Modified Gram-Schmidt-based methods for block downdating the Cholesky factorization (Q629427)

Let \(A\) be an \(n\times n\) symmetric positive definite matrix and \(A = R^T R\) its Cholesky factorization, that is, \(R\) is upper triangular and has positive diagonal entries. Consider the modified matrix \(\tilde A = A - X^T X = R^T R - X^T X\), where \(X\) is an \(t\times n\) matrix, and assume that \(\tilde A\) remains positive definite. The paper is concerned with computing the Cholesky factorization of \(\tilde A\) by appropriately downdating the Cholesky factorization of \(A\) rather than computing the factorization of \(\tilde A\) explicitly from scratch. A hyperbolic modified Gram-Schmidt method is proposed for this purpose. Several numerical experiments with random matrices and an academic example illustrate the accuracy properties of the newly proposed method.