Numerical Computation Homework 1:

Description

1. Execute the following lines in an interpreter (Matlab or Python).

1. • format long

1. • x = 9.4

1. • y = x – 9

1. • z = y – 0.4

What did you get for z? What should it be in exact arithmetic? Why is it not what it should be? (Hint: For a detailed description, see Chapter 0.3.3 in Sauer’s Numerical Analysis.)

Convince yourself that p₁(x) = p₂(x) in exact arithmetic. (No need to show your work on this in the write-up).

Given a polynomial expressed as

n
Xi
p(x) =ai xi;	(2)
=0

Horner’s algorithm for evaluating the polynomial at some given x is: (i) initialize p = a_n, (ii) for i = n 1 down to 0, do p = p x + a_i. Implement Horner’s algorithm. (I’m using Matlab.)

Note that x = 2 is a root of p₁(x) and p₂(x). Generate 8000 equally spaced points in the interval [1:92; 2:08]. (In Matlab, you can do this with linspace.) Evaluate and plot p₁(x) at each point in the interval. In a separate gure, evaluate and plot p₂(x) using Horner’s algorithm. In exact arithmetic, these should be the same. What’s going on in these plots? (Hint: For a detailed description, see Chapter 0.1 in Sauer’s Numerical Analysis or Chapter 1.4 and surrounding text in Demmel’s Applied Numerical Linear Algebra.)

3. Consider the functions

1		cos(x)			1
f1(x) =			;	f2(x) =		:	(3)
		sin2(x)			1 + cos(x)

Using trig identities, show that f₁(x) = f₂(x). (Please show your work on this one.) Implement f₁ and f₂. Make a table of your implementations evaluated at the points x_k = 10 ^k for k = 0; 1; : : : ; 12. You should see that f₁ loses all accuracy as k increases (that is, as x_k approaches zero), while f₂ retains its accuracy. Explain why. (Hint: For a detailed description, see Chapter 0.4 in Sauer’s Numerical Analysis.)

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