The least-squares solution of [itex]A x = b[/itex] using Tikhonov regularization with a matrix [itex]\mu^2 I[/itex] has the solution:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

x = \sum_i \left( \frac{\sigma_i^2}{\sigma_i^2 + \mu^2} \right) \left( \frac{u_i^T b}{\sigma_i} \right) v_i

[/tex]

where [itex]A = U S V^T[/itex] is the SVD of [itex]A[/itex] and [itex]u_i,v_i[/itex] are the columns of [itex]U,V[/itex].

For ill-conditioned matrices, the singular values [itex]\sigma_i[/itex] could be tiny leading to problems in computing the quantity [itex]\left( \frac{\sigma_i^2}{\sigma_i^2 + \mu^2} \right)[/itex] since [itex]\sigma_i^2[/itex] could underflow.

Does anyone know how to compute this solution safely and efficiently in IEEE double precision?

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# Tikhonov regularization

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