{VERSION 3 0 "SUN SPARC SOLARIS" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Error" 7 8 1 {CSTYLE "" -1 -1 "" 0 1 255 0 255 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 571 "restart;with(DEtool s):\nwith(linalg):\nVdot:= (theta,v,x,h) -> piecewise( h>0 and v>0, \+ -sin(theta) - R*v^2, 0);\nThetadot:=(theta,v,x,h) -> piecewise(h>0 and v>0, (v^2 - cos(theta))/v, 0);\nXdot := (theta, v,x,h) -> piecewise(h >0 and v>0, v*cos(theta));\nHdot := (theta, v,x,h) -> piecewise(h>0 an d v>0, v*sin(theta));\nGlider := \{diff(v(t),t) = Vdot(theta(t),v(t), \+ x(t), h(t)), \n diff(theta(t),t) = Thetadot(theta(t),v(t), x (t), h(t)), \n diff(x(t),t) = Xdot(theta(t),v(t), x(t), h(t) ), \n diff(h(t),t) = Hdot(theta(t),v(t), x(t), h(t))\};\nR:= 0.2;" }}{PARA 7 "" 1 "" {TEXT -1 35 "Warning, new definition for adjoi nt" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" } }{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%VdotGR6&%&thetaG%\"vG%\"xG%\"hG6\"6 $%)operatorG%&arrowGF+-%*piecewiseG6%32\"\"!9'2F49%,&-%$sinG6#9$!\"\"* &%\"RG\"\"\")F7\"\"#\"\"\"F=F4F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%)ThetadotGR6&%&thetaG%\"vG%\"xG%\"hG6\"6$%)operatorG%&arrowGF+-%*p iecewiseG6%32\"\"!9'2F49%*&,&*$)F7\"\"#\"\"\"\"\"\"-%$cosG6#9$!\"\"F=F 7!\"\"F4F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%XdotGR6&%&thetaG% \"vG%\"xG%\"hG6\"6$%)operatorG%&arrowGF+-%*piecewiseG6$32\"\"!9'2F49%* &F7\"\"\"-%$cosG6#9$F9F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Hdo tGR6&%&thetaG%\"vG%\"xG%\"hG6\"6$%)operatorG%&arrowGF+-%*piecewiseG6$3 2\"\"!9'2F49%*&F7\"\"\"-%$sinG6#9$F9F+F+F+" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'GliderG<&/-%%diffG6$-%\"vG6#%\"tGF--%*PIECEWISEG6$7$ ,&-%$sinG6#-%&thetaGF,!\"\"*&%\"RG\"\"\")F*\"\"#\"\"\"F832,$-%\"hGF,F8 \"\"!2,$F*F8FD7$FD%*otherwiseG/-F(6$F6F--F/6$7$*&,&*$FF;-%$cosGF5F8 F>F*!\"\"F?FG/-F(6$-%\"xGF,F--F/6$7$*&F*F;FRF;F?FG/-F(6$FBF--F/6$7$*&F *F>F3F;F?FG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG$\"\"#!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 228 "xcrash := proc(v0)\n local \+ sol, xsol;\n sol:=dsolve(\{ op(Glider),theta(0)=0, v(0)=v0, x(0)=0, h( 0)=1\},\n [theta(t), v(t), x(t), h(t)],type=numeric, outpu t=listprocedure);\n xsol :=subs(sol,x(t));\n RETURN(xsol(15));\nend;\n " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'xcrashGR6#%#v0G6$%$solG%%xsolG6 \"F+C%>8$-%'dsolveG6&<'-%#opG6#%'GliderG/-%&thetaG6#\"\"!F;/-%\"vGF:9$ /-%\"xGF:F;/-%\"hGF:\"\"\"7&-F96#%\"tG-F>FI-FBFI-FEFI/%%typeG%(numeric G/%'outputG%.listprocedureG>8%-%%subsG6$F.FL-%'RETURNG6#-FU6#\"#:F+F+F +" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot( xcrash(v), v=1.5 ..2.5);" }}{PARA 8 "" 1 "" {TEXT -1 78 "Error, (in dsolve/numeric/init _y0) invalid specification of initial conditions" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot( sin(x^2), x=0..4);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7[s7$\"\"!F(7$$\"1mmmm;')=()!#<$\"1G*4-m\"y,w! 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lo));\n\n while (wid > epsi lon) do\n mid := evalf( (hi+lo)/2);\n fmid := f(mid);\npri nt(mid,fmid);\n if (sign(fmid) = sign(fhi)) then\n hi \+ := mid;\n fhi := fmid;\n else\n lo := mid;\n flo := fmid;\n fi;\n wid := evalf(abs(hi - lo)) ;\n od;\n RETURN(mid);\nend:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "bisect( x-> x^2-2, 1 , 2, .0005);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+++++:!\"*$\"*+++]# F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"++++]7!\"*$!*++]P%F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$$\"++++v8!\"*$!*+]P4\"F%" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$$\"+++]P9!\"*$\")]iSmF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+++D19!\"*$!)P4YAF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"++](=U\"!\"*$\");&G<#F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"++ D199!\"*$!'YsUF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+](ozT\"!