I want to solve system of 2 equations, were all variables below are given except 2 unknowns: $V_0, \sigma_V$. It should be easy to solve numerically system of 2 non-linear equations with 2 unknowns: \begin{cases} E_0 = V_0 N(d_1) - L e^{-rt}N(d_2) \\ \sigma_E E_0 = N(d_1) \sigma_V V_0 \end{cases} where $$d_2 = d_1 - \sigma_V \sqrt{t} = \frac{\log\frac{V_0}{K} + \frac{1}{2}\sigma^2t}{\sigma_V \sqrt{t}} - \sigma_V \sqrt{t}$$ But, using matlabs' fsolve function, even with feeding analytical Jacobian for different data inputs sometimes returns me:
No solutions found. or Equation solved, fsolve stalled.
This is even worse if I use more complex equations. Could somebody give some hint how to solve this system?
or Could somebody give tips, ideas or reference how to choose good initial conditions for fsolve? Finally, I eager to be able to solve more complex model than the written above.
Update: I have been thinking to search for reference how to work with ill-posed problems, or trying to identify if condition number is good, this is still in progress.
Also I have been restating the problem as the minimisation of sum of squares, with some boundary conditions, but this performs not better.
Also I have pictured( but lost somewhere, so can not attach) the plot of the sum of squares of the above functions. Plot looks very flat, which I believe explains why this is difficult for solver to find global minimum.
Using fminsearch first and then feeding this result as initial condition to fmincon or fmin or fsolve will not work as fminsearch returns unreal numbers. Update: I did use function $$\min_x[y-f(x) ]$$ to minimize. I missed that it is not convex problem, squaring it I get reasonable results. Now I am trying further. Thanks!