MATH 441/541 - Numerical Analysis
First Meeting: Computer Number Representation and Arithmetic
Thursday, August 23, 2007
Outline
1. Internal Number Representation
(a) Bases (Binary, Decimal, Hexidecimal, Converting)
(b) Computer (Floating-Point) Representation of Numbers (IEEE Standard) – BINARY
y= (−1)s2(c−shift)(1 + f)
where
s= sign bit
c−shift = shifted exponent (characteristic)
f= fractional part (mantissa)
with double precision-floating point numbers using 64 bit representation:
1 bit for sign, s
11 bits for exponent, c
52 bits for fraction, f
sexponent, cfraction, f
· · ·
i. Determining the value of shift.
ii. The number of numbers we can represent exactly.
iii. The range of the numbers we can represent.
iv. Plotting computer numbers (to see the meshing pattern).
v. Determining the pattern:
A. Increasing/decreasing number of bits for c=⇒?
B. Increasing/decreasing number of bits for f=⇒?
vi. Definitions:
A. Underflow
B. Overflow
C. Machine Epsilon,
D. Unit Roundoff, u
(c) Normalized k-Digit DECIMAL Floating-Point Representation of Numbers
±0.d1d2d3· · · dk×10n, d16= 0, di∈ {0,1,· · · ,9}
i. Rounding vs. Chopping
ii. Absolute and Relative Error
