Matlab Solve function solutions don't satisfy equation

Question

I have an equation that theoretically has to only have one positive solution and the rest must be negative. So I used matlab solve function but two of the solutions are positive instead of one. Then as expected, by testing the solutions one of them does not satisfy the equation which means it is not the proper results. Here is the code:

n=4
c = 10.3343;
theta = [51.7924 ; 4.3564 ; 0.0103 ; 1.3505];
syms x
eqn = sum(sqrt(x.^2 + theta))  - n.*x == c;
sol = solve(eqn,x);
% find pi
xdouble = double(sol);
xdouble(xdouble<0) = [];

cnew1  = sum(sqrt(xdouble(1).^2 + theta)) - n.*xdouble(1)
cnew2  = sum(sqrt(xdouble(2).^2 + theta)) - n.*xdouble(2)

The results are

cnew1 =

   10.5373


cnew2 =

   10.3343

which clearly second one is the proper answer, but I am confused why matlab generates the first incorrect solution? Any help much appreciated.

Answer 1

It could happens that solve produce spurious solutions (a.k.a incorrect solutions). Check the wikipedia article about spurious solutions.

Why ?

Because solve try to solve your problem analytically. During this process matlab can transform your initial equation to something easier to solve.

For example if we have the following equation:

x+1=0

It is obvious that the solution is x = -1.

But matlab could do something like:

x   = -1
x^2 = (-1)^2 
x^2 = 1

And now we have two solutions: -1 and 1. 1 is a spurious solution.

So you have 2 choices to avoid this problem:

1. Check that the solutions returned by solve are correct.
2. Solve the equation with vpasolve, vpasolve use numerical analysis. So, strictly speaking,vpasolve do not solve your equation but instead determine by iteration the possible solutions.

So:

sol = vpasolve(eqn,x);

will do the trick