x0 = zeros(N-1,1);
so x0 is a vector.
x = x0 - ((F_springs(x0))./(J_springs(x0)));
x0 = x;
with x0 being a vector, x will be a vector; and then you assign it to x0 so x0 will stay a vector.
error = max(abs(x-F_springs(x0)));
with x and x0 being vectors, error is going to be a vector.
while ((error > tol) && (niter < nmax))
error is a vector, so error > tol is a vector. You now have a vector on the left side of && . But the && operator is strictly for scalars, never for vectors. For vectors you would need to use the & operator, as in
while ((error > tol) & (niter < nmax))
As niter<nmax is a scalar, the effect of that would be to calculate (niter < nmax) and to "and" it with each element of (error > tol), giving a vector of results.
The definition of if or while with a vector is to consider the condition to be true only of all of the items in the vector are non-zero. This would be equivalent to
while all((error > tol) & (niter < nmax))
which would stop as soon as any location in error was not greater than tol.
You should re-think why you want x0 to be a vector. If you have a good reason for it being a vector, then think about whether you truly want the entire vector to be involved in the calculations each time.
