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2 2 2 2 1 2 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot " -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 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 31 "Brent's method for root-finding" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 24.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 40 "load root-finding procedures including: " }{TEXT 0 5 "b rent" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 303 7 " roots.m" }{TEXT -1 37 " contains the code for the procedure " }{TEXT 0 5 "brent" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 " " {TEXT -1 122 "It can be read into a Maple session by a command simil ar to the one that follows, where the file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\\\Maple/procdrs /roots.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 22 "load inverse functions" }}{PARA 0 "" 0 "" {TEXT -1 84 " Typical file path to read Maple m-file containing the code for the inv erse function " }{TEXT 303 1 "K" }{TEXT -1 6 " for " }{XPPEDIT 18 0 " f(x)=x+exp(x)" "6#/-%\"fG6#%\"xG,&F'\"\"\"-%$expG6#F'F)" }{TEXT -1 29 " used in one of the examples." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "read \"K:\\\\Maple/procdrs/invfcns.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 50 "Inverse quadratic interpolat ion and Brent's method" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 49 "Suppose that we are trying to solve th e equation " }{XPPEDIT 18 0 "f(x)=0" "6#/-%\"fG6#%\"xG\"\"!" }{TEXT -1 44 ", and that we have three approximate values " }{TEXT 300 1 "a" }{TEXT -1 2 ", " }{TEXT 301 1 "b" }{TEXT -1 5 " and " }{TEXT 302 1 "c " }{TEXT -1 57 " for a particular solution. Assume that the three poin ts " }{XPPEDIT 18 0 "``(a, f(a)), ``(b, f(b))" "6$-%!G6$%\"aG-%\"fG6#F &-F$6$%\"bG-F(6#F," }{TEXT -1 5 " and " }{XPPEDIT 18 0 "``(c, f(c))" " 6#-%!G6$%\"cG-%\"fG6#F&" }{TEXT -1 37 ", are not in the same straight \+ line. " }}{PARA 0 "" 0 "" {TEXT -1 56 "Then it is possible to construc t a parabola of the form " }{XPPEDIT 18 0 "x=A*y^2+B*y+C" "6#/%\"xG,(* &%\"AG\"\"\"*$%\"yG\"\"#F(F(*&%\"BGF(F*F(F(%\"CGF(" }{TEXT -1 6 ", say " }{XPPEDIT 18 0 "x=g(y)" "6#/%\"xG-%\"gG6#%\"yG" }{TEXT -1 30 ", to \+ fit through these points." }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 402 320 320 {PLOTDATA 2 "63-%'CURVESG6$7R7$$\"\"!F)$!3`[GVPU 7.M!#<7$$\"3')*****fT:(z@!#=$!36I=n]uW C$F,7$$\"3&******4z_\"4iF0$!3CUnN$*e#f:$F,7$$\"3C+++S&phN)F0$!3![HQdR1 \"oIF,7$$\"3%*******)=)H\\5F,$!3#)o4F1QUzHF,7$$\"31+++[!3uC\"F,$!3eUNI \">7g*GF,7$$\"35+++J$RDX\"F,$!3!Q?WC;#Q3GF,7$$\"3'******zR'ok;F,$!3J/I (z=Ujr#F,7$$\"31+++1J:w=F,$!3FU:4)H)4BEF,7$$\"3))*****zgsO4#F,$!3j2E*> /Wb_#F,7$$\"3!)*****R!RE&G#F,$!3uAcd`)o\"QCF,7$$\"3\")*****\\K]4]#F,$! 32M\"4q2%4QBF,7$$\"3;+++vB_/#F,7$$\"3;+++347TLF,$!3%e.%p:X$ *G>F,7$$\"3()*****HjM?`$F,$!3m)*)=*458J=F,7$$\"31+++\"o7Tv$F,$!3EhCgX. \"[r\"F,7$$\"3%)*****HQ*o]RF,$!3)o`fIM)R4;F,7$$\"3u*****4=lj;%F,$!3Y3W qH!**3\\\"F,7$$\"3M+++V&RY2aF,$! 3-&eD(pNomtF07$$\"3#)*****zdWZh&F,$!3q#o,X9KJ&fF07$$\"3,+++\\y))GeF,$! 3(GXTm!\\!>V%F07$$\"3D+++i_QQgF,$!3[)Q.O)[jvGF07$$\"3c******zZ3TiF,$!3 g^y]zuG'H\"F07$$\"3))*****f.[hY'F,$\"3#fjAp*=$Rc&!#>7$$\"31+++#Qx$omF, $\"3yK_BXSqEBF07$$\"3Q+++u.I%)oF,$\"3//;sRpT^VF07$$\"3O+++(pe*zqF,$\"3 I3q/HXCNjF07$$\"3=+++C\\'QH(F,$\"3F!3:$*z;Yr)F07$$\"3Q+++8S8&\\(F,$\"3 ]J#f\">,oA6F,7$$\"3t*****\\?=bq(F,$\"3.L3=D/hF9F,7$$\"3g*****fqi$3yF,$ \"3Q**p#=a*\\s#F,7$$\"3')*****Hgb8>)F,$\"3%*G&G@-,@z#F,7$$\"3w+++6 n$y>)F,$\"3qY*)fgy*f*GF,7$%*undefinedGF`z-%'COLOURG6&%$RGBG$\"#5!\"\"F (F(-F$6$7$7$F(F(7$$FfzF)F(-Fbz6&FdzF)F)F)-F$6&7%7$$\"\"\"F)$!\"$F)7$$ \"\"&F)$FgzF)7$$\"\")F)$\"\"#F)-%'SYMBOLG6#%'CIRCLEGF^[l-%&STYLEG6#%&P OINTG-F$6&Fb[l-Fb\\l6#%(DIAMONDGF^[lFe\\l-F$6&Fb[l-Fb\\l6#%&CROSSGF^[l Fe\\l-F$6&7#7$$\"3M+++++++kF,F(Fa\\l-Fbz6&FdzF(F($\"*++++\"!\")Fe\\l-F $6&Fe]lF[]lFi]lFe\\l-F$6&Fe]lF`]lFi]lFe\\l-%%TEXTG6$7$$\"#=FgzFf[lQ)(a ,f(a))6\"-Fc^l6$7$$\"#eFgzF[\\lQ)(b,f(b))Fi^l-Fc^l6$7$$\"#()FgzF_\\lQ) (c,f(c))Fi^l-Fc^l6$7$$\"#'*Fgz$!\"#FgzQ\"xFi^l-Fc^l6%7$$\"#oFgzF[`lQ&( v,0)Fi^lFi]l-Fc^l6%7$$\"#uFgz$\"#FFgzQ)x~=~g(y)Fi^l-Fbz6&FdzF[^lF(F(-% *AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%F]`lQ!Fi^l-%%FONTG6#%(DEFAULTG-%%V IEWG6$;F(F][lFial" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu rve 13" "Curve 14" }}{TEXT -1 1 " " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Then the curve " }{XPPEDIT 18 0 "x=g(y) " "6#/%\"xG-%\"gG6#%\"yG" }{TEXT -1 108 " may provide at least a local approximation for the curve in the neighbourhood of the three points. In fact " }{XPPEDIT 18 0 "x=g(y)" "6#/%\"xG-%\"gG6#%\"yG" }{TEXT -1 24 " may be thought of as a " }{TEXT 270 51 "local approximation for t he inverse of the function" }{TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)" "6#- %\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 40 "A new ap proximation for the solution of " }{XPPEDIT 18 0 "f(x) = 0;" "6#/-%\"f G6#%\"xG\"\"!" }{TEXT -1 18 " is then given by " }{XPPEDIT 18 0 "v = g (0);" "6#/%\"vG-%\"gG6#\"\"!" }{TEXT -1 7 ". Thus " }{TEXT 272 1 "v" } {TEXT -1 53 " will be the constant coefficient C of the quadratic " } {XPPEDIT 18 0 "A*y^2+B*y+C" "6#,(*&%\"AG\"\"\"*$%\"yG\"\"#F&F&*&%\"BGF &F(F&F&%\"CGF&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 70 "In term s of the coordinates of the three original points the equation " } {XPPEDIT 18 0 "x=g(y)" "6#/%\"xG-%\"gG6#%\"yG" }{TEXT -1 24 ", the new approximation " }{TEXT 273 1 "v" }{TEXT -1 31 " can be calculated as \+ follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "First compute the ratios:" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "r = f(b)/f(c), s = f(b)/f(a), t = f(a)/f(c)" "6%/%\"rG* &-%\"fG6#%\"bG\"\"\"-F'6#%\"cG!\"\"/%\"sG*&-F'6#F)F*-F'6#%\"aGF./%\"tG *&-F'6#F6F*-F'6#F-F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 9 "Th en let " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p = s*(t*( r-t)*(c-b)-(1-r)*(b-a))" "6#/%\"pG*&%\"sG\"\"\",&*(%\"tGF',&%\"rGF'F*! \"\"F',&%\"cGF'%\"bGF-F'F'*&,&F'F'F,F-F',&F0F'%\"aGF-F'F-F'" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "q = (t-1)*(r-1)*(s-1)" "6#/%\"qG*(,&%\" tG\"\"\"F(!\"\"F(,&%\"rGF(F(F)F(,&%\"sGF(F(F)F(" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 32 "Then the new approximation is: " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v = b+p/q" "6#/%\"vG, &%\"bG\"\"\"*&%\"pGF'%\"qG!\"\"F'" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "Ve rification" }}{PARA 0 "" 0 "" {TEXT -1 89 "The symbolic expression for the new approximation v given by the formulas above is . . . " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 218 "unassign('f','a','b','c','r','s','t','p','q','v');\nr := f(b)/f(c ); s := f(b)/f(a); t := f(a)/f(c);\np := s*(t*(r-t)*(c-b)-(1-r)*(b-a)) ;\np := normal(p);\nq := (t-1)*(r-1)*(s-1);\nq := normal(q);\nv := b+p /q;\nv := normal(v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG*&-%\"fG 6#%\"bG\"\"\"-F'6#%\"cG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG *&-%\"fG6#%\"bG\"\"\"-F'6#%\"aG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"tG*&-%\"fG6#%\"aG\"\"\"-F'6#%\"cG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG*(-%\"fG6#%\"bG\"\"\"-F'6#%\"aG!\"\",&**F+F*-F'6# %\"cGF.,&*&F&F*F1F.F**&F+F*F1F.F.F*,&F3F*F)F.F*F**&,&F*F*F5F.F*,&F)F*F -F.F*F.F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG,$**-%\"fG6#%\"bG\" \"\",2*(-F(6#%\"aGF+F'F+%\"cGF+!\"\"*(F.F+F'F+F*F+F+*&)F.\"\"#F+F1F+F+ *&F5F+F*F+F2*&)-F(6#F1F6F+F*F+F+*&F9F+F0F+F2*(F:F+F'F+F*F+F2*(F:F+F'F+ F0F+F+F+F.F2F:!\"#F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG*(,&*&-% \"fG6#%\"aG\"\"\"-F)6#%\"cG!\"\"F,F,F0F,,&*&-F)6#%\"bGF,F-F0F,F,F0F,,& *&F3F,F(F0F,F,F0F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG*,,&-%\"fG 6#%\"aG\"\"\"-F(6#%\"cG!\"\"F+,&F,F/-F(6#%\"bGF+F+,&F1F+F'F/F+F'F/F,! \"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG,&%\"bG\"\"\"*,-%\"fG6#F& F',2*(-F*6#%\"aGF'F)F'%\"cGF'!\"\"*(F.F'F)F'F&F'F'*&)F.\"\"#F'F1F'F'*& F5F'F&F'F2*&)-F*6#F1F6F'F&F'F'*&F9F'F0F'F2*(F:F'F)F'F&F'F2*(F:F'F)F'F0 F'F'F',&F.F'F:F2F2,&F:F2F)F'F2,&F)F'F.F2F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG,$**,.*(%\"bG\"\"\")-%\"fG6#%\"aG\"\"#F*-F-6#%\"c GF*!\"\"*(F)F*F,F*)F1F0F*F**(F,F*)-F-6#F)F0F*F3F*F4*(F9F*F+F*F3F*F**(F 9F*F6F*F/F*F4*(F1F*F8F*F/F*F*F*,&F,F*F1F4F4,&F1F4F9F*F4,&F9F*F,F4F4F4 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "On \+ the other hand, the inverse interpolating quadratic can be obtained by using the Maple procedure " }{TEXT 0 6 "interp" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 44 "The new approximation v for the solution \+ of " }{XPPEDIT 18 0 "f(x)=0" "6#/-%\"fG6#%\"xG\"\"!" }{TEXT -1 38 " ca n then be obtained by substituting " }{XPPEDIT 18 0 "y=0" "6#/%\"yG\" \"!" }{TEXT -1 42 ", which extracts the constant coefficient." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "interp([f(a),f(b),f(c)],[a,b,c],y):\nu := subs(y=0,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG,$**,.*(%\"bG\"\"\")-%\"fG6#%\"aG\"\"# F*-F-6#%\"cGF*!\"\"*(F)F*F,F*)F1F0F*F**(F,F*)-F-6#F)F0F*F3F*F4*(F9F*F+ F*F3F*F**(F9F*F6F*F/F*F4*(F1F*F8F*F/F*F*F*,&F,F*F1F4F4,&F1F4F9F*F4,&F9 F*F,F4F4F4" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "is(u=v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "Example" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "We can us e the method outlined above to estimate the solution of the equation \+ " }{XPPEDIT 18 0 "cos(x)=x" "6#/-%$cosG6#%\"xGF'" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 20 "The solution is the " }{TEXT 297 1 "x" } {TEXT -1 27 " intercept of the graph of " }{XPPEDIT 18 0 "f(x)=cos(x)- x" "6#/-%\"fG6#%\"xG,&-%$cosG6#F'\"\"\"F'!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "f := x -> cos(x)-x;\nplot(f(x),x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%$cosG6#9$ \"\"\"F0!\"\"F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"\"!F)$\"\"\"F)7$$\"3emmm;arz@!#>$\"3 yF$RF(Hlz(*!#=7$$\"3[LL$e9ui2%F/$\"37N&z-tlSe*F27$$\"3nmmm\"z_\"4iF/$ \"3W7d3CT\")f$*F27$$\"3[mmmT&phN)F/$\"3!GGV9d!\\H\"*F27$$\"3CLLe*=)H\\ 5F2$\"3qWJ)F27$$ \"3w***\\i5`h(=F2$\"3%=.Rm,l$[zF27$$\"3WLLL3En$4#F2$\"3mXmV)e`zo(F27$$ \"3qmm;/RE&G#F2$\"3_(>-;(*[ZX(F27$$\"3\")*****\\K]4]#F2$\"3OIr=M(Qz=(F 27$$\"3$******\\PAvr#F2$\"3^$y#HF!)\\:pF27$$\"3)******\\nHi#HF2$\"33wd U\"pv'[mF27$$\"3jmm\"z*ev:JF2$\"3NxnV4:w-kF27$$\"3?LLL347TLF2$\"3XhDUO w*e5'F27$$\"3,LLLLY.KNF2$\"3-2**Q3(f1&eF27$$\"3w***\\7o7Tv$F2$\"3!*=d \"))Qc%\\bF27$$\"3'GLLLQ*o]RF2$\"3=zM,e4,z_F27$$\"3A++D\"=lj;%F2$\"3wl 72'\\(=y\\F27$$\"31++vV&Rp=yxo%F27$$\"3WLL$e9Ege%F2$\"3qH G'\\u$p!Q%F27$$\"3GLLeR\"3Gy%F2$\"3?iuT>o1&4%F27$$\"3cmm;/T1&*\\F2$\"3 G<#H,%o7$y$F27$$\"3&em;zRQb@&F2$\"3C7%yU?A\\X$F27$$\"3\\***\\(=>Y2aF2$ \"3I;tPmgylJF27$$\"39mm;zXu9cF2$\"3)Gs&32_'*\\GF27$$\"3l******\\y))GeF 2$\"3zf*yh`s)>DF27$$\"3'*)***\\i_QQgF2$\"3=?2p'=OK>#F27$$\"3@***\\7y%3 TiF2$\"37EmZg%fP(=F27$$\"35****\\P![hY'F2$\"3wiMfU58::F27$$\"3kKLL$Qx$ omF2$\"3!e0;RDP%*=\"F27$$\"3!)*****\\P+V)oF2$\"3Eqvf6mV\"Q)F/7$$\"3?mm \"zpe*zqF2$\"3_.b:3U3n^F/7$$\"3%)*****\\#\\'QH(F2$\"31YoJ5**o>;F/7$$\" 3GKLe9S8&\\(F2$!31.%*\\zMH\\2))F/7$$\"3a***\\7`Wl7)F2$!3U[E7M.\"3D\"F27$ $\"3#pmmm'*RRL)F2$!3ra+;alE5;F27$$\"3Qmm;a<.Y&)F2$!3clhj]G&3)>F27$$\"3 =LLe9tOc()F2$!3;S>%HA(G^BF27$$\"3u******\\Qk\\*)F2$!3On>f=y<%p#F27$$\" 3CLL$3dg6<*F2$!3O1h.L&R+4$F27$$\"3ImmmmxGp$*F2$!3aBZ8eMjYMF27$$\"3A++D \"oK0e*F2$!3wesuMiRHQF27$$\"3A++v=5s#y*F2$!3Wq]0DP:)>%F27$F*$!3M-'=8%p (pf%F2-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!Fa[l-%% VIEWG6$;F(F*%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 71 "First apply the formulas given above with a = 0.6, b = 0.7 and c = 0.8." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "f := x -> cos(x)-x;\na := 0.6; b := 0.7; c := 0.8;\nr := f(b)/f(c); s := f(b)/f(a); t := f(a)/f(c);\np := s*(t *(r-t)*(c-b)-(1-r)*(b-a));\nq := (t-1)*(r-1)*(s-1);\nv := b+p/q;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,&-%$cosG6#9$\"\"\"F0!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"aG$\"\"'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG$\"\"(!\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG$\"\")!\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"rG$!+nH[xi!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG$\"+EFexG!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"tG$!+) f7:=#!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG$!+K4xV9!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG$!+au[)o$!\"*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"vG$\"+#GE9R(!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 79 "To obtain a better approximation now \+ let a = 0.7, b = 0.7391426282 and c = 0.8." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "a := 0.7;\nb := 0 .7391426282;\nc := 0.8;\nr := f(b)/f(c); s := f(b)/f(a); t := f(a)/f(c );\np := s*(t*(r-t)*(c-b)-(1-r)*(b-a));\nq := (t-1)*(r-1)*(s-1);\nv := b+p/q;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG$\"\"(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG$\"+#GE9R(!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG$\"\")!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"rG$\"+/^v:$*!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG$!+#G&*R[\" !#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"tG$!+nH[xi!#5" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"pG$\"+$)f_n$*!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG$!+Z_kG;!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"vG$\"+46&3R(!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 110 "A third iteration with a = 0.7, b = 0.7390851109 and c = 0.7391426282 gives the solution correct to 10 digits." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "a := 0.7;\nb := 0.7390851109;\nc := 0.7391426282;\nr := f(b)/f(c); s := f( b)/f(a); t := f(a)/f(c);\np := s*(t*(r-t)*(c-b)-(1-r)*(b-a));\nq := (t -1)*(r-1)*(s-1);\nv := b+p/q;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"a G$\"\"(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG$\"+46&3R(!#5" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG$\"+#GE9R(!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG$!+t8JwQ!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"sG$\"+(yFCv&!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"tG$!++fcQn !\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG$!+o]k/:!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG$!+qz<^n!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG$\"+K8&3R(!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 19 "The value given by " }{TEXT 0 6 "fsolve " }{TEXT -1 18 " agrees with this." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "evalf(fsolve(f(x)=0,x=0.74)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+K8&3R(!#5" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 95 "Brent's root-finding method combines inverse quadratic interpol ation with the bisection method." }}{PARA 0 "" 0 "" {TEXT -1 202 "To o btain each new approximation an attempt is first made to use inverse q uadratic interpolation. If the value given by this method is not judge d to be acceptable, then a bisection step is made instead." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "A procedure implementing Brent' s method: " }{TEXT 0 5 "brent" }}{PARA 0 "" 0 "" {TEXT -1 120 "The rou tine for Brent's root-finding is adapted from code given in \"Numerica l Recipes in C\", Cambridge University press." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "brent: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }{TEXT 260 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 261 2 " \+ " }{TEXT -1 22 " brent( eqn, start ) " }{TEXT 264 1 "\n" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 4 " \+ " }}{PARA 0 "" 0 "" {TEXT 23 1 " " }{TEXT -1 76 "eqn - an e quation or expression involving a single variable, say x," }}{PARA 0 " " 0 "" {TEXT -1 22 " " }{TEXT 270 2 "OR" }{TEXT -1 73 " a function of the form x -> f(x), where f(x) evaluates to a re al number," }}{PARA 0 "" 0 "" {TEXT -1 22 " " } {TEXT 270 2 "OR" }{TEXT -1 71 " a numerical procedure which evaluates \+ to a real floating point number." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 18 " start - " }{TEXT 263 86 "a rang e x=a..b (or simply a..b when the1st argument is a procedure), where a and b are" }}{PARA 0 "" 0 "" {TEXT 266 69 " two di stinct initial approximations for the root " }{TEXT 270 22 "which brac ket the root" }{TEXT -1 1 "," }{TEXT 267 22 " and x is the variable" } }{PARA 0 "" 0 "" {TEXT 265 49 " appearing in the 1s t argument" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 " \+ " }{TEXT 270 2 "OR" }{TEXT -1 49 " a single initial approximat ion x=a for the root." }}{PARA 0 "" 0 "" {TEXT 268 67 " \+ (Omit \"x=\" when the1st argument is a procedure)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 12 "Description:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The proce dure " }{TEXT 0 5 "brent" }{TEXT -1 28 " attempts to find a root of " }{XPPEDIT 18 0 "f(x) = 0;" "6#/-%\"fG6#%\"xG\"\"!" }{TEXT -1 21 " by a Brent's method." }}{PARA 0 "" 0 "" {TEXT -1 53 "When given a single s tarting value in the form x=a, " }{TEXT 0 5 "brent" }{TEXT -1 56 " at tempts to find a root of f(x) = 0 located near x = a." }}{PARA 0 "" 0 "" {TEXT -1 79 "When given two strarting values in the form of an inte rval in the form x=a..b, " }{TEXT 0 5 "brent" }{TEXT -1 59 " attempts \+ to find a root located inside the given interval." