Re: Plotting symbolic representation of numerica function

*To*: mathgroup at smc.vnet.net*Subject*: [mg54470] Re: Plotting symbolic representation of numerica function*From*: "Carl K. Woll" <carlw at u.washington.edu>*Date*: Mon, 21 Feb 2005 03:44:39 -0500 (EST)*Organization*: University of Washington*References*: <cv6rkr$6hd$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Mukhtar, The reason you are getting error messages is because the right hand side of your equation is not a list of two objects, like the left hand side is. On the other hand, when using Set, the right hand side is evaluated, and becomes a list of two objects if there are no errors. Of course, in your example below Mathematica will not be able to evaluate the right hand side since x will not be numerical. At any rate, it seems to me that you ought to investigate the function ImplicitPlot. If you can turn your system of two equations into a single equation for either f1 or f2, then ImplicitSolve ought to be able to plot it. If you can't or don't want to reduce the number of equations, then you might be interested in a function I presented in the past called ImplicitSolve. The usage message for ImplicitSolve is: "ImplicitSolve[eqns, {x->x0,y->y0,...}, {x, xmin, xmax}, opts] finds a numerical solution to the implicit equations eqns for the functions y, ... with the independent variable x in the range xmin to xmax. The root {x0,y0,...} should satisfy the equations, or should provide a good starting point for finding a solution when using FindRoot. Options which can be supplied with opts include those which can be provided to FindRoot or NDSolve." In your example, you would use ImplicitSolve as follows: ImplicitSolve[{g1[f1,f2,x]==0,g2[f1,f2,x]==0},{x->x0,f1->f10,f2->f20},{x,a,b}] where {x0,f10,f20} is a reasonable approximation to a root of the supplied equations, and a and b are the endpoints of the range of x that you are interested in. ImplicitSolve returns InterpolatingFunctions which you can then plot. The code for ImplicitSolve is given below. Carl Woll When copying the code below into mathematica, it's probably a good idea to change your DefaultInputFormatType to InputForm before you cut and paste. When I cut and paste the code below into a StandardForm input cell, I get extraneous input which Mathematica treats as multiplication, so that running the code produces output like Null^6. Cutting and pasting into an InputForm input cell avoids this problem. Clear[ImplicitSolve] ImplicitSolve[eqn_List, rt_, {x_, min_, max_}, opts___?OptionQ] := Module[{x0, y, y0, ag}, ag = AccuracyGoal /. {opts} /. AccuracyGoal -> 6; x0 = Cases[rt, (x -> a_) -> a]; If[x0 == {}, Message[ImplicitSolve::badroot, x];Return[], x0 = x0[[1]] ]; {y, y0} = Transpose[DeleteCases[rt, x -> x0] /. Rule -> List]; If[!(And @@ Thread[Abs[eqn /. Equal -> Subtract /. rt] < 10^-ag]), Message[ImplicitSolve::inaccurate]; y0 = FindRoot[Evaluate[eqn /. x->x0], Evaluate[Sequence@@Transpose[{y, y0}]],opts][[All,2]], Null, Message[ImplicitSolve::incomplete]; Return[] ]; NDSolve[Flatten[{D[eqn /. Thread[Rule[y, #[x]& /@ y]], x], Thread[#[x0]& /@ y == y0]}], y, {x, min, max}, opts, PrecisionGoal -> ag] ] ImplicitSolve::usage = "ImplicitSolve[eqns, {x->x0,y->y0,...}, {x, xmin, xmax}, opts] finds a numerical solution to the implicit equations eqns for the functions y, ... with the independent variable x in the range xmin to xmax. The root {x0,y0,...} should satisfy the equations, or should provide a good starting point for finding a solution when using FindRoot. Options which can be supplied with opts include those which can be provided to FindRoot or NDSolve."; ImplicitSolve::badroot = "Supplied root is missing value for `1`"; ImplicitSolve::incomplete = "Supplied root is incomplete"; ImplicitSolve::inaccurate = "Supplied root is inaccurate, using FindRoot to improve accuracy"; "mbekkali" <mbekkali at gmail.com> wrote in message news:cv6rkr$6hd$1 at smc.vnet.net... > Hello. I need to plot solutions to system of equations solved using > numerical functions. I tried: > > {f1[(x_)?NumericQ],f2[(x_)?NumericQ]}:={f1,f2}/.FindRoot[{g1[f1,f2,x]==0,g2[f1,f2,x]==0},{f1,0},{f2,0}][[1]] > Plot[Evaluate[{f1[x],f2[x]},{x,x0,x1}] > > where g1,g2 are some functions and x0,x1 some starting variables. I > get an error message SetDelayed::shape : Lists > {f1[x_?NumericQ],f2[x_?NumericQ]} and {f1,f2} are not the same shape. > I suspect that there is something wrong with SetDelayed because using > Set and Solve plots functions, yet SetDelayed and Solve generates the > same message as SetDelayed and FindRoot. > > I do not understand what is wrong and help file is not useful as it > does not contain examples with SetDelayed for several variables. > > Mukhtar Bekkali >

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