Section 2.1 Question 10

10. There are 21 roots of the function f(x) = cos(x) on the
    interval [0, 65]. To which root will the bisection method
    converge, starting with a = 0 and b = 65?

Solution to Section 2.1 Question 10

%matplotlib inline
import numpy as np
from matplotlib import pyplot as plt
def f(x): return np.cos(x)
x=np.linspace(0, 65, 1000)
y=f(x)
plt.plot(x, y)

import numpy as np
def bisection(f,a,b,return_cnt = False):
    y1 = f(a)
    if y1 == 0:
        return a
    y2 = f(b)
    if y2 == 0:
        return b
    if np.sign(y1)*np.sign(y2) > 0:
        raise Exception('function has same signs at the endpoints')
    cnt = 0
    while abs(a-b) > a*10**(-15) and cnt < 100:
        c = (a+b)/2
        y3 = f(c)
        if y3 == 0:
            if return_cnt:
                return c,cnt
            else:
                return c
        if np.sign(y1)*np.sign(y3) < 0:
            b = c
            y2 = y3
        elif np.sign(y2)*np.sign(y3) < 0:
            a = c
            y1 = y3
        cnt = cnt+1
    if return_cnt:
        return c,cnt
    else:
        return c

bisection(f, 0, 65, True)

(58.11946409141122, 50)

From the code above it appears that, using values a=0 and b=65, the bisection method converges to the root 58.11946409141122. It does so in 50 iterations.