MATH 4329
Homework 2
Fall 2014
Due: 9:00 a.m. on Wednesday, Sept. 17, 2014
Note: Show all important steps to solve a given problem to receive full credits. If you are
asked to write a MATLAB program, please provide your code, output as well as explanations
of your results (Also, email your code to your TA at [email protected]). You may
adopt and modify the sample codes in the text book.
(15 pts) 1. Take the last two digits of your UTEP ID number(80 number) and call it m (for example,
m=78).
(a) Find the binary floating-point representation and machine representation in IEEE
double precision format for m by hand.
(b) Using MATLAB command format hex, confirm your answer in (a).
(Hint: see the example of (2.6)–(2.7) on page 37. )
(c) To generate overflow and underflow errors on your computer, write a MATLAB
program to evaluate mn and m1n for n = 1, 2, 3, 4, · · · . Eventually, you will exceed
your machine’s exponent limit for floating-point numbers. For what n do you get
overflow error and underflow errors?
Note: write your program in such a way that it prints only the number n that
gives you overflow and underflow errors. Do not print all the intermediate values
mn or m1n .
(15 pts) 2. A certain microcomputer uses a binary floating-point format with 4 bits for the exponent
e and 1 bit for the sign σ. The normalized mantissa x̄ contains 4 bits. The arithmetic
uses rounding.
(a) Find the machine epsilon, i.e., the distance between 1 and the next larger floatingpoint number.
(b) Let x = (7.125)10 . Find its floating-point approximation fl(x). Give fl(x) in decimal.
(c) What is the relative error in fl(x)
(20 pts) 3. In some cases, loss-of-significance errors can be avoided by reformulating functions to
avoid the subtraction of nearly equal quantities. Consider the following functions:
√
4+x−2
x − sin x
f (x) =
,
g(x) =
x
x3
For each function, f (x) and g(x),
1
x
10−1
..
.
f(x) (or g(x))
10−15
(a) Use Calculus to find the limit as x → 0.
(b) Write a MATLAB program to evaluate the function as it is written at x = 10−n , n =
1, 2, · · · , 15. Print the results in the above tabular format and describe your observation.
(c) Reformulate the function to eliminate loss-of-significance errors and evaluate it at
x = 10−n , n = 1, 2, · · · , 15. Print the results in the same format as (b)
2