"\"\"!" }{TEXT -1 103 ".\nThe solution we seek is between 1 a nd 2, in fact it is easy to check that it is between 1.4 and 1.5.\n" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "1.4^5;\n1.5^5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"'Cy`!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"'v$f(!\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "r := evalf(bisect(x^5-7,x=1.4..1.5),30);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG$\"?j&p;pFp?b%fhJxv9!#H" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "We can ch eck that the result given is the best that we can do with 30 digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "saveDigits := Digits: Digits := 30:\n1.475773161594552069276916 69562^5;\n1.47577316159455206927691669563^5;\n1.4757731615945520692769 1669564^5;\nDigits := saveDigits:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"?r**************************p!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"?&***************************p!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"?=+++++++++++++q!#H" }}}{PARA 0 "" 0 "" {TEXT -1 55 "Of course we can calculate the solution more simply as " }{XPPEDIT 18 0 "7^(1/5); " "6#)\"\"(*&\"\"\"F&\"\"&!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "evalf(7^(1/5 ),30);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?j&p;pFp?b%fhJxv9!#H" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 6" }{TEXT 338 49 " .. a difficult problem for the bisection method " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The equation " } {XPPEDIT 18 0 "sin(x)=Pi-x" "6#/-%$sinG6#%\"xG,&%#PiG\"\"\"F'!\"\"" } {TEXT -1 18 " has the solution " }{XPPEDIT 18 0 "x=Pi" "6#/%\"xG%#PiG " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "The solution can be \+ interpreted as the " }{TEXT 332 1 "x" }{TEXT -1 55 " coordinate of the point of intersection of the graphs " }{XPPEDIT 18 0 "y=sin(x)" "6#/% \"yG-%$sinG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=Pi-x" "6#/% \"yG,&%#PiG\"\"\"%\"xG!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot([sin(x) ,Pi-x],x=0..5,y=-1.2..1.2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$\"\"!F)F(7$$\"3GLLL3x&)*3\"!#=$\"3s-w# GX,x3\"F-7$$\"3umm\"H2P\"Q?F-$\"3H'37.kbS-#F-7$$\"3MLL$eRwX5$F-$\"35Z_ @+V%\\0$F-7$$\"33ML$3x%3yTF-$\"3ilpL,UedSF-7$$\"3emm\"z%4\\Y_F-$\"3]5s VrK44]F-7$$\"3`LLeR-/PiF-$\"3n?(*4JzXSeF-7$$\"3]***\\il'pisF-$\"3)G![! *=D&3k'F-7$$\"3>MLe*)>VB$)F-$\"3?Mn0?r5&R(F-7$$\"3Y++DJbw!Q*F-$\"3#)[5 J8=Ak!)F-7$$\"3%ommTIOo/\"!#<$\"3qt\\lPsWe')F-7$$\"3YLL3_>jU6Fen$\"33: IVkVH(4*F-7$$\"37++]i^Z]7Fen$\"37AQ\"eTV8\\*F-7$$\"33++](=h(e8Fen$\"3e 9$QoWYgx*F-7$$\"3/++]P[6j9Fen$\"3%yqSR[z?%**F-7$$\"3UL$e*[z(yb\"Fen$\" 3;TY<*el\"****F-7$$\"3wmm;a/cq;Fen$\"3OF?pboF]**F-7$$\"3%ommmJ#yY`*F-7$$\"3gmmm\"pW`(>F en$\"3-GYOhD!G>*F-7$$\"3K+]i!f#=$3#Fen$\"34ZlIU5x:()F-7$$\"3?+](=xpe=# Fen$\"3!)Q1L*p5t;)F-7$$\"37nm\"H28IH#Fen$\"3g2^^Z>U.vF-7$$\"3um;zpSS\" R#Fen$\"39lgnDtwpx 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YE\")zeN%ph#Fen7$FE$\"3)*f9jH'))y^#Fen7$FJ$\"3G$zkz)HK:CFen7$FO$\"3#*f 9ja%\\#4BFen7$FT$\"3'Hzk/5;N?#Fen7$FY$\"3EEJU\\jv%4#Fen7$Fin$\"3mfk],2 '*)*>Fen7$F^o$\"3)Hz*3\"\\<6*=Fen7$Fco$\"3/$z*3m9$Gy\"Fen7$Fho$\"31$z* 3;yZy;Fen7$F]p$\"3of9j/Zr$e\"Fen7$Fbp$\"3OEJU*>K5Z\"Fen7$Fgp$\"3GEJ#pL vbP\"Fen7$F\\q$\"33$zkHJOXE\"Fen7$Faq$\"3_EJ#>'zCm6Fen7$Ffq$\"3y#zkH15 %e5Fen7$F[r$\"3;Hz9<)Gsb*F-7$F`r$\"3+g7t1ez&[)F-7$Fer$\"3jj7)z$e)=](F- 7$Fjr$\"3H)fk],11W'F-7$F_s$\"3E&f9jaM#Q`F-7$Fds$\"3SIz9UphyVF-7$Fis$\" 3\"Qfk+k.AM$F-7$F^t$\"3qIz*eG([rAF-7$Fct$\"3pHzRB-+C7F-7$Fht$\"3I%GzkH E]5#F\\u7$F^u$!3Y)o?g^O\"[\"*F\\u7$Fcu$!3*>to2Qgf#>F-7$Fhu$!3?r?5R`d0I F-7$F]v$!3!