# I Tikhonov regularization and SVD to compute condition number

1. Apr 7, 2016

### vibe3

I am solving linear least squares problems with generalized Tikhonov regularization, minimizing the function:
$$\chi^2 = || b - A x ||^2 + \lambda^2 || L x ||$$
where $L$ is a diagonal regularization matrix and $\lambda$ is the regularization parameter. I am solving this system using the singular value decomposition, and I use the normal trick to convert the system into "Tikhonov standard form":
$$\tilde{A} = A L^{-1}, \tilde{x} = L x$$
so that
$$\chi^2 = || b - \tilde{A} \tilde{x} ||^2 + \lambda^2 || \tilde{x} ||$$
This system can now be solved easily with the SVD of $\tilde{A}$. My problem is that I want to compute the condition number of $A$ while solving the system, but I don't see any easy way to compute this since I calculate only the SVD of $\tilde{A}$. It seems that there is no simple relationship between the singular values of $A$ and the singular values of $\tilde{A}$, even when $L$ is a diagonal matrix.

Does anyone know of a solution to this problem, without having to separately compute the SVD of $A$ itself?

2. Apr 12, 2016