Floating point numbers

It is important to understand how computers make small errors during a computation. As a result, some algorithms may yield completely incorrect errors because these errors may be allowed to grow out of control. This happens for unstable algorithms. The first step is understanding how a computer stores numbers in memory. For this purpose, it uses floating point numbers.

What is a floating point number?

Floating point numbers with base 2 have the form

$$\pm (1+\sum _{i=1}^{p-1}{d}_i{2}^{-i})2^e,$$

where $e$ is called the exponent, $p$ is the precision, and

$$1+\sum _{i=1}^{p-1}{d}_i{2}^{-i}$$

is the significand.

Example of floating-point number: 3.140625.

It is positive, so the sign is $+$.

It’s between $2$ (aka, $2^1$) and $4$ (aka, $2^2$). So, the exponent $e$ is equal to $1$.

The significand is 1.5703125. Decomposition of the significand:

$$1.5703125=\frac{1}{2}+\frac{1}{16}+\frac{1}{128}=2^{-1}+2^{-4}+2^{-7}$$

Bits for significand: $d_1=1$, $d_4=1$, $d_7=1$.

Stability of the LU factorization