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 8 "Options:" }{TEXT -1 1 "\n " }}{PARA 0 "" 0 "" {TEXT -1 149 "maxiterations=n or maxiter=n\nThis o ption can be used to override the default value of Digits*10 for the m aximum number of iterations to be performed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "info=true/false/0/1/2/3\n\"inf o=0\" is the same as \"info=false\" and \"info=1\" is the same as \"in fo=true\"." }}{PARA 0 "" 0 "" {TEXT -1 119 "This option allows the pro gress of the procedure to be monitored by printing 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 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 16 "How to activate:" }{TEXT -1 134 "\nTo \+ make the procedure active place the cursor anywhere after the prompt [ > and press [Enter].\nYou can then close up the subsection." }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "brent: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14451 "b rent := proc(ff,start)\n local Options,x,a,b,c,fa,fb,fc,e,d,p,q,r,s, t,h,\n eps,tol,i,maxit,prntflg,f,fn,rs,vars,\n lmr,sf, delta,savea,noroot,numpts,proctype,m,n,\n diverg,tiny,small,tr iedzero,f0,u,prevu,quad,\n sgn,sgna,sgnb,sgnc,digtinc,alt,fpre vb,saveDigits,\n extraDigits,startDigits,minute,val,printg;\n \n if nargs<2 then\n error \"at least 2 arguments are required; the basic syntax is: 'brent(f(x),x=a)' or 'brent(f(x),x=a..b)'.\"\n \+ end if;\n \n if type(ff,procedure) then\n if nops([op(1,eval( ff))])<>1 then\n error \"the 1st argument, %1, is invalid .. i t should be a procedure with a single argument\",ff;\n end if;\n \+ proctype := true;\n if type(start,realcons) or type(start,re alcons..realcons) then\n rs := start\n else\n err or \"the 2nd argument, %1, is invalid .. when the 1st argument is a pr ocedure, the 2nd argument should be a real constant or a range of real constants\",start;\n end if;\n elif type(ff,algebraic) or type (ff,equation) then\n if type(ff,equation) then\n lmr := l hs(ff)-rhs(ff);\n sf := traperror(simplify(lmr));\n if sf<>lasterror then\n f := sf;\n else\n \+ f := lmr;\n end if;\n else\n f := ff;\n end \+ if;\n vars := indets(f,name) minus indets(f,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 a n expression or an equation which depends only on a single variable\", ff;\n end if;\n end if;\n if type(start,name=realcon s) or\n type(start,name=realcons..realcons) then\n \+ proctype := false;\n x := op(1,start);\n if not m ember(x,vars) then\n error \"the 1st argument, %1, is inval id .. it should be an expression or an equation which depends only on \+ the variable %2\",ff,x;\n end if;\n rs := op(2,start); \n else\n error \"the 2nd argument, %1, is invalid .. it \+ should have the form 'x=a' or 'x=a..b', to provide one or two real num ber starting values\",start;\n end if;\n else\n error \"th e 1st argument, %1, is invalid .. it should be an algebraic expression in a single variable, an equation in a single variable, or a procedur e with a single real argument\",ff;\n end if;\n \n # Get the opt ions\n # Set the default values to start with.\n maxit := Digits*1 0:\n prntflg := 0;\n if nargs>2 then\n Options:=[args[3..narg s]];\n if not type(Options,list(equation)) then\n error \+ \"each optional argument must be an equation\"\n end if;\n i f hasoption(Options,'maxiterations','maxit','Options') then\n \+ if not type(maxit,posint) then\n error \"\\\"maxiterations \\\" must be a positive integer\"\n end if;\n elif hasopt ion(Options,'maxiter','maxit','Options') then\n if not type(ma xit,posint) then\n error \"\\\"maxiter\\\" must be a positi ve integer\"\n end if;\n end if;\n if hasoption(Opti ons,'info','prntflg','Options') then\n if not member(prntflg, \{true,false,0,1,2\}) then\n error \"\\\"info\\\" must be f alse <-> 0, true <-> 1 or 2\"\n end if;\n if prntflg=f alse then prntflg := 0\n elif prntflg=true then prntflg := 1 e nd if; \n end if;\n if nops(Options)>0 then\n error \+ \"%1 is not a valid option for %2 .. the recognised options are \\\"ma xiterations\\\",(or \\\"maxiter\\\") and \\\"info\\\"\",op(1,Options), procname;\n end if;\n end if;\n\n # Increase precision for th e computation\n saveDigits := Digits;\n extraDigits := iquo(Digits ,3);\n Digits := Digits + min(max(extraDigits,5),10);\n startDigit s := 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 if type(rs,realcons..realcons) then\n a := evalf( min(op(rs)));\n b := evalf(max(op(rs)));\n if a=b then\n \+ error \"when two starting values are given, they must be distinct \"\n end if;\n numpts := 2;\n else # type(rs,realcons)\n \+ a := evalf(rs);\n numpts := 1;\n end if;\n\n # procedure \+ to print a float with d+7 characters\n printg := proc(x::float,d::po sint)\n local lg,wdth,dcpl,fmt,f,e;\n if x<>0 then lg := ilo g10(x) else lg := 0 end if;\n wdth := convert(d+7,string);\n \+ if lg>-7 and lg0 then \n delta := abs(a)* 10^(-m);\n n := ceil(12*m);\n else\n delta := 10^ (-2*m);\n n := ceil(20*m);\n end if;\n sgn := proc(_ u)\n if _u > 0 then 1 elif _u < 0 then -1 else 0 end if;\n \+ end proc;\n alt := false;\n for i to n do\n if i=1 then\n c := a\n elif alt then\n c := a \+ + delta\n else\n c := b - delta\n end if;\n fc := traperror(evalf(fn(c)));\n if fc=lasterror or n ot type(fc,numeric) then\n error \"evaluation failed at %1 \",c;\n end if;\n sgnc := sgn(fc);\n if sgnc=0 then # increase precision\n while digtinc<=saveDigits do\n Digits := Digits + extraDigits;\n if irem (prntflg,2)=1 then\n printf(\" ** increasing work ing precision to %d digits **\\n\",Digits);\n elif prntf lg=2 then\n print(`** increasing working precision to `,Digits,` digits **`);\n end if;\n if not proctype then\n fn := unapply(evalf(f),x)\n \+ end if;\n fc := traperror(evalf(fn(c)));\n \+ if fc=lasterror or not type(fc,numeric) then\n \+ error \"evaluation failed at %1\",evalf[saveDigits](c);\n \+ end if;\n sgnc := sgn(fc);\n digtinc := digtinc + 1;\n if sgnc<>0 then break end if;\n \+ end do; \n end if;\n if alt then\n b \+ := c;\n fb := fc;\n sgnb := sgnc;\n else \n a := c;\n fa := fc;\n sgna := sgnc ;\n end if;\n if sgnc=0 then break end if;\n i f i>1 then\n if prntflg=1 then\n printf(\" \" );\n printg(a,startDigits);\n if sgna>0 th en\n printf(\" -> [+] \")\n else\n \+ printf(\" -> [-] \")\n end if; \n \+ printg(b,startDigits);\n if sgnb>0 then\n \+ printf(\" -> [+] \\n\")\n else\n \+ printf(\" -> [-] \\n\")\n end if; \n \+ elif prntflg=2 then\n if sgna>0 and sgnb>0 then \n \+ print(evalf[startDigits](a),` -> [+] `,evalf[startDi gits](b),` -> [+] `);\n elif sgna>0 and sgnb<0 then \n print(evalf[startDigits](a),` -> [+] `,evalf[start Digits](b),` -> [-] `);\n elif sgna<0 and sgnb>0 then \+ \n print(evalf[startDigits](a),` -> [-] `,evalf[sta rtDigits](b),` -> [+] `);\n else # sgna>0 and sgnb<0 \+ \n print(evalf[startDigits](a),` -> [-] `,evalf[sta rtDigits](b),` -> [-] `);\n end if;\n end i f;\n if sgna*sgnb < 0 then\n noroot := false; \n if prntflg=1 then\n printf(cat(\" \" $35));\n printf(\"pair OK\\n\\n\");\n e lif prntflg=2 then\n print(`pair OK`); print(``);\n \+ end if;\n break;\n end if;\n \+ end if;\n delta := delta*1.587401052;\n alt := no t alt;\n end do;\n if noroot then\n if i=1 then\n \+ return evalf[saveDigits](a);\n else\n erro r \"could not construct an interval containing a root; try a different starting criterion\"\n end if;\n end if;\n else\n \+ fa := traperror(evalf(fn(a)));\n if fa=lasterror or not type(fa, numeric) then\n error \"evaluation failed at %1\",a;\n en d if;\n fb := traperror(evalf(fn(b)));\n if fb=lasterror or \+ not type(fb,numeric) then\n error \"evaluation failed at %1\", b;\n end if;\n if (fa>0 and fb>0) or (fa<0 and fb<0) then\n \+ \011\011 error \"interval does not appear to bracket a root\"\n \+ \011 end if;\n end if;\n\n eps := Float(1,-saveDigits-min(iquo( Digits,10),2));\n tiny := abs(b-a)*Float(5,-saveDigits);\n minute \+ := Float(1,-2*saveDigits);\n diverg := 0;\n triedzero := false;\n \+ \n c := b;\n fc := fb;\n\n for i from 1 to maxit do\n if (f b>0 and fc>0) or (fb<0 and fc<0) then\n # rename and adjust er ror bound\n c := a;\n fc := fa;\n d := b-a;\n \+ e := d;\n end if;\n if abs(fc)1 then \n if prntflg=1 then\n printf(\" best value so far : \");\n printg(b,startDigits); \n prin tf(\"\\n root-bracketing interval: \\n \");\n if b `, evalf[startDigits](b));\n if \+ b `,evalf[star tDigits](b..c))\n else\n print(`root-bracketing \+ interval --> `,evalf[startDigits](c..b))\n end if;\n \+ print(``);\n end if;\n end if;\n tol := eps*abs (b);\n h := (c-b)/2;\n\n if i>6 and not triedzero \n \+ and abs(b)0 then break end if;\n end do; \n \+ end if;\n\011 if fb=0 then\n if not proctype then\n \+ val := `convert/real_rat`(b,Digits);\n if eval(f,x=v al)=0 then return evalf[saveDigits](b) end if;\n else\n \+ WARNING(\"the accuracy of the result is uncertain.\");\n \+ return evalf[saveDigits](b)\n end if;\n end if;\n \+ quad := true;\n if abs(e)>= tol and abs(fa)>abs(fb) then\n \+ # attempt quadratic interpolation\n s := fb/fa;\n i f a=c then\n p := 2*h*s;\n q := 1-s;\n e lse\n t := fa/fc;\n r := fb/fc;\n p : = s*(2*h*t*(t-r)-(b-a)*(r-1));\n q := (t-1)*(r-1)*(s-1);\n \+ end if;\n if p>0 then q := -q end if;\n p := a bs(p);\n if 2*ptol then\n \+ b := b + d;\n else\n # perturb b towards c\n \+ if h>=0 then\n b := b + tol;\n else\n b := b - tol;\n end if;\n end if;\n if prntflg=1 then \n if quad then\n printf(\" step %d inverse quad ratic interpolation\\n\",i);\n else\n printf(\" st ep %d bisection\\n\",i);\n end if;\n printf(\" curr ent value: \");\n printg(b,startDigits);\n printf( \"\\n\"); \n elif prntflg=2 then\n if quad then \n print(`step `||i||` -- inverse quadratic interpolation`) ;\n else\n print(`step `||i||` -- bisection`);\n \+ end if;\n print(``);\n print(`current valu e ------------> `,evalf[startDigits](b));\n end if;\n \n \+ fprevb := fb;\n # get new function value\n fb := traperro r(evalf(fn(b)));\n if fb=lasterror or not type(fb,numeric) then\n error \"evaluation failed at %1\",evalf[saveDigits](b);\n \+ end if;\n\n prevu := u;\n u := abs(fb-fprevb);\n if i >6 then\n if abs(u)>abs(prevu) then \n diverg := di verg+1\n else\n diverg := 0;\n end if;\n \+ if diverg>3 then\n error \"there appears to be a disc ontinuity near %1; try a different starting criterion\",evalf[iquo(sav eDigits,2)](b);\n end if;\n end if;\n end do;\n if me mber(prntflg,\{0,1\}) then\n printf(\" last iteration gives \"); \n printg(evalf[saveDigits](b),saveDigits);\n else\n print (`last iteration gives `,evalf[saveDigits](b));\n end if;\n error \+ \"reached max, %1, iterations without convergence\",maxit;\nend proc: " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Exa mples 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 5 "brent" }{TEXT -1 10 ": examples" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examp le 1" }}{PARA 0 "" 0 "" {TEXT -1 214 "A single starting value can be g iven. It should not be too far away from the desired root, but there i s some leeway as a search will be made for a suitable root bracketing \+ interval in the neighbourhood of the root." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "brent(cos(x)=x,x=0 .75,info=true);" }}{PARA 6 "" 1 "" {TEXT -1 61 " 0.75000000000000 0 -> [-] 0.762460192365855 -> [-] " }}{PARA 6 "" 1 "" {TEXT -1 61 " 0.742680869896175 -> [-] 0.762460192365855 -> [-] " }} {PARA 6 "" 1 "" {TEXT -1 61 " 0.742680869896175 -> [-] 0.774 078587192393 -> [-] " }}{PARA 6 "" 1 "" {TEXT -1 61 " 0.724237817 725978 -> [+] 0.774078587192393 -> [-] " }}{PARA 6 "" 1 "" {TEXT -1 42 " pair OK" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 42 " step 1 inverse quadra tic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: \+ 0.738971620603820" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value s o far: 0.738971620603820" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-b racketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 0.7389716 20603820 .. 0.774078587192393" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 42 " step 2 inverse quadratic interpolatio n" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 0.73908514 2836735" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 0.73 9085142836735" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interv al: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 0.738971620603820 .. \+ 0.739085142836735" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 42 " step 3 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 0.739085133214919" }} {PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 0.7390851332149 19" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }} {PARA 6 "" 1 "" {TEXT -1 51 " 0.739085133214919 .. 0.73908 5142836735" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 42 " step 4 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 0.739085133222310" }}{PARA 6 " " 1 "" {TEXT -1 43 " best value so far: 0.739085133214919" }} {PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 " " 1 "" {TEXT -1 51 " 0.739085133214919 .. 0.73908513322231 0" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+K8&3R(!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "It is more efficient to give a root-bracketing interval, \+ even if it is fairly wide." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "brent(cos(x)=x,x=0.5..0.8,info=2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Jstep~1~--~inverse~quadratic~inte rpolationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>current~value~~------------>~G$\"0\"ejyFfbt!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%>best~value~so~far~~-------->~G$\"0\" ejyFfbt!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>root-bracketing~interv al~-->~G;$\"0\"ejyFfbt!#:$\"\")!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Jstep~2~--~inverse~quadratic ~interpolationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>current~value~~------------>~G$\"0F*y=7R!R(!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>best~value~so~far~~-------->~G$\"0 F*y=7R!R(!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>root-bracketing~inte rval~-->~G;$\"0F*y=7R!R(!#:$\"\")!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Jstep~3~--~inverse~quadr atic~interpolationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>current~value~~------------>~G$\"0j6![8&3R (!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>best~value~so~far~~--------> ~G$\"0j6![8&3R(!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>root-bracketin g~interval~-->~G;$\"0F*y=7R!R(!#:$\"0j6![8&3R(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Jstep~4~--~inve rse~quadratic~interpolationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%>current~value~~------------>~G$\"0W ^@L^3R(!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>best~value~so~far~~--- ----->~G$\"0W^@L^3R(!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>root-brac keting~interval~-->~G;$\"0W^@L^3R(!#:$\"0j6![8&3R(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Jstep~5~--~i nverse~quadratic~interpolationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>current~value~~------------>~G$ \"0NDAL^3R(!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>best~value~so~far~ ~-------->~G$\"0W^@L^3R(!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>root- bracketing~interval~-->~G;$\"0W^@L^3R(!#:$\"0NDAL^3R(F'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+K8&3R (!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 90 "Giving a single starting value close to the root can still produce efficient root-finding." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "evalf(bren t(sin(1/x),x=0.0102,info=true),15);" }}{PARA 6 "" 1 "" {TEXT -1 71 " \+ 0.010200000000000000000 -> [-] 0.010275921304666481649 -> [+] \+ " }}{PARA 6 "" 1 "" {TEXT -1 42 " pa ir OK" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 42 " step 1 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 48 " current value: 0.010267605243770446177" }}{PARA 6 "" 1 "" {TEXT -1 48 " best value so far: 0.010267605243770446177" }} {PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 " " 1 "" {TEXT -1 61 " 0.010267605243770446177 .. 0.0102759213 04666481649" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 42 " step 2 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 48 " current value: 0.010268061590923471477" }} {PARA 6 "" 1 "" {TEXT -1 48 " best value so far: 0.01026806159092 3471477" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 61 " 0.010267605243770446177 .. 