**R&o`p'Q)RF-7$Fbv$!3mu?5*3)R`]F-7$Fgv$!3eL(=q`V(fgF-7$F\\w $!3Ys?N&\\k;6(F-7$Faw$!3PL(od]4,9)F-7$Ffw$!3_m?N?hz;#*F-7$F[x$!3oSNuHt PD5Fen7$F`x$!3&3a$\\BKUJ6Fen7$Fex$!39u=q.5fO7Fen7$Fjx$!3T2-Tr#HKL\"Fen 7$F_y$!3Guo#=j()RW\"Fen7$Fdy$!3\"3aV(H70V:Fen7$Fiy$!3+2_.(ot'[;Fen7$F^ z$!3W2_ybyw\\ " 0 "" {MPLTEXT 1 0 48 "plot(sin(x)-Pi+x,x=0..5,y=-1.2..1.2,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7S7$$\"\"!F)$!37$z *e`EfTJ!#<7$$\"3GLLL3x&)*3\"!#=$!3a*ptutOQ#HF,7$$\"3umm\"H2P\"Q?F0$!3t F,7$$\"3]***\\il'pisF0$!3A8V2OxB^MLe*)>VB$)F0$!3]'yDEuQ(p: F,7$$\"3Y++DJbw!Q*F0$!3=)oL\">R4(R\"F,7$$\"3%ommTIOo/\"F,$!3!*GwlD;\"* G7F,7$$\"3YLL3_>jU6F,$!39eJ1l7B*3\"F,7$$\"37++]i^Z]7F,$!3x2T3&\\J)>%*F 07$$\"33++](=h(e8F,$!3o9'fS@oA0)F07$$\"3/++]P[6j9F,$!3+Cs&pn)pUoF07$$ \"3UL$e*[z(yb\"F,$!3vb*RrX\")z$eF07$$\"3wmm;a/cq;F,$!3GP#R&Q^/gZF07$$ \"3%ommmJF,$!3;Pm'y0x'pCF07$$\"3K+]i!f#=$3#F,$!3)=QTteH$o=F07$$\"3?+ ](=xpe=#F,$!3e#H7$$\"3u m;zpSS\"R#F,$!3pi>0B^=ToFfr7$$\"3GLL3_?`(\\#F,$!3=R(*)o758O%Ffr7$$\"3f L$e*)>pxg#F,$!3Qu4n\"H\"\\*\\#Ffr7$$\"33+]Pf4t.FF,$!3_ufA;Py&Q\"Ffr7$$ \"3uLLe*Gst!GF,$!3iS#yiC!f(='!#?7$$\"30+++DRW9HF,$!3E*[,v9:$[>F`t7$$\" 3:++DJE>>IF,$!3#>I!*e5#*R0$!#@7$$\"3F+]i!RU07$F,$!3O$3e')3vXb\"!#B7$$ \"3+++v=S2LKF,$\"3;M4[&QaaF\"F[u7$$\"3Jmmm\"p)=MLF,$\"3Cw#)fw>Y)=\"F`t 7$$\"3B++](=]@W$F,$\"3Q;bY=2u/XF`t7$$\"35L$e*[$z*RNF,$\"3c;?B;6[X5Ffr7 $$\"3e++]iC$pk$F,$\"3W3?-W/]B@Ffr7$$\"3[m;H2qcZPF,$\"3\"\\Kd(eR6TOFfr7 $$\"3O+]7.\"fF&QF,$\"3,[N/IJ&[%eFfr7$$\"3Ymm;/OgbRF,$\"3k1.IalS'p)Ffr7 $$\"3w**\\ilAFjSF,$\"3/oC&)=dh]7F07$$\"3yLLL$)*pp;%F,$\"3!p)Qq'HtYq\"F 07$$\"3)RL$3xe,tUF,$\"35>0I!=ZSE#F07$$\"3Cn;HdO=yVF,$\"3,>HF07$$ \"3a+++D>#[Z%F,$\"3K5i$eaeJh$F07$$\"3SnmT&G!e&e%F,$\"3WMijf3\\'e-5OMaF07$$\"37+]iSjE!z%F,$\"3z?)pM'f/ " 0 "" {MPLTEXT 1 0 68 "f := x -> sin(x)-Pi+x:\n'f(x )'=f(x);\nbisect(f(x),x=3..3.2,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,(-%$sinGF&\"\"\"%#PiG!\"\"F'F+" }}{PARA 6 "" 1 "" {TEXT -1 71 " start [-] 3.0000000000000 \+ 3.2000000000000 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 1 [ -]-> 3.1000000000000 3.2000000000000 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 2 [-] 3.1000000000000 3 .1500000000000 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 3 [-] -> 3.1250000000000 3.1500000000000 [+]" }}{PARA 6 " " 1 "" {TEXT -1 71 " step 4 [-]-> 3.1375000000000 3. 1500000000000 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 5 [-] \+ 3.1375000000000 3.1437500000000 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 6 [-]-> 3.1406250000000 3.14 37500000000 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 7 [-] \+ 3.1406250000000 3.1421875000000 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 8 [-]-> 3.1414062500000 3.1421 875000000 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 9 [-] \+ 3.1414062500000 3.1417968750000 <-[+]" }}{PARA 6 "" 1 " " {TEXT -1 52 " ** increasing working precision to 17 digits **" } }{PARA 6 "" 1 "" {TEXT -1 71 " step 10 [-] 3.1414062500000 \+ 3.1416015625000 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 11 [-]-> 3.1415039062500 3.1416015625000 [+]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 12 [-]-> 3.1415527343750 \+ 3.1416015625000 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 13 [-]-> 3.1415771484375 3.1416015625000 [+]" }} {PARA 6 "" 1 "" {TEXT -1 52 " ** increasing working precision to 2 0 digits **" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 14 [-]-> 3. 