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# Fixed Point of E^x=Cos(x)

Posted 10 years ago
 I do not understand how to find the fixed point of a problem in mathematica. I need to find the fixed point of e^x=cosx and I am stuck. If anyone understands fixed points that would be very helpful. thanks
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Posted 10 years ago
 Funny thing: by accident I typed in the following Plot[Exp[-x] == Cos[x], {x, -1, 10}, PlotLabel -> "Plot[Exp[-x]==Cos[x]]"] Which is different from Plot[{Exp[-x], Cos[x]}, {x, -1, 10}, PlotLabel -> "Plot[{Exp[-x],Cos[x]}]"] Normally I would expect a == b to be a logical expression which is either True or False.Any explanation for the strange behaviour discovered by accident?Regards, Wolfgang
Posted 10 years ago
 I get an empty plot in Mathemaca 10 running under Linux.
Posted 10 years ago
 Must be a peculiarity of $Version (* Out[40]= "8.0 for Microsoft Windows (64-bit) (October 7, 2011)" *) In 5.2 for Microsoft Windows x86 (64 bit) (June 20, 2005), which I still keep on my PC (you will remember; it is much faster doing Integrate than Version 8) I get the error message "Plot::plnr : Exp[-x] == Cos[x] is not a machine size real number at x = -1 ..." and an empty Plot which is correct. Posted 10 years ago  Thanks, Daniel, for pointing out this site to me.It seems, however, that in contrast to StackExchangeMMA Latex is not supported here. Example:$\sqrt{1+x^2}\$Or am I doing something wrong?I have gradually become acustomed to using it for better readability in texts.Regards, Wolfgang
Posted 10 years ago
 Are you looking for solutions to that equation? Solve[Exp[x] == Cos[x] && -10 <= x <= 10, x, Reals] (* Out[90]= {{x -> 0}, {x -> Root[{-E^#1 + Cos[#1] &, -7.8535932799712482001}]}, {x -> Root[{-E^#1 + Cos[#1] &, -4.7212927588476862166}]}, {x -> Root[{-E^#1 + Cos[#1] &, -1.29269571937339838117}]}} *) Or an iterative process that finds a solution from a starting point. For that one can use e.g. Newton iterations. FixedPointList[{#[[1]] - f[#[[1]]]/f'[#[[1]]], f[#[[1]]]} &, {-.7, f[-.2]}] (* Out[112]= {{-0.7, -0.1613358247632598}, {-2.51705989723243, \ -0.268256883493079}, {-0.7474281702634069, 0.8919329283107287}, {-2.007816769456799, -0.2598565433504163}, \ {-1.285388510652165, 0.5575233413007432}, {-1.292717416706861, -0.005005670608255908}, \ {-1.2926957195615, 0.00001490725795189052}, {-1.292695719373398, 1.292351226034327*10^-10}, {-1.292695719373398, \ -5.551115123125783*10^-17}, {-1.292695719373398, \ -5.551115123125783*10^-17}} *)