public class NormOpsextends Object
Norms are a measure of the size of a vector or a matrix. One typical application is in error analysis.
Vector norms have the following properties:
- ||x|| > 0 if x ≠ 0 and ||0|| = 0
- ||αx|| = |α| ||x||
- ||x+y|| ≤ ||x|| + ||y||
Matrix norms have the following properties:
- ||A|| > 0 if A ≠ 0 where A ∈ ℜ m × n
- || α A || = |α| ||A|| where A ∈ ℜ m × n
- ||A+B|| ≤ ||A|| + ||B|| where A and B are ∈ ℜ m × n
- ||AB|| ≤ ||A|| ||B|| where A and B are ∈ ℜ m × m
Matrix norms can be induced from vector norms as is shown below:
||A||M = maxx≠0||Ax||v/||x||v
where ||.||M is the induced matrix norm for the vector norm ||.||v.
By default implementations that try to mitigate overflow/underflow are used. If the word fast is found before a function's name that means it does not mitigate those issues, but runs a bit faster.