1415893554688 3.1416015625000 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 15 [-] 3.1415893554688 3.1415954 589844 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 52 " ** increasing work ing precision to 23 digits **" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 1 6 [-]-> 3.1415924072266 3.1415954589844 [+]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 17 [-] 3.1415924072266 \+ 3.1415939331055 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 18 [-] 3.1415924072266 3.1415931701660 <-[+]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 19 [-] 3.1415924072266 \+ 3.1415927886963 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 52 " ** increasing working precision to 26 digits **" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 20 [-]-> 3.1415925979614 3.1415927 886963 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 21 [-] \+ 3.1415925979614 3.1415926933289 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 22 [-]-> 3.1415926456451 3.1415926 933289 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 23 [-] \+ 3.1415926456451 3.1415926694870 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 52 " ** increasing working precision to 29 digits **" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 24 [-] 3.1415926456451 \+ 3.1415926575661 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 25 [-]-> 3.1415926516056 3.1415926575661 [+]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 26 [-] 3.1415926516056 \+ 3.1415926545858 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 52 " ** increasing working precision to 32 digits **" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 27 [-]-> 3.1415926530957 3.1415926 545858 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 28 [-] \+ 3.1415926530957 3.1415926538408 <-[+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 29 [-]-> 3.1415926534683 3.1415926 538408 [+]" }}{PARA 6 "" 1 "" {TEXT -1 52 " ** increasing work ing precision to 35 digits **" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 3 0 [-] 3.1415926534683 3.1415926536545 <-[+]" }} {PARA 6 "" 1 "" {TEXT -1 71 " step 31 [-]-> 3.1415926535614 \+ 3.1415926536545 [+]" }}{PARA 6 "" 1 "" {TEXT -1 71 " step \+ 32 [-] 3.1415926535614 3.1415926536079 <-[+]" }} {PARA 6 "" 1 "" {TEXT -1 52 " ** increasing working precision to 3 8 digits **" }}{PARA 6 "" 1 "" {TEXT -1 71 " step 33 [-]-> 3. 1415926535847 3.1415926536079 [+]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+aEfTJ!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 21 "At higher precisions " }{TEXT 0 6 "fsolve " }{TEXT -1 100 " gives no result when given the same starting interva l, but works fine with a single starting value." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "Digits := 3 0:\nbisect(sin(x)-Pi+x=0,x=3..3.2);\nfsolve(sin(x)-Pi+x=0,x=3..3.2);\n fsolve(sin(x)-Pi+x=0,x=3);\nevalf(Pi);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?G$QVEYQKz*e`EfTJ!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'fsolveG6%/,(-%$sinG6#%\"xG\"\"\"%#PiG!\"\"F+F,\"\"!F +;\"\"$$\"#KF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?G$QVEYQKz*e`EfTJ !#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?G$QVEYQKz*e`EfTJ!