LU Decomposition of PSD Matrix + Diagonal Matrix

Question

If I have a psd, symmetric matrix $\mathbf{A}$ and I need to do LU decomps on $\mathbf{B_i}= \mathbf{A} + \mathbf{D_i}$ (where $\mathbf{D_i}$ is a diagonal psd matrix, where $\mathbf{D_i}$ changes randomly). Is there an easy way to go from the LU of $\mathbf{A}$ + some function of $\mathbf{D_i}$ to get the LU of $\mathbf{B_i}$?

Any references in the literature would be greatly appreciated.

Answer 1

Use a Cholesky decomposition or an LDL decomposition instead of LU.

Judging from answers to:

• Diagonal update of a symmetric positive definite matrix
• Solving a system with a small rank diagonal update
• Can diagonal plus fixed symmetric linear systems be solved in quadratic time after precomputation?

there's no good way to update any of those decompositions. You'd do no worse to recompute the decomposition from scratch each time unless the rank of the diagonal matrix is really low, in which case, maybe you could do something similar to Sherman-Morrison(-Woodbury)-type updates.