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We present theory and practice for robust implementations of bivariate Jacobi set and Reeb space algorithms. Robustness is a fundamental topic in computational geometry that deals with the issues of numerical errors and degenerate cases in…
A novel method of summing asymptotic series is advanced. Such series repeatedly arise when employing perturbation theory in powers of a small parameter for complicated problems of condensed matter physics, statistical physics, and various…
We introduce a new iterative root-finding method for complex polynomials, dubbed {\it Newton-Ellipsoid} method. It is inspired by the Ellipsoid method, a classical method in optimization, and a property of Newton's Method derived in…
In computational practice, most attention is paid to rational approximations of functions and approximations by the sum of exponents. We consider a wide enough class of nonlinear approximations characterized by a set of two required…
We investigate solutions to the functional equation $f(f(x)) = e^x$, which can be interpreted as the problem of finding a half iterate of the exponential map. While no elementary solution exists, we construct and analyze non-elementary…
We propose a numerical method to solve the Monge-Ampere equation which admits a classical convex solution. The Monge-Ampere equation is reformulated into an equivalent first-order system. We adopt a novel reconstructed discontinuous…
The Newton-Raphson method is a fundamental root-finding technique with numerous applications in physics. In this study, we propose a parameterized variant of the Newton-Raphson method, inspired by principles from physics. Through analytical…
We study the geometry of algebraic numbers in the complex plane, and their Diophantine approximation, aided by extensive computer visualization. Motivated by these images, called algebraic starscapes, we describe the geometry of the map…
We present a novel certified and complete algorithm to compute arrangements of real planar algebraic curves. It provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in…
Nowadays, analysing data from different classes or over a temporal grid has attracted a great deal of interest. As a result, various multiple graphical models for learning a collection of graphical models simultaneously have been derived by…
Broyden's method is a general method commonly used for nonlinear systems of equations, when very little information is available about the problem. We develop an approach based on Broyden's method for nonlinear eigenvalue problems. Our…
Given any polynomial system with fixed monomial term structure, we give explicit formulae for the generic number of roots with specified coordinate vanishing restrictions. For the case of affine space minus an arbitrary union of coordinate…
A problem concerning the shift of roots of a system of homogeneous algebraic equations is investigated. Its conservation and decomposition of a multiple root into simple roots are discussed.
As showed in (Fiedler, 1990), any polynomial can be expressed as a characteristic polynomial of a complex symmetric arrowhead matrix. This expression is not unique. If the polynomial is real with only real distinct roots, the matrix can be…
In various areas of applied numerics, the problem of calculating the logarithm of a matrix A emerges. Since series expansions of the logarithm usually do not converge well for matrices far away from the identity, the standard numerical…
In supervised learning using kernel methods, we often encounter a large-scale finite-sum minimization over a reproducing kernel Hilbert space (RKHS). Large-scale finite-sum problems can be solved using efficient variants of Newton method,…
This paper proposes a novel algorithm for the problem of structural image segmentation through an interactive model-based approach. Interaction is expressed in the model creation, which is done according to user traces drawn over a given…
We study the numerical approximation of a class of degenerate parabolic stochastic partial differential equations on non-compact metric graphs, which naturally arise in the asymptotic analysis of Hamiltonian flows under small noise…
In this article, numerical integration is formulated as evaluation of a matrix function of a matrix that is obtained as a projection of the multiplication operator on a finite-dimensional basis. The idea is to approximate the continuous…
The $k$-core decomposition is a widely studied summary statistic that describes a graph's global connectivity structure. In this paper, we move beyond using $k$-core decomposition as a tool to summarize a graph and propose using $k$-core…