Abstract

Perturbation theory is developed for the Cholesky decomposition of an n × n symmetric positive semi-definite matrix A of rank r. The matrix W = A −1 11 A12 is found to play a key role in the perturbation bounds, where A11 and A12 are r × r and r × (n − r) submatrices of A respectively. A backward error analysis is given; it shows that the computed Cholesky fac-tors are the exact ones of a matrix whose distance from A is bounded by 4r(r + 1) � �W �2+1 � 2 u�A�2+O(u 2), where u is the unit roundoff. For the complete pivot-ing strategy it is shown that �W � 2 2 ≤ 1 3 (n −r)(4r −1), and empirical evidence that �W �2 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. Key words. Cholesky decomposition, positive semi-definite matrix, per-turbation theory, backward error analysis, QR decomposition, rank estimation, LINPACK.

Citations