Bidiagonal decompositions of oscillating systems of vectors

Bidiagonal Decompositions of Oscillating Systems of Vectors

We establish necessary and sufficient conditions, in the language of bidiagonal decompositions, for a matrix V to be an eigenvector matrix of a totally positive matrix. Namely, this is the case if and only if V and V −T are lowerly totally positive. These conditions translate into easy positivity requirements on the parameters in the bidiagonal decompositions of V and V −T . Using these decompo...

Bidiagonal decompositions, minors and applications

Ela Bidiagonal Decompositions, Minors and Applications

Abstract. Matrices, called ε-BD matrices, that have a bidiagonal decomposition satisfying some sign constraints are analyzed. The ε-BD matrices include all nonsingular totally positive matrices, as well as their matrices opposite in sign and their inverses. The signs of minors of ε-BD matrices are analyzed. The zero patterns of ε-BD matrices and their triangular factors are studied and applied ...

Butterflies Solve Bidiagonal Toeplitz Systems

Here and hereafter T = (ti,j) n−1 i,j=0, ti,j = 1 for i − j = 0, ti,j = c 6= 0 for i − j = 1, and ti,j = 0 otherwise, x = (xi) n−1 i=0 , and b = (bi) n−1 i=0 . Note that the system is scaled so that the main diagonal is composed exclusively of ones, with no loss of generality. These systems are at the heart of problems as diverse as cubic spline and Bspline curve fitting [3], [11], precondition...

NORMAL FORM SOLUTION OF REDUCED ORDER OSCILLATING SYSTEMS

This paper describes a preliminary investigation into the use of normal form theory for modelling large non-linear dynamical systems. Limit cycle oscillations are determined for simple two-degree-of-freedom double pendulum systems. The double pendulum system is reduced into its centre manifold before computing normal forms. Normal forms are obtained using a period averaging method which is appl...