Homework Problem#1:

• Corrected 07 Sep 2004, in Exercise#4: "exercise 18" --> "Exercise #3".

• Modified from Source: Gerald & Wheatley Background Text p. 33ff:
  (All problem statements are here, modified for MCS 471 and independent of the background text.)

• Most of these problems will require a calculator.

1.    Express the quantities below as normalized floating–point numbers rounded to 4 significant digits in the form

x.yyy e zz

where the first digit before the decimal point is nonzero and e stands for "times 10 to the power" and zz is an signed integer and x is a signed decimal digit.

a. 1234.5678
b. –0.001020304
c. 1234567890
d. 0.000000001

2.    Parallel the simplified examples of normalized floating–point numbers of the form x.yyy e zz(see your class lecture notes if you need more information) with base b = 10, precision p = 4, and exponent range –4 < exp < 5.

a. How many distinct numbers are there in this system, counting just one zero? Give the exact value.
b. What is the largest positive number?
c. What is the smallest (most negative) number?
d. What is the positive number of least magnitude?
f. What is the machine epsilon when rounding to 4 decimal digits?

3.    Using the exam/black board normalized floating point number system (b = 10, p = 4 rounded; see your class lecture notes for more information), determine the results exactly with rounding for each of the following.

a. 12.3 + 0.0234
b. –0.0321 + 0.000136
c. 12.3 – 0.0234
d. –321 +32.1
e. 132 * 0.987
f. –2.14/0.000137
g. (–0.111 + 0.222) * 0.00111/999    (in left to right order)

4.    Compute the absolute and relative errors for each result of Exercise #3 (not the old number 18). (b = 10, p = 4 rounded only; caution: this means a lot of cases: parts a–g or 7×2 cases. Make a table of all results. See class notes for a proper definition of absolute and relative error. Asolute error is not the same as absolute value.)

5.    Evaluate the following polynomial for x = 1.07, using rounding (b = 10, p = 4) proceeding through the polynomial term by term from left to right, truncating each binary operation. What are the absolute and relative errors of your results?

PLR(x) = 2.75x³ – 2.95x² + 3.16x – 4.67   .

6.    Repeat Exercise 5, but this time proceed from right to left, term by term, i.e.,

PRL(x) = – 4.67 + 3.16x – 2.95x² +2.75x³ ,

6.    Evaluating a polynomial in "nested form" is more efficient. The nested form of the polynomial in Exercise 5 is

((2.75x – 2.95)x + 3.16)x – 4.67

Repeat Exercise 5, but this time do it with this nested form.