相关论文: Computing sharp and scalable bounds on errors in a…
We describe an approximate rational arithmetic with round-off errors (both absolute and relative) controlled by the user. The rounding procedure is based on the continued fraction expansion of real numbers. Results of computer experiments…
We introduce remarkable upper bounds for the interpolation error constants on triangles, which are sharp and given by simple formulas. These constants are crucial in analyzing interpolation errors, particularly those associated with the…
Roundoff errors cannot be avoided when implementing numerical programs with finite precision. The ability to reason about rounding is especially important if one wants to explore a range of potential representations, for instance for FPGAs…
In this paper, we improve the usual relative error bound for the computation of x^n through iterated multiplications by x in binary floating-point arithmetic. The obtained error bound is only slightly better than the usual one, but it is…
This paper is concerned with certifying that a given point is near an exact root of an overdetermined or singular polynomial system with rational coefficients. The difficulty lies in the fact that consistency of overdetermined systems is…
We revisit the method of Carleman linearization for systems of ordinary differential equations with polynomial right-hand sides. This transformation provides an approximate linearization in a higher-dimensional space through the exact…
We use a recently developed method \cite{Costinetal}, \cite{Dubrovin} to find accurate analytic approximations with rigorous error bounds for the classic similarity solution of Blasius of the boundary layer equation in fluid mechanics, the…
In this paper, we study the arithmetics of skew polynomial rings over finite fields, mostly from an algorithmic point of view. We give various algorithms for fast multiplication, division and extended Euclidean division. We give a precise…
Let $F$ be a univariate polynomial or rational fraction of degree $d$ defined over a number field. We give bounds from above on the absolute logarithmic Weil height of $F$ in terms of the heights of its values at small integers: we review…
It is often useful to have polynomial upper or lower bounds on a one-dimensional function that are valid over a finite interval, called a trust region. A classical way to produce polynomial bounds of degree $k$ involves bounding the range…
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…
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,…
This paper studies the approximation capacity of neural networks with an arbitrary activation function and with norm constraint on the weights. Upper and lower bounds on the approximation error of these networks are computed for smooth…
This paper considers a probabilistic model for floating-point computation in which the roundoff errors are represented by bounded random variables with mean zero. Using this model, a probabilistic bound is derived for the forward error of…
The development of randomized algorithms for numerical linear algebra, e.g. for computing approximate QR and SVD factorizations, has recently become an intense area of research. This paper studies one of the most frequently discussed…
We study limitations of polynomials computed by depth two circuits built over read-once polynomials (ROPs) and depth three syntactically multi-linear formulas. We prove an exponential lower bound for the size of the $\Sigma\Pi^{[N^{1/30}]}$…
The paper is concerned with sharp estimates of constants in Poincare type inequalities for functions having zero mean value on the boundary of a Lipschitz domain or on a measurable part of it. These estimates are useful for various…
We consider multilinear Littlewood polynomials, polynomials in $n$ variables in which a specified set of monomials $U$ have $\pm 1$ coefficients, and all other coefficients are $0$. We provide upper and lower bounds (which are close for $U$…
Extending recent work of Corrado, we derive an algorithm that computes rigorous upper and lower bounds for rectangle scan probabilities for Markov increments. We experimentally examine the closeness of the bounds computed by the algorithm…
Stable multivariate Eulerian polynomials were introduced by Br\"and\'en. Particularizing some variables, it is possible to extract real zero multivariate Eulerian polynomials from them. These real zero multivariate Eulerian polynomials can…