0 .010268061590923471477" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 42 " step 3 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 48 " current value: 0.01026806084466920306 3" }}{PARA 6 "" 1 "" {TEXT -1 48 " best value so far: 0.010268060 844669203063" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interva l: " }}{PARA 6 "" 1 "" {TEXT -1 61 " 0.010267605243770446177 .. \+ 0.010268060844669203063" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 42 " step 4 inverse quadratic interpolation" }} {PARA 6 "" 1 "" {TEXT -1 48 " current value: 0.01026806084463 8408759" }}{PARA 6 "" 1 "" {TEXT -1 48 " best value so far: 0.010 268060844638408759" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing i nterval: " }}{PARA 6 "" 1 "" {TEXT -1 61 " 0.0102680608446384087 59 .. 0.010268060844669203063" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 42 " step 5 inverse quadratic interpolatio n" }}{PARA 6 "" 1 "" {TEXT -1 48 " current value: 0.010268060 844638408862" }}{PARA 6 "" 1 "" {TEXT -1 48 " best value so far: \+ 0.010268060844638408759" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracket ing interval: " }}{PARA 6 "" 1 "" {TEXT -1 61 " 0.01026806084463 8408759 .. 0.010268060844638408862" }}{PARA 6 "" 1 "" {TEXT -1 0 " " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0%QY%31o-\"!#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 13 "The equation " }{XPPEDIT 18 0 "sin(x)^2 = 0;" "6#/*$-%$si nG6#%\"xG\"\"#\"\"!" }{TEXT -1 26 " clearly has the solution " } {XPPEDIT 18 0 "x = Pi;" "6#/%\"xG%#PiG" }{TEXT -1 6 ", but " } {XPPEDIT 18 0 "sin(x)^2;" "6#*$-%$sinG6#%\"xG\"\"#" }{TEXT -1 25 " doe s not change sign as " }{TEXT 298 1 "x" }{TEXT -1 29 " increases throu gh the value " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 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Sj[>$*FD7$$\"3,z&[2')pc^%Fjq$\"3UXe)R3#)zh*FD7$$\"3oh/x$[qGe%Fjq$\"3na 6^,b=L)*FD7$$\"3>](o$ed[9YFjq$\"3?S%QMWbW!**FD7$$\"3qQq'H.,hk%Fjq$\"3! ynBFSBh&**FD7$$\"3Q$=m-n3>m%Fjq$\"3!)H%45-RX(**FD7$$\"33G`c2jrxYFjq$\" 3!HV$RuG)z)**FD7$$\"3)=Zk[%R_$p%Fjq$\"3i]%>l_Tk***FD7$$\"3e;O;#eJ$4ZFj q$\"3!)RCXBl!*****FD7$$\"3ns[!z6bes%Fjq$\"3;9/,Tn=)***FD7$$\"3lHhk`'yB u%Fjq$\"3#[qP4*)35***FD7$$\"3k'Q(Q*=-*eZFjq$\"3#H5::\"3Qy**FD7$$\"3sU' G^sDax%Fjq$\"3#f)es#H;.'**FD7$$\"3zb6h'zs%3[Fjq$\"3;3uleF'z!**FD7$$\"3 &)oO4o)>:%[Fjq$\"3r[rb(>Cu* )FD7$$\"3*zX#[4&eg5&Fjq$\"3*Q>q::l'G&)FD7$$\"3#Q_\")pq87<&Fjq$\"3z())[ 3!HWQ!)FD7$$\"3n*e![/*ojB&Fjq$\"37mM8J%)p'\\(FD7$$\"3Dtq\\/&**HI&Fjq$ \"3[,!H2wx*)*oFD7$$\"3#ob8X5I'p`Fjq$\"3g6wB!)QeniFD7$$\"3fR3_p*3dV&Fjq $\"3*[/T/5)=>cFD7$$\"3PA\"GX$yy,bFjq$\"3WNlU>N**f\\FD7$$\"3wxVy[u]ibFj q$\"3Kv^=PFgaVFD7$$\"39L1/jqABcFjq$\"39IXt]#=(ePFD7$$\"3WaK=$f=Gp&Fjq$ \"3C)\\%y\\qw)4$FD7$$\"3mweKB,TidFjq$\"3[S)39(pevCFD7$$\"3w\"4#\\SRI#f>FD7$$\"3'oIe;6(*o)eFjq$\"3opVfIe$**[\"FD7$$\"3yp>qe;E `fFjq$\"3wav^k2d\\5FD7$$\"3nKcu0ii>gFjq$\"35bZM$[$)py'F-7$$\"3G:=>Xb9$ 3'Fjq$\"3Gw<@]))\\[RF-7$$\"3w)*zj%)[mYhFjq$\"3-GwOhBB_=F-7$$\"31*\\Lr) \\z!='Fjq$\"3RQ7:G#>Z/\"F-7$$\"3P***G'*3D\\@'Fjq$\"3_hY3Z\\A_YF67$$\"3 '*\\n(39!*>B'Fjq$\"3])f)[ogl=EF67$$\"3o*\\C@>b!\\iFjq$\"3IC\">1_7W;\"F 67$$\"3Q\\APV-7miFjq$\"3eLP[[8)=\"HF07$$\"3)****>YH&=$G'Fjq$\"3[z@sJ@V x:!#L-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!Ffdm-%%V IEWG6$;F($\"+3`=$G'!\"*%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 5 "brent" } {TEXT -1 117 " cannot be used to determine a root in a situation like \+ this where the function does not change sign across the root." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "brent(sin(x)^2,x=3..3.2);" }}{PARA 8 "" 1 "" {TEXT -1 61 "Error, ( in brent) interval does not appear to bracket a root\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "One of the procedure s " }{TEXT 0 7 "findmin" }{TEXT -1 4 " or " }{TEXT 0 7 "findmax" } {TEXT -1 25 " can be employed instead." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "evalf(findmin(sin(x)^2,x =3..3.2),15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"0z*e`EfTJ!#9$\"0 Q>[PDP,\"!#W" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "The problem arises because the derivative of the function is zero as well as the function itself." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "df := diff(sin(x)^2,x );\nbrent(df,x=3);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dfG,$*(\"\" #\"\"\"-%$sinG6#%\"xGF(-%$cosGF+F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"+aEfTJ!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure " }{TEXT 0 6 "fsolve" }{TEXT -1 20 " \+ can find this root." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "evalf(fsolve(sin(x)^2,x=3..3.2),15);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0z*e`EfTJ!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT -1 87 "The derivative can be zero at a root with a sign change also occur ring across the root." }}{PARA 0 "" 0 "" {TEXT -1 9 "The root " } {XPPEDIT 18 0 "x = Pi;" "6#/%\"xG%#PiG" }{TEXT -1 5 " of " }{XPPEDIT 18 0 "sin(x)^3;" "6#*$-%$sinG6#%\"xG\"\"$" }{TEXT -1 29 " provides an \+ example of this." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 25 "plot(sin(x)^3,x=0..2*Pi);" }}{PARA 13 "" 1 "" {GLPLOT2D 264 141 141 {PLOTDATA 2 "6%-%'CURVESG6$7eo7$\"\"!F(7$$\"1^cC &eb&p8!#;$\"18t%ePd[a#!#=7$$\"16)>63x`'>F,$\"1X%)eU]FYuF/7$$\"1qR*pd)> hDF,$\"1Fy8%f^di\"!#<7$$\"1@%HT;i7B$F,$\"1y1LECe,KF:7$$\"1s[E^dK,RF,$ \"1@WWD\\o+bF:7$$\"1c^NehL]_F,$\"15;Ue)Q$f7F,7$$\"1*QhW&\\$Hf'F,$\"1=m 4N7\\)H#F,7$$\"1T11.fpPyF,$\"1ILg$Qu#=NF,7$$\"1qr'fwtl7*F,$\"1R#\\K,8; &\\F,7$$\"1%Ra&4L&f/\"!#:$\"1\"=g]!y=\"['F,7$$\"1jXwg<#)y6Ffn$\"1\"y[@ o+H*yF,7$$\"1qGW%H$\\:8Ffn$\"1Osv&G#ye!*F,7$$\"1H22vMov8Ffn$\"1\\k\\$> 1:W*F,7$$\"1*e)pbO(eV\"Ffn$\"16gfoC#)H(*F,7$$\"1>$*Q*f`(p9Ffn$\"1Q\"o$ y[wZ)*F,7$$\"1]+3VNj.:Ffn$\"18kF;_^K**F,7$$\"1\"yqn[8v`\"Ffn$\"1F!3@_% R$)**F,7$$\"16:YIMRr:Ffn$\"0G4?l%******Ffn7$$\"1ahX)[7ag\"Ffn$\"1uGyz$ Q?)**F,7$$\"1'z]kaJ%R;Ffn$\"1B665?`H**F,7$$\"1RaW/1Xt;Ffn$\"14rE%=+H%) *F,7$$\"1#3SCmpuq\"Ffn$\"1J\\j)eUGs*F,7$$\"1u?N%*p.tFfn$\"1Ym35rXUzF,7$ $\"1U(ye<)G*4#Ffn$\"1*\\0&)3n+W'F,7$$\"1a[#o!GC>AFfn$\"1n%GW;aI1&F,7$$ \"1YwJf&y(eBFfn$\"10u\\y7;3NF,7$$\"1OrpV8H#[#Ffn$\"1U'4hwg&)H#F,7$$\"1 zY)QW/yh#Ffn$\"1wM.Z2B^7F,7$$\"1v)>8'\\%ou#Ffn$\"12gAiyz(o&F:7$$\"1z?n #3lT\"GFfn$\"1%zwQqckK$F:7$$\"1%GCS?&[\")GFfn$\"1%**yl_:6q\"F:7$$\"1%3 $\\!41L%HFfn$\"1n]b'e!=WwF/7$$\"1%)='p(p70IFfn$\"1fa3d`\"y^#F/7$$\"1B% )\\K8\\QJFfn$\"1D9p%)f*G)H!#B7$$\"1'fJlT>qF$Ffn$!1G01;c4hCF/7$$\"1\"=@ (oRJPLFfn$!1pJ!))*y/btF/7$$\"1l2\"4_3wR$Ffn$!1(z?Lb:Qi\"F:7$$\"1^()4*G GFY$Ffn$!1Ux'4C2[9$F:7$$\"1OnGd![y_$Ffn$!1i4(*RKVY`F:7$$\"1>4JU#)RiOFf n$!17U)[)3$>B\"F,7$$\"18!f%[$HSz$Ffn$!1r_***)4'yB#F,7$$\"1oE+7#*Q@RFfn $!14+knyGwMF,7$$\"1)f0ii+G1%Ffn$!1&)Q2YI\"*\\]F,7$$\"1fORr]')*=%Ffn$!1 'p(*3,zs]'F,7$$\"1&p%*z[LbK%Ffn$!1aKOA-!G%zF,7$$\"1'pExBp%[WFfn$!1i&=z )yz'**)F,7$$\"1z&[2')pc^%Ffn$!1l:YU6[K%*F,7$$\"1i/x$[qGe%Ffn$!1Os<(oC3 v*F,7$$\"1](o$ed[9YFfn$!1FK$3/Eq&)*F,7$$\"1Rq'H.,hk%Ffn$!1ZzI]tDM**F,7 $$\"1G`c2jrxYFfn$!18$*4G(z>)**F,7$$\"1kRGiA')*F,7$$\"1pO4o)>:%[Ffn$!1p>9z$*H_(*F,7$$\"1@9vt* Qh!\\Ffn$!1q))p(y5\"\\%*F,7$$\"1u\"4%z!e2(\\Ffn$!18o_BY#p.*F,7$$\"1eC[ 4&eg5&Ffn$!1FRmP\")GwyF,7$$\"1!f![/*ojB&Ffn$!1'3^m8-4\\'F,7$$\"1dN^/,j p`Ffn$!1W!=.UD>'\\F,7$$\"1A\"GX$yy,bFfn$!1v\">#>c=$\\$F,7$$\"1L1/jqABc Ffn$!1ysA6&4WI#F,7$$\"1xeKB,TidFfn$!1a?F,]tJ7F,7$$\"12$e;6(*o)eFfn$!1: *3(fc5^dF:7$$\"1q>qe;E`fFfn$!1*R.qv-.S$F:7$$\"1Lcu0ii>gFfn$!1g:4`S8oXb9$3'Ffn$!1,'*3U\\*f%yF/7$$\"1**zj%)[mYhFfn$!1PoRFz#3_#F/7$ $\"1++i%H&=$G'Ffn$!1YdQJ&*=\")>!#R-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+A XESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F($\"+3`=$G'!\"*%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "The convergence t owards the root " }{XPPEDIT 18 0 "x = Pi;" "6#/%\"xG%#PiG" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "sin(x)^3;" "6#*$-%$sinG6#%\"xG\"\"$" }{TEXT -1 7 " using " }{TEXT 0 5 "brent" }{TEXT -1 9 " is slow." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "brent (sin(x)^3,x=3,info=true);" }}{PARA 6 "" 1 "" {TEXT -1 61 " 3.000 00000000000 -> [+] 3.04984076946342 -> [+] " }}{PARA 6 "" 1 "" {TEXT -1 61 " 2.97072347958470 -> [+] 3.04984076946342 -> \+ [+] " }}{PARA 6 "" 1 "" {TEXT -1 61 " 2.97072347958470 -> [+] \+ 3.09631434876957 -> [+] " }}{PARA 6 "" 1 "" {TEXT -1 61 " 2. 89695127090391 -> [+] 3.09631434876957 -> [+] " }}{PARA 6 "" 1 " " {TEXT -1 61 " 2.89695127090391 -> [+] 3.21342043043782 - > [-] " }}{PARA 6 "" 1 "" {TEXT -1 42 " \+ pair OK" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 42 " step 1 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.20539685901178" }}{PARA 6 " " 1 "" {TEXT -1 43 " best value so far: 3.20539685901178" }} {PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 " " 1 "" {TEXT -1 51 " 2.89695127090391 .. 3.2053968590117 8" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 42 " st ep 2 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.18689584342309" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.18689584342309" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 2.89695127090391 .. 3.18689584342309" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 20 " step 3 bisection " }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.04192355 716351" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.18 689584342309" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interva l: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.04192355716351 .. \+ 3.18689584342309" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 42 " step 4 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.17440552015865" }} {PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.174405520158 65" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }} {PARA 6 "" 1 "" {TEXT -1 51 " 3.04192355716351 .. 3.1744 0552015865" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 42 " step 5 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.16701024908250" }}{PARA 6 " " 1 "" {TEXT -1 43 " best value so far: 3.16701024908250" }} {PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 " " 1 "" {TEXT -1 51 " 3.04192355716351 .. 3.1670102490825 0" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 20 " st ep 6 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: \+ 3.10446690312301" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.16701024908250" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-br acketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.1044669 0312301 .. 3.16701024908250" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 42 " step 7 inverse quadratic interpolatio n" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.1518115 1944736" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.1 5181151944736" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interv al: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.10446690312301 .. \+ 3.15181151944736" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 42 " step 8 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.15077649769680" }} {PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.150776497696 80" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }} {PARA 6 "" 1 "" {TEXT -1 51 " 3.10446690312301 .. 3.1507 7649769680" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 42 " step 9 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14807750844600" }}{PARA 6 " " 1 "" {TEXT -1 43 " best value so far: 3.14807750844600" }} {PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 " " 1 "" {TEXT -1 51 " 3.10446690312301 .. 3.1480775084460 0" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 21 " st ep 10 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: \+ 3.12627220578450" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value s o far: 3.14807750844600" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-b racketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.126272 20578450 .. 3.14807750844600" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 43 " step 11 inverse quadratic interpolati on" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.146540 27239395" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3. 14654027239395" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing inter val: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.12627220578450 .. \+ 3.14654027239395" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 " " {TEXT -1 43 " step 12 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14535216226874" }} {PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.145352162268 74" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }} {PARA 6 "" 1 "" {TEXT -1 51 " 3.12627220578450 .. 3.1453 5216226874" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 21 " step 13 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.13581218402662" }}{PARA 6 "" 1 "" {TEXT -1 43 " b est value so far: 3.14535216226874" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " \+ 3.13581218402662 .. 3.14535216226874" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 14 inverse quadr atic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: \+ 3.14329386799857" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value \+ so far: 3.14329386799857" }}{PARA 6 "" 1 "" {TEXT -1 28 " root- bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.13581 218402662 .. 3.14329386799857" }}{PARA 6 "" 1 "" {TEXT -1 0 "" } }{PARA 6 "" 1 "" {TEXT -1 43 " step 15 inverse quadratic interpolat ion" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14308 889763117" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3 .14308889763117" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing inte rval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.13581218402662 .. \+ 3.14308889763117" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 " " {TEXT -1 43 " step 16 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14266040165561" }} {PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.142660401655 61" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }} {PARA 6 "" 1 "" {TEXT -1 51 " 3.13581218402662 .. 3.1426 6040165561" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 21 " step 17 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.13923629284112" }}{PARA 6 "" 1 "" {TEXT -1 43 " b est value so far: 3.14266040165561" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " \+ 3.13923629284112 .. 3.14266040165561" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 18 inverse quadr atic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: \+ 3.14236893259960" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value \+ so far: 3.14236893259960" }}{PARA 6 "" 1 "" {TEXT -1 28 " root- bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.13923 629284112 .. 3.14236893259960" }}{PARA 6 "" 1 "" {TEXT -1 0 "" } }{PARA 6 "" 1 "" {TEXT -1 43 " step 19 inverse quadratic interpolat ion" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14219 330233365" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3 .14219330233365" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing inte rval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.13923629284112 .. \+ 3.14219330233365" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 " " {TEXT -1 21 " step 20 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " \+ current value: 3.14071479758739" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.14219330233365" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14071479758739 .. 3.14219330233365" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 21 inverse \+ quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current valu e: 3.14183460052784" }}{PARA 6 "" 1 "" {TEXT -1 43 " best v alue so far: 3.14183460052784" }}{PARA 6 "" 1 "" {TEXT -1 28 " \+ root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3. 14071479758739 .. 3.14183460052784" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 22 inverse quadratic inte rpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3 .14181003146735" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: \+ 3.14181003146735" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketin g interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14071479758739 .. 3.14181003146735" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 23 inverse quadratic interpolation" }} {PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.141746172330 51" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.141746 17233051" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: \+ " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14071479758739 .. 3. 14174617233051" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 21 " step 24 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " c urrent value: 3.14123048495895" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.14174617233051" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14123048495895 .. 3.14174617233051" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 25 inverse \+ quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current valu e: 3.14170967512412" }}{PARA 6 "" 1 "" {TEXT -1 43 " best v alue so far: 3.14170967512412" }}{PARA 6 "" 1 "" {TEXT -1 28 " \+ root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3. 14123048495895 .. 3.14170967512412" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 26 inverse quadratic inte rpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3 .14168160548158" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: \+ 3.14168160548158" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketin g interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14123048495895 .. 