Related papers: On solution of tropical discrete best approximatio…
We consider the problem of computing the nearest matrix polynomial with a non-trivial Smith Normal Form. We show that computing the Smith form of a matrix polynomial is amenable to numeric computation as an optimization problem.…
Uniform polynomial approximation, also called minimax approximation or Chebyshev approximation, consists in searching polynomial approximation that minimizes the worst case error. Optimality conditions for the uniform approximation of…
We introduce the notion of tropical defects, certificates that a system of polynomial equations is not a tropical basis, and provide two algorithms for finding them in affine spaces of complementary dimension to the zero set. We use these…
Building on introducing exponentially clustered poles, Trefethen and his collaborators introduced lightning algorithms for approximating functions of singularities. These schemes may achieve root-exponential convergence rates. In…
We investigate location problems whose optimum lies in the tropical convex hull of the input points. Firstly, we study geodesically star-convex sets under the asymmetric tropical distance and introduce the class of tropically quasiconvex…
We exploit the idea to use the maximal-entropy method, successfully tested in information theory and statistical thermodynamics, to determine approximating function's coefficients and squared errors' weights simultaneously as output of one…
Weighted least squares polynomial approximation uses random samples to determine projections of functions onto spaces of polynomials. It has been shown that, using an optimal distribution of sample locations, the number of samples required…
We examine a new optimization problem formulated in the tropical mathematics setting as a further extension of certain known problems. The problem is to minimize a nonlinear objective function, which is defined on vectors over an idempotent…
This paper introduces an efficient algorithm for computing the best approximation of a given matrix onto the intersection of linear equalities, inequalities and the doubly nonnegative cone (the cone of all positive semidefinite matrices…
This paper presents a novel method for generating a single polynomial approximation that produces correctly rounded results for all inputs of an elementary function for multiple representations. The generated polynomial approximation has…
This paper considers the approximation of a monomial $x^n$ over the interval $[-1,1]$ by a lower-degree polynomial. This polynomial approximation can be easily computed analytically and is obtained by truncating the analytical Chebyshev…
We describe an approach for finding upper bounds on an ODE dynamical system's maximal Lyapunov exponent among all trajectories in a specified set. A minimization problem is formulated whose infimum is equal to the maximal Lyapunov exponent,…
We study best approximation to a given function, in the least square sense on a subset of the unit circle, by polynomials of given degree which are pointwise bounded on the complementary subset. We show that the solution to this problem, as…
The maximum function, on vectors of real numbers, is not differentiable. Consequently, several differentiable approximations of this function are popular substitutes. We survey three smooth functions which approximate the maximum function…
We present a numerical algorithm for finding real non-negative solutions to polynomial equations. Our methods are based on the expectation maximization and iterative proportional fitting algorithms, which are used in statistics to find…
The algorithms of Pan (1995) and(2002) approximate the roots of a complex univariate polynomial in nearly optimal arithmetic and Boolean time but require precision of computing that exceeds the degree of the polynomial. This causes…
We consider a decision-making problem to evaluate absolute ratings of alternatives from the results of their pairwise comparisons according to two criteria, subject to constraints on the ratings. We formulate the problem as a bi-objective…
A fully implementable filtered polynomial approximation on spherical shells is considered. The method proposed is a quadrature-based version of a filtered polynomial approximation. The radial direction and the angular direction of the…
We derive efficient algorithms for coarse approximation of algebraic hypersurfaces, useful for estimating the distance between an input polynomial zero set and a given query point. Our methods work best on sparse polynomials of high degree…
We propose a fast and scalable optimization method to solve chance or probabilistic constrained optimization problems governed by partial differential equations (PDEs) with high-dimensional random parameters. To address the critical…