Related papers: How to hunt wild constants
Numerical approximate computation can solve large and complex problems fast. It has the advantage of high efficiency. However it only gives approximate results, whereas we need exact results in many fields. There is a gap between…
Many interesting and useful symbolic computation algorithms manipulate mathematical expressions in mathematically meaningful ways. Although these algorithms are commonplace in computer algebra systems, they can be surprisingly difficult to…
The typical processors used for scientific computing have fixed-width data-paths. This implies that mathematical libraries were specifically developed to target each of these fixed precisions (binary16, binary32, binary64). However, to…
Static analyses overwhelmingly trade precision for soundness and automation. For this reason, their use-cases are restricted to situations where imprecision isn't prohibitive. In this paper, we propose and specify a static analysis that…
This paper introduces some methods to determine the simultaneous approximation constants of a class of well approximable numbers $\zeta_{1},\zeta_{2},...,\zeta_{k}$. The approach relies on results on the connection between the set of all…
Interval arithmetic libraries provide the four elementary arithmetic operators for operand intervals bounded by floating-point numbers. Actual implementations need to make a large case analysis that considers, e.g., magnitude relations…
We use the martingale-theoretic approach of game-theoretic probability to incorporate imprecision into the study of randomness. In particular, we define a notion of computable randomness associated with interval, rather than precise,…
In this paper we present a convergence rate analysis of inexact variants of several randomized iterative methods. Among the methods studied are: stochastic gradient descent, stochastic Newton, stochastic proximal point and stochastic…
Logarithmic Number Systems (LNS) hold considerable promise in helping reduce the number of bits needed to represent a high dynamic range of real-numbers with finite precision, and also efficiently support multiplication and division.…
Lagrangian relaxation and approximate optimization algorithms have received much attention in the last two decades. Typically, the running time of these methods to obtain a $\epsilon$ approximate solution is proportional to…
Multidimensional Continued Fraction Algorithms are generalizations of the Euclid algorithm and find iteratively the gcd of two or more numbers. They are defined as linear applications on some subcone of $\mathbb{R}^d$. We consider…
High dimensional covariance estimation and graphical models is a contemporary topic in statistics and machine learning having widespread applications. An important line of research in this regard is to shrink the extreme spectrum of the…
We describe a framework in which is possible to develop and implement algorithms for the approximation of invariant measures of dynamical systems with a given bound on the error of the approximation. Our approach is based on a general…
A very simple example of an algorithmic problem solvable by dynamic programming is to maximize, over sets A in {1,2,...,n}, the objective function |A| - \sum_i \xi_i 1(i \in A,i+1 \in A) for given \xi_i > 0. This problem, with random…
In mutation testing the question whether a mutant is equivalent to its program is important in order to compute the correct mutation score. Unfortunately, answering this question is not always possible and can hardly be obtained just by…
We propose new estimates for the frontier of a set of points. They are defined as kernel estimates covering all the points and whose associated support is of smallest surface. The estimates are written as linear combinatio- ns of kernel…
A calculator program has been written to give confidence intervals on branching ratios for rare decay modes (or similar quantities) calculated from the number of events observed, the acceptance factor, the background estimate and the…
We completely describe a new domain for abstract interpretation of numerical programs. Fixpoint iteration in this domain is proved to converge to finite precise invariants for (at least) the class of stable linear recursive filters of any…
Many modern solvers and program analyzers rely on non-monotone reasoning (e.g. negation-as-failure, speculative updates, backtracking) for which classical monotone fixed-point methods do not apply. The general problem of finding the fixed…
We derive an exact and efficient Bayesian regression algorithm for piecewise constant functions of unknown segment number, boundary location, and levels. It works for any noise and segment level prior, e.g. Cauchy which can handle outliers.…