Related papers: A Contribution to the Numerics of Polynomials and …
Solving a quadratic equation $P(x)=ax^2+bx+c=0$ with real coefficients is known to middle school students. Solving the equation over the quaternions is not straightforward. Huang and So \cite{Huang} give a complete set of formulas, breaking…
In this note, we provide a wide range of upper bounds for the moduli of the zeros of a complex polynomial. The obtained bounds complete a series of previous papers on the location of zeros of polynomials.
Poles of a multi-input multi-output (MIMO) linear system can be computed by solving an eigenvalue problem; however, the problem of computing its invariant zeros is equivalent to a generalized eigenvalue problem. This paper revisits the…
In this paper we consider monic polynomials such that their coefficients coincide with their zeros. These polynomials were first introduced by S. Ulam. We combine methods of algebraic geometry and dynamical systems to prove several results.…
We present some upper and lower bounds for the numerical radius of a bounded linear operator defined on complex Hilbert space, which improves on the existing upper and lower bounds. We also present an upper bound for the spectral radius of…
There have been some effective tools for solving (constant/parametric) semi-algebraic systems in Maple's library RegularChains since Maple 13. By using the functions of the library, e.g., RealRootClassfication, one can prove and discover…
We propose a general methodology for testing whether a given polynomial with integer coefficients is identically zero. The methodology evaluates the polynomial at efficiently computable approximations of suitable irrational points. In…
An algorithm is presented for the efficient and accurate computation of the coefficients of the characteristic polynomial of a general square matrix. The algorithm is especially suited for the evaluation of canonical traces in determinant…
We study algebraic properties of full rank 1 algebras in a general framework and derive a method to verify if one such matrix polynomial sub-algebra is bispectral. We give two examples illustrating the method. In the first one, we consider…
Given an approximation to a multiple isolated solution of a polynomial system of equations, we have provided a symbolic-numeric deflation algorithm to restore the quadratic convergence of Newton's method. Using first-order derivatives of…
Complexity bounds for many problems on matrices with univariate polynomial entries have been improved in the last few years. Still, for most related algorithms, efficient implementations are not available, which leaves open the question of…
This paper provides an accurate method to obtain the bidiagonal factorization of many generalized Pascal matrices, which in turn can be used to compute with high relative accuracy the eigenvalues, singular values and inverses of these…
In this paper we report on an application of computer algebra in which mathematical puzzles are generated of a type that had been widely used in mathematics contests by a large number of participants worldwide. The algorithmic aspect of our…
We introduce the concept of multiplication matrices for ideals of projective dimension zero. We discuss various applications and in particular, we give a new algorithm to compute the variety of an ideal of projective dimension zero.
We give an easy algorithm which calculates the generating function of the cocharacter sequence of the T-ideal of the polynomial identities of the algebra of upper triangular matrices over a field of characteristic zero. Applying this…
This note presents the multivariate Hermite criterion: a practical and powerful algorithm for determining the number of distinct real and complex roots of a zero-dimensional system of polynomials in any finite number of variables. The final…
Factorization of polynomials is one of the foundations of symbolic computation. Its applications arise in numerous branches of mathematics and other sciences. However, the present advanced programming languages such as C++ and J++, do not…
Given a multivariate real (or complex) polynomial $p$ and a domain $\cal D$, we would like to decide whether an algorithm exists to evaluate $p(x)$ accurately for all $x \in {\cal D}$ using rounded real (or complex) arithmetic. Here…
An improved characteristic set algorithm for solving Boolean polynomial systems is proposed. This algorithm is based on the idea of converting all the polynomials into monic ones by zero decomposition, and using additions to obtain…
We present randomized algorithms to compute the sumset (Minkowski sum) of two integer sets, and to multiply two univariate integer polynomials given by sparse representations. Our algorithm for sumset has cost softly linear in the combined…