Related papers: New Progress in Classic Area: Polynomial Root-squa…

We revisit the notion of root polynomials, thoroughly studied in [F. Dopico and V. Noferini, Root polynomials and their role in the theory of matrix polynomials, Linear Algebra Appl. 584:37--78, 2020] for general polynomial matrices, and…

Let $\mathbb{Z}_p[x]$ be the set of all functions whose coefficients are in the field of $p$-adic integers $\mathbb{Z}_p$. This work considers a problem of finding a root of a polynomial equation $P(x)=0$ where $P(x)\in\mathbb{Z}_p[x]$. The…

In the first part of the paper, we address an invertible matrix polynomial $L(z)$ and its inverse $\hat{L}(z) := -L(z)^{-1}$. We present a method for obtaining a canonical set of root functions and Jordan chains of $L(z)$ through elementary…

In this paper, we provide a new method to find all zeros of polynomials with quaternionic coefficients located on only one side of the powers of the variable (these polynomials are called simple polynomials). This method is much more…

For an odd prime $p$, we say $f(X) \in {\mathbb F}_p[X]$ computes square roots in $\mathbb F_p$ if, for all nonzero perfect squares $a \in \mathbb F_p$, we have $f(a)^2 = a$. When $p \equiv 3 \mod 4$, it is well known that $f(X) =…

Incremental Newton (IN) iteration, proposed by Iannazzo, is stable for computing the matrix $p$th root, and its computational cost is $\mathcal{O}(n^3p)$ flops per iteration. In this paper, a cost-efficient variant of IN iteration is…

Computing the regularized solution of Bayesian linear inverse problems as well as the corresponding regularization parameter is highly desirable in many applications. This paper proposes a novel iterative method, termed the Projected Newton…

In recent years many efforts have been devoted to finding bidiagonal factorizations of nonsingular totally positive matrices, since their accurate computation allows to numerically solve several important algebraic problems with great…

Let $p(x)=a_0 + a_1 x + \ldots + a_n x^n$ be a polynomial with all roots real and satisfying $x \leq -\delta$ for some $0<\delta <1$. We show that for any $0 < \epsilon <1$, the value of $p(1)$ is determined within relative error $\epsilon$…

Arthur Cohn's irreducibility criterion for polynomials with integer coefficients and its generalization connect primes to irreducibles, and integral bases to the variable $x$. As we follow this link, we find that these polynomials are ready…

The Newton-Raphson (N-R) method is useful to find the roots of a polynomial of degree n. However, this method is limited since it diverges for the case in which polynomials only have complex roots if a real initial condition is taken. In…

We consider the problem of learning high dimensional polynomial transformations of Gaussians. Given samples of the form $p(x)$, where $x\sim N(0, \mathrm{Id}_r)$ is hidden and $p: \mathbb{R}^r \to \mathbb{R}^d$ is a function where every…

The reciprocal square root is an important computation for which many sophisticated algorithms exist (see for example \cite{Moroz,863046,863031} and the references therein). A common theme is the use of Newton's method to refine the…

Polynomial factorization and root finding are among the most standard themes of computational mathematics. Yet still, little has been done for polynomials over quaternion algebras, with the single exception of Hamiltonian quaternions for…

To study a Dirichlet polynomial $f(s)=\frac{a_{m}}{m^{s}}+\cdots +\frac{a_{n}}{n^{s}}$ by regarding it as a multivariate polynomial via the canonical map $\phi$ sending $p_i^{-s}$ to an indeterminate $X_i$, with $p_i$ the $i$th prime…

Suppose $p$ is a prime, $t$ is a positive integer, and $f\!\in\!\mathbb{Z}[x]$ is a univariate polynomial of degree $d$ with coefficients of absolute value $<\!p^t$. We show that for any fixed $t$, we can compute the number of roots in…

Suppose $k,p\!\in\!\mathbb{N}$ with $p$ prime and $f\!\in\!\mathbb{Z}[x]$ is a univariate polynomial with degree $d$ and all coefficients having absolute value less than $p^k$. We give a Las Vegas randomized algorithm that computes the…

Bernstein-Sato polynomial of a hypersurface is an important object with numerous applications. It is known, that it is complicated to obtain it computationally, as a number of open questions and challenges indicate. In this paper we propose…

Cardinal's factorization algorithm of 1996 splits a univariate polynomial into two factors with root sets separated by the imaginary axis, which is an important goal itself and a basic step toward root-finding. The novelty of the algorithm…

In this paper, we develop two new randomized block-coordinate optimistic gradient algorithms to approximate a solution of nonlinear equations in large-scale settings, which are called root-finding problems. Our first algorithm is…