Coding the Jacobian efficiently in MATLAB

How the Jacobian of F(t,u)= Au - u/1+u be computed in MATLAB? A is a square matrix. How best to compute the Jacobian in Exponential Rosenbrock integrator?

function Jn = J(u)

Jn = -A-1./(1+u).^2;

end

Answer 1

[Matt J]

minusA = -A;

J=@(u) JacobianEngine(u,minusA);

function Jn = JacobianEngine(u,Jn)

       p=numel(u);
       
       Jn(1:p+1:end)= Jn(1:p+1:end) + (-1./(1+u).^2); 
end