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Homework 1 Numerical Computation

1. Execute the following lines in an interpreter (Matlab or Python).
>> format long
>> x = 9.4
>> y = x – 9
>> 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.)

2. Consider the following two polynomials:

p1(x) = (x − 2)9
p2(x) = x
9 − 18x
8 + 144x
7 − 672x
6 + 2016x
5 − 4032x
4 + 5376x
3 − 4608x
2 + 2304x − 512
(1)

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

p(x) = Xn
i=0
ai x
i
, (2)

Horner’s algorithm for evaluating the polynomial at some given x is: (i) initialize p = an, (ii) for
i = n − 1 down to 0, do p = p ∗ x + ai

. 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 figure, 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

f1(x) = 1 − cos(x)
sin2
(x)
, f2(x) = 1
1 + cos(x)
. (3)

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 xk = 10−k
for k =
0, 1, . . . , 12. You should see that f1 loses all accuracy as k increases (that is, as xk approaches
zero), while f2 retains its accuracy. Explain why.

Hint:

For a detailed description, see Chapter
0.4 in Sauer’s Numerical Analysis.)
1