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It's often stated (eg: in Numerical Recipes in C) that Cholesky factorization is numerically stable even without column pivoting, unlike LU decomposition, which usually need pivoting schemes.

But I've never seen a proof or argument for why that's true. Does it have to do with the matrix class it works on being Positive Definite? If you LU decompose Positive Definite matrices without pivoting is it still numerically unstable?

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André-Louis Cholesky :) –  t.b. Apr 25 '12 at 23:15
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The fact that one can have a Cholesky decomposition without pivoting is precisely a consequence of the theorem that you can compute the LU decomposition of a symmetric positive definite matrix without pivoting. Here is one paper that tackles the issue of when not to pivot for unsymmetric positive definite systems; note that the criterion they give there vastly simplifies in the symmetric case. Have a look here as well. –  J. M. Apr 26 '12 at 1:08
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