I have an equation F[x,y]=0. I need to plot y as a function of x. Due to the properties of the function F[x,y], it is impossible to just use ContourPlot for this problem. Also, FindRoot (I don't know why) doesn't work in this situation.
However, NMinimize[...,Method -> "NelderMead"] works well. I need to find all yi for every xi starting from some point {x0,y0} which I know a priori. I can do it step by step by hand and it takes a lot of time, but due to my low knowledge in Mathematica to date I can't automotize this process.
So, please, could anyone help me to to write such a code in Mathematica:
put starting point {x0,y0}.
for i from 1 to N
do
xi=x{i-1}+deltaX,
NMinimize[{F[xi, yi] y{i-1}-deltaY < yi < y{i-1}+deltaY}, {yi}, Method -> "NelderMead"]
(*so, for every xi we find yi and look for y{i+1} in the vicinity of yi;
deltaX is just a step, deltaY is a small constant (much less then y_i)*)
- Plot the points {yi,xi}.
For example, two steps of this algorithm:
{x0,y0}=(0,0)
x_1=0+0.05
NMinimize[{F[x1, y1] 0-0.1 < yi < 0+0.1}, {yi}, Method -> "NelderMead"]
we get y_1=0.05.
x_2=0.05+0.05
NMinimize[{F[x2, y2] 0.05-0.1 < yi < 0.05+0.1}, {yi}, Method -> "NelderMead"]
we get y_1=0.75.
...
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