Root finding

Root finding is the process of finding the x for a function h(x), so that h(x) == 0.

1. Use root finding using the bisection method (implemented in scipy.optimize.bisect) to find the value of x_c at which f(x_c)=g(x_c) for f and g as defined in lab 9.

   (You may have done this in the training part of lab10 already.)

2. Use a more efficient root finding (implemented in scipy.optimize.fsolve) to find the value of x_c at which f(x_c)=g(x_c).

   Notes:

   • both algorithms (bisect and fsolve) work fine here, and should give very similar results for the default xtol values.
   • In general, fsolve is more efficient (better convergence), but bisect is more robust (but we must know a bracket that encloses a sign-changing root).

3. Use the bisection method to determine the square root of two. Hint: find the root for f(x)=2-x².

   Vary the xtol parameter of the bisection method, and check the relation between xtol and the true error of the approximated root.

Optimization

Optimization: for a function f(x) we are looking for x_m such that f(x_m) is minimal (locally or globally).

You may want to read section 14.7 Optimisation in the lecture notes for a brief introduction to function optimisation using fmin.

The program dips.py can create the following contour plot of a function f(x,y) (pdf, png):

and can use scipy.optimize.fmin to find a minimum:

Optimization terminated successfully.
    Current function value: -0.310155
    Iterations: 63
    Function evaluations: 116
[-2.13332808  0.53330643]

The minimum fmin finds a local minimum at (x,y) = (-2.13332808,0.53330643) with a function value (f(x,y)=-0.310155.