solve nonlinear equations with fsolve - exitflag is 2 and the roots are complex numbers

Hi Matlab experts,

I am trying to solve a system of highly nonlinear equations below: f(x) = .01*(2*(1.04+x).^(-4.8).*(.1+x).^(-1.9)+(1.04+x).^(-3.8).*(.1+x).^(-2.9)).*A-3*B; where both A and B are vectors, say of size 100 by 1.

I tried to solve this nonlinear equation by fsolve and I run into 3 issues: 1) the results from using fsolve in vector form (solve all equations in one go) are different from in scalar form (each time solve one equation) 2) I got complex results, no matter what starting values I tried, particularly in vector form (almost all x estimated are complex). Results from the 3 procedures are different too! 3) when I try to solve the nonlinear problem equation by equation, I get exitflag = 2 for some equations, is the result with exitflag = 2 correct? In this case I also try different starting values and it does not work.

I add in 'JacobPattern' in options as suggested by one of your colleague but it does not seem to help much.

Thank you so much for your help!

Below is my matlab codes:

////////////////////////////////////////////////////////////////////////////////////////////

function main()

% assign values for B and A (skipped here)

........

% Solve nonlinear equations one by one%

options=optimset('Display','off');

sb = length(B);

for i = 1:sb

    fmiu = @(x)miu(B(i),A(i),x);

    [xs(i,1),fval(i,1),exitflag(i,1)] = fsolve(@(x)fmiu(x),.05,options);

    while exitflag(i,1) <= 0

        [xs(i,1),fval(i,1),exitflag(i,1)] = fsolve(@(x)fmiu(x),rand(1)*.1,options);

    end

end

if sum(xs ~= real(xs)) disp('!!!Warning: xs is complex!!!'); mu end

[xs mu]

% solve nonlinear equations in one go - vector form%

options = optimset('Display', 'notify', 'JacobPattern', speye(sb));

         fx = @(x)miu(B,A,x);

         [xz, fval, exitflag] = fsolve(@(x)fx(x),rand(sb,1), options);

    % Private subfunction
     function fun = miu(B,A,x)
     fun = .01*(2*(1.04+x).^(-4.8).*(.1+x).^(-1.9)+(1.04+x).^(-3.8).*(.1+x).^(-2.9)).*A-3*B;
    end      
end

[Walter Roberson]

miu() would go complex as soon as x < -0.1 . fsolve() does not know that x should not be permitted to go negative.

You could have your function check for negative x and return a large value ("this is not close to a solution") for those cases.

You might wish to consider turning on the FunValCheck option.

[Sargondjani]

and you get different results every time because you take different starting values (random draws). fsolve only finds a local minimum, so it can get stuck in different local optima

if you use the jacobpattern and use the same starting values, the solutions should be the same for each sinlge equation (or at least very close to each other)

[Sargondjani]

pls note that with walter's solution the problem becomes non-differentiable at the break. this will make derivative based methods problematic