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
(1)
p2(x) = x9
18x8 + 144x7 672x6 + 2016x5
4032x4 + 5376x3
4608x2
+ 2304x 512
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 = 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 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 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.)
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