3.14168160548158" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 21 " step 27 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14145604522027" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.14168160548158" }}{PARA 6 " " 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14145604522027 .. 3.14168160548158" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 28 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " c urrent value: 3.14163280561734" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.14163280561734" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14145604522027 .. 3.14163280561734" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 29 inverse \+ quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current valu e: 3.14162798517814" }}{PARA 6 "" 1 "" {TEXT -1 43 " best v alue so far: 3.14162798517814" }}{PARA 6 "" 1 "" {TEXT -1 28 " \+ root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3. 14145604522027 .. 3.14162798517814" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 30 inverse quadratic inte rpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3 .14161786098769" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: \+ 3.14161786098769" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketin g interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14145604522027 .. 3.14161786098769" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 21 " step 31 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14153695310399" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.14161786098769" }}{PARA 6 " " 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14153695310399 .. 3.14161786098769" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 32 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " c urrent value: 3.14161099815895" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.14161099815895" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14153695310399 .. 3.14161099815895" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 33 inverse \+ quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current valu e: 3.14160684266779" }}{PARA 6 "" 1 "" {TEXT -1 43 " best v alue so far: 3.14160684266779" }}{PARA 6 "" 1 "" {TEXT -1 28 " \+ root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3. 14153695310399 .. 3.14160684266779" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 21 " step 34 bisection" }}{PARA 6 " " 1 "" {TEXT -1 43 " current value: 3.14157189788589" }} {PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.141606842667 79" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }} {PARA 6 "" 1 "" {TEXT -1 51 " 3.14157189788589 .. 3.1416 0684266779" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 35 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14159838154322" }}{PARA 6 " " 1 "" {TEXT -1 43 " best value so far: 3.14159838154322" }} {PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 " " 1 "" {TEXT -1 51 " 3.14157189788589 .. 3.1415983815432 2" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " st ep 36 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14159779798458" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.14159779798458" }}{PARA 6 " " 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14157189788589 .. 3.14159779798458" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 37 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " c urrent value: 3.14159628738364" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.14159628738364" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14157189788589 .. 3.14159628738364" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 21 " step 38 bisectio n" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.1415840 9263476" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.1 4159628738364" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interv al: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14158409263476 .. \+ 3.14159628738364" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 39 inverse quadratic interpolation" }}{PARA 6 " " 1 "" {TEXT -1 43 " current value: 3.14159542105336" }} {PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.141595421053 36" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }} {PARA 6 "" 1 "" {TEXT -1 51 " 3.14158409263476 .. 3.1415 9542105336" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 40 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14159475794937" }}{PARA 6 " " 1 "" {TEXT -1 43 " best value so far: 3.14159475794937" }} {PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 " " 1 "" {TEXT -1 51 " 3.14158409263476 .. 3.1415947579493 7" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 21 " st ep 41 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: \+ 3.14158942529207" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value s o far: 3.14159475794937" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-b racketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.141589 42529207 .. 3.14159475794937" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 43 " step 42 inverse quadratic interpolati on" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.141593 60130271" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3. 14159360130271" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing inter val: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14158942529207 .. \+ 3.14159360130271" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 " " {TEXT -1 43 " step 43 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14159348790087" }} {PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.141593487900 87" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }} {PARA 6 "" 1 "" {TEXT -1 51 " 3.14158942529207 .. 3.1415 9348790087" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 44 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14159324870236" }}{PARA 6 " " 1 "" {TEXT -1 43 " best value so far: 3.14159324870236" }} {PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 " " 1 "" {TEXT -1 51 " 3.14158942529207 .. 3.1415932487023 6" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 21 " st ep 45 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: \+ 3.14159133699722" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value s o far: 3.14159324870236" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-b racketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.141591 33699722 .. 3.14159324870236" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 43 " step 46 inverse quadratic interpolati on" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.141593 08707989" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3. 14159308707989" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing inter val: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159133699722 .. \+ 3.14159308707989" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 " " {TEXT -1 43 " step 47 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14159298877154" }} {PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.141592988771 54" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }} {PARA 6 "" 1 "" {TEXT -1 51 " 3.14159133699722 .. 3.1415 9298877154" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 21 " step 48 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14159216288438" }}{PARA 6 "" 1 "" {TEXT -1 43 " b est value so far: 3.14159298877154" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " \+ 3.14159216288438 .. 3.14159298877154" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 49 inverse quadr atic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: \+ 3.14159278917450" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value \+ so far: 3.14159278917450" }}{PARA 6 "" 1 "" {TEXT -1 28 " root- bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159 216288438 .. 3.14159278917450" }}{PARA 6 "" 1 "" {TEXT -1 0 "" } }{PARA 6 "" 1 "" {TEXT -1 52 " ** increasing working precision to \+ 18 digits **" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 50 inverse quadr atic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: \+ 3.14159277531905" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value \+ so far: 3.14159277531905" }}{PARA 6 "" 1 "" {TEXT -1 28 " root- bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159 216288438 .. 3.14159277531905" }}{PARA 6 "" 1 "" {TEXT -1 0 "" } }{PARA 6 "" 1 "" {TEXT -1 52 " ** increasing working precision to \+ 21 digits **" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 51 inverse quadr atic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: \+ 3.14159273958930" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value \+ so far: 3.14159273958930" }}{PARA 6 "" 1 "" {TEXT -1 28 " root- bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159 216288438 .. 3.14159273958930" }}{PARA 6 "" 1 "" {TEXT -1 0 "" } }{PARA 6 "" 1 "" {TEXT -1 52 " ** increasing working precision to \+ 24 digits **" }}{PARA 6 "" 1 "" {TEXT -1 21 " step 52 bisection" }} {PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.141592451236 84" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.141592 73958930" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: \+ " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159245123684 .. 3. 14159273958930" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 52 " ** increasing working precision to 27 digits **" }} {PARA 6 "" 1 "" {TEXT -1 43 " step 53 inverse quadratic interpolati on" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.141592 71903218" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3. 14159271903218" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing inter val: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159245123684 .. \+ 3.14159271903218" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 " " {TEXT -1 52 " ** increasing working precision to 30 digits **" } }{PARA 6 "" 1 "" {TEXT -1 43 " step 54 inverse quadratic interpolat ion" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14159 270336765" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3 .14159270336765" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing inte rval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159245123684 .. \+ 3.14159270336765" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 " " {TEXT -1 52 " ** increasing working precision to 33 digits **" } }{PARA 6 "" 1 "" {TEXT -1 21 " step 55 bisection" }}{PARA 6 "" 1 " " {TEXT -1 43 " current value: 3.14159257730224" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.14159270336765" }} {PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 " " 1 "" {TEXT -1 51 " 3.14159257730224 .. 3.1415927033676 5" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 52 " \+ ** increasing working precision to 36 digits **" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 56 inverse quadratic interpolation" }}{PARA 6 " " 1 "" {TEXT -1 43 " current value: 3.14159267595965" }} {PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.141592675959 65" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }} {PARA 6 "" 1 "" {TEXT -1 51 " 3.14159257730224 .. 3.1415 9267595965" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 52 " ** increasing working precision to 39 digits **" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 57 inverse quadratic interpolation" }} {PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.141592673291 17" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.141592 67329117" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: \+ " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159257730224 .. 3. 14159267329117" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 52 " ** increasing working precision to 42 digits **" }} {PARA 6 "" 1 "" {TEXT -1 43 " step 58 inverse quadratic interpolati on" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.141592 66763992" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3. 14159266763992" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing inter val: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159257730224 .. \+ 3.14159266763992" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 " " {TEXT -1 52 " ** increasing working precision to 45 digits **" } }{PARA 6 "" 1 "" {TEXT -1 21 " step 59 bisection" }}{PARA 6 "" 1 " " {TEXT -1 43 " current value: 3.14159262247108" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.14159266763992" }} {PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 " " 1 "" {TEXT -1 51 " 3.14159262247108 .. 3.1415926676399 2" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 52 " \+ ** increasing working precision to 48 digits **" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 60 inverse quadratic interpolation" }}{PARA 6 " " 1 "" {TEXT -1 43 " current value: 3.14159266383298" }} {PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.141592663832 98" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }} {PARA 6 "" 1 "" {TEXT -1 51 " 3.14159262247108 .. 3.1415 9266383298" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 61 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14159266150751" }}{PARA 6 " " 1 "" {TEXT -1 43 " best value so far: 3.14159266150751" }} {PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 " " 1 "" {TEXT -1 51 " 3.14159262247108 .. 3.1415926615075 1" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 21 " st ep 62 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: \+ 3.14159264198930" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value s o far: 3.14159266150751" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-b racketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.141592 64198930 .. 3.14159266150751" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 43 " step 63 inverse quadratic interpolati on" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.141592 65679873" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3. 14159265679873" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing inter val: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159264198930 .. \+ 3.14159265679873" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 " " {TEXT -1 43 " step 64 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14159265646987" }} {PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.141592656469 87" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }} {PARA 6 "" 1 "" {TEXT -1 51 " 3.14159264198930 .. 3.1415 9265646987" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 65 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14159265562484" }}{PARA 6 " " 1 "" {TEXT -1 43 " best value so far: 3.14159265562484" }} {PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 " " 1 "" {TEXT -1 51 " 3.14159264198930 .. 3.1415926556248 4" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 21 " st ep 66 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: \+ 3.14159264880707" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value s o far: 3.14159265562484" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-b racketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.141592 64880707 .. 3.14159265562484" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 43 " step 67 inverse quadratic interpolati on" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.141592 65513719" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3. 14159265513719" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing inter val: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159264880707 .. \+ 3.14159265513719" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 " " {TEXT -1 43 " step 68 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14159265476715" }} {PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.141592654767 15" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }} {PARA 6 "" 1 "" {TEXT -1 51 " 3.14159264880707 .. 3.1415 9265476715" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 21 " step 69 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14159265178711" }}{PARA 6 "" 1 "" {TEXT -1 43 " b est value so far: 3.14159265476715" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " \+ 3.14159265178711 .. 3.14159265476715" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 70 inverse quadr atic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: \+ 3.14159265411783" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value \+ so far: 3.14159265411783" }}{PARA 6 "" 1 "" {TEXT -1 28 " root- bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159 265178711 .. 3.14159265411783" }}{PARA 6 "" 1 "" {TEXT -1 0 "" } }{PARA 6 "" 1 "" {TEXT -1 43 " step 71 inverse quadratic interpolat ion" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14159 265405502" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3 .14159265405502" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing inte rval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159265178711 .. \+ 3.14159265405502" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 " " {TEXT -1 43 " step 72 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14159265392151" }} {PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.141592653921 51" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }} {PARA 6 "" 1 "" {TEXT -1 51 " 3.14159265178711 .. 