Related papers: Backward Error Measures for Roots of Polynomials
A new measure called min-max elementwise backward error is introduced for approximate roots of scalar polynomials $p(z)$. Compared with the elementwise relative backward error, this new measure allows for larger relative perturbations on…
The eigenvalues of a matrix polynomial can be determined classically by solving a generalized eigenproblem for a linearized matrix pencil, for instance by writing the matrix polynomial in companion form. We introduce a general scaling…
We examine several of the normal-form multivariate polynomial rootfinding methods of Telen, Mourrain, and Van Barel and some variants of those methods. We analyze the performance of these variants in terms of their asymptotic temporal…
In backward error analysis, an approximate solution to an equation is compared to the exact solution to a nearby modified equation. In numerical ordinary differential equations, the two agree up to any power of the step size. If the…
We derive computable expressions of structured backward errors of approximate eigenelements of *-palindromic and *-anti-palindromic matrix polynomials. We also characterize minimal structured perturbations such that approximate…
For the pure biharmonic equation and a biharmonic singular perturbation problem, a residual-based error estimator is introduced which applies to many existing nonconforming finite elements. The error estimator involves the local…
Residual-based a~posteriori error estimators are derived for the modified Morley FEM, proposed by Wang, Xu, Hu [J. Comput. Math, 24(2), 2006], for the singularly perturbed biharmonic equation and the nonlinear von K\'arm\'an equations. The…
We derive computable formulas for the structured backward errors of a complex number $\lambda$ when considered as an approximate eigenvalue of rational matrix polynomials that carry a symmetry structure. We consider symmetric,…
Backwards analysis, first popularized by Seidel, is often the simplest most elegant way of analyzing a randomized algorithm. It applies to incremental algorithms where elements are added incrementally, following some random permutation,…
We prove sharp, computable error estimates for the propagation of errors in the numerical solution of ordinary differential equations. The new estimates extend previous estimates of the influence of data errors and discretisation errors…
Roots of random polynomials have been studied exclusively in both analysis and probability for a long time. A famous result by Ibragimov and Maslova, generalizing earlier fundamental works of Kac and Erdos-Offord, showed that the…
Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a real coefficient polynomial. They can be approximated at a low computational cost if the…
Inverse problems use physical measurements along with a computational model to estimate the parameters or state of a system of interest. Errors in measurements and uncertainties in the computational model lead to inaccurate estimates. This…
The paper is concerned with parabolic time-periodic boundary value problems which are of theoretical interest and arise in different practical applications. The multiharmonic finite element method is well adapted to this class of parabolic…
The numerical solution of an ordinary differential equation can be interpreted as the exact solution of a nearby modified equation. Investigating the behaviour of numerical solutions by analysing the modified equation is known as backward…
The recently introduced backward Monte-Carlo method [Johan Carlsson, arXiv:math.NA/0010118] is validated, benchmarked, and compared to the conventional, forward Monte-Carlo method by analyzing the error in the Monte-Carlo solutions to a…
In this work, we propose an a pointwise a posteriori error estimator for conforming finite element approximations of eigenfunctions corresponding to multiple and clustered eigenvalues of elliptic operators. It is proven that the pointwise a…
We consider mixed finite element approximation of a singularly perturbed fourth-order elliptic problem with two different boundary conditions, and present a new measure of the error, whose components are balanced with respect to the…
The Durand-Kerner algorithm is a widely used iterative technique for simultaneously finding all the roots of a polynomial. However, its convergence heavily depends on the choice of initial approximations. This paper introduces two novel…
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…