Sherman-Morrison

Copyright (C) 2020 Andreas Kloeckner

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Let's set up some matrices and data for the rank-one modification:

Let's start by computing the "base" factorization.

We'll use lu_factor from scipy, which stuffs both L and U into a single matrix (why can it do that?) and also returns pivoting information:

Next, we set up a subroutine to solve using that factorization and check that it works:

As a last step, we try the Sherman-Morrison formula:

$$(A+uv^T)^{-1} = A^{-1} - {A^{-1}uv^T A^{-1} \over 1 + v^T A^{-1}u}$$

To see that we got the right answer, we first compute the right solution of the modified system:

Next, apply Sherman-Morrison to find xhat2: