相关论文: A Note on Singularities and Polynomial Zeros

A new one-parameter family of iterative method for solving nonlinear equations is constructed and studied. Two variants, both with cubic convergence, are developed, one for finding simple zeros and other for multiple zeros of known…

A problem based on the Extended Euclidean Algorithm applied to a class of polynomials with many factors is presented and believed to be hard. If so, it is a one-way function well suited for applications in digital signicatures.

Finding roots of equations is at the heart of most computational science. A well-known and widely used iterative algorithm is the Newton's method. However, its convergence depends heavily on the initial guess, with poor choices often…

This paper presents a comprehensive survey of methods which can be utilized to search for solutions to systems of nonlinear equations (SNEs). Our objectives with this survey are to synthesize pertinent literature in this field by presenting…

The computation of matrix functions is a well-studied problem. Of special importance are the exponential and the logarithm of a matrix, where the latter also raises existence and uniqueness questions. This is particularly relevant in the…

This thesis settles a number of questions related to computational complexity and algebraic, semidefinite programming based relaxations in optimization and control.

The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients. These fundamental solutions are directly connected with multiple…

A simple iteration methodology for the solution of a set of a linear algebraic equations is presented. The explanation of this method is based on a pure geometrical interpretation and pictorial representation. Convergence using this method…

In this paper we study some fundamental algebraic properties of slice functions and slice regular functions over an alternative $^*$-algebra $A$ over $\mathbb{R}$. These recently introduced function theories generalize to higher dimensions…

There are four division algebras over $\mathbb{R}$, namely real numbers, complex numbers, quaternions, and octonions. Lack of commutativity and associativity make it difficult to investigate algebraic and geometric properties of octonions.…

Equations arising in General Relativity are usually too complicated to be solved analytically and one has to rely on numerical methods to solve sets of coupled partial differential equations. Among the possible choices, this paper focuses…

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…

Many proofs of the fundamental theorem of algebra rely on the fact that the minimum of the modulus of a complex polynomial over the complex plane is attained at some complex number. The proof then follows by arguing the minimum value is…

We simplify the proof of some widely used theoretical theorems, extending their applicability, while correcting some erroneous results. We also generalize key results and present new results that contribute to the development of the theory.…

A new method of root finding is formulated that uses a numerical iterative process involving three points. A given function y = f(x) whose roots are desired is fitted and approximated by a polynomial function of the form P(x)= a(x-b)^N that…

This paper presents analytical solvers for four common types of algebraic equations encountered in robot kinematics: single trigonometric equations, single-angle trigonometric systems, two-angle trigonometric systems, and bilinear two-angle…

Real algebraic geometry adapts the methods and ideas from (complex) algebraic geometry to study the real solutions to systems of polynomial equations and polynomial inequalities. As it is the real solutions to such systems modeling…

Techniques for the evaluation of complex polynomials with one and two variables are introduced. Polynomials arise in may areas such as control systems, image and signal processing, coding theory, electrical networks, etc., and their…

Modelling real world systems frequently requires the solution of systems of nonlinear equations. A number of approaches have been suggested and developed for this computational problem. However, it is also possible to attempt solutions…

There is no general existence theorem for solutions for nonlinear difference equations, so we must prove the existence of solutions in accordance with models one by one. In our work, we found theorems for the existence of analytic solutions…