# SVD approximation error

1. Jul 1, 2013

### yiorgos

It is well known that the approximation error of the SVD is
$\left\| \textbf{A}-\textbf{A}_{\rm SVD} \right\| = σ_k$,
where $σ_k$, is the k-th greatest singular value of matrix Ʃ in the SVD.

Yes this tells nothing about the accuracy of the approximation.
E.g. is it $10^{-3}, 10^{-6}, 0.1, >1?$.

I need to solve a system of equations $\textbf{A}x =b$ and I want to estimate the approximation error ε
$\left\| \textbf{A}-\tilde{\textbf{A}} \right\| = \epsilon$,
introduced by several different methods.

How is ε related with the rank k of matrix instead of the obscure $σ_k$?