CS 594 - Spring 2002

Understanding Parallel Architectures:

From Theory to Practice

Assignment 4

(Typo corrected)

Due: February 13, 2002

The accuracy of a floating point system can be characterized by a quantity variously known as the unit roundoff, machine precision, machine epsilon, or macheps. Its value, which we denote by ò _mach , depends on the particular rounding rules used. With rounding by chopping,

ò _mach = b ^{1 - p},

(where b is the base and p is the number of digits in the mantissa) whereas with rounding to nearest

ò _mach = ½ b ^{1 - p}.

The unit roundoff is important because it determines the maximum possible relative error in representing a nonzero real number x in a floating point system.

A characterization of the unit roundoff that you may sometimes see is that it is the smallest number ò such that

fl( 1 + ò ) > 1.

Here fl(x) is the floating point approximation to x.

Part 1:

Compute the machine precision on your computer. In addition compute the overflow and underflow threshold. Explain how you did the computation and your results.

Part 2:

Explain why an alternating infinite series, such as

for x < 0, is difficult to evaluate accurately in floating point arithmetic.

Part 3:

What happens when you evaluate the infinite series?

Explain why summing the series in floating point arithmetic yields a finite sum.