Row Modifications of a Sparse Cholesky Factorization

Abstract

Given a sparse, symmetric positive definite matrix C and an associated sparse Cholesky factorization LDL$\tr$, we develop sparse techniques for updating the factorization after a symmetric modification of a row and column of C. We show how the modification in the Cholesky factorization associated with this rank-2 modification of C can be computed efficiently using a sparse rank-1 technique developed in [T. A. Davis and W. W. Hager, SIAM J. Matrix Anal. Appl., 20 (1999), pp. 606--627]. We also determine how the solution of a linear system Lx = b changes after changing a row and column of C or after a rank-r change in C.