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What is a pathological matrix?


Say you’re writing some MATLAB and you encounter a matrix you want to LU regularize very accurately.

How do you know if a matrix is “pathological” (slide 134) so that you can use complete pivoting?


This is really hard to answer, or the correct answer is you don’t know,
and you just research. There are certain matrices that are known that
LU complete pivoting would work better on them, than LU partial pivoting.
One such is the so-called Hilbert matrix with entries 1/(i+j-1).
But for practical purposes, you use LU partial pivoting, and if you notice
weird results, then you could consider complete pivoting.
Of course, you could also consider changing the problem so that
you get a better matrix.