CSC 420 TEST #1 Due: 10:30 a.m., 9/22/08
(20) 1. A new subatomic storage device has been discovered that has three states (positive, negative, no
charge) instead of our usual two state devices. This leads naturally to a base three machine instead of a
binary, octal, or hexadecimal machine. Each of these trinary digits are called trits, much as binary digits
are called bits. These trits are grouped into six-trit groups called trytes. Design a single precision
floating-point number scheme using four trytes (twenty-four trits). Be explicit about how the sign,
exponent, and mantissa are each to be represented and whether your design uses rounded or chopped
arithmetic. Determine the machine epsilon, underflow, and overflow for your design.
(20) 2. Write a program that will allow the input of values N and F and an integer value B. N represents
the whole number portion and F the fractional portion of the number, N.F, to be converted to base B. For
example, if N = 255.0, F = 0.125, and B = 16, your program should produce FF.2 as output. You should
probably use double precision variables to store N and F in order to accommodate up to 15 decimal digits
on each side of the decimal point. You may assume that 2 < B < 16.
(15) 3. Stirling's formula to approximate n! is given by:
2
n
n
/
e
n
n
!
2
n
n
/
e
n
1
1
12n
1
Investigate the accuracy of this approximation by calculating both the absolute and relative error of the
approximation for increasing values of n. Be sure that the values that you use for
and e are as accurate as
possible.
(15) 4. Find all solutions to x = tan x, with 0 < x < 15.
(15) 5. Given a sequence
{
x
n
n
1
}
, Aitken's
2process generates a new sequence
{
x '
n
n
3
}
defined by the formula:
x ' n
=
xn
xn
xn
1
2
xn
2xn
1
xn
2
Modify your secant method to incorporate one application of Aitken's
2process at each iteration and
compare the rate of convergence of the two sequences to the solution of
x
1
3
=
0.
You may need to
experiment with your starting values and your convergence tolerance in order to obtain enough iterations for
a reasonable comparison.
(15) 6. A chain or perfectly flexible cable supported at its ends takes a shape called a
catenary
. If the
coordinates are chosen such that (x, y) = (0, a) at the low point on this curve, then the equation of the
catenary is y = a cosh (x/a). A cable is suspended from supports at equal height and 20 feet apart. The
middle of the cable sags exactly 10 feet below the level of the supports. What is the value of
a
?
This is a take-home exam. Your answers should be submitted on standard-sized paper, one side only, in a
file folder or other binder. NEATNESS COUNTS !! Also turn in ALL of the source code that you use in
preparing your answers to the test questions. This should be on a diskette, CD, or email attachment as a
zip file named YourLastName.zip. You may use your textbook, course notes, computer hardware and
software, and the library as you see fit, as long as appropriate bibliographical references are cited.
Plagiarism is turning in another person's work as if it were your own, so be sure to acknowledge your
sources. You should not discuss this test with any living person other than your instructor.
