Question

Really short question! Also need the R code for implementing the bisection method. Thank you!

(30%)Q1: Please use the bisection method to find all zero points of the following function, f(x)7.82 - 28.33 39.27

Answer 1

BISECTION METHOD:-

Consider a equation f (x) = 0 which has a zero in the interval [a,b] it means sign of f(x) changes

i.e. f (a) * f (b) < 0.

       Bisection scheme computes the zero, say c, by repeatedly halving the interval [a,b].

       c = (a+b) / 2

the interval [a,b] is replaced either with [c,b] or with [a,c] depending on the sign of f (a) * f (c) . This process is continued until the zero is obtained.

Here f(x) = x³ + 7.8x² - 28.33x - 39.27

To get basic idea about the range in which the roots can be exist

f'(x) = 3x² + 15.6x - 28.33 ( slope of f(x) )

roots( f'(x) = -6.6253, 1.4253

When slope of a curve = 0 (minima/maxima) => both side of that point roots exist ( not always true)

Hence root roots lies (-∞, -6.6), (-6.6, 1.4) , (1.4, ∞)

Otherwise we can use hit-trial method

So for first root we will pick-up an interval from (-∞, -6.6)

[-15, -6]   

f(-15) = -1234.32 , f(-6) = 195.51 ===> valid interval

f(-21/2) = -39.48

Hence first root = -10.2

So for 2nd root we will pick-up an interval from (-6.6, 1.4)

[-6, 2]   

f(-6) = 195.51, f(2) = -56.73 , f(0) = -39.27 so we prefer

f(-3) = 88.92

2nd root = -1.1

So for 3rd root we will pick-up an interval from (1.4, ∞)

[2,10]

f(2) = -56.73, f(10) = 1457.43 ===> valid interval

x   f(x)
6   287.66
4   36.21
3   -27.06
3.5   0

3rd root = 3.5