{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "He lp Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "M aple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {SECT 0 {SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "Code for NewtonSum" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3053 "NewtonSum := proc()\n########################################### ##############\n# \+ #\n# Variable List #\n# \+ #\n# f is the poly nomial input #\n# p is the power to which \+ the roots of f are raised #\n# x is the variable of polynomial f \+ #\n# N is the variable of polynomial f \+ #\n# c[j] is the coefficient of x^(N-j) #\n# \+ i,j,m index variables #\n# S[p] is t he return value: the sum of the roots of #\n# f raised to th e p'th power #\n# \+ #\n############################################## ########### \n\nlocal f, N, p, x, S, m, c, i, j;\n\n################## #######################################\n# \+ #\n# Begin Checking and Assignment of Proce dure Arguments #\n# \+ #\n#########################################################\n\ni f nargs <> 2 then\nERROR(`NewtonSum requires two arguments`):\nelif no t type(args[1],polynom) then\nERROR(`NewtonSum requires a polynomial a s its first argument`):\nelif not type(args[2],nonnegint) then\nERROR( `NewtonSum requires a nonnegint as its second argument`):\nfi:\n\nf := args[1]: \np := args[2]:\n\nif nops(indets(f)) <> 1 then\nERROR(`Newt onSum expects its first argument to have one indeterminate`):\nfi:\n\n #########################################################\n# \+ #\n# End Checking and A ssignment of Procedure Arguments #\n# \+ #\n############################################ #############\n\n\n################################################### ######\n# #\n# \+ Begin Assignment of Polynomial data #\n# \+ #\n######################### ################################\n\nx := op(indets(f)):\nN := degree(f ,x):\n\n for i from 1 to N-1 do\n c[i] := coeff(f,x^(N-i)):\n od :\n\nc[N] := subs(x=0,f):\n\n######################################### ################\n# \+ #\n# End Assignment of Polynomial data #\n# \+ #\n############### ##########################################\n\n\n###################### ###################################\n# \+ #\n# Calculate and Return S[p] \+ #\n# \+ #\n#########################################################\n\nS[0] \+ := N:\nfor m from 1 to min(p,N) do\nS[m] := -m*c[m]-sum(c[j]*S[m-j],j= 1..m-1):\nod:\nif p <= N then RETURN(S[p]) fi:\nfor m from N+1 to p do \nS[m] := -sum(c[j]*S[m-j],j=1..N):\nod:\nRETURN(S[p]);\nend;\n" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%*NewtonSumG:6\"6+%\"fG%\"NG%\"pG%\"x G%\"SG%\"mG%\"cG%\"iG%\"jGF&F&C/@(09#\"\"#-%&ERRORG6#%ANewtonSum~requi res~two~argumentsG4-%%typeG6$&9\"6#\"\"\"%(polynomG-F76#%VNewtonSum~re quires~a~polynomial~as~its~first~argumentG4-F<6$&F?6#F5%*nonnegintG-F7 6#%VNewtonSum~requires~a~nonnegint~as~its~second~argumentG>8$F>>8&FI@$ 0-%%nopsG6#-%'indetsG6#FPFA-F76#%inNewtonSum~expects~its~first~argumen t~to~have~one~indeterminateG>8'-%#opGFW>8%-%'degreeG6$FPFin?(8+FAFA,&F ]oFA!\"\"FA%%trueG>&8*6#Fbo-%&coeffG6$FP)Fin,&F]oFAFboFdo>&Fho6#F]o-%% subsG6$/Fin\"\"!FP>&8(6#FfpF]o?(8)FAFA-%$minG6$FRF]oFeo>&Fip6#F\\q,&*& F\\qFA&FhoFbqFAFdo-%$sumG6$*&&Fho6#8,FA&Fip6#,&F\\qFAF\\rFdoFA/F\\r;FA ,&F\\qFAFdoFAFdo@$1FRF]o-%'RETURNG6#&Fip6#FR?