Description
1. (50 pts.) Based on the material in Chapter 3, develop Python code which writes each of the following
numbers in the form a = b × 2
k
, where b ∈ (1/4, 1] and k is even. Then, use Newton’s method to find
the square root of b and hence of a.
(a) a = 1/10
(b) a = 1/1000
(c) a = 1000
(d) a = 0
Be sure to use the initial value as generated by linear interpolation on (1/4, 1], and doing only as
many iterations as are necessary for 10−16 accuracy. Compare your values to the intrinsic square root
function in the math library for Python.
2. (50 pts.) Consider the function f(x) = x/2−sin(x). Code Newton’s Method in Python to show that f
has a root on the interval [1,3]. Also, write Python code which uses the Newton error estimate formula
to help you to determine how close x0 has to be to the root to guarantee convergence.
3. (50 pts.) For the function f(x) = arctan(x), write a program that estimates the (only) real root of f
by computing 10 iterations of the bisection method (on the interval [-1,1]), as well as 10 iterations of
Newton’s method (with an initial value of x0 = 1).
Additionally, for Newton’s method, formulate an equation that must be satisfied by the value x = β,
in order to have the Newton iteration cycle back and forth between β and −β. Run Newton’s Method
again for f(x) with 20 iterations and x0 = β.
