Financial Modelling in Python

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Basic Mathematical Tools 35

4.4 ROOT FINDING

We will present two schemes in this section for finding the roots of a function y = f (x) with

f : R → R.

4.4.1 Bisection Method

The bisection method is a classiscal root-finding routine that does not require derivative infor-

mation. The ppf.math.root

finding module provides the following implementation

derived from the Boost.Math

Toolkit library:

import math

from special

functions import sign, max flt

def bisect(f, min, max, tol, max

its):

"""Bisection method

"""

fmin, fmax = f(min), f(max)

if fmin == 0: return (min, min, 0)

if fmax == 0: return (max, max, 0)

if min >= max: raise RuntimeError, "Arguments in wrong order"

if fmin*fmax >= 0: raise RuntimeError, "Root not bracketed"

count = max

its

if count < 3:

count = 0

else: count -= 3

while count and tol(min, max) == 0:

mid = (min + max)/2.

fmid = f(mid)

if mid == max or mid ==min:

break

if fmid == 0:

min = max = mid

break

elif sign(fmid)*sign(fmin) < 0:

max, fmax = mid, fmid

else:

min, fmin = mid, fmid

--count

max

its -= count

return (min, max, max

its)

The quadratic y(x) = x

2

+ 2x −1 has roots

−1 −

√

2 =−2.4142135623730950488016887242097

−1 +

√

2 =+0.4142135623730950488016887242097.

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36 Financial Modelling ...