We review strategies for differentiating matrix-based computations, and derive symbolic and algorithmic update rules for differentiating expressions containing the Cholesky decomposition. We recommend new ‘blocked’ algorithms, based on differentiating the Cholesky algorithm DPOTRF in the LAPACK library, which uses ‘Level 3’ matrix-matrix operations from BLAS, and so is cache-friendly and easy t...

We discuss a multilinear generalization of the singular value decomposition. There is a strong analogy between several properties of the matrix and the higher-order tensor decomposition; uniqueness, link with the matrix eigenvalue decomposition, first-order perturbation effects, etc., are analyzed. We investigate how tensor symmetries affect the decomposition and propose a multilinear generaliz...

One of the most important and well studied problems related to sparse Cholesky factorization is to compute elimination orderings that give as few nonzero entries as possible in the resulting factors. We study the problem of modifying a given elimination ordering through local reorderings. We present new theoretical results on equivalent orderings, including a new characterization of such orderi...

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In this paper we give a combinatorial proof of an addition formula for weighted partial Motzkin paths. The addition formula allows us to determine the LDU decomposition of a Hankel matrix of the polynomial sequence defined by weighted partial Motzkin paths. As a direct consequence, we get the determinant of the Hankel matrix of certain combinatorial sequences. In addition, we obtain an addition...

We extend, in the context of a connected real semisimple Lie group, some results on the QR iteration and the Cholesky iteration of a nonsingular matrix. A group theoretic understanding of the abstract mechanisms of the iterations is obtained.

We introduce a generalized Lilbert [Lucas-Hilbert] matrix. Explicit formulæ are derived for the LU-decomposition and their inverses, as well as the Cholesky decomposition. The approach is to use q-analysis and to leave the justification of the necessary identities to the q-version of Zeilberger’s celebrated algorithm.

Dedicated to our friends Beresford and Velvel on the occasion of their sixtieth birthdays. ABSTRACT We show that a certain matrix norm ratio studied by Parlett has a supremum that is O(p n) when the chosen norm is the Frobenius norm, while it is O(log n) for the 2-norm. This ratio arises in Parlett's analysis of the Cholesky decomposition of an n by n matrix.