Analysis of the Cholesky Decomposition of a Semi-definite Matrix

Perturbation theory is developed for the Cholesky decomposition of an $n \times n$ symmetric positive semidefinite matrix $A$ of rank~$r$. The matrix $W=\All^{-1}\A{12}$ is found to play a key role in the perturbation bounds, where $\All$ and $\A{12}$ are $r \times r$ and $r \times (n-r)$ submatrices of $A$ respectively. A backward error analysis is given; it shows that the computed Cholesky factors are the exact ones of a matrix whose distance from $A$ is bounded by $4r(r+1)\bigl(\norm{W}+1\bigr)^2u\norm{A}+O(u^2)$, where $u$ is the unit roundoff. For the complete pivoting strategy it is shown that $\norm{W}^2 \le {1 \over 3}(n-r)(4^r- 1)$, and empirical evidence that $\norm{W}$ is usually small is presented. The overall conclusion is that the Cholesky algorithm with complete pivoting is stable for semi-definite matrices. Similar perturbation results are derived for the QR decomposition with column pivoting and for the LU decomposition with complete pivoting. The results give new insight into the reliability of these decompositions in rank estimation