1. Roots of equations
Class IX

2. Review of Class VI Bisection Method finding roots of function
Function f(x) on the interval [a,b] and f(a)*f(b) < f(x) has a root(s) on [a,b]
The method produces a sequence of the intervals [𝑎 𝑛 , 𝑏 𝑛 ] with each containing the desired root of the function estimated as
𝑟=( 𝑎 𝑛 + 𝑏 𝑛 )/2
Error (accuracy) of root estimate after n steps
𝜀≤ (𝑏 𝑛 − 𝑎 𝑛 )/2 or 𝜀≤(𝑏−𝑎)/ 2 𝑛+1
If one requires that error 𝜀<𝛿 (tolerance of an error) , the number of steps required in the bisection method is
n > [ log(b-a) - log (2𝛿) ] / log 2

3. Review of Class VII Newton’s Method finding roots of function
Newton’s method requires that function f(x) is differentiable implying that the graph of f(x) has a definite slope at each point.
𝑥 𝑛+1 = 𝑥 𝑛 − 𝑓( 𝑥 𝑛 ) 𝑓 ′ ( 𝑥 𝑛 )
The method evaluates (numerically or analytically) f(x) and f’(x) at each step
If 𝑓 ′ 𝑥 𝑛 ≈0 or 𝑓 ′ 𝑥 𝑛 =0, the method diverges
It requires an initial value 𝑥 0
The method converges quadratically to the desired root r if 𝑥 0 is sufficiently close to r :
| 𝑟−𝑥 𝑛+1 | ≤𝑐 | 𝑟−𝑥 𝑛 | 2

4. Pitfalls of Newton’s method
a) Runaway
Each successive point 𝑥 𝑛 in Newton’s iteration recedes from r instead of converging to r. Pure choice of the initial point 𝑥 0 .

5. Pitfalls of Newton’s method
b) Flat spot
The tangent to the curve is parallel to the x-axis resulting in 𝑥 1 =±∞.

6. Mathematica functions for finding roots