This paper presents several new results on the inversion of full normal rank nonsquare polynomial matrices. New analytical right/left inverses of polynomial matrices are introduced, including the so-called τ -inverses, σ-inverses and, in particular, S-inverses, the latter providing the most general tool for the design of various polynomial matrix inverses. The applicationoriented problem of sel...

Cardinal’s matrix version of the Sebastiao e Silva polynomial root-finder rapidly approximates the roots as the eigenvalues of the associated Frobenius matrix. We preserve rapid convergence to the roots but amend the algorithm to allow input polynomials with multiple roots and root clusters. As in Cardinal’s algorithm, we repeatedly square the Frobenius matrix in nearly linear arithmetic time p...

We consider the following problem in the theory of (virtual) species of structures: Given a polynomial species F such that its derivative dF equals 1, will its inverse w.r.t substitution F h?1i also be polynomial species? We shall see that this is in general not the case. From this result it follows that an analogue of the jacobian conjecture in the context of species does not hold.

The Bertrand-Darboux integrability condition for a certain class of perturbed harmonic oscillators is studied from the viewpoint of the Birkhoff-Gustavson(BG)-normalization: By solving an inverse problem of the BG-normalization on computer algebra, it is shown that if the perturbed harmonic oscillator with a homogeneous cubic-polynomial potential and the perturbed harmonic oscillator with a hom...

We present a general method of operational nature to analyze and obtain solutions for a variety of equations of mathematical physics and related mathematical problems. We construct inverse differential operators and produce operational identities, involving inverse derivatives and families of generalised orthogonal polynomials, such as Hermite and Laguerre polynomial families. We develop the me...

We prove that the Jordan Normal Form of a rational matrix can be computed exactly in polynomial time. We obtain the transformation matrix and its inverse exactly, and we show how to apply the basis transformation to any commuting matrices.

We present the asymptotically fastest known algorithms for some basic problems on univariate polynomial matrices: rank, nullspace, determinant, generic inverse, reduced form [8, 9, 16, 17]. We show that they essentially can be reduced to two computer algebra techniques, minimal basis computations and matrix fraction expansion/reconstruction, and to polynomial matrix multiplication. Such reducti...

For a large indefinite linear system, there exists the option to directly precondition for the normal equations. Matrix nearness problems are formulated to assess the attractiveness of this alternative. Polynomial preconditioning leads to polynomial approximation problems involving lemniscate-like sets, both in the plane and in Cn×n. A natural matrix analytic extension for lemniscates is introd...

An analysis on the algebraic decoding of the (31, 16, 7) quadratic residue (QR) code with reducib le generator polynomial that uses the inverse-free Berlekamp-Massey (IFBM) algorithm to determine the error-locator polynomial is presented in this paper. The primary known syndrome S1 will be equal to zero for some weight-3 error patterns. However, the zero S1 does not cause a decoding failure whi...

We reprove a result of Boppana and Lagarias: If Π P 2 Σ P 2 then there exists a partial function f that is computable by a polynomial-size family of circuits, but no inverse of f is computable by a polynomial-size family of circuits. We strengthen this result by showing that there exist length-preserving total functions that are one-way by circuit size and that are computable in uniform polynom...