The concept of comprehensive triangular decomposition (CTD) was first introduced by Chen et al. in their CASC’2007 paper and could be viewed as an analogue of comprehensive Gröbner systems for parametric polynomial systems. The first complete algorithm for computing CTD was also proposed in that paper and implemented in the RegularChains library in Maple. Following our previous work on generic ...

We show that the characters of all highest weight modules over an affine Lie algebra with the highest weight away from the critical hyperplane are meromorphic functions in the positive half of the Cartan subalgebra, their singularities being at most simple poles at zeros of real roots. We obtain some information about these singularities. 0. Introduction 0.0.1. Let g be a simple finite-dimensio...

The number of multiplications required for matrix multiplication, for the triangular decomposition of a matrix with partial pivoting, and for the Cholesky decomposition of a positive definite symmetric matrix, can be roughly halved if Winograd’s identity is used to compute the inner products involved. Floating-point error bounds for these algorithms are shown to be comparable to those for the n...

We introduce the concept of comprehensive triangular decomposition (CTD) for a parametric polynomial system F with coefficients in a field. In broad words, this is a finite partition of the the parameter space into regions, so that within each region the “geometry” (number of irreducible components together with their dimensions and degrees) of the algebraic variety of the specialized system F ...

Deterministic recursive algorithms for the computation of matrix triangular decompositions with permutations like LU and Bruhat decomposition are presented for the case of commutative domains. This decomposition can be considered as a generalization of LU and Bruhat decompositions, because they both may be easily obtained from this triangular decomposition. Algorithms have the same complexity a...

Mastronardi and Van Dooren [this journal, 34 (2013) pp. 173–196] recently introduced the block antitriangular (“Batman”) decomposition for symmetric indefinite matrices. Here we show the simplification of this factorisation for saddle point matrices and demonstrate how it represents the common nullspace method. We show that rank-1 updates to the saddle point matrix can be easily incorporated in...

In this paper a Levinson-like algorithm is derived for solving symmetric positive definite semiseparable plus diagonal systems of equations. In a first part we solve a Yule-Walker-like system of equations. Based on this O(n) solver an algorithm for a general right-hand side is derived. The new method has a linear complexity and takes 19n − 13 operations. The relation between the algorithm and a...

This paper describes triangular risk decomposition, which provides a useful, geometric view of the relationship between the risk of a position and that of the portfolio. We review triangular decomposition for the case of the parametric, or delta-normal, Value-at-Risk (VaR), which assumes that changes in a portfolio’s value are normally distributed with mean zero. We then generalize it for the c...