#H" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 53 "Some general remarks concerning the bisec tion method " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The bisection method is guaranteed to give a numerical solution to an equation " }{XPPEDIT 18 0 "f(x)=0" "6#/-%\"fG6#%\"xG\"\"!" } {TEXT -1 16 " in an interval " }{XPPEDIT 18 0 "a <=x" "6#1%\"aG%\"xG" }{XPPEDIT 18 0 "``<=b" "6#1%!G%\"bG" }{TEXT -1 33 " under the followin g conditions: " }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) " "6#-%\"fG6#%\"xG" }{TEXT -1 32 " is continuous on the interval " } {XPPEDIT 18 0 "a <=x" "6#1%\"aG%\"xG" }{XPPEDIT 18 0 "``<=b" "6#1%!G% \"bG" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)=0" "6#/-%\"fG6#%\"xG\"\"!" }{TEXT -1 40 " has a single solutio n in the interval " }{XPPEDIT 18 0 "a <=x" "6#1%\"aG%\"xG" }{XPPEDIT 18 0 "``<=b" "6#1%!G%\"bG" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(a)" "6#-%\"fG6#%\"aG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "f(b)" "6#-%\"fG6#%\"bG" }{TEXT -1 21 " have opposite si gns." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "D isadvantages of the method are: " }}{PARA 15 "" 0 "" {TEXT -1 58 " A r elatively large number of evaluations of the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 73 " are needed compared to other methods. \n To obtain a solution correct to " }{TEXT 327 1 "d" } {TEXT -1 23 " digits requires about " }{XPPEDIT 18 0 "log[2](10)*d;" " 6#*&-&%$logG6#\"\"#6#\"#5\"\"\"%\"dGF+" }{TEXT -1 1 " " }{TEXT 328 1 " ~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "33/10;" "6#*&\"#L\"\"\"\"#5!\"\"" } {TEXT -1 1 " " }{TEXT 329 1 "." }{TEXT -1 18 " d evaluations of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 63 "The method does not cope well in a situation where t he zero of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 47 " oc curs where higher derivatives are also zero." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 25 "Find the single root \+ of " }{XPPEDIT 18 0 "cos(x) = x" "6#/-%$cosG6#%\"xGF'" }{TEXT -1 52 " correct to 10 digits by using the bisection method." }}{PARA 0 "" 0 " " {TEXT -1 47 "_______________________________________________" }} {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 47 "_______________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 16 "Plot a graph of " } {XPPEDIT 18 0 "f(x) = x^3-3*x^2+1;" "6#/-%\"fG6#%\"xG,(*$F'\"\"$\"\"\" *&F*F+*$F'\"\"#F+!\"\"F+F+" }{TEXT -1 50 " to locate the roots of this cubic approximately. " }}{PARA 0 "" 0 "" {TEXT -1 73 "Calculate these roots correct to 10 digits by using the bisection method." }}{PARA 0 "" 0 "" {TEXT -1 47 "_______________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 47 "_______________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 19 "(a) Plot graphs of " } {XPPEDIT 18 0 "y = exp(x)" "6#/%\"yG-%$expG6#%\"xG" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "y = 2-x" "6#/%\"yG,&\"\"#\"\"\"%\"xG!\"\"" }{TEXT -1 92 " in the same picture in order to find an approximate value for \+ the solution of the equation " }{XPPEDIT 18 0 "exp(x)=2-x" "6#/-%$expG 6#%\"xG,&\"\"#\"\"\"F'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 28 "(b) Compute the solution of " }{XPPEDIT 18 0 "exp(x)=2-x" "6#/- %$expG6#%\"xG,&\"\"#\"\"\"F'!