<div dir="ltr"><div class="gmail_extra"><br><div class="gmail_quote">On Mon, Jan 28, 2013 at 1:27 PM, Ling Zou <span dir="ltr"><<a href="mailto:lingzou80@gmail.com" target="_blank">lingzou80@gmail.com</a>></span> wrote:<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div>Ahh.. that's true! <br>In case M is not A (as you pointed out earlier), does PCLU provide the approximated inverse matrix of M^{-1} using LU factorization on M?</div>
</blockquote></div><br>Not really, preconditioners are based on inexact algorithms applied to A, not explicit formation of an M that is easier to factor exactly. Since the preconditioner P ("=M^{-1}") is non-singular, there _exists_ an M such that P=M^{-1}, but M is not explicitly computed and it's not used in the solve. Only P is used, and only in special cases (like incomplete factorization) is there even a practical algorithm available to compute M if you wanted to. (For many interesting algorithms, M is dense even though A is sparse.)</div>
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