Modified Cholesky Decomposition and Applications
Thomas J. McSweeney · MIMS EPrints (University of Southampton) · 2017
The modified Cholesky decomposition is one of the standard tools in various areas of mathematics for dealing with symmetric indefinite matrices that are required to be positive definite. We survey the literature and determine which of the existing modified Cholesky algorithms is most suitable for inclusion in the Numerical Algorithms Group (NAG) software library, focussing in particular on the algorithms of Gill, Murray and Wright, Schnabel and Eskow, Cheng and Higham, and Moré and Sorensen. In order to make this determination we consider how best to take advantage of modern computer architectures and existing numerical software. We create an efficient implementation of the chosen algorithm and perform extensive numerical testing to ensure that it works as intended. We then discuss various applications of the modified Cholesky decomposition and show how the new implementation can be used for some of these. In particular, significant attention is devoted to describing how the modified Cholesky decomposition can be used to compute an upper bound on the distance to the nearest correlation matrix.