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We present a combination of two algorithms that accurately calculate multiple roots of general polynomials. Algorithm I transforms the singular root-finding into a regular nonlinear least squares problem on a pejorative manifold, and…
We consider the problem of finding for a given $N$-tuple of polynomials (real or complex) the closest $N$-tuple that has a common divisor of degree at least $d$. Extended weighted Euclidean seminorm of the coefficients is used as a measure…
In this paper we revisit the greatest common right divisor (GCRD) extraction from a set of polynomial matrices $P_i(\lambda)\in \F[\la]^{m_i\times n}$, $i=1,\ldots,k$ with coefficients in a generic field $\F$, and with common column…
We present a new GCD algorithm of two integers or polynomials. The algorithm is iterative and its time complexity is still $O(n \\log^2 n ~ log \\log n)$ for $n$-bit inputs.
In the realm of statistical learning, the increasing volume of accessible data and increasing model complexity necessitate robust methodologies. This paper explores two branches of robust Bayesian methods in response to this trend. The…
We propose a new algorithm solving the extended gcd problem, which provides a solution minimizing one of the two coordinates. The algorithm relies on elementary arithmetic properties.
Dual Bernstein polynomials of one or two variables have proved to be very useful in obtaining B\'{e}zier form of the $L^2$-solution of the problem of best polynomial approximation of B\'{e}zier curve or surface. In this connection, the…
A new algorithm is presented for computing the largest degree invariant factor of the Sylvester matrix (with respect either to $x$ or $y$) associated to two polynomials $a$ and $b$ in $\mathbb F_q[x,y]$ which have no non-trivial common…
Subresultant of two univariate polynomials is a fundamental object in computational algebra and geometry with many applications (for instance, parametric GCD and parametric multiplicity of roots). In this paper, we generalize the theory of…
Given two polynomials, we find a convergence property of the GCD of the rising factorial and the falling factorial. Based on this property, we present a unified approach to computing the universal denominators as given by Gosper's algorithm…
We present an optimized algorithm calculating determinant for multivariate polynomial matrix on GPU. The novel algorithm provides precise determinant for input multivariate polynomial matrix in controllable time. Our approach is based on…
We study polynomial-time approximation algorithms for the Quantum Max-Cut (QMC) problem. Given an edge-weighted graph $G$ on n vertices, the QMC problem is to determine the largest eigenvalue of a particular $2^n \times 2^n$ matrix that…
The problem of finding the distance from a given $n \times n$ matrix polynomial of degree $k$ to the set of matrix polynomials having the elementary divisor $(\lambda-\lambda_0)^j, \, j \geqslant r,$ for a fixed scalar $\lambda_0$ and $2…
The inverse of a large matrix can often be accurately approximated by a polynomial of degree significantly lower than the order of the matrix. The iteration polynomial generated by a run of the GMRES algorithm is a good candidate, and its…
In this paper, an original reduction algorithm for solving simultaneous multivariate polynomial equations is presented. The algorithm is exponential in complexity, but the well-known algorithms, such as the extended Euclidean algorithm and…
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…
In April 2025 GMV announced a competition for finding the best method to solve a particular polynomial system over a finite field. In this paper we provide a method for solving the given equation system significantly faster than what is…
In this paper we develop an optimisation based approach to multivariate Chebyshev approximation on a finite grid. We consider two models: multivariate polynomial approximation and multivariate generalised rational approximation. In the…
Complex polynomial optimization has recently gained more and more attention in both theory and practice. In this paper, we study the optimization of a real-valued general conjugate complex form over various popular constraint sets including…
We propose an algorithm that approximates a given matrix polynomial of degree $d$ by another skew-symmetric matrix polynomial of a specified rank and degree at most $d$. The algorithm is built on recent advances in the theory of generic…