(F\\q,&F]oFAFAFAFAFRFeo>F aq,$-Fgq6$Fiq/F\\r;FAF]oFdoFerF&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Description of NewtonSum" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Function: " }{TEXT -1 60 "NewtonSum - Cal culate sum of powers of roots of a polynomial" }}{PARA 0 "" 0 "usage" {TEXT 26 17 "Calling Sequence:" }{TEXT -1 21 "\n NewtonSum(poly,p)" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 4 "\n " } {TEXT 23 7 "poly - " }{TEXT -1 41 "any polynomial with one indetermina te\n " }{TEXT 23 30 "p - any nonnegative integer" }}}{SECT 0 {PARA 0 "" 0 "synopsis" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 127 "NewtonSum( f , p) returns z[1]^p + z[2]^p + ... + z[ N]^p where z[1], z[2], ... , z[N] are the roots of polynomial p. " }}}{SECT 0 {PARA 0 "" 0 "examples" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f := x^3-2*x^2-2*x-3; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,**$%\"xG\"\"$\"\"\"*$F'\"\"#!\"#F'F,!\"$ F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(f(x),x=0..3.5); " }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7U7$\"\"!$!\"$F(7$$\"+fR+ Hw!#6$!+*RwP;$!\"*7$$\"+^fpE9!#5$!+LW9BLF17$$\"+xM?t@F5$!+GL$)=NF17$$ \"+S$fY#HF5$!+`!))4t$F17$$\"+kOasOF5$!+dlsaRF17$$\"+o\"GfO%F5$!+<=>rTF 17$$\"+gw)Q3&F5$!+&3)H-WF17$$\"+$R-k#eF5$!+*4Jkk%F17$$\"+se`mlF5$!+)** \\D*[F17$$\"+IT&yK(F5$!+(pKg9&F17$$\"+lOU)*zF5$!+_H[n`F17$$\"+QhK`()F5 $!+gMR7cF17$$\"+8$G8^*F5$!+*=F6&eF17$$\"+'Q!=C5F1$!+HB%>2'F17$$\"+kX^! 4\"F1$!+I#4EE'F17$$\"+=BRp6F1$!+%pAYZ'F17$$\"+A@@O7F1$!+p/mRmF17$$\"+Q %RRJ\"F1$!+5\">B\"oF17$$\"+%GTFQ\"F1$!+\\[mXpF17$$\"+9yAe9F1$!+p`\\oqF 17$$\"+S)3,`\"F1$!+*RU.;(F17$$\"+^\"4^g\"F1$!+n`gFsF17$$\"+\\G)Rn\"F1$ !+;q`hsF17$$\"+PCF[F1$!+[f'[1(F17$$\"+[26S?F1$!+#[yK\"pF17$$\"+U[V 8@F1$!+V3??nF17$$\"+u'zV=#F1$!+[!*)*)['F17$$\"+8=:jAF1$!+1zZyhF17$$\"+ %3KRL#F1$!+3j&)[eF17$$\"+J^]4CF1$!+#>S:W&F17$$\"+Wb)zZ#F1$!+[F%4-&F17$ $\"+CF&Gb#F1$!+=Ft-XF17$$\"+0pHBEF1$!+?)fs&RF17$$\"+s8$pp#F1$!+W]xCLF1 7$$\"+BD#*oFF1$!+'RoDk#F17$$\"+'e!HWGF1$!+s'z#e=F17$$\"+))*yo\"HF1$!+E PyK5F17$$\"+966\"*HF1$!*GJ+:\"F17$$\"+g&GZ1$F1$\"*!*42r)F17$$\"+[`PKJF 1$\"+%zhe%=F17$$\"++i!*4KF1$\"+!\\`k/$F17$$\"+=2DzKF1$\"+CB!z>%F17$$\" +Rk=`LF1$\"+_]m3bF17$$\"+*Rp&)Q$F1$\"+US(p;'F17$$\"+dB&RU$F1$\"+'=Hd%o F17$$\"+zh(>Y$F1$\"*9S#)f(!\")7$$\"#N!\"\"$\"%v$)F*-%'COLOURG6&%$RGBG$ \"#5Fa[lF(F(-%+AXESLABELSG6$%\"xG%!G-%%VIEWG6$;F(F_[l%(DEFAULTG" 2 365 365 365 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 232 1 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "r := [solve(f = 0, x )];" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"rG7%\"\"$,&#!\"\"\"\"#\"\"\"*&%\"IGF+F&#F+F*F.,&F (F+F,F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "for p from 0 to \+ 3 do\nNewtonSum(f,p);\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#H" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "simplify(sum(r[k]^3, k=1..3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#H" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "R := NewtonSum(f,51)/NewtonSum(f,50);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG#\":\\2JmxbvI'Rp`@\"9[-x)e_=p()z*yr" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalf(R); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++I!