3.1415 9265392151" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 21 " step 73 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14159265285431" }}{PARA 6 "" 1 "" {TEXT -1 43 " b est value so far: 3.14159265392151" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " \+ 3.14159265285431 .. 3.14159265392151" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 74 inverse quadr atic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: \+ 3.14159265383183" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value \+ so far: 3.14159265383183" }}{PARA 6 "" 1 "" {TEXT -1 28 " root- bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159 265285431 .. 3.14159265383183" }}{PARA 6 "" 1 "" {TEXT -1 0 "" } }{PARA 6 "" 1 "" {TEXT -1 43 " step 75 inverse quadratic interpolat ion" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14159 265377682" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3 .14159265377682" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing inte rval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159265285431 .. \+ 3.14159265377682" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 " " {TEXT -1 21 " step 76 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " \+ current value: 3.14159265331557" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.14159265377682" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159265331557 .. 3.14159265377682" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 77 inverse \+ quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current valu e: 3.14159265366573" }}{PARA 6 "" 1 "" {TEXT -1 43 " best v alue so far: 3.14159265366573" }}{PARA 6 "" 1 "" {TEXT -1 28 " \+ root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3. 14159265331557 .. 3.14159265366573" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 78 inverse quadratic inte rpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3 .14159265363431" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: \+ 3.14159265363431" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketin g interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159265331557 .. 3.14159265363431" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 79 inverse quadratic interpolation" }} {PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.141592653602 90" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.141592 65360290" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: \+ " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159265331557 .. 3. 14159265360290" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 21 " step 80 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " c urrent value: 3.14159265345923" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.14159265360290" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159265345923 .. 3.14159265360290" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 81 inverse \+ quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current valu e: 3.14159265357148" }}{PARA 6 "" 1 "" {TEXT -1 43 " best v alue so far: 3.14159265360290" }}{PARA 6 "" 1 "" {TEXT -1 28 " \+ root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3. 14159265357148 .. 3.14159265360290" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+aEfTJ!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }}{PARA 0 "" 0 "" {TEXT -1 40 "This example is similar to the last one." }}{PARA 0 "" 0 "" {TEXT -1 23 "The derivative at root " }{XPPEDIT 18 0 "x = Pi;" "6#/ %\"xG%#PiG" }{TEXT -1 5 " of " }{XPPEDIT 18 0 "sin(x)^5;" "6#*$-%$sin G6#%\"xG\"\"&" }{TEXT -1 14 " is also zero." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot(sin(x)^5,x=0. .2*Pi);" }}{PARA 13 "" 1 "" {GLPLOT2D 264 141 141 {PLOTDATA 2 "6%-%'CU RVESG6$7dp7$\"\"!F(7$$\"1^cC&eb&p8!#;$\"1B4MBZdVZ!#?7$$\"1qR*pd)>hDF,$ \"1$z-+pML/\"!#=7$$\"1@%HT;i7B$F,$\"132bNm0GKF57$$\"1s[E^dK,RF,$\"1H9e )4(*f&zF57$$\"19+\"[&4$ed%F,$\"1x*f.)H\"Go\"!#<7$$\"1c^NehL]_F,$\"1I)) fM(3S;$FE7$$\"1s#3kbN;#fF,$\"19*\\&o>JG0TnF, 7$$\"1qGW%H$\\:8F\\o$\"1Hp_i:0\"[)F,7$$\"1H22vMov8F\\o$\"1w(>q?:m3*F,7 $$\"1*e)pbO(eV\"F\\o$\"10O/F$*Q*f`(p9F\\o$\"1v)Qe^kvu*F, 7$$\"1]+3VNj.:F\\o$\"1#)\\\\h&yx))*F,7$$\"1:a#\\^t0_\"F\\o$\"18t>pP6P* *F,7$$\"1\"yqn[8v`\"F\\o$\"1.cLD&RB(**F,7$$\"1YhheMXa:F\\o$\"12&4#)pCL ***F,7$$\"16:YIMRr:F\\o$\"0t$o'3\"******F\\o7$$\"1K)e%fHS)e\"F\\o$\"1d oDkFD#***F,7$$\"1ahX)[7ag\"F\\o$\"1Ai?$*=3q**F,7$$\"1vMXF\\o$\"1v)e01T<\"oF,7$$\"1[,\"*pw[G?F\\o$\"1 wO&)4\"o&4eF,7$$\"1U(ye<)G*4#F\\o$\"1/^k1Po-[F,7$$\"1)z^8\\l#f@F\\o$\" 1ivo0It\")RF,7$$\"1a[#o!GC>AF\\o$\"1#3G6D$G;KF,7$$\"1YwJf&y(eBF\\o$\"1 :wN[`,X8'\\%ou#F\\o$\"1C/pY$[@T)F57$$\"1z?n#3lT\"GF\\o$\"1*3GB ,=1W$F57$$\"1%GCS?&[\")GF\\o$\"1xJFqF$F\\o$!1i aZADA'[%F/7$$\"1l2\"4_3wR$F\\o$!1]J/QZET5F57$$\"1^()4*GGFY$F\\o$!1Edr[ R@LJF57$$\"1OnGd![y_$F\\o$!1L[uIMk(e(F57$$\"1G))z\\J7&f$F\\o$!1C88VBj9 ;FE7$$\"1>4JU#)RiOF\\o$!1U.vv`2]IFE7$$\"1m\\Q&z8#GPF\\o$!1*[lk1)Q(>&FE 7$$\"18!f%[$HSz$F\\o$!10zS&*pv[#)FE7$$\"1oE+7#*Q@RF\\o$!1=[r\\:n=m%F\\o$!1`*\\*\\!pk$**F,7$$\"1G`c2jrxYF\\o$!1N)zwD%)*p**F,7$$\"1sW '[%R_$p%F\\o$!1T.U!>16***F,7$$\"1:%[F\\o$!1z*\\ft$e!f*F,7$$\"1@9v t*Qh!\\F\\o$!1*pDzR;))4*F,7$$\"1u\"4%z!e2(\\F\\o$!1vV2zG(pW)F,7$$\"1eC [4&eg5&F\\o$!1/?M;CUqe;E`fF\\o$!12#fUOe)oNF57$$\"1Lcu0ii>gF\\o$!1%=n 9mH+?\"F57$$\"1**zj%)[mYhF\\o$!12Pck!f\"pYF/7$$\"1++i%H&=$G'F\\o$!1$[R e-#>DJ!#b-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"%!G-%% VIEWG6$;F($\"+3`=$G'!\"*%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 33 "The convergence towards the root " } {XPPEDIT 18 0 "x = Pi;" "6#/%\"xG%#PiG" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "sin(x)^5;" "6#*$-%$sinG6#%\"xG\"\"&" }{TEXT -1 7 " using " } {TEXT 0 5 "brent" }{TEXT -1 39 " is again slow, as in the last example ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "brent(sin(x)^5,x=3,info=true);" }}{PARA 6 "" 1 "" {TEXT -1 61 " 3.00000000000000 -> [+] 3.04984076946342 -> \+ [+] " }}{PARA 6 "" 1 "" {TEXT -1 61 " 2.97072347958470 -> [+] \+ 3.04984076946342 -> [+] " }}{PARA 6 "" 1 "" {TEXT -1 61 " 2. 97072347958470 -> [+] 3.09631434876957 -> [+] " }}{PARA 6 "" 1 " " {TEXT -1 61 " 2.89695127090391 -> [+] 3.09631434876957 - > [+] " }}{PARA 6 "" 1 "" {TEXT -1 61 " 2.89695127090391 -> [+] \+ 3.21342043043782 -> [-] " }}{PARA 6 "" 1 "" {TEXT -1 42 " \+ pair OK" }}{PARA 6 "" 1 "" {TEXT -1 0 "" } }{PARA 6 "" 1 "" {TEXT -1 42 " step 1 inverse quadratic interpolati on" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.212699 34071637" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3. 21269934071637" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing inter val: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 2.89695127090391 .. \+ 3.21269934071637" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 " " {TEXT -1 42 " step 2 inverse quadratic interpolation" }}{PARA 6 " " 1 "" {TEXT -1 43 " current value: 3.19876939987115" }} {PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.198769399871 15" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }} {PARA 6 "" 1 "" {TEXT -1 51 " 2.89695127090391 .. 3.1987 6939987115" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 20 " step 3 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " current \+ value: 3.04786033538753" }}{PARA 6 "" 1 "" {TEXT -1 43 " be st value so far: 3.19876939987115" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " \+ 3.04786033538753 .. 3.19876939987115" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 42 " step 4 inverse quadra tic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: \+ 3.18696636557688" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value s o far: 3.18696636557688" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-b racketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.047860 33538753 .. 3.18696636557688" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 42 " step 5 inverse quadratic interpolatio n" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.1816790 7786988" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.1 8167907786988" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interv al: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.04786033538753 .. \+ 3.18167907786988" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 20 " step 6 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " cu rrent value: 3.11476970662870" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.11476970662870" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.11476970662870 .. 3.18167907786988" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 42 " step 7 inverse q uadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value : 3.12268835056136" }}{PARA 6 "" 1 "" {TEXT -1 43 " best va lue so far: 3.12268835056136" }}{PARA 6 "" 1 "" {TEXT -1 28 " r oot-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.1 2268835056136 .. 3.18167907786988" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 42 " step 8 inverse quadratic interpo lation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.12 431771377963" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: \+ 3.12431771377963" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing i nterval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.12431771377963 .. 3.18167907786988" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 42 " step 9 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.12713376525135" }} {PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.127133765251 35" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }} {PARA 6 "" 1 "" {TEXT -1 51 " 3.12713376525135 .. 3.1816 7907786988" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 21 " step 10 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.15440642156061" }}{PARA 6 "" 1 "" {TEXT -1 43 " b est value so far: 3.15440642156061" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " \+ 3.12713376525135 .. 3.15440642156061" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 11 inverse quadr atic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: \+ 3.14476689936470" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value \+ so far: 3.14476689936470" }}{PARA 6 "" 1 "" {TEXT -1 28 " root- bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.12713 376525135 .. 3.14476689936470" }}{PARA 6 "" 1 "" {TEXT -1 0 "" } }{PARA 6 "" 1 "" {TEXT -1 43 " step 12 inverse quadratic interpolat ion" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14475 790203391" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3 .14475790203391" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing inte rval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.12713376525135 .. \+ 3.14475790203391" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 " " {TEXT -1 43 " step 13 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14412875530798" }} {PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.144128755307 98" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }} {PARA 6 "" 1 "" {TEXT -1 51 " 3.12713376525135 .. 3.1441 2875530798" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 21 " step 14 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.13563126027967" }}{PARA 6 "" 1 "" {TEXT -1 43 " b est value so far: 3.14412875530798" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " \+ 3.13563126027967 .. 3.14412875530798" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 15 inverse quadr atic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: \+ 3.14401197064075" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value \+ so far: 3.14401197064075" }}{PARA 6 "" 1 "" {TEXT -1 28 " root- bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.13563 126027967 .. 3.14401197064075" }}{PARA 6 "" 1 "" {TEXT -1 0 "" } }{PARA 6 "" 1 "" {TEXT -1 43 " step 16 inverse quadratic interpolat ion" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14357 740399265" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3 .14357740399265" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing inte rval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.13563126027967 .. \+ 3.14357740399265" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 " " {TEXT -1 21 " step 17 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " \+ current value: 3.13960433213617" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.14357740399265" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.13960433213617 .. 3.14357740399265" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 18 inverse \+ quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current valu e: 3.14159979562188" }}{PARA 6 "" 1 "" {TEXT -1 43 " best v alue so far: 3.14159979562188" }}{PARA 6 "" 1 "" {TEXT -1 28 " \+ root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3. 13960433213617 .. 3.14159979562188" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 52 " ** increasing working precisi on to 18 digits **" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 19 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current val ue: 3.14159979559046" }}{PARA 6 "" 1 "" {TEXT -1 43 " best \+ value so far: 3.14159979559046" }}{PARA 6 "" 1 "" {TEXT -1 28 " \+ root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3 .13960433213617 .. 3.14159979559046" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 52 " ** increasing working precisi on to 21 digits **" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 20 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current val ue: 3.14159836720290" }}{PARA 6 "" 1 "" {TEXT -1 43 " best \+ value so far: 3.14159836720290" }}{PARA 6 "" 1 "" {TEXT -1 28 " \+ root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3 .13960433213617 .. 3.14159836720290" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 52 " ** increasing working precisi on to 24 digits **" }}{PARA 6 "" 1 "" {TEXT -1 21 " step 21 bisecti on" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.140601 34966953" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3. 14159836720290" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing inter val: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14060134966953 .. \+ 3.14159836720290" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 " " {TEXT -1 52 " ** increasing working precision to 27 digits **" } }{PARA 6 "" 1 "" {TEXT -1 43 " step 22 inverse quadratic interpolat ion" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14159 836717148" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3 .14159836717148" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing inte rval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14060134966953 .. \+ 3.14159836717148" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 " " {TEXT -1 52 " ** increasing working precision to 30 digits **" } }{PARA 6 "" 1 "" {TEXT -1 43 " step 23 inverse quadratic interpolat ion" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14159 722446771" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3 .14159722446771" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing inte rval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14060134966953 .. \+ 3.14159722446771" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 " " {TEXT -1 52 " ** increasing working precision to 33 digits **" } }{PARA 6 "" 1 "" {TEXT -1 21 " step 24 bisection" }}{PARA 6 "" 1 " " {TEXT -1 43 " current value: 3.14109928706862" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.14159722446771" }} {PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 " " 1 "" {TEXT -1 51 " 3.14109928706862 .. 3.1415972244677 1" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 52 " \+ ** increasing working precision to 36 digits **" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 25 inverse quadratic interpolation" }}{PARA 6 " " 1 "" {TEXT -1 43 " current value: 3.14159722443629" }} {PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.141597224436 29" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }} {PARA 6 "" 1 "" {TEXT -1 51 " 3.14109928706862 .. 3.1415 9722443629" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 52 " ** increasing working precision to 39 digits **" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 26 inverse quadratic interpolation" }} {PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.141596310279 56" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.141596 31027956" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: \+ " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14109928706862 .. 3. 14159631027956" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 52 " ** increasing working precision to 42 digits **" }} {PARA 6 "" 1 "" {TEXT -1 21 " step 27 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14134779867409" }}{PARA 6 " " 1 "" {TEXT -1 43 " best value so far: 3.14159631027956" }} {PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 " " 1 "" {TEXT -1 51 " 3.14134779867409 .. 