\"*$\" )w`j5F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+Dc,;9!\"*$\"(]-5&F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+j!R]T\"!\"*$\"(]bL#F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+#y]XT\"!\"*$\"':R&*F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+#y]XT\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "bisect( v -> xcrash(v) - 8, 1.9, 2.0, .00005);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"++++]>!\"*$\")MK&R#F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"++++D>!\"*$!)[a5:F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+++]P>!\"*$\"(iR7%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+++DJ >!\"*$!(,wb&F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"++]PM>!\"*$!'q^tF %" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"++v$f$>!\"*$\"(&3!p\"F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+]i:N>!\"*$\"']wZF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+DcwM>!\"*$!'A'H\"F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+Q4'\\$>!\"*$\"'QM!\"*$\"&Y=#F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+a W\"[$>!\"*$!&kA&F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+aW\"[$>!\"* " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Let's try a secant method" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 585 "secant := proc ( f, xlow, \+ xhi, epsilon) \n local lo, flo, hi, fhi, mid, fmid, wid;\n\n lo : = xlow;\n flo := f(lo);\n hi := xhi;\n fhi := f(hi);\n wid := \+ evalf(abs(hi - lo));\n fmid := 1;\n\n while (wid > epsilon and abs (fmid) > epsilon^2) do\n mid := evalf(hi -fhi*(hi -lo)/(fhi - fl o));\n fmid := f(mid);\nprint(wid,[lo,flo],[mid,fmid], [hi,fhi]) ;\n if (sign(fmid) = sign(fhi)) then\n hi := mid;\n \+ fhi := fmid;\n else\n lo := mid;\n f lo := fmid;\n fi;\n wid := evalf(abs(hi - lo));\n od;\n \+ RETURN(mid);\nend:\n\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "secant( x-> x^2-2, 1, 2, .0005);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6&$\"\"\"\"\"!7$F$!\"\"7$$\"+LLLL8!\"*$!*BAAA#F+7$\"\"#F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"*nmmm'!\"*7$$\"+LLLL8F%$!*BAAA#F%7$$\"+++++9 F%$!)+++SF%7$\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"*++++'!\"*7 $$\"+++++9F%$!)+++SF%7$$\"+1Zw69F%$!(:/#pF%7$\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"*%HN#)e!\"*7$$\"+1Zw69F%$!(:/#pF%7$$\"+/Jz89F%$!( f!*=\"F%7$\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"*'*o?'e!\"*7$$ \"+/Jz89F%$!(f!*=\"F%7$$\"+99999F%$!'hS?F%7$\"\"#F1" }}{PARA 11 "" 1 " " {XPPMATH 20 "6&$\"*'eeee!\"*7$$\"+99999F%$!'hS?F%7$$\"+$=,UT\"F%$!&9 ]$F%7$\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"*<))z&e!\"*7$$\"+$ =,UT\"F%$!&9]$F%7$$\"+Q9@99F%$!%4gF%7$\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"*i&)y&e!\"*7$$\"+Q9@99F%$!%4gF%7$$\"+)>8UT\"F%$!%J5F %7$\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"*-oy&e!\"*7$$\"+)>8UT \"F%$!%J5F%7$$\"++N@99F%$!$w\"F%7$\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++N@99!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "secant( v -> xcrash(v) - 8, 1.9, 2.0, .00005);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"\"\"!\"\"7$$\"#>F%$!)H97_!\"*7$$\"+B)H>$>F+$!(eB^ %F+7$$\"#?F%$\"*@g66\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\")x,2o! \"*7$$\"+B)H>$>F%$!(eB^%F%7$$\"+QieM>F%$!'`)3%F%7$$\"#?!\"\"$\"*@g66\" F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\")iPTl!\"*7$$\"+QieM>F%$!'`)3% F%7$$\"+Zg#[$>F%$!&ed$F%7$$\"#?!\"\"$\"*@g66\"F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\")`RF%$!&ed$F%7$$\"+9q%[$>F%$!%S@F% 7$$\"#?!\"\"$\"*@g66\"F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\")')H:l! \"*7$$\"+9q%[$>F%$!%S@F%7$$\"+p#[[$>F%$\"$n(F%7$$\"#?!\"\"$\"*@g66\"F% " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+p#[[$>!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "17 0 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 }