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Quadratic eigenvalue problems (QEP) and more generally polynomial eigenvalue problems (PEP) are among the most common types of nonlinear eigenvalue problems. Both problems, especially the QEP, have extensive applications. A typical approach…
The singularity structure of solutions of a class of Hamiltonian systems of ordinary differential equations in two dependent variables is studied. It is shown that for any solution, all movable singularities, obtained by analytic…
Homotopy perturbation method is used for solving the multi-point boundary value problems. The approximate solution is found in the form of a rapidly convergent series. Several numerical examples have been considered to illustrate the…
Given a real closed polytope $P$, we first describe the Fourier transform of its indicator function by using iterations of Stokes' theorem. We then use the ensuing Fourier transform formulations, together with the Poisson summation formula,…
Discovering "good" algorithms for an operation is often considered an art best left to experts. What if there is a simple methodology, an algorithm, for systematically deriving a family of algorithms as well as their cost analyses, so that…
We consider discrete best approximation problems in the setting of tropical algebra, which is concerned with the theory and application of algebraic systems with idempotent operations. Given a set of input--output pairs of an unknown…
Locating the zeros of quaternionic polynomials is a fundamental problem with significant implications across scientific and engineering disciplines, yet the noncommutative nature of quaternion multiplication makes it fundamentally more…
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…
A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. A much harder version under recent consideration is to enforce bounds constraints on the approximating polynomial. In…
In this paper we study the subset sum problem with real numbers. Starting from the given problem, we formulate a quadratic maximization problem over a polytope, P, which is eventually written as a distance maximization to a fixed point over…
This paper formulates an elementary algorithm for resolution of singularities in a neighborhood of a singular point over a field of characteristic zero. The algorithm is composed of finite sequences of Newton polyhedra and monomial…
In this paper we propose a method that uses Lagrange multipliers and numerical algebraic geometry to find all critical points, and therefore globally solve, polynomial optimization problems. We design a polyhedral homotopy algorithm that…
The problem of finding superintegrable Hamiltonians and their integrals of motion can be reduced to solving a series of compatibility equations that result from the overdetermination of the commutator or Poisson bracket relations. The…
A deep approximation is an approximating function defined by composing more than one layer of simple functions. We study deep approximations of functions of one variable using layers consisting of low-degree polynomials or simple conformal…
The homotopy continuation method has been widely used in solving parametric systems of nonlinear equations. But it can be very expensive and inefficient due to singularities during the tracking even though both start and end points are…
The real type of a finite family of univariate polynomials characterizes the combined sign behavior of the polynomials over the real line. We derive an explicit formula for the number of real types subject to given degree bounds. For the…
We present a new method, called the pixel array method, for approximating all solutions in a bounding box for an arbitrary nonlinear system of relations. In contrast with other solvers, our approach requires that the user must specify which…
We describe fast algorithms for approximating the connection coefficients between a family of orthogonal polynomials and another family with a polynomially or rationally modified measure. The connection coefficients are computed via…
Certain monotonicity properties of the Poisson approximation to the binomial distribution are established. As a natural application of these results, exact (rather than approximate) tests of hypotheses on an unknown value of the parameter…
We propose an Extended Hybrid High-Order scheme for the Poisson problem with solution possessing weak singularities. Some general assumptions are stated on the nature of this singularity and the remaining part of the solution. The method is…