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Given a set of linear equations $AX=B$, say $A$ is an ill posed matrix (has a few singular values equal or very close to zero), which numerical algorithm (conjugate gradient, least squares or steepest decent etc ) should be used to obtain the best solution? More specifically, is there a concrete comparison between these methods?

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While your question is on topic here, for future reference you may get better/quicker/more complete answers if you ask it at scicomp.stackexchange.com/questions/tagged/condition-number –  Willie Wong Jan 3 '13 at 14:10
    
Preconditioning will also be helpful. –  Fixed Point Jan 7 '13 at 22:08
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1 Answer

What do you mean by "best solution"?

If your problem is ill-posed, then you need regularization. In this case, conjugate gradient provides a form of regularization. Also, you may wish to consider using Truncated Singular Value Decomposition, consisting into filtering out, during the matrix inversion, the less relevant singular values which are responsible for the ill-posedness. You may also wish to consider Tikhonov regularization (leading, under the SVD approach, to a particular shaping of the singular values).

As reference on inverse problems (especially linear inverse problems) and related (regularized) solution strategies, a good book is M.Bertero, P.Boccacci, Introduction to Inverse Problems in Imaging, IOP Publishing, Bristol.

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