3.1415963102795 6" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 52 " \+ ** increasing working precision to 45 digits **" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 28 inverse quadratic interpolation" }}{PARA 6 " " 1 "" {TEXT -1 43 " current value: 3.14159631024814" }} {PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.141596310248 14" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }} {PARA 6 "" 1 "" {TEXT -1 51 " 3.14134779867409 .. 3.1415 9631024814" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 52 " ** increasing working precision to 48 digits **" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 29 inverse quadratic interpolation" }} {PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.141595578929 04" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.141595 57892904" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: \+ " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14134779867409 .. 3. 14159557892904" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 21 " step 30 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " c urrent value: 3.14147168880157" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.14159557892904" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14147168880157 .. 3.14159557892904" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 31 inverse \+ quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current valu e: 3.14159557889762" }}{PARA 6 "" 1 "" {TEXT -1 43 " best v alue so far: 3.14159557889762" }}{PARA 6 "" 1 "" {TEXT -1 28 " \+ root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3. 14147168880157 .. 3.14159557889762" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 32 inverse quadratic inte rpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3 .14159499384863" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: \+ 3.14159499384863" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketin g interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14147168880157 .. 3.14159499384863" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 21 " step 33 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14153334132510" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.14159499384863" }}{PARA 6 " " 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14153334132510 .. 3.14159499384863" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 34 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " c urrent value: 3.14159499381721" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.14159499381721" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14153334132510 .. 3.14159499381721" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 35 inverse \+ quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current valu e: 3.14159452578434" }}{PARA 6 "" 1 "" {TEXT -1 43 " best v alue so far: 3.14159452578434" }}{PARA 6 "" 1 "" {TEXT -1 28 " \+ root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3. 14153334132510 .. 3.14159452578434" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 21 " step 36 bisection" }}{PARA 6 " " 1 "" {TEXT -1 43 " current value: 3.14156393355472" }} {PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.141594525784 34" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }} {PARA 6 "" 1 "" {TEXT -1 51 " 3.14156393355472 .. 3.1415 9452578434" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 37 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14159452574833" }}{PARA 6 " " 1 "" {TEXT -1 43 " best value so far: 3.14159452574833" }} {PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 " " 1 "" {TEXT -1 51 " 3.14156393355472 .. 3.1415945257483 3" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " st ep 38 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14159415133147" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.14159415133147" }}{PARA 6 " " 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14156393355472 .. 3.14159415133147" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 21 " step 39 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: \+ 3.14157904244309" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far : 3.14159415133147" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracke ting interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14157904244 309 .. 3.14159415133147" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 43 " step 40 inverse quadratic interpolati on" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.141594 15108772" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3. 14159415108772" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing inter val: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14157904244309 .. \+ 3.14159415108772" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 " " {TEXT -1 43 " step 41 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14159385169044" }} {PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.141593851690 44" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }} {PARA 6 "" 1 "" {TEXT -1 51 " 3.14157904244309 .. 3.1415 9385169044" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 21 " step 42 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14158644706677" }}{PARA 6 "" 1 "" {TEXT -1 43 " b est value so far: 3.14159385169044" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " \+ 3.14158644706677 .. 3.14159385169044" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 43 inverse quadr atic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: \+ 3.14159384970613" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value \+ so far: 3.14159384970613" }}{PARA 6 "" 1 "" {TEXT -1 28 " root- bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14158 644706677 .. 3.14159384970613" }}{PARA 6 "" 1 "" {TEXT -1 0 "" } }{PARA 6 "" 1 "" {TEXT -1 43 " step 44 inverse quadratic interpolat ion" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14159 361133864" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3 .14159361133864" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing inte rval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14158644706677 .. \+ 3.14159361133864" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 " " {TEXT -1 21 " step 45 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " \+ current value: 3.14159002920270" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.14159361133864" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159002920270 .. 3.14159361133864" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 46 inverse \+ quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current valu e: 3.14159358829991" }}{PARA 6 "" 1 "" {TEXT -1 43 " best v alue so far: 3.14159358829991" }}{PARA 6 "" 1 "" {TEXT -1 28 " \+ root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3. 14159002920270 .. 3.14159358829991" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 47 inverse quadratic inte rpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3 .14159341136311" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: \+ 3.14159341136311" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketin g interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159002920270 .. 3.14159341136311" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 21 " step 48 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14159172028290" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.14159341136311" }}{PARA 6 " " 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159172028290 .. 3.14159341136311" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 49 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " c urrent value: 3.14159297030796" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.14159297030796" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159172028290 .. 3.14159297030796" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 50 inverse \+ quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current valu e: 3.14159296463531" }}{PARA 6 "" 1 "" {TEXT -1 43 " best v alue so far: 3.14159296463531" }}{PARA 6 "" 1 "" {TEXT -1 28 " \+ root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3. 14159172028290 .. 3.14159296463531" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 51 inverse quadratic inte rpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3 .14159290490016" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: \+ 3.14159290490016" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketin g interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159172028290 .. 3.14159290490016" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 21 " step 52 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14159231259153" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.14159290490016" }}{PARA 6 " " 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159231259153 .. 3.14159290490016" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 53 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " c urrent value: 3.14159279912095" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.14159279912095" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159231259153 .. 3.14159279912095" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 54 inverse \+ quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current valu e: 3.14159279185542" }}{PARA 6 "" 1 "" {TEXT -1 43 " best v alue so far: 3.14159279185542" }}{PARA 6 "" 1 "" {TEXT -1 28 " \+ root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3. 14159231259153 .. 3.14159279185542" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 55 inverse quadratic inte rpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3 .14159276723486" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: \+ 3.14159276723486" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketin g interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159231259153 .. 3.14159276723486" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 21 " step 56 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14159253991320" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.14159276723486" }}{PARA 6 " " 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159253991320 .. 3.14159276723486" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 57 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " c urrent value: 3.14159265365284" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.14159265365284" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159253991320 .. 3.14159265365284" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 58 inverse \+ quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current valu e: 3.14159265362142" }}{PARA 6 "" 1 "" {TEXT -1 43 " best v alue so far: 3.14159265362142" }}{PARA 6 "" 1 "" {TEXT -1 28 " \+ root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3. 14159253991320 .. 3.14159265362142" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 59 inverse quadratic inte rpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3 .14159265359001" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: \+ 3.14159265359001" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketin g interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159253991320 .. 3.14159265359001" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 21 " step 60 bisection" }}{PARA 6 "" 1 "" {TEXT -1 43 " current value: 3.14159259675160" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.14159265359001" }}{PARA 6 " " 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159259675160 .. 3.14159265359001" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 " step 61 inverse quadratic interpolation" }}{PARA 6 "" 1 "" {TEXT -1 43 " c urrent value: 3.14159265355859" }}{PARA 6 "" 1 "" {TEXT -1 43 " best value so far: 3.14159265359001" }}{PARA 6 "" 1 "" {TEXT -1 28 " root-bracketing interval: " }}{PARA 6 "" 1 "" {TEXT -1 51 " 3.14159265355859 .. 3.14159265359001" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+aEfTJ!\"*" }} }{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 30 "General root-finding examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 256 "" 0 "" {TEXT 276 8 "Question" }{TEXT 289 3 ": " }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 62 "Find all t he solutions (correct to 10 digits) of the equation " }{XPPEDIT 18 0 " J[1](x)=sin(x)" "6#/-&%\"JG6#\"\"\"6#%\"xG-%$sinG6#F*" }{TEXT -1 17 " \+ in the interval " }{XPPEDIT 18 0 "[0,10]" "6#7$\"\"!\"#5" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "J[1](x)" "6#-&%\"JG6#\"\"\"6#%\"xG" }{TEXT -1 40 " is the Bessel function of the 1st kind." }}{PARA 0 "" 0 "" {TEXT 270 4 "Note" }{TEXT -1 14 ": The fuction " }{XPPEDIT 18 0 "J[1]( x)" "6#-&%\"JG6#\"\"\"6#%\"xG" }{TEXT -1 22 " is given in Maple by " } {TEXT 0 12 "BesselJ(1,x)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 277 8 "Solution" }{TEXT 290 3 ": " } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 32 "The solutions correspon d to the " }{TEXT 291 1 "x" }{TEXT -1 61 " coordinates of the points o f intersection of the two graphs " }{XPPEDIT 18 0 "y=J[1](x)" "6#/%\"y G-&%\"JG6#\"\"\"6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=sin(x) " "6#/%\"yG-%$sinG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot([BesselJ(1,x),s in(x)],x=0..10);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7io7$$\"\"!F)F(7$$\"+3x&)*3\"!#5$\"+\"y,7W&!#67$$\"+; arz@F-$\"+**yR$3\"F-7$$\"+!y%*z7$F-$\"+Ro%\\a\"F-7$$\"+XTFwSF-$\"+0q4' *>F-7$$\"+oMrU^F-$\"+-4G([#F-7$$\"+\"z_\"4iF-$\"+Z\\MdHF-7$$\"+m6m#G(F -$\"+y&*>0MF-7$$\"+S&phN)F-$\"+!3qQ#QF-7$$\"+*=)H\\5!\"*$\"+C7!ob%F-7$ $\"+[!3uC\"FW$\"+ll1+^F-7$$\"+J$RDX\"FW$\"+B^X3bF-7$$\"+kGhe:FW$\"+akR acF-7$$\"+)R'ok;FW$\"+e)eXv&F-7$$\"+vIb<sy&F-7$$\"+_(>/x\"FW $\"+BKO3eF-7$$\"+HkGB=FW$\"+4\\*z\"eF-7$$\"+1J:w=FW$\"+Oo9;eF-7$$\"+#) H`I>FW$\"+t'HB!eF-7$$\"+dG\"\\)>FW$\"+7*=lx&F-7$$\"+KFHR?FW$\"+QW$)QdF 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GQ)>&zr\"[F-7$$\"3G****\\-!pU7&F*$\"3aAw'Rn%)[1%F-7$$\"3)))*****>`9V_F *$\"35,g\"*[N'\\?$F-7$$\"3[***\\(oA*)p`F*$\"39'oY*eZ/j@F-7$$\"3/++]<#R m\\&F*$\"3_=OA&zh&F*$!3)zdR\\%oEkFFG7$$\"3G++D@(3'ycF*$!37Lg%f;:Re*FG7$$\"3g*** *\\A_ERdF*$!3K6(QeRobn\"F-7$$\"3?+](o\"*[W!eF*$!3/*[y:p_k[#F-7$$\"3o** *\\7hK'peF*$!3#[D=x1_*RLF-7$$\"3=**\\i0j\"[$fF*$!3_vV%HU&3PUF-7$$\"3w* *************fF*F+-%'COLOURG6&%$RGBG$\"\"!FgglFfgl$\"*++++\"!\")-%+AXE SLABELSG6$Q\"x6\"Q!F_hl-%%VIEWG6$;$!\"'!\"\"$\"\"'Fghl%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "f := x -> sin(x)^2:\n'f(x)'=f(x);\ng := x -> 9*x^2/10:\n'g(x)'=g( x);\nx1 := brent(f(x)=g(x),x=0.56);\nx2 := brent(f(x)=g(x),x=-0.56);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*$)-%$sinGF&\"\"#\"\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,$*(\"\"*\"\"\"\"# 5!\"\"F'\"\"#F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"+=yW#f&!#5 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x2G$!+=yW#f&!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "The corresponding " }{TEXT 292 1 "y" }{TEXT -1 39 " coordinates can be computed from . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f(x1);\nf(x2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+MDz9G!#5 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+MDz9G!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 " . . or . ." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "g(x1) ;\ng(x2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+MDz9G!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+MDz9G!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 256 "" 0 "" {TEXT 304 8 "Question " }{TEXT 307 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 9 "Find the " }{TEXT 306 1 "x" }{TEXT -1 57 " coordinates of the points of intersection of \+ the graphs " }{XPPEDIT 18 0 "y = x^4;" "6#/%\"yG*$%\"xG\"\"%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = exp(x);" "6#/%\"yG-%$expG6#%\"xG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 305 8 "Solution" }{TEXT 308 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 40 "Looking at the graphs over the interval " }{XPPEDIT 18 0 "[-2,2]" "6#7$,$\"\"#!\"\"F%" }{TEXT -1 50 ", it looks as though there are only two solutions." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 35 "plot([x^4,exp(x)],x=-2..2,y=0..10);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7gn7$$!\"# \"\"!$\"#;F*7$$!3ymmm\"p0k&>!#<$\"3[SU\"zS$*\\Y\"!#;7$$!3MLLL$Q6G\">F0 $\"3#o!3-@krQ8F37$$!31++v3-)[(=F0$\"3?ic\\)3YcB\"F37$$!3bmm;M!\\p$=F0$ \"3IWxDA_kQ6F37$$!3#)***\\7Y\"H%z\"F0$\"3=*>0pL1l.\"F37$$!3MLLL))Qj^VkB'F07$$!3OLL$3yO5]\"F0$\"3;hNGf5^w]F07$$!3&*****\\nU)*=9F0$\"3O_ bS(4[U0%F07$$!3SLL$3WDTL\"F0$\"33ml.v3,oJF07$$!35++]d(Q&\\7F0$\"3A(>fp y/yV#F07$$!3gmmmc4`i6F0$\"3:'***!#=$\"3%H4KxY.[)**F\\q7$$!3E++++0\"*H\"*F\\q$\" 3=1VDE.5[pF\\q7$$!35++++83&H)F\\q$\"3'zxo8I#fMZF\\q7$$!3\\LLL3k(p`(F\\ q$\"3(f(H5cJ#pA$F\\q7$$!3Anmmmj^NmF\\q$\"3@VcX?xkQ>F\\q7$$!3)zmmmYh=(e 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(p4&F07$$\"3!*******Rv&)z:F0$\"3.^g@PcwHiF07$$\"3gmm;%)3;C;F0$\"3?y]+1 *3&epF07$$\"3ILLLGUYo;F0$\"3KO,Ea9R\\xF07$$\"3\"*****\\n'*33+L*F07$$\"3ILLe*3k**y\"F0$\"3x/Q 6+LaE5F37$$\"34++]sI@K=F0$\"37K#zW3[p7\"F37$$\"33+++S2ls=F0$\"3T$*)Q\" 3#z(H7F37$$\"34++]2%)38>F0$\"33M\\X6@\\R8F37$$\"3/++v.Uac>F0$\"3eCtzL$ 3aY\"F37$$\"\"#F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*Ff^l-F$6$7S7$F($\" 3-FhOKGN`8F\\q7$F5$\"3mL4@0mkw9F\\q7$F?