The generalized triangular decomposition

The generalized triangular decomposition

Given a complex matrix H, we consider the decomposition H = QRP∗, where R is upper triangular and Q and P have orthonormal columns. Special instances of this decomposition include the singular value decomposition (SVD) and the Schur decomposition where R is an upper triangular matrix with the eigenvalues of H on the diagonal. We show that any diagonal for R can be achieved that satisfies Weyl’s...

Hierarchical Comprehensive Triangular Decomposition

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 ...

Comprehensive Triangular Decomposition

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 ...

Technical Note - Generalized Upper Bounds and Triangular Decomposition in the Simplex Method

Triangular Decomposition Methods for Solving

This paper generalizes to the nonlinear case a standard way to solve general sparse systems of linear equations. In particular, Duu 1977] has suggested that row and column interchanges be applied to permute the coeecient matrix of a linear system into block lower triangular form. The linear system then can be solved by using the associated block Gauss-Seidel or forward block substitution scheme...