a question on fsolve

Dear Madam/Sir,

I am trying to solve a system of nonlinear equations with fsolve.

For example, to solve f(x) = x.^3 - b*10 = 0, where b is a known vector, say 100 by 1.

I tried to solve the equation in vector form or in scalar form (with loops), the results are not always the same from the 2 approaches, depending on the starting values chosen. I am wondering why this is the case? Could you please give me some hints? Thanks a lot!

Problem: sz and sx are not always equal to each other, why?!

please see the codes below:

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
rand('state',2);
b = rand(100,1);

fun = @(x)(x.^3 + b*10);
options=optimset('Display','off');

start = -ones(100,1);
sx = fsolve(fun,start,options);
for i = 1:100
    ifun = @(x)(x.^3 + 10*b(i));
    sz(i,1) = fsolve(@(x)ifun(x), -2,options);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Answer 1

[Sargondjani]

the problem in the first case is that matlab assumes that every x will show up in every equation. In other words, it will try to estimate a full Jacobian matrix. If you set: 'JacobPattern',speye(length(start)) in the options, the results should be the same

In this case i definitely advice to include the analytical jacobian, because that will be faster and better