$\"3q(zuc7FIf\"F\\q7$FI$\"33E,e ![-\\t\"F\\q7$FS$\"3%*R$Rw4#[!*=F\\q7$Fgn$\"3A\"pB8%G;Te'f>CF\\q7$Ffo$\"3>&e(=NU)Qj#F\\q7$F[p$ \"3OD#)Hb(pj'GF\\q7$F`p$\"3DqR_Tw$p7$F\\q7$Fep$\"3=$*H3()o*fP$F\\q7$Fj p$\"3'*[&ew4$>!o$F\\q7$F`q$\"31/%os_?K,%F\\q7$Feq$\"3e#z1k=QEO%F\\q7$F jq$\"3%*3#z8OJiq%F\\q7$F_r$\"3'Hf#)HA*=]^F\\q7$Fdr$\"3o![.*pF*)ebF\\q7 $Fir$\"3KmY(yn#HvgF\\q7$F_s$\"372'*[w-GslF\\q7$Fds$\"397`(G1\\W;(F\\q7 $Fis$\"3smIfnu&yx(F\\q7$F_t$\"3))R?QF)RRZ)F\\q7$Fet$\"3Kv$eR)3!z;*F\\q 7$F[u$\"3A!4+V*eF!)**F\\q7$Fau$\"3I4?O`5/!4\"F07$Ffu$\"3'Rk81w=q<\"F07 $F[v$\"3e$Ryr\"*o(y7F07$F`v$\"3g:?^hH8$R\"F07$Fev$\"3RF\")*=32\\^\"F07 $Fjv$\"3@9IJdA&Gk\"F07$F_w$\"3O(f&Qv@h(z\"F07$Fdw$\"31MZg0t1\\>F07$Fiw $\"3Rf[`T-*[7#F07$F^x$\"3i$[nV+syH#F07$Fcx$\"3B(Qnc1SJ]#F07$Fhx$\"3)QX ji7'*Hr#F07$F]y$\"3K'Go$pk=^HF07$Fby$\"3MP8ivaE/KF07$Fgy$\"3%o$fA,+]# \\$F07$F\\z$\"3+9#RDb)e%z$F07$Faz$\"3))**e)[!)e08%F07$Ffz$\"3YrqL[\"=J \\%F07$F[[l$\"3WGb\\BUEa[F07$Fe[l$\"3uo!4=y:SI&F07$F_\\l$\"3\\_g\"H.p9 u&F07$Fi\\l$\"3qQjuyzpZiF07$Fc]l$\"3;#Gn\"Gt(Rx'F07$F]^l$\"3S]1$*)4c!* Q(F0-F`^l6&Fb^lFf^lFc^lFf^l-%+AXESLABELSG6$Q\"x6\"Q\"yFchl-%%VIEWG6$;F (F]^l;Ff^l$Fd^lF*" 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 3 "As " }{TEXT 311 1 "x" }{TEXT -1 15 " incre ases the " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" }{TEXT -1 20 " e ventually exceeds " }{XPPEDIT 18 0 "x^10" "6#*$%\"xG\"#5" }{TEXT -1 41 ", so there is a third intersection point." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 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))*\\P%eWA-*)FI$\"31w(3JO,$[tFeu7$Fgz$\"3J$Qvv#R3.\")Feu-F\\[l6&F^[lF( F_[lF(-%+AXESLABELSG6$Q\"x6\"Q!Fbfl-%%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" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 309 1 "x" }{TEXT -1 46 " coordinates of the points of intersect ion of " }{XPPEDIT 18 0 "y=x^4" "6#/%\"yG*$%\"xG\"\"%" }{TEXT -1 5 " a nd " }{XPPEDIT 18 0 "y=exp(x)" "6#/%\"yG-%$expG6#%\"xG" }{TEXT -1 31 " are the zeros of the function " }{XPPEDIT 18 0 "f(x)=exp(x)-x^4" "6#/ -%\"fG6#%\"xG,&-%$expG6#F'\"\"\"*$F'\"\"%!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "f := x -> x^4:\n'f(x)'=f(x);\ng := x -> exp(x):\n'g(x)'=g(x);\nx1 := brent(f(x)=g(x),x=-0.8);\nx2 := brent(f(x)=g(x),x=1.4);\nx3 := bre nt(f(x)=g(x),x=8.7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG *$)F'\"\"%\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG-%$e xpGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$!+)=Mb:)!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x2G$\"+D=hH9!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x3G$\"+c%pJh)!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "The corresponding " }{TEXT 310 1 "y" } {TEXT -1 39 " coordinates can be computed from . . ." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f(x1);\nf( x2);\nf(x3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+-V%RU%!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Uu2xT!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)okO]&!\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 11 " . . or . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "g(x1);\ng(x2);\ng(x3) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+-V%RU%!#5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+Su2xT!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ (okO]&!\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4 " }}{PARA 256 "" 0 "" {TEXT 274 8 "Question" }{TEXT 284 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 65 "Find the smallest and largest positive so lutions of the equation " }{XPPEDIT 18 0 "sqrt(x)+exp(-x)-1=4*sin(x*sq rt(x))" "6#/,(-%%sqrtG6#%\"xG\"\"\"-%$expG6#,$F(!\"\"F)F)F.*&\"\"%F)-% $sinG6#*&F(F)-F&6#F(F)F)" }{TEXT -1 24 " correct to 10 digits. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 275 8 "Solution " }{TEXT 285 2 ": " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "We n eed to find solutions of the equation " }{XPPEDIT 18 0 "f(x)=g(x)" "6# /-%\"fG6#%\"xG-%\"gG6#F'" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "f(x)=s qrt(x)+exp(-x)-1" "6#/-%\"fG6#%\"xG,(-%%sqrtG6#F'\"\"\"-%$expG6#,$F'! \"\"F,F,F1" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(x)=4*sin(x*sqrt(x)) " "6#/-%\"gG6#%\"xG*&\"\"%\"\"\"-%$sinG6#*&F'F*-%%sqrtG6#F'F*F*" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "f := x -> sqrt(x)+exp(-x)-1:\n'f(x)' =f(x);\ng : = x -> 4*sin(x*sqrt(x)):\n'g(x)'=g(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,(*$F'#\"\"\"\"\"#F+-%$expG6#,$F'!\"\"F+F+F1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,$*&\"\"%\"\"\"-%$sinG6# *$)F'#\"\"$\"\"#F+F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot([f(x),g(x)],x=0..30,numpoints= 75);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVE SG6$7[p7$$\"\"!F)F(7$$\"3m%f%f%4-/1\"!#=$\"3#[#fx1kE]AF-7$$\"3M*=*=*=/ 37#F-$\"3Q'o*HDa@%p#F-7$$\"3+%y$y$G17=$F-$\"3z%Q)4u&o`\"HF-7$$\"3ny$y$ y$3;C%F-$\"3'z0yv[\\f0$F-7$$\"3S9N^j(3p3'F-$\"3%GbpE&3\\UKF-7$$\"3,\\' ['[\"4A$zF-$\"3^Jx$Hc[,V$F-7$$\"3Kq-FqvE37!#<$\"3?e+&)GeEzRF-7$$\"3mCV KCa1E;FL$\"3w2pghpu=ZF-7$$\"363\"3\"ev(=/#FL$\"3uef9[hF(e&F-7$$\"3y$y$ yL')QFCFL$\"33%)*[)\\$\\FY'F-7$$\"3-tH(H-jl#GFL$\"3.w(o$oca/uF-7$$\"3X (H(HZ(*QRKFL$\"3u$*QV'z#=!R)F-7$$\"39\"3\"3JD*3l$FL$\"3o\\&RZI))pO*F-7 $$\"3+DVKCPN'*Q\"FL7$$\"3K3\"3\"eD4jgFL$\"3QM!)fq>mk9 FL7$$\"3I(['['oS;]'FL$\"3]K-8wCL^:FL7$$\"3Awcnv%[J(oFL$\"3cO=SPUqA;FL7 $$\"3R.Fq_.I0tFL$\"3Dzl;\")Q].FL7$$\"3U(['[')*eT#*)FL$\"3s5=qAiY()>FL7$$\"38g%f%4h32$*FL$\"3g!\\j s]U30#FL7$$\"3;VKCVZ7?(*FL$\"31VWx@:x<@FL7$$\"3>N^8!e:\\,\"!#;$\"3k)4! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "pl ot([f(x),g(x)],x=0..0.3);" }}{PARA 13 "" 1 "" {GLPLOT2D 227 275 275 {PLOTDATA 2 "6&-%'CURVESG6$7Y7$$\"\"!F)F(7$$\"3y)***\\7G$R<)!#@$\"3kdo kIJIxF!#>7$$\"3w****\\ilyM;!#?$\"3]\\Dt'G/*zQF07$$\"3k***\\P%)z@X#F4$ \"3tt/w#QJqq%F07$$\"3_*****\\7t&pKF4$\"33%eGMt%f\"R&F07$$\"3F****\\(of V!\\F4$\"3h1z$\\2yQ^'F07$$\"3.******\\i9RlF4$\"3$)[D%G_lH%)*F07$$\"3 %*****\\Peui=F0$\"3WsPCwSF!=\"!#=7$$\"3%*****\\i3&o]#F0$\"3F&*[i_ZtN8F Z7$$\"3e***\\(oX*y9$F0$\"37XUzyUMk9FZ7$$\"3l***\\P9CAu$F0$\"3)Gr_i3wrc \"FZ7$$\"3S***\\P*zhdVF0$\"3%yivR.(3h;FZ7$$\"3Q***\\P>fS*\\F0$\"3_*Hyn a)fZ7JT(*=FZ7$$\"3E****\\7F^?Z]d&>FZ7$$\"3)******\\(4&G](F0$ \"3(y<;Ai.j,#FZ7$$\"3q)****\\7nD:)F0$\"33`c#HagB2#FZ7$$\"3%******\\-*o y()F0$\"3d3TM/eWA@FZ7$$\"3])**\\PpnsM*F0$\"3w)>spN*Rs63H#FZ7$$\"3u*****\\\"o?&=\"FZ$\"3?vxGKx,DBF Z7$$\"3'***\\Pa&4*\\7FZ$\"3c\"=!QQ[XgBFZ7$$\"3-+]7j=_68FZ$\"3_nx!z\"oL #R#FZ7$$\"3')***\\P%y!eP\"FZ$\"36V/sWy#QU#FZ7$$\"3****\\(=WU[V\"FZ$\"3 '*fun:fF^CFZ7$$\"3!)***\\7B>&)\\\"FZ$\"3I2=/ybUzCFZ7$$\"3#***\\P>:mk:F Z$\"3#*y\\V()[=2DFZ7$$\"3!***\\iv&QAi\"FZ$\"3_p&GRx5-`#FZ7$$\"3+++vtLU %o\"FZ$\"3iC9X!esRb#FZ7$$\"3y*****\\Nm'[1Kz#*f#FZ7$$\"3#)**\\PMaKs=FZ$\"3U8Hdlja>EFZ7$$\"3!* ***\\7TW)R>FZ$\"31/S7yO2TEFZ7$$\"3i*****\\@80+#FZ$\"3MqkmmqffEFZ7$$\"3 y****\\7,Hl?FZ$\"3)Hv/'QHdyEFZ7$$\"3f**\\P4w)R7#FZ$\"3'pUa&*y&4&p#FZ7$ $\"3!*****\\x%f\")=#FZ$\"3xjtuSDZ7FFZ7$$\"3o**\\P/-a[AFZ$\"3)RH]SU0#GF FZ7$$\"3()**\\(=Yb;J#FZ$\"3!yZ()**e[Su#FZ7$$\"3a****\\i@OtBFZ$\"3ed\"G f5x*eFFZ7$$\"3q**\\PfL'zV#FZ$\"3[A.RUe/uFFZ7$$\"3(********)>=+DFZ$\"3c Ctt%3[!)y#FZ7$$\"3#****\\i_4Qc#FZ$\"3VT=*3hy=!GFZ7$$\"31+]P%>5pi#FZ$\" 3Q4@%f!Q8:GFZ7$$\"3#******\\:$*[o#FZ$\"3D'33!y8$p#GFZ7$$\"3')***\\7<[8 v#FZ$\"3#)zF<#yE+%GFZ7$$\"3n******Hjy5GFZ$\"3;y&eeiu8&GFZ7$$\"3'***\\P /)fT(GFZ$\"31?1Kf]6jGFZ7$$\"3%)**\\i0j\"[$HFZ$\"3c`[4\"pBS(GFZ7$$\"3)) **************HFZ$\"3gS)o=y2a)GFZ-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-F$6$ 7SF'7$FG$\"3nXj.G=::@F47$FQ$\"3STaceqC4aF47$FV$\"3)eqvcMFp,\"F07$Ffn$ \"3aZ$)*3TQwe\"F07$F[o$\"3m0bPc[-MAF07$F`o$\"3)zmoP<#o&*GF07$Feo$\"39m B>t[aQOF07$Fjo$\"3W'RKKAvSY%F07$F_p$\"3%p)*[cF36M&F07$Fdp$\"3g_^#)F07$Fcq$\"3H`!G*y9F5$*F 07$Fhq$\"3+^!*GAEHS5FZ7$F]r$\"3\\Sg'*\\<&H9\"FZ7$Fbr$\"3a$4_UQL\"p7FZ7 $Fgr$\"3[Aj$pT4%z8FZ7$F\\s$\"3=+fmG'p9^\"FZ7$Fas$\"3d<,^JQmJ;FZ7$Ffs$ \"3Q+u#3*)**pw\"FZ7$F[t$\"3S$e$fiJ:**=FZ7$F`t$\"3;SZoKjOS?FZ7$Fet$\"3w B)R*3P'H<#FZ7$Fjt$\"3R)pp^u[!>BFZ7$F_u$\"3cX%oD%FZ7$F[y$\"3CAr$=q)fOWFZ7$F`y$\"3l*olU&4k9YFZ7$Fey$\"3+p$*p(=H M![FZ7$Fjy$\"3orL'\\0Jv)\\FZ7$F_z$\"3ec$)Q3E2y^FZ7$Fdz$\"3so-jyyoa&FZ7$F^[l$\"3_ " 0 "" {MPLTEXT 1 0 26 "brent(f(x)=g(x),x=0.1..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+'*pZ\"e\"!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "The following picture suggests that the largest positive \+ solution is approximately 24.5. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plot([f(x),g(x)],x=24.3..25. 4,y=3..4.2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6& -%'CURVESG6$7S7$$\"32++++++IC!#;$\"3IiWduvJ%RM>$RF-7$$\"3!pTg:!R[MCF*$\"3*og`u%*[S$RF-7$$\"3QL32o+$oV#F*$\"3o pl\\,fUORF-7$$\"3PLe\\'y\">RCF*$\"3z$)Q:cu\")QRF-7$$\"3l;a3!GU:W#F*$\" 3%Rq))[\\'>TRF-7$$\"3O$3F&)[@PW#F*$\"35D$=b9,M%RF-7$$\"3A]PkKz(fW#F*$ \"3)eP4z*GoXRF-7$$\"3G$3x.b6$[CF*$\"3Kjro#eT![RF-7$$\"39](oToP1X#F*$\" 3QiWq%e\"R]RF-7$$\"3wm;p)RIIX#F*$\"3OWcF(o2G&RF-7$$\"3_Le%H!z8bCF*$\"3 Ega)e\"[$\\&RF-7$$\"3/+]d`/^dCF*$\"3)*)ynnOGt&RF-7$$\"3)***\\7YF*)fCF* $\"3!f\\bNeI(fRF-7$$\"3/+]UE&)=iCF*$\"3mS^E$\\N()RF-7$$\"3#)\\i5\"3#[*[#F*$\"3K&GF^*4Z*)RF-7$$\"3OL3 P!>i<\\#F*$\"3U#[JIRb<*RF-7$$\"3'****\\jwl)\\#F*$\"3QS-$3v^')*RF-7$$ \"3'**\\7%Gw7,DF*$\"39<.I,v7,SF-7$$\"3]mm@^@N.DF*$\"3]W5HG5N.SF-7$$\"3 $***\\7/ts0DF*$\"3^J))oFSs0SF-7$$\"3S$3xcazy]#F*$\"3@l5tYL(y+%F-7$$\"3 1+]<9DB5DF*$\"3)*=g7l?A5SF-7$$\"3W;/;ukW7DF*$\"3gSS9@5V7SF-7$$\"32](o- qgZ^#F*$\"3jHy`w*QZ,%F-7$$\"3Vm;HzK-R>DF*$\"35/=2IXN>SF-7$$\"3?LLjRLn@DF*$\"3$=x'zoli@SF-7$$\"37LeH\\j +CDF*$\"3xTg(Q**[R-%F-7$$\"3l;/YS+KEDF*$\"3CeuHG6DESF-7$$\"3#)***\\B3Y %GDF*$\"3^J@^@cOGSF-7$$\"3Um\"ziw#)3`#F*$\"3^62MwzyISF-7$$\"39LLVl@1LD F*$\"3*\\*[lrN&H.%F-7$$\"3\"*\\P\\feQNDF*$\"3n?,,A:ENSF-7$$\"3.]i?J*4w `#F*$\"3FTMhM&pu.%F-7$$\"3')************RDF*$\"3#)p4Nn7%)RSF--%'COLOUR G6&%$RGBG$\"#5!\"\"$\"\"!F`[lF_[l-F$6$7^o7$F($\"3)yr0thH7e\"F-7$F/$\"3 29`*=$[e/AF-7$F4$\"3Gp#pfg`:p#F-7$F9$\"3kjL^sQSiJF-7$F>$\"3yaBBE,SSNF- 7$FC$\"3rA'z\\DP%4QF-7$$\"3+]iI%)=jUCF*$\"3]4rE%pTa*QF-7$FH$\"33-]J_'e g&RF-7$$\"3)*\\ib*f&GWCF*$\"3g!Re*yWKxRF-7$$\"3h;ae5(\\[W#F*$\"35t`$Rk K;*RF-7$$\"3e$e9;#QTXCF*$\"3e_+5%pb*)*RF-7$FM$\"3#o4#>\"Ry#**RF-7$$\"3 e;/^TZ9ZCF*$\"3q\"4d\"eDtxRF-7$FR$\"3ZMoTq(yj#RF-7$$\"3a;HFF-7$Fjo$\"3)=)4[H_:h8 F-7$F_p$\"3pF`rLikLw!#=7$Fdp$\"3=?h!fUf=&H!#>7$Fip$!3bXKotp=UfFb`l7$F^ q$!3K`-h&*=z+8F-7$Fcq$!3;&[naI:8*=F-7$Fhq$!3KxD$QUeC[#F-7$F]r$!3;22^!) =qtHF-7$Fbr$!3!RtN3%\\H'R$F-7$Fgr$!3!p4D&[#fAp$F-7$$\"3)[iStHyP[#F*$!3 !=(RCI\")H7QF-7$F\\s$!3EyBG0[M.RF-7$$\"3Ue9w8$eh[#F*$!3s'=3*)pbk'RF-7$ Fas$!3)fYdR)\\(p*RF-7$$\"3km;u,lU)[#F*$!3UkU)o5(y'*RF-7$Ffs$!3C,*eP/v; (RF-7$$\"3eT&Qd8A1\\#F*$!3)HiqLQBn\"RF-7$F[t$!3p(Q%eucCLQF-7$F`t$!3Q8g ZVJ;tNF-7$Fet$!3a(\\*[Z$=6@$F-7$Fjt$!3?\"fgX\"QcpFF-7$F_u$!3]EkZPb;*=# F-7$Fdu$!3C-u[aFa-;F-7$Fiu$!3g!GY8;19F*Fb`l7$F^v$!3%z\"ec\\gZ))GFb`l7$ Fcv$\"3;u`NA$zI<%Fb`l7$Fhv$\"3a=$eMF]02\"F-7$F]w$\"3uXY&H>bLykoA&3BF-7$Fgw$\"3o5@ctqF^GF-7$F\\x$\"3y&)e*Qs1*)G$F-7$Fax$\"3c V>t\\\\MOOF-7$$\"3*[7y[>j^_#F*$\"3K?4_)fUww$F-7$Ffx$\"3/QH)*\\sLqQF-7$ $\"313_ShIQFDF*$\"3o(**p(H^zQRF-7$F[y$\"3sgqmzx'>)RF-7$$\"3k\"HKLDb!HD F*$\"3wTg^#H\">&*RF-7$$\"37$e9VUk'HDF*$\"3Qq;I!)4)***RF-7$$\"3fuoH&ft- `#F*$\"3/Q%>6oBj*RF-7$F`y$\"3GmTfqSA%)RF-7$$\"3'*\\i&eYs>`#F*$\"3\\W5z hQgTRF-7$Fey$\"3)34\"3,DKsQF-7$$\"3qTNY7SAMDF*$\"3A.'p*[$)fpPF-7$Fjy$ \"3\"pq(3P6&yj$F-7$F_z$\"3#)ys&RR(\\3LF-7$Fdz$\"3[i0CSUx]GF--Fiz6&F[[l F_[lF\\[lF_[l-%+AXESLABELSG6$Q\"x6\"Q\"yFdjl-%%VIEWG6$;$\"$V#F^[l$\"$a #F^[l;$\"\"$F`[l$\"#UF^[l" 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 -1 206 "We need to obtain a starting approximati on which is sufficiently close to the largest solution or an interval \+ which is small enough to contain only the largest solution. Another pi cture is useful for this. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot([f(x),g(x)],x=24.35..24.55,y=3 .6..4);" }}{PARA 13 "" 1 "" {GLPLOT2D 272 279 279 {PLOTDATA 2 "6&-%'CU RVESG6$7S7$$\"3:++++++NC!#;$\"3nLnh;>dMR!#<7$$\"3SLL3VfVNCF*$\"3c8H:?O ,NRF-7$$\"3om\"H[D:eV#F*$\"3B%e:>]EORF-7$$\"3!o;zjf)4PCF*$\"3r&[s0)ypO RF-7$$\"3jLe4;[\\PCF*$\"3W1Ahk\"*4PRF-7$$\"36+Dmy]!zV#F*$\"3\\+WiOY^PR F-7$$\"3bLezs$H$QCF*$\"3jkEg\"GWz$RF-7$$\"3>+D@1BvQCF*$\"3\\uR]6DPQRF- 7$$\"3smm@Xt=RCF*$\"3jzO&)fH\")QRF-7$$\"3PL$3y_q&RCF*$\"3$)zoYu3?RRF-7 $$\"3F++l+>+SCF*$\"3%f/$*z`P'RRF-7$$\"31++vW]VSCF*$\"3.ffmbf2SRF-7$$\" 36++NfC&3W#F*$\"3k?#>bT)\\SRF-7$$\"3gLez6:BTCF*$\"3@UQC=?)3%RF-7$$\"3u mm\"=C#oTCF*$\"3#HTF-7Q8%RF-7$$\"3&pmm#pS1UCF*$\"3WG!*\\lWsTRF-7$$\"3* **\\i`A3DW#F*$\"3j\"4'*)QQHzOzYRF-7$$\"3)** \\i&p@[ZCF*$\"3G.zr@M?ZRF-7$$\"3/+]2'HKzW#F*$\"3Mnu1J$ew%RF-7$$\"3qmmw anL[CF*$\"3+gvn^q1[RF-7$$\"3E++v+'o([CF*$\"3u\"4/4T.&[RF-7$$\"3[LeR<*f \"\\CF*$\"3)z#4*Ry)*)[RF-7$$\"3=++&)Hxe\\CF*$\"3=mm1&*4L\\RF-7$$\"3!p; 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" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 4 "Note" }{TEXT -1 24 ": The special procedure " }{TEXT 0 8 "allroots" }{TEXT -1 40 " fin ds all 41 solutions of the equation " }{XPPEDIT 18 0 "f(x)=g(x)" "6#/- %\"fG6#%\"xG-%\"gG6#F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "allroots(f(x)=g(x),x=0. .25);\nnops([%]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6K$\"\"!F$$\"+'*pZ \"e\"!#5$\"+/yxz?!\"*$\"+WRK'[$F*$\"+#HfJP%F*$\"+ZvS.bF*$\"+;:OqhF*$\" +A&f5>(F*$\"+he.RxF*$\"+26c&p)F*$\"+WDKk\"*F*$\"+w7u25!\")$\"+&3f([5F= $\"+Mz*o8\"F=$\"+3NHt6F=$\"+iA,f7F=$\"+CZf\"H\"F=$\"+UQTv8F=$\"+3Zu/9F =$\"+r\"[q[\"F=$\"+(oKN^\"F=$\"+u$=Yf\"F=$\"+oib=;F=$\"+Tem)p\"F=$\"+y 6G?'*z\"F=$\"+$fy!>=F=$\"+Qp#y*=F=$\"+8%\\_\">F=$\"+ysd$*>F= $\"+\"RS!4?F=$\"+jd6(3#F=$\"+jYl+@F=$\"+culy@F=$\"+mvD!>#F=$\"+&G,%oAF =$\"+Sj(zF#F=$\"+cfbcBF=$\"+UY)QO#F=$\"+5`WVCF=$\"+Q^!zW#F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#T" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 5" }}{PARA 0 "" 0 "" {TEXT 282 8 "Question" } {TEXT 293 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 25 "The \"black box functi on\" " }{TEXT 303 2 "Kn" }{TEXT -1 41 " is defined in the following su bsection. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "code for \"black box\" numerical function " }{TEXT 0 2 "Kn" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1749 "Kn := proc(x::realcons)\n \+ local xx,eps,saveDigits,doK,val,p,q,maxit;\n option `Copyright (c) 2003 Peter Stone.`;\n \n doK := proc(x,eps,maxit)\n local p, s,t,u,h,i; \n # set up a starting approximation\n if x<2.7 a nd x>-.61712 then\n s := -.5671393500+(.6380737305+(-.73577880 86e-1+\n (.1152886285e-2+(.1861572266e-2+\n (- .4028320312e-3+.3187391493e-4*x)*x)*x)*x)*x)*x\n elif x>0 then\n \+ if x<8.34856 then\n s := (-.6142682630+(.5994669286 +.4904857562e-2*x)*x)/\n (1.+.1835614373*x) \n el se\n s := ln(x)+(-.4962008873-.2828832687e-2*x)/\n \+ (1.+(.1229792074+.3481393331e-3*x)*x)\n end if;\n el se\n if x>-8.5 then\n s := (-.5549339971+(.78140844 60+\n (-.2064924826+.1309215565*x)*x)*x)/\n \+ (1.+(-.1767242104+.1323023216*x)*x)\n else\n s \+ := x\n end if;\n end if;\n\n # solve the equation y+ exp(y)=x for y by Halley's method \n for i to maxit do\n \+ p := exp(s);\n t := s+p-x;\n u := 1+p;\n h := \+ t/(u-1/2*p*t/u);\n s := s-h;\n if abs(h)<=eps*abs(s) t hen break end if;\n end do;\n s;\n end proc;\n\n p := il og10(Digits);\n q := Float(Digits,-p);\n maxit := trunc((p+(.02331 061386+.1111111111*q))*2.095903274)+2;\n saveDigits := Digits;\n D igits := Digits+min(iquo(Digits,3),5);\n xx := evalf(x);\n eps := \+ Float(3,-saveDigits-1);\n if Digits<=trunc(evalhf(Digits)) and \n \+ ilog10(xx)trunc(evalhf(DBL_MIN_10_EXP)) then\n val := evalhf(doK(xx,eps, maxit))\n else\n val := doK(xx,eps,maxit)\n end if;\n evalf [saveDigits](val);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "xx := 1.3;\nKn(xx) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"#8!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+6x " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "Find numerical approximations for the solutions of the eq uation " }{XPPEDIT 18 0 "Kn(x) = 1/x;" "6#/-%#KnG6#%\"xG*&\"\"\"F)F'! \"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 283 8 "Solution" }{TEXT 294 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 92 "Once the function has been assigned or loaded, its graph can be pl otted, but the expression " }{TEXT 303 5 "Kn(x)" }{TEXT -1 31 " must b e enclosed in quotes as " }{TEXT 0 7 "'Kn(x)'" }{TEXT -1 4 " or " } {TEXT 0 7 "'Kn'(x)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "The graph " }{XPPEDIT 18 0 "y = Kn(x);" "6#/%\"yG-%#KnG6#%\"xG" }{TEXT -1 43 " is plotted below along with the graph of " }{XPPEDIT 18 0 "y \+ = 1/x;" "6#/%\"yG*&\"\"\"F&%\"xG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(['Kn '(x),1/x],x=-3..5,y=-3..3);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 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k)Q9#Fi[l7$$\"3=+++:o " 0 "" {MPLTEXT 1 0 37 "brent('Kn(x)'=1/x,x=1.8..2.2,info=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Jstep~1~--~inverse~quadratic~interpolationG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>current~value~~------------>~G$\"0=qFZ.