Related papers: Numerical Instability of Algebraic Rootfinding Met…
The usual definition of the stability region of implicit multistep methods often implies that there are some isolated points of stability within the region of instability of the numerical method. These isolated stable points may appear when…
This paper presents a guided tour of some specific problems encountered in the stability analysis of linear dynamical systems including delays in their systems' representation. More precisely, we will address the characterization of…
The usual methods for root finding of polynomials are based on the iteration of a numerical formula for improvement of successive estimations. The unpredictable nature of the iterations prevents to search roots inside a pre-specified region…
The purpose of this article is to study the algorithmic complexity of the Besicovitch stability of noisy subshifts of finite type, a notion studied in a previous article. First, we exhibit an unstable aperiodic tiling, and then see how it…
Given a diagonalizable matrix $A$, we study the stability of its invariant subspaces when its matrix of eigenvectors is ill-conditioned. Let $\mathcal{X}_1$ be some invariant subspace of $A$ and $X_1$ be the matrix storing the right…
We show in this paper that the roots $x_1$ and $x_2$ of a scalar quadratic polynomial $ax^2+bx+c=0$ with real or complex coefficients $a$, $b$ $c$ can be computed in a element-wise mixed stable manner, measured in a relative sense. We also…
Loop invariants are properties of a program loop that hold before and after each iteration of the loop. They are often employed to verify programs and ensure that algorithms consistently produce correct results during execution.…
After reconsidering the theorem of continuity of the roots of a polynomial in terms of its coefficients in the deformation framework, we study the stability of the greater common divisor of two polynomials compared to perturbations on their…
Existing structural analysis methods may fail to find all hidden constraints for a system of differential-algebraic equations with parameters if the system is structurally unamenable for certain values of the parameters. In this paper, for…
Functional iterations such as Newton's are a popular tool for polynomial root-finding. We consider realistic situation where some (e.g., better-conditioned) roots have already been approximated and where further computations is directed to…
Projection methods provide an appealing way to construct reduced-order models of large-scale linear dynamical systems: they are intuitively motivated and fairly easy to compute. Unfortunately, the resulting reduced models need not inherit…
We develop a framework to give upper bounds on the "practical" computational complexity of stability problems for a wide range of nonlinear continuous and hybrid systems. To do so, we describe stability properties of dynamical systems using…
A new criterion for A-stability of peer two-step methods is presented which is verifiable exactly in exact arithmetic by checking semi-definiteness of a certain test matrix. It depends on the existence of two positive definite weight…
Focusing on the bipartite Stable Marriage problem, we investigate different robustness measures related to stable matchings. We analyze the computational complexity of computing them and analyze their behavior in extensive experiments on…
The Nystr\"om method is a widely used technique for improving the scalability of kernel-based algorithms, including kernel ridge regression, spectral clustering, and Gaussian processes. Despite its popularity, the numerical stability of the…
We investigate errors in tangents and adjoints of implicit functions resulting from errors in the primal solution due to approximations computed by a numerical solver. Adjoints of systems of linear equations turn out to be unconditionally…
Given a polynomial P in several variables over an algebraically closed field, we show that except in some special cases that we fully describe, if one coefficient is allowed to vary, then the polynomial is irreducible for all but at most…
The symmetric multistep methods developed by Quinlan and Tremaine (1990) are shown to suffer from resonances and instabilities at special stepsizes when used to integrate nonlinear equations. This property of symmetric multistep methods was…
Numerical analysis for linear constant-coefficients Finite Difference schemes was developed approximately fifty years ago. It relies on the assumption of scheme stability and in particular -- for the $L^2$ setting -- on the absence of…
In this paper, we consider a matroid generalization of the stable matching problem. In particular, we consider the setting where preferences may contain ties. For this generalization, we propose a polynomial-time algorithm for the problem…