\"*" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "Code for BernoulliRootApprox\n" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2172 "BernoulliRootApprox := pr oc()\n#########################################################\n# \+ #\n# \+ Variable List #\n# \+ #\n# f is the polynomial input \+ #\n# epsilon is the parameter that triggers termin ation #\n# q is an index variable #\n# breakFlag is used to exit loop #\n# formerVa lue is the former root aproximation #\n# currentValue is the current root aproximation #\n# \+ #\n############################################# ############ \nlocal f, epsilon, q, breakFlag, formerValue, currentVal ue;\n\n#########################################################\n# \+ #\n# Begin Checkin g and Assignment of Procedure Arguments #\n# \+ #\n##################################### ####################\n\nif nargs <> 2 then\nERROR(`BernoulliRootApprox requires two arguments`):\nelif not type(args[1],polynom) then\nERROR (`BernoulliRootApprox requires a polynomial as its first argument`):\n elif not type(args[2],positive) then\nERROR(`BernoulliRootApprox requi res a positive real number as its second argument`):\nfi:\n\nif nops(i ndets(args[1])) <> 1 then\nERROR(`BernoulliRootApprox expects its firs t argument to have one indeterminate`):\nfi:\n\nf := args[1]:\nepsilon := args[2]:\n\n###################################################### ###\n# #\n# En d Checking and Assignment of Procedure Arguments #\n# \+ #\n############################ #############################\n\n\nbreakFlag := 0:\nformerValue := eva lf(NewtonSum(p,1)/NewtonSum(p,0)):\n\n\n\nfor q from 2 to infinity whi le breakFlag < 2 do\ncurrentValue := evalf(NewtonSum(f,q)/NewtonSum(f, q-1)): \nif abs(currentValue-formerValue) < epsilon then\nbreakFlag := breakFlag+1:\nelse breakFlag := 0:\nfi:\nformerValue := currentValue \nod:\nRETURN(currentValue);\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# >%4BernoulliRootApproxG:6\"6(%\"fG%(epsilonG%\"qG%*breakFlagG%,formerV alueG%-currentValueGF&F&C*@(09#\"\"#-%&ERRORG6#%KBernoulliRootApprox~r equires~two~argumentsG4-%%typeG6$&9\"6#\"\"\"%(polynomG-F46#%jnBernoul liRootApprox~requires~a~polynomial~as~its~first~argumentG4-F96$&F<6#F2 %)positiveG-F46#%eoBernoulliRootApprox~requires~a~positive~real~number ~as~its~second~argumentG@$0-%%nopsG6#-%'indetsG6#F;F>-F46#%coBernoulli RootApprox~expects~its~first~argument~to~have~one~indeterminateG>8$F;> 8%FF>8'\"\"!>8(-%&evalfG6#*&-%*NewtonSumG6$%\"pGF>F>-F_o6$FaoFgn!\"\"? (8&F2F>%)infinityG2FfnF2C%>8)-F[o6#*&-F_o6$FXFfoF>-F_o6$FX,&FfoF>FdoF> Fdo@%2-%$absG6#,&F[pF>FinFdoFZ>Ffn,&FfnF>F>F>>FfnFgn>FinF[p-%'RETURNG6 #F[pF&F&" }}}{PARA 3 "" 0 "" {TEXT -1 1 "\n" }}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 35 " Description of BernoulliRootApprox" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 10 "Function: " } {TEXT -1 72 " BernoulliRootApprox - Calculate sum of powers of roots \+ of a polynomial" }}{PARA 0 "" 0 "usage" {TEXT 26 17 "Calling Sequence: " }{TEXT -1 37 "\n BernoulliRootApprox(poly,epsilon)" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 4 "\n " }{TEXT 23 9 "poly \+ - " }{TEXT -1 48 "any polynomial with one indeterminate\n epsilon" }{TEXT 23 23 " - any positive number" }}}{SECT 0 {PARA 0 "" 0 "synops is" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 159 "Bernou lliRootApprox(poly,epsilon) returns NewtonSum(n+1)/NewtonSum(n) wh ere the value of n is determined by epsilon and increases as epsilo n decreases. " }}}{SECT 0 {PARA 0 "" 0 "examples" {TEXT 26 9 "Examples :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f := x -> x^3-2*x^2-2*x -3; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"xG6\"6$%)operatorG %&arrowGF(,**$9$\"\"$\"\"\"*$F.