I am solving linear least squares problems with generalized Tikhonov regularization, minimizing the function:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\chi^2 = || b - A x ||^2 + \lambda^2 || L x ||

[/tex]

where [itex]L[/itex] is a diagonal regularization matrix and [itex]\lambda[/itex] 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":

[tex]

\tilde{A} = A L^{-1}, \tilde{x} = L x

[/tex]

so that

[tex]

\chi^2 = || b - \tilde{A} \tilde{x} ||^2 + \lambda^2 || \tilde{x} ||

[/tex]

This system can now be solved easily with the SVD of [itex]\tilde{A}[/itex]. My problem is that I want to compute the condition number of [itex]A[/itex] while solving the system, but I don't see any easy way to compute this since I calculate only the SVD of [itex]\tilde{A}[/itex]. It seems that there is no simple relationship between the singular values of [itex]A[/itex] and the singular values of [itex]\tilde{A}[/itex], even when [itex]L[/itex] is a diagonal matrix.

Does anyone know of a solution to this problem, without having to separately compute the SVD of [itex]A[/itex] itself?

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# I Tikhonov regularization and SVD to compute condition number

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