Can someone please explain what is the Sherman-Morrison formula and it's specialities when it comes to matrix calculations? I'm a little bit confused on understanding how the preconditioning works with Sherman-Morrison formula.

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    $\begingroup$ What do you expect that you cannot trivially find by searching the web? $\endgroup$ – Christian Waluga Nov 6 '18 at 18:07
  • $\begingroup$ Please edit the post with: what do you do not understand? So people may be can help you $\endgroup$ – Mauro Vanzetto Nov 7 '18 at 11:51
  • $\begingroup$ This question is more suited to math.stackexchange, there in the linear-algebra and numerical mathematics section. $\endgroup$ – Nox Nov 7 '18 at 13:25

Sherman-Morrison formula helps to find the inverse of the matrix cheap after a rank-1 update. Sherman-Morrison-Woodbury does it for a low-rank update (not necessarily rank-1).

In short,

  1. You have a matrix $A$ and you already have a computed $A^{-1}$.
  2. For some reason, you need to do a rank-1 update to $A$. This rank-1 update can be described as an outer-product of two vectors $u$ and $v$.
  3. So, you want an inverse of $(A+uv^T)$, but you don't want to redo all the calculations. Can you reuse your knowledge of $A^{-1}$? Yes.
  4. Sherman-Morrison tells you that $$ (A+uv^T)^{-1} = A^{-1}-\frac{A^{-1}uv^TA^{-1}}{1+v^tA^{-1}u} $$ which allows huge savings since you are just performing relatively cheap manipulations with an already obtained $A^{-1}$.

I strongly encourage you to read at least the Wikipedia articles regarding this topic in order to get an understanding about

  • numerical stability
  • requirements for the formula to be applicable

Important notes:

  • calculations of the inverse should be avoided at all costs
  • same ideas (SM, SMW) usually can be applied to factorizations, where they are extremely useful

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