Matlab: Fsolve giving incorrect roots

Question

I'm trying to solve this system:

x = a + e(c - e*x/((x^2+y^2)^(3/2)))
y = b + c(d - e*y/((x^2+y^2)^(3/2)))

I'm using fsolve, but not matter what I put in as the starting points for the iteration, I get the answer that the starting points are the roots of the equation.

close all, clear all, clc

a = 1;
b = 2;
c = 3;
d = 4;
e = 5;

fsolve(@u1FsolveFUNC, [1,2])

Function:

function outvar = u1FsolveFUNC(invar)

    global a b c d e

    outvar = [...
        -invar(1) + a + e*(c - e*(invar(1) / ((invar(1)^2 + invar(2)^2)^(3/2)))) ; 
        -invar(2) + b + e*(d - e*(invar(2) / ((invar(1)^2 + invar(2)^2)^(3/2))))]

end

I could try with [1,2] as invariables, and it will say that that is a root to the equation, alltough the correct answer for [1,2] is [12.76,15.52]

Ideas?

You are aware that your implemented and intended equation differ? The leading e in the implementation is a c in your description... — Rody Oldenhuis

Rody Oldenhuis Rody Oldenhuis · Accepted Answer · 2013-06-10T20:00:25

If you write your script like this

a = 1;
b = 2;
c = 3;
d = 4;
e = 5;

f = @(XY) [...
        -XY(1) + a + e*(c - e*(XY(1) ./ (XY(1).^2 + XY(2).^2).^(3/2))) 
        -XY(2) + b + e*(d - e*(XY(2) ./ (XY(1).^2 + XY(2).^2).^(3/2)))];

fsolve(f, [1,2])

it is a lot clearer and cleaner. Moreover, it works :)

It works because you haven't declared a,b,c,d and e to be global before you assigned values to them. You then try to import them in your function, but at that time, they are still not defined as being global, so MATLAB thinks you just initialized a bunch of globals, setting their initial values to empty ([]).

And the solution to an empty equation is the initial value (I immediately admit, this is a bit counter-intuitive).

So this equation involves some inverse-square law...Gravity? Electrodynamics?

Note that, depending on the values of a-e, there may be multiple solutions; see this figure:

enter image description here

Solutions are those points [X,Y] where Z is simultaneously zero for both equations. for the values you give, there is a point like that at

[X,Y] = [15.958213798693690  13.978039302961506]

but also at

[X,Y] = [0.553696225634946   0.789264790080377]

(there's possibly even more...)

Matlab: Fsolve giving incorrect roots

2 Answers