UTK - Math 171 - Alexiades
** Floating Point Numbers - Roundoff **

** Machine numbers **

There are **finitely many** floating point numbers on any computer.
They are NOT evenly spaced (same # between consecutive powers of 2)
so spread out further and further apart.
Any given real number x must be approximated by a nearest
machine number, fl(x), resulting in a ** roundoff error**.

** *** Important consequencies:*

1. There is an inherent error in representing real numbers,
called ** round-off**.

2. Machine arithmetic is not necessarily commutative,
i.e. **the order of operations may affect the result!**

3. ** Loss of significant digits in arithmetic**, especially in:

• ** *** subtractive cancellation:*
when subtracting nearly equal reals

• ** *** unbalanced multiplication:*
when nultiplying by a very small number
(or dividing by a very large number).

4. Number of decimal digits that can be trusted:
7 in single precision, 15 in double precision.

** Machine epsilon ε**
ε is the smallest positive number such that ** 1+ε > 1**
ε = 2^{−p} , p=precision = # of bits in mantissa.
In 32 bit double precision, p=52, so ε = 2^{−52} ≈ 10^{−15}
For any real x, ** fl(x) = x ( 1 + ε )**,
so ** ε = ( fl(x) − x ) / x**
= *relative error in representing x by fl(x)*
Thus, the * absolute error * in representing x by fl(x)
is ** roundoff = fl(x)−x = εx ** ,
so it is proportional to x .

So, the larger the number the larger the roundoff error!
In Matlab: **eps(x)** gives the machine number nearest to x
for that computer.

** eps(1) ** or just ** eps** gives ε itself.
For example: eps(1) = 2.2204e-16 , eps(100) = 1.4211e-14 ,
eps(1000) = 1.1369e-13 , eps(1.e6) = 1.1642e-10 , ...

losing more and more accuracy...
For example: 1+eps == 1 gives false, 1+eps/2 == 1 gives true

So, any (positive) number less than ε is effectively zero
in computation!