\"\"" }{TEXT -1 52 " correct to 10 digits by using the bisection method." }}{PARA 0 "" 0 "" {TEXT -1 47 "______ _________________________________________" }}{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 47 "____________________ ___________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 5 "Find " }{TEXT 275 3 "all" }{TEXT -1 31 " the solutions of \+ the equation " }{XPPEDIT 18 0 "1-exp(-x^2)=1/(x^2+x+2)" "6#/,&\"\"\"F% -%$expG6#,$*$%\"xG\"\"#!\"\"F-*&F%F%,(*$F+F,F%F+F%F,F%F-" }{TEXT -1 54 " correct to 10 digits by using the bisection method. " }}{PARA 0 "" 0 "" {TEXT -1 47 "_______________________________________________" }}{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 47 "_______________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5" }}{PARA 0 "" 0 "" {TEXT -1 5 "Find " }{TEXT 275 3 "all " }{TEXT -1 31 " the solutions of the equation " }{XPPEDIT 18 0 "3-7*x ^2+x^3 = exp(-x);" "6#/,(\"\"$\"\"\"*&\"\"(F&*$%\"xG\"\"#F&!\"\"*$F*F% F&-%$expG6#,$F*F," }{TEXT -1 53 " correct to 10 digits by using the bi section method. " }}{PARA 0 "" 0 "" {TEXT -1 47 "_____________________ __________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 47 "_______________________________________ ________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q6" }}{PARA 0 "" 0 "" {TEXT -1 9 "Find the " }{TEXT 275 28 "first two positive solutions" }{TEXT -1 41 " (in ord er of magnitude) of the equation " }{XPPEDIT 18 0 "3*x-x^2=tan(x)" "6# /,&*&\"\"$\"\"\"%\"xGF'F'*$F(\"\"#!\"\"-%$tanG6#F(" }{TEXT -1 53 " cor rect to 10 digits by using the bisection method. " }}{PARA 0 "" 0 "" {TEXT -1 47 "_______________________________________________" }}{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 47 "__ _____________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q7" }} {PARA 0 "" 0 "" {TEXT -1 75 "Use the bisection method to find the cube root of 50 correct to 10 digits. " }}{PARA 0 "" 0 "" {TEXT -1 47 "___ ____________________________________________" }}{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 47 "____________________ ___________________________" }}{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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 16 "Cod e for picture" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 474 "a := 'a': \+ b := 'b': c := 'c':\np1 := plot(x^3+x-1,x=0..1):\np2 := plot([[[0,0],[ 1,0]],[[0,-0.05],[0,0.05]],\n[[0.5,-0.05],[0.5,0.05]],[[1,-0.05],[1,0. 05]]],color=black):\nt1 := plots[textplot]([[0,0.15,`a`],[0.41,0.19,`m =`],\n [0.5,0.24,`a+b`],[0.5,0.23,`____`],[1,0.15,`b`]],font=[HELVE TICA,10]):\nt2 := plots[textplot]([0.5,0.12,`2`],font=[HELVETICA,9]): \nt3 := plots[textplot]([0.82,0.72,`y = f(x)`],color=red,font=[HELVETI CA,10]):\nplots[display]([p1,p2,t1,t2,t3],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 260 "" 0 "" {TEXT -1 34 "Test examples for the printout \+ of " }{TEXT 267 6 "bisect" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "evalf(bisect(arcta n(x-10^(-9)),x=-10^(-20)..2*10^(-6),info=1),20);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "evalf(bisect (sin(1/x),x=1.000002e-6..1.000003e-6,info=1),20);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "evalf(bisect (sin(10*x),x=-1.000005e+5..-1.000002e+5,info=1),20);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "evalf(bis ect(sin(10^(-18)*x),x=-1.000005e+24..-1.000002e+24,info=1),20);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "evalf(bisect(sin(10^(-250)*x),\n x=-1.000005e+256 ..-1.000002e+256,info=1),20);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {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 }