plotting - automatic processing of numerical results in `Plot`

First I want to solve an equation $F(x,y)=0$ for $y$ by supplying a value of $x$ . (suppose obtaining the analytic form of $y(x)$ is too difficult) Then I want to plot root $y$ (numerically calculated) as a function of $x$ by using the following:

Plot[y /. FindRoot[.../.{x->x0},{y, 0.2}],{x0, 0, 1}]

and I got something like the following

enter image description here

I omit ... here since it is terribly complicated. Update: F(x,y) is of the form

$\sum_{n,m}a_{m,n}x^ny^m$

and $m$ and $n$ can be as high as 19, which basically makes Solve impractical.

The result is satisfying except a small number of points near 1.0. Setting another initial starting value of $y$ in FindRoot might be an option but it is very tedious and often I cannot find a value of $y$ that fits the whole range of $x$ .

My question is: suppose I stick with the initial value of $y$ , is there a way to just eliminate that anomaly point after I Plot the numerical results? Or is there a better way to deal with this kind of numerical problem in general?

Answer

Using NDSolve to create an interpolation often works well. Hard to tell if it will work with your function.

With[{f = x^2 y + y^3 - 1/3},
 sol = NDSolveValue[{Dt[f == 0, x] /. y -> y[x], 
    y[0] == (y /. FindRoot[f /. x -> 0, {y, 0.6}])}, y, {x, 0, 1}]]
(* InterpolatingFunction[{{0., 1.}}, <>] *)


Plot[sol[x], {x, 0, 1}]

Mathematica graphics

Comments

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