\"\"#!\"#F.F3!\"$F0F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "r := [solve(f = 0, x )];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "BernoulliRootApprox(f(x),10^(-5));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+7&*****H!\"*" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "p := x -> x^4-31.02*x^3+231. 070*x^2+230.2926*x-2102.72504;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" pG:6#%\"xG6\"6$%)operatorG%&arrowGF(,,*$9$\"\"%\"\"\"*$F.\"\"$$!%-J!\" #*$F.\"\"#$\"'q5B!\"$F.$\"(EHI#!\"%$!*/DF5#!\"&F0F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(p(x),x=0..18);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7hn7$\"\"!$!1+++S]s-@!#77$$\"1+++]x[BR!#; $!1\"H9I\\\\'y>F+7$$\"1,+]iMHPtF/$!16J77uJ@=F+7$$\"1++]-vk<6!#:$!1;**) RtZ%)f\"F+7$$\"1++]<06/:F:$!1f+3$)f,M8F+7$$\"1++DTnt))=F:$!1$)\\s5(R(R 5F+7$$\"1++D'[M`C#F:$!1V5A%QAUY(!#87$$\"1++D'zqXh#F:$!1'GixlLrG%FL7$$ \"1++D;bV'*HF:$!1>O*=mZ>>*!#97$$\"1++D\"fvqP$F:$\"1/:b-$=hX#FL7$$\"1++ +&p5'oPF:$\"1bm7%e`K)eFL7$$\"1++]F]Z8TF:$\"1W))p)H'p;))FL7$$\"1,++&e5< ]%F:$\"1Y\"[eDvu>\"F+7$$\"1+++v-a\"*[F:$\"1+d\"pkF+7$$\"1+++Nw,9gF:$\"1 gs31aX+AF+7$$\"1+++SBmdjF:$\"1i0e*)3fjBF+7$$\"1++DEGSdnF:$\"1W&f;^B#=D F+7$$\"1*******)3C6rF:$\"1E4hej1)F:$\"1&4\"**=?dUFF+7$$\"1,+]iq%[D)F:$\"1*[>PUf%QFF+7$$\"1-](o& 3&>V)F:$\"1i:uQ%yis#F+7$$\"1++D^Y04')F:$\"10%e\"H781FF+7$$\"1****\\(Q: 6**)F:$\"1^FG3G0OEF+7$$\"1++D;\"pzQ*F:$\"1p5l)e]g_#F+7$$\"1,+v`9VL(*F: $\"1kCm\"\\n5S#F+7$$\"1++DCSl55FW$\"1sT_!=!oPAF+7$$\"1+++8)*>\\5FW$\"1 bXN\"==2/#F+7$$\"1++DZ$4p3\"FW$\"1$)fhCMlB=F+7$$\"1+]ig_RB6FW$\"1&H;'G &\\Uf\"F+7$$\"1++vYm!R;\"FW$\"1)4tX))H>K\"F+7$$\"1+++HzI+7FW$\"1X=hX96 m5F+7$$\"1++]nSi#FL7$$\"1+]iAT7\\8FW$\"1R*eydic'=FW7$$\"1+]7x K*pQ\"FW$!1a:#zkhy;#FL7$$\"1++](H4,]\"FW$!16M#*=J(QD(FL7$$\"1+]([X(>>:FW$!1: &3PU41r(FL7$$\"1++v:dGQ:FW$!16;yNx0?!)FL7$$\"1](of\"3vZ:FW$!1`#e%\\#4] 6)FL7$$\"1+v=;f@d:FW$!1]ZtTv4p\")FL7$$\"1]iS;5om:FW$!1B3ip8u!=)FL7$$\" 1+]i;h9w:FW$!1g(\\wAS$[\")FL7$$\"1+D\"[+TNf\"FW$!1nT7AO_ozFL7$$\"1+++$ *e$4h\"FW$!1+JgU'QUi(FL7$$\"1+](yRs3j\"FW$!1epJy!**Q,(FL7$$\"1++v-*33l \"FW$!16u(pc9x:'FL7$$\"1+++)zrko\"FW$!1:6/mz%3'RFL7$$\"1+]i#)e\\C " 0 "" {MPLTEXT 1 0 32 "BernoulliRootApprox (p(x),0.001);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+$=o(H " 0 "" {MPLTEXT 1 0 17 "fsolve(p(x)=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$!+++++H!\"*$\"+++++JF%$\"++++_8!\")$\"++++I " 0 "" {MPLTEXT 1 0 60 "r := 19.02: #time increases \+ greatly as r approaches 20.01 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "p := x -> expand((x-20.01)*(x+r)*(x^2-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG:6#%\"xG6\"6$%)operatorG%&arrowGF(-%'e xpandG6#*(,&9$\"\"\"$!%,?!\"#F2F2,(F1F2%\"aGF2$\"\"#F5F2F2,&*$F1F9F2! \"\"F2F2F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "p(x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,,*$%\"xG\"\"%\"\"\"*$F%\"\"#$!(-f\"Q! \"%*$F%\"\"$$!#**!\"#F%$\"#**F1$\"(-f!QF,F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "setTime := time():" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "BernoulliRootApprox(p(x),1);\n`elapsed time` := time( ) - setTime;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+qthc>!\")" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%-elapsed~timeG$\"&1e\"!\"$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "3" 0 }{VIEWOPTS 1 1 0 1 1 1803 }