Dixon 1908] introduces three distinct determinant formulations for the resultant of three bivariate polynomials of bidegree (m; n). The rst technique applies Sylvester's dialytic method to construct the resultant as the determinant of a matrix of order 6mn. The second approach uses Cayley's determinant device to form a more compact representation for the resultant as the determinant of a matrix...

We present a general block formulation of the least mean square (LMS) algorithm for adaptive filtering. This formulation has an exact equivalence with the original LMS algorithm, hence retaining the same convergence properties, while allowing a reduction in arithmetic complexity, even for very small block lengths. Working with small block lengths is very interesting from an implementation point...

Cache oblivious algorithms are designed to benefit from any existing cache hierarchy—regardless of cache size or architecture. In matrix computations, cache oblivious approaches are usually obtained from block-recursive approaches. In this article, we extend an existing cache oblivious approach for matrix operations, which is based on Peano space-filling curves, for multiplication of sparse and...

This paper presents a framework for synthesizing eecient out-of-core programs for block recursive algorithms such as the fast Fourier transform (FFT) and Batcher's bitonic sort. The block recursive algorithms considered in this paper are described using tensor (Kronecker) product and other matrix operations. The algebraic properties of the matrix representation are used to derive eecient out-of...

ABSTRACT A Toeplitz-block-Toeplitz (TBT) matrix is block Toeplitz with Toeplitz blocks. TBT systems of equations arise in 2D interpolation, 2-D linear prediction and 2-D least-squares deconvolution problems. Although the doubly Toeplitz structure should be exploitable in a fast algorithm, existing fast algorithms only exploit the block Toeplitz structure, not the Toeplitz structure of the block...

A Las Vegas type probabilistic algorithm is presented for nding the Frobenius canonical form of an n n matrix T over any eld K. The algorithm requires O~(MM(n)) = MM(n) (logn) O(1) operations in K, where O(MM(n)) operations in K are suucient to multiply two n n matrices over K. This nearly matches the lower bound of (MM(n)) operations in K for this problem, and improves on the O(n 4) operations...

The relationship between the Jordan forms of the matrix products AB and BA for some given A and B was first described by Harley Flanders in 1951. Their non-zero eigenvalues and non-singular Jordan structures are the same, but their singular Jordan block sizes can differ by 1. We present an elementary proof that owes its simplicity to a novel use of the Weyr characteristic.

By elementary linear algebra involving row operations (replacing row i by row i plus row j), and column exchanges, we can write G as (Ik|A) where A is some k×n−k matrix. Now the matrix H = ( −A In−k ) satisfies GH = 0. Thus every vector y = xG satisfies yH = 0. Furthermore standard linear algebra implies that the space of vectors that satisfies yH = 0 is at most k and so this must be the space ...

A new polynomial approach is proposed to analysis of the standard and positive descriptor electrical circuits composed of resistors, coils, capacitors and voltage (current) sources. It is shown that for given descriptor fractional electrical circuit the equivalent standard fractional electrical circuit can be found by premultiplication of the equation of the descriptor electrical circuit by sui...

Starting from the first elementary canonical state model, the new simple state model of the thirdorder dynamical systems that belong to Class C is derived. A typical property of this new model is a very simple form of its partial transformation matrix in the conditions of linear topological conjugacy, which represents the unity matrix. Complete state equations and the corresponding integrator-b...