&*4#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>best~value~so~far~~-------->~G$\"0=qFZ.&*4#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>root-bracketing~interval~-->~G;$\" #=!\"\"$\"0=qFZ.&*4#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%Jstep~2~--~inverse~quadratic~interpol ationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>current~value~~------------>~G$\"0;]Py'G\"4#!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%>best~value~so~far~~-------->~G$\"0;] Py'G\"4#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>root-bracketing~inter val~-->~G;$\"0;]Py'G\"4#!#9$\"0=qFZ.&*4#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Jstep~3~--~inve rse~quadratic~interpolationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%>current~value~~------------>~G$\"0v 7P(fH\"4#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>best~value~so~far~~- ------->~G$\"0v7P(fH\"4#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>root- bracketing~interval~-->~G;$\"0;]Py'G\"4#!#9$\"0v7P(fH\"4#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Jste p~4~--~inverse~quadratic~interpolationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>current~value~~-------- ---->~G$\"0u.Z&fH\"4#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>best~val ue~so~far~~-------->~G$\"0u.Z&fH\"4#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>root-bracketing~interval~-->~G;$\"0;]Py'G\"4#!#9$\"0u.Z&fH\"4# F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Jstep~5~--~inverse~quadratic~interpolationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>cur rent~value~~------------>~G$\"0$GoafH\"4#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>best~value~so~far~~-------->~G$\"0u.Z&fH\"4#!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%>root-bracketing~interval~-->~G;$\"0$ GoafH\"4#!#9$\"0u.Z&fH\"4#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+bfH\"4#!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "brent('Kn(x) '=1/x,x=-1..-.7,info=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Jstep~1~- -~inverse~quadratic~interpolationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>current~value~~------------>~ G$!0Z\"3^7jG()!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>best~value~so~f ar~~-------->~G$!0Z\"3^7jG()!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>r oot-bracketing~interval~-->~G;$!0Z\"3^7jG()!#:$!\"(!\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Jstep~2~ --~inverse~quadratic~interpolationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>current~value~~------------> ~G$!03rO<\"4k&)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>best~value~so~ far~~-------->~G$!03rO<\"4k&)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%> root-bracketing~interval~-->~G;$!0Z\"3^7jG()!#:$!03rO<\"4k&)F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%Jstep~3~--~inverse~quadratic~interpolationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>current~value~ ~------------>~G$!0Of)o**ok&)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%> best~value~so~far~~-------->~G$!0Of)o**ok&)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>root-bracketing~interval~-->~G;$!0Of)o**ok&)!#:$!03rO <\"4k&)F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Jstep~4~--~inverse~quadratic~interpolationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>cur rent~value~~------------>~G$!0l*>vIok&)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>best~value~so~far~~-------->~G$!0l*>vIok&)!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%>root-bracketing~interval~-->~G;$!0l* >vIok&)!#:$!03rO<\"4k&)F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%Jstep~5~--~inverse~quadratic~interpol ationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>current~value~~------------>~G$!0U2\\2$ok&)!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%>best~value~so~far~~-------->~G$!0U2 \\2$ok&)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>root-bracketing~inter val~-->~G;$!0U2\\2$ok&)!#:$!03rO<\"4k&)F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%C**~increasing~ working~precision~toG\"#=%+~digits~**G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Jstep~6~--~inverse~quadratic~interpolationG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>current~valu e~~------------>~G$!0x@[2$ok&)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% >best~value~so~far~~-------->~G$!0U2\\2$ok&)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%>root-bracketing~interval~-->~G;$!0U2\\2$ok&)!#:$!0x@[ 2$ok&)F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+vIok&)!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 4 "Note" }{TEXT -1 2 ": " }{XPPEDIT 18 0 "Kn(x)" " 6#-%#KnG6#%\"xG" }{TEXT -1 41 " is a numerical inverse for the functio n " }{XPPEDIT 18 0 "f(x)=x+exp(x)" "6#/-%\"fG6#%\"xG,&F'\"\"\"-%$expG6 #F'F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 42 "The code for th e special inverse function " }{XPPEDIT 18 0 "K(x)" "6#-%\"KG6#%\"xG" } {TEXT -1 4 " of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 118 " can be obtained from another worksheet. This code includes infor mation for Maple to obtain a symbolic derivative for " }{XPPEDIT 18 0 "K(x)" "6#-%\"KG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Diff(K(x),x);\n``=value (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$-%\"KG6#%\"xGF)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&\"\"\"F&,(F&F&%\"xGF&-%\"KG6#F(! \"\"F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Newton's method may now be used." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "newton(K(x)=1/x,x=2,info=t rue);\nnewton(K(x)=1/x,x=-1,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\"/Ut7A9*3#!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\"/@a\"y%H\"4#!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$\"/MqafH\"4#!#8" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\"/PqafH\"4 #!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+bfH\"4#!\"*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$!/AQWL^P%)!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$!/b%H'*4Nc)! #9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$!/vKwH ok&)!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$! /t!\\2$ok&)!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~5~~- >~~~G$!/w!\\2$ok&)!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+vIok&)!#5 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 6" }}{PARA 0 "" 0 "" {TEXT 278 8 "Question" }{TEXT 295 2 ": " }}{PARA 0 " " 0 "" {TEXT -1 78 "Find, correct to 10 digits, the first two and last two zeros of the function " }{XPPEDIT 18 0 "f(x)=sin(x^3)*(x-10^(-9) )" "6#/-%\"fG6#%\"xG*&-%$sinG6#*$F'\"\"$\"\"\",&F'F.)\"#5,$\"\"*!\"\"F 4F." }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[0,6]" "6#7$\"\" !\"\"'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 279 8 "Solution" }{TEXT 296 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 35 "This function clearly has zeros at " }{XPPEDIT 18 0 "x = 0" "6# /%\"xG\"\"!" }{TEXT -1 8 " and at " }{XPPEDIT 18 0 "x=10^(-9)" "6#/%\" xG)\"#5,$\"\"*!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "f := x -> sin(x^3)*(x-10^(- 9)):\n'f(x)'=f(x);\nplot(f(x),x=-0.5*10^(-9)..1.2*10^(-9));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&-%$sinG6#*$)F'\"\"$\"\"\"F/, &F'F/#F/\"+++++5!\"\"F/" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7gn7$$!3K+++++++]!#F$\"3u++++++v=!#a7$$!3#p mm\"z$[%HYF*$\"37ah.#[&\\^9F-7$$!3gL$3_RLqI%F*$\"3$)Hq!*>-5V6F-7$$!3#p m;a-WW%RF*$\"3#**oNpIIxb)!#b7$$!3!om;z<^%zNF*$\"30W)[)4avFiF=7$$!3XL$3 x2$>;KF*$\"3a2#Ho_fnR%F=7$$!33n;a=jSzGF*$\"3c\"3qD&QruIF=7$$!3Q+](o8$o IDF*$\"3Sm#e^w(*3.#F=7$$!35n;aBJ.q@F*$\"38:lUC!HOC\"F=7$$!3X+]P>(R0\"= F*$\"3FV0)z&4h4q!#c7$$!3gLL$ecc2W\"F*$\"3#)Hxoj*y:U$Ffn7$$!3rmm\"HO^]6 \"F*$\"3I'pmCxw4a\"Ffn7$$!3i)****\\ZWQ[(!#G$\"3T2`\"f.N_]%!#d7$$!3%=++ ]i>@!QFdo$\"3l<(4*flO0d!#e7$$!3%[-++Db4a#!#H$\"3')o/(>aCZk\"!#h7$$\"33 LLek-&y'HFdo$!3P)Q@2RRl`#F]p7$$\"3!Hmm;Wb!*z'Fdo$!30ETn0\\JHHFgo7$$\"3 !pmmm()eW+\"F*$!3@dAy#[3k6*Fgo7$$\"3u**\\7e:*>Q\"F*$!3Oj$*fUXpuAFfn7$$ \"3Xmmm^><;V#F*$!3%[^f$)\\d&)3\"F=7$$\"3mmm\"zWWiz#F*$!3eR(3-$Q,v:F=7$$\" 3/n;HPQxIJF*$!3AWT44E'z5#F=7$$\"3sKL3x*3;\\$F*$!3e[Ye(z]/x#F=7$$\"3SL$ ekF:k'QF*$!3!))olQ$G>XNF=7$$\"3c**\\(=E&o#>%F*$!33pJ%=f#3!G%F=7$$\"3CL Le%yl]a%F*$!3(fyR\"o;l@^F=7$$\"3l)****\\M4\"4\\F*$!3194#ewZG-'F=7$$\"3 w***\\i%\\Dl_F*$!3DbuJ@:A6pF=7$$\"3o+]7GT%)4cF*$!3mL**RYka]xF=7$$\"3S* **\\Pm^C*fF*$!3i.D-U^nB')F=7$$\"3'emm;bTiL'F*$!3EbM(G#38?$*F=7$$\"37,+ ]P1J.nF*$!3GvcxpHd`5F-7$$\"3?Lek)4\\1d(F*$!3EHPx,*=T0\"F-7$$\"3)[m\"zCys TxF*$!3!fc7G&z#y/\"F-7$$\"3&HL3xQa0#zF*$!3hbw\")e6FL5F-7$$\"3-,]i]4Q*4 )F*$!3;`)f-KN)45F-7$$\"3[lm;aA0\\%)F*$!3qg\"31Q7XN*F=7$$\"3.**\\7.d7:) )F*$!3D1'))zp%H;\")F=7$$\"3?MLLVzpn\"*F*$!3J`'*e$34IT'F=7$$\"3.NL3#)RD G&*F*$!3E@[>?t#33%F=7$$\"3in;zMW#e))*F*$!3BhyY$H#4.6F=7$$\"3]$e*)*[4,0 5!#E$\"3opdK=fl'3&Ffn7$$\"3B++]a%R9-\"F^z$\"3AlH`N!=[G#F=7$$\"3bL3xv$o -/\"F^z$\"3JfE&H?_J`%F=7$$\"3)omTqH(4f5F^z$\"3-]#=Lj:1-(F=7$$\"33+vo3 \"Qf2\"F^z$\"3!**[YNY'[e%*F=7$$\"3GLLL?*yF4\"F^z$\"3$R\"4[Pqs57F-7$$\" 3o;H2QZt56F^z$\"3.%[:W%>X<:F-7$$\"32+D\"eb!pG6F^z$\"3kME?:WU]=F-7$$\"3 +++]kl(e9\"F^z$\"3sa5xb1#[>#F-7$$\"3%**\\(=tD1j6F^z$\"3/Gzer)\\ac#F-7$ $\"3'\\i!*)HpHs6F^z$\"3].rR$Q9ex#F-7$$\"3'*\\Pf'GJ:=\"F^z$\"35<6$\\$QC %*HF-7$$\"3)\\(oHVcw!>\"F^z$\"3qkRg+M\"4A$F-7$$\"3)**************>\"F^ z$\"3C***********fX$F--%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fj^lFi^l-%+AXE SLABELSG6$Q\"x6\"Q!F__l-%%VIEWG6$;$!+++++]!#>$\"+++++7!#=%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Although \+ we don't need to do this, both of these solutions can be found using \+ " }{TEXT 0 5 "brent" }{TEXT -1 69 " with starting values which are rel atively close to the desired root." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "x1 := brent(f(x),x=2e-10);\n x2 := brent(f(x),x=1.2e-9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$ \"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x2G$\"+++++5!#=" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "There are numerous solutions crowded together near " }{XPPEDIT 18 0 "x=6" "6#/% \"xG\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot(f(x),x=0..6,numpoints=75);" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7fgn 7$$\"\"!F)F(7$$\"3Ccnvcn@$[)!#>$\"3!fNcm-c*y^!#A7$$\"3uH(H(H=W'e\"!#=$ \"3p@@X(o`UL'!#@7$$\"3!3aS09NlT#F4$\"3pC7*HxG+T$!#?7$$\"3))['['[38_KF4 $\"3#fX)R6BP=6F-7$$\"37;i@;^v$3%F4$\"3EiP2uB4zFF-7$$\"3onvcnsxa[F4$\"3 #GL)\\N&*zUbF-7$$\"3[Yf%f/EJl&F4$\"3D=ADglv:5F47$$\"3O&f%f%\\z(ykF4$\" 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(f(x),x=5.94.. 6);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVES G6$7en7$$\"3R++++++Sf!#<$\"3m&=A*[JzgYF*7$$\"3M++DHyITfF*$\"3xRuxPB#)3 TF*7$$\"3<+v[kdWUfF*$\"3'Gwcf)zqjNF*7$$\"3(***\\n\"\\DP%fF*$\"3A$o2#\\ 1x))GF*7$$\"3@+]s,P,XfF*$\"3[@FLa<>b@F*7$$\"3M+v8*y&HYfF*$\"3#*4lp'**p XQ\"F*7$$\"3m+vG[W[ZfF*$\"3'opc#>OWjk!#=7$$\"3i+v)fB:([fF*$!3+a0lg>X$H \"FK7$$\"33+vQ=\"))*\\fF*$!3u)ofjh'e(H*FK7$$\"3=+vj=pD^fF*$!3r#3Wx339r \"F*7$$\"3k++lN?c_fF*$!3uz(o$***oN[#F*7$$\"31+]U$e6P&fF*$!3%=$4e\"zr^7 $F*7$$\"3A++&>q0]&fF*$!3qh4h\\fw\"z$F*7$$\"3'****\\U80j&fF*$!3Y['\\?6H %*Q%F*7$$\"3i++0ytbdfF*$!3SUs$GO]m)[F*7$$\"3/+vQNXpefF*$!3-\")>'*4tbj_ F*7$$\"3q++XDn/gfF*$!3BW-:^4H6cF*7$$\"3#*****z2A>hfF*$!3`$*)*Gf%)H:eF* 7$$\"3O]P%>We='fF*$!3K,MGLAK%*eF*7$$\"3#**\\(3wY_ifF*$!3#4o#y\\*QO%fF* 7$$\"3y\\P>1W6jfF*$!3O'>!yU?GifF*7$$\"3_++IOTqjfF*$!3Wj[$Q`Qt&fF*7$$\" 3u]Ppj6NkfF*$!3Ciq>:JyCfF*7$$\"33+v3\">)*\\'fF*$!3EchQ`e!fK$F*7$$\"3;+]Zn%)otfF*$!3wl1\"ytK'QEF*7$$\"3%)****4FL(\\ (fF*$!3yFbx.))3z=F*7$$\"3u**\\d6.BwfF*$!3XF#*\\uV*35\"F*7$$\"3s*\\(o3l WxfF*$!3aKZ*)o`%>G$FK7$$\"3A+]A))ozyfF*$\"3'p\"zcg/]k`FK7$$\"3s****Hk- ,!)fF*$\"3F#y.^#H(\\I\"F*7$$\"3_++D-eI\")fF*$\"3Izh#p)z^,@F*7$$\"3S+v= _(zC)fF*$\"3%\\$G#o,b#*y#F*7$$\"3I++b*=jP)fF*$\"3M2=e**fr!\\$F*7$$\"3E +v3/3(\\)fF*$\"3'=ED=S)*44%F*7$$\"3I+vB4JB')fF*$\"3!)o`l36EXYF*7$$\"3C ++DVsY()fF*$\"3&\\^$y^'f]5&F*7$$\"3_+v=n#f())fF*$\"3-Z#)[7%\\/\\&F*7$$ \"3G++!)RO+!*fF*$\"3!4w\\&f(>=w&F*7$$\"3G+D;:*R1*fF*$\"3'fA*\\S&45'eF* 7$$\"3E+]_!>w7*fF*$\"37f;\"\\&>uKfF*7$$\"3w]i?(>2>*fF*$\"35CVO]#3k(fF* 7$$\"3O+v)Q?QD*fF*$\"3=hD!R+\"\\#*fF*7$$\"3#3v$\\L!=J*fF*$\"3a4//A\\'G )fF*7$$\"3U++5jyp$*fF*$\"3SGwnPQ))\\fF*7$$\"3S+DE8CO%*fF*$\"397sfiy]$) eF*7$$\"3S+]Ujp-&*fF*$\"3=j%o()R:py&F*7$$\"3-++gEd@'*fF*$\"3gn'e3wI.a& F*7$$\"3#**\\(3'>$[(*fF*$\"3gFP)4](3x^F*7$$\"3O+D6Ejp)*fF*$\"3,R+oM%>% QZF*7$$\"\"'\"\"!$\"3Y5MPH4NwTF*-%'COLOURG6&%$RGBG$\"#5!\"\"$F^]lF^]lF h]l-%+AXESLABELSG6$Q\"x6\"Q!F]^l-%%VIEWG6$;$\"$%f!\"#F\\]l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 " . . . an d use " }{TEXT 0 5 "brent" }{TEXT -1 32 " to locate them accurately . \+ . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "x3 := brent(f(x),x=5.94..5.95);\nx4 := brent(f(x),x=5 .97..5.98);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x3G$\"+*R5&[f!\"*" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x4G$\"+]*ez(f!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The required zeros a re . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "x1,x2,x3,x4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"\" !F$$\"+++++5!#=$\"+*R5&[f!\"*$\"+]*ez(fF*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 270 4 "Note" }{TEXT -1 55 ": These problems can all be solved using the proc edure " }{TEXT 0 6 "fsolve" }{TEXT -1 2 ", " }{TEXT 0 5 "brent" } {TEXT -1 51 " or another other numerical root-finding procedure." }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 45 "Find all the real solutions of the equation " }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "(9/5+x)^x = 7-13/10;" "6#/),&*&\"\"*\"\"\"\"\"&!\"\"F(% \"xGF(F+,&\"\"(F(*&\"#8F(\"#5F*F*" }{TEXT -1 1 " " }{TEXT 269 1 "x" } {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 21 "correct to 10 digits. " }}{PARA 0 "" 0 "" {TEXT -1 57 "_____________________________________ ____________________" }}{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 57 "_______________________________________________ __________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 63 "Calcula te, correct to 10 digits, the least positive solution of" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-x) = sin(1/x);" "6#/-%$exp G6#,$%\"xG!\"\"-%$sinG6#*&\"\"\"F.F(F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 27 "which is greater than 0.01." }}{PARA 0 "" 0 "" {TEXT -1 57 "_________________________________________________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 57 "__ _______________________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 61 "Calculate, correct to 10 digits, the solution of the equation" }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/(t+exp(t)),t = \+ 0 .. x) = 1-x;" "6#/-%$IntG6$*&\"\"\"F(,&%\"tGF(-%$expG6#F*F(!\"\"/F*; \"\"!%\"xG,&F(F(F2F." }{TEXT -1 2 ", " }}{PARA 258 "" 0 "" {TEXT -1 27 "which lies between 0 and 1." }}{PARA 0 "" 0 "" {TEXT -1 57 "______ ___________________________________________________" }}{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 57 "____________________ _____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }} {PARA 0 "" 0 "" {TEXT -1 54 "Find the first two and last two zeros of \+ the function" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = sin(x^3)*(exp(1/(3-2*sin(x)))-1845/1322);" "6#/-%\"fG6#%\"xG*&-%$si nG6#*$F'\"\"$\"\"\",&-%$expG6#*&F.F.,&F-F.*&\"\"#F.-F*6#F'F.!\"\"F9F.* &\"%X=F.\"%A8F9F9F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 16 "in the interval " }{XPPEDIT 18 0 "[0,8]" "6#7$\"\"!\"\")" }{TEXT -1 23 " , correct to 20 digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "____________________________________________________ _____" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 57 "_________________________________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q5" }}{PARA 0 "" 0 "" {TEXT -1 35 "Find all the zeros of the function " }{XPPEDIT 18 0 "f(x) = sin(x)^2+sin(20*x)^3/10;" "6#/- %\"fG6#%\"xG,&*$-%$sinG6#F'\"\"#\"\"\"*&-F+6#*&\"#?F.F'F.\"\"$\"#5!\" \"F." }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[0,4]" "6#7$\" \"!\"\"%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 57 "____________ _____________________________________________" }}{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 57 "____________________ _____________________________________" }}{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 258 "" 0 "" {TEXT -1 16 "Code for picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 576 "a := 'a': b := 'b': c := 'c ':\np1 := plot(3-sqrt(41-5*x),x=0..10):\np2 := plot([[0,0],[10,0]],col or=black):\np3 := plot([[[1,-3],[5,-1],[8,2]]$3],style=point,\n \+ symbol=[circle,diamond,cross],color=black):\np4 := plot([[[6.4 ,0]],[[6.4,0]],[[6.4,0]]],style=point,\n symbol=[circle,diamond,cr oss],color=blue):\nt1 := plots[textplot]([[1.8,-3,`(a,f(a))`],[5.8,-1, `(b,f(b))`],\n[8.7,2,`(c,f(c))`],[9.6,-.2,`x`]]):\nt2 := plots[textplo t]([6.8,-.2,`(v,0)`],color=blue):\nt3 := plots[textplot]([7.4,2.7,`x = g(y)`],color=red):\nplots[display]([p1,p2,p3,p4,t1,t2,t3],axes=none); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }