Row modifications of a sparse cholesky factorization

• Published in: SIAM journal on matrix analysis and applications Vol. 26; no. 3; pp. 621 - 639
• Main Authors: DAVIS, Timothy A, HAGER, William W
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 2005
• Subjects: Algorithms; Applied sciences; Computer science; control theory; systems; Exact sciences and technology; Linear algebra; Linear programming; Mathematics; Methods of scientific computing (including symbolic computation, algebraic computation); Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Sciences and techniques of general use; Software; Software engineering; Variables; Modification; direct methods; matrix updates 65F05; Numerical linear algebra; Factorization; mathematical software; Direct method; 65F50. 65Y20; Linear system; Symmetric matrix; Cholesky method; Positive definite matrix; Sparse matrix; Software; Cholesky factorization; sparse matrices
• ISSN: 0895-4798; 1095-7162
• DOI: 10.1137/s089547980343641x

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.