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Roots Multiplicity without Companion Matrices

Koprowski P.

Fundamenta Informaticae

2017

Vol. 153, nr 3

265--270

We show a method for constructing a polynomial interpolating roots' multiplicities of another polynomial, that does not use companion matrices. This leads to a modification to Guersenzvaig-Szechtman square-free decomposition algorithm that is more efficient both in theory and in practice.

About a certain relation between two polynomials of the same degree

Piwowarczyk T.

Archives of Control Sciences

2011

Vol. 21, no. 1

85-103

This article presents several different methods for solving the problem of how to find a certain relation defined in chapter 2. The first method deals with the identities known in the theory of symmetric polynomials as the elements of a certain vector space. The second method is designed around the matrix transformations between symmetric polynomials. The third method is designed around the property of a linear operator and its characteristic polynomial. The fourth method is designed in the area of complex numbers, and introduces the multiplication group of 'complex roots of one'. Significant improvement in the third and fourth method is made by introducing so called 'block method'. It facilitates all calculations by making them much shorter. The article ends with an example showing symmetry and regularity of all procedures. Finally, the article shows how to solve the problem for any degree n of the polynomial, and for any degree k. At the end of the paper solutions for n < 5 and k < 5 are tabulated.

A uniform quantitative stiff stability estimate for BDF schemes

Auzinger W., Herfort W.

Opuscula Mathematica

2006

Vol. 26, no. 2

203-227

The concepts of stability regions, A- and A(α)-stability - albeit based on scalar models - turned out to be essential for the identification of implicit methods suitable for the integration of stiff ODEs. However, for multistep methods, knowledge of the stability region provides no information on the quantitative stability behavior of the scheme. In this paper we fill this gap for the important class of Backward Differentiation Formulas (BDF). Quantitative stability bounds are derived which are uniformly valid in the stability region of the method. Our analysis is based on a study of the separation of the characteristic roots and a special similarity decomposition of the associated companion matrix.