Fast computation of linear equation with row and column removed

Given $$A, x, b$$ and a linear system $$Ax=b$$, where $$A\in\mathbb{R}^{n\times n}$$ is full rank. Denote $$A_{\backslash i}$$ as a $$(n-1)\times(n-1)$$ matrix where $$A$$ removes its $$i^{th}$$ column and row, similarly $$b_{\backslash i}$$. What's the most efficient way to find a $$y$$ for each $$i\in[n]$$ s.t $$A_{\backslash i}y=b_{\backslash i}$$? (So in total these are $$n$$ linear systems, for different $$i$$ I want a different solution $$y$$).

I know one way is to use Sherman-Morrison formula, but is there any way that doesn't involve calculating $$A^{-1}$$? That is, suppose I have $$x$$ calculated, can $$y$$ be derived just using $$A, x, b$$?

• When you write, "Given a linear system $Ax=b$," do you mean, given all three of $A,x,b$? When you write "find $A_{\backslash i}y=b_{\backslash i}$", do you mean "find $y$"? Jul 14, 2020 at 3:03
• Yes to both questions, I have modified the question. Jul 14, 2020 at 3:08
• Why don't you want to use $A^{-1}$ (or an equivalent factorization)? It seems the right tool for the job here. Jul 14, 2020 at 6:26
• How would you use Sherman-Morrison formula? Jul 14, 2020 at 16:40
• @FedericoPoloni it's roughly 80k*80k and dense. The issue is mostly with memory, I assume existing linear system solvers use iterative methods? Taking inverse seems to load the whole matrix into memory. Jul 14, 2020 at 18:16