Related papers: Real-number Computability from the Perspective of …
Real number calculations on elementary functions are remarkably difficult to handle in mechanical proofs. In this paper, we show how these calculations can be performed within a theorem prover or proof assistant in a convenient and highly…
System identification is an important area of science, which aims to describe the characteristics of the system, representing them by mathematical models. Since many of these models can be seen as recursive functions, it is extremely…
Arithmetic constraints on integer intervals are supported in many constraint programming systems. We study here a number of approaches to implement constraint propagation for these constraints. To describe them we introduce integer interval…
In this article, we consider a simple representation for real numbers and propose top-down procedures to approximate various algebraic and transcendental operations with arbitrary precision. Detailed algorithms and proofs are provided to…
Reasoning about real number expressions in a proof assistant is challenging. Several problems in theorem proving can be solved by using exact real number computation. I have implemented a library for reasoning and computing with complete…
We propose here a number of approaches to implement constraint propagation for arithmetic constraints on integer intervals. To this end we introduce integer interval arithmetic. Each approach is explained using appropriate proof rules that…
We examine the convergence properties of sequences of nonnegative real numbers that satisfy a particular class of recursive inequalities, from the perspective of proof theory and computability theory. We first establish a number of results…
An original approach to solving rather difficult probabilistic problems arising in studying the readout of random discrete fields and having no exact analytical solutions at the moment is proposed. Several algorithms for direct, iterative,…
Although the Turing-machine model of computation is widely used in computer science it is fundamentally inadequate as a foundation for the theory of modern scientific computation. The real-number model is described as an alternative.…
In this article, for modelling numeral systems, the operator approach, which is introduced in [25], is generalized for a certain case. An example of such numeral systems is introduced and considered.
Interval arithmetic is a simple way to compute a mathematical expression to an arbitrary accuracy, widely used for verifying floating-point computations. Yet this simplicity belies challenges. Some inputs violate preconditions or cause…
The work is devoted to the development of numerical methods for computing "formal solutions" of interval systems of linear algebraic equations. These solutions are found in Kaucher interval arithmetic, which extends and completes the…
In this work, we show a methodology aimed to improve the quality of the assessment process for subjects related to basic programming. The method takes into account the relevance of the items and the students answers to follow different…
The article gives a survey of mathematical proofs that rely on computer calculations and formal proofs.
We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such…
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…
What is called "numerical reproducibility" is the problem of getting the same result when the scientific computation is run several times, either on the same machine or on different machines, with different types and numbers of processing…
We introduce and study expansions of real numbers with respect to two integer bases.
We present a comprehensive survey of constructions of the real numbers (from either the rationals or the integers) in a unified fashion, thus providing an overview of most (if not all) known constructions ranging from the earliest attempts…
We give a number of formal proofs of theorems from the field of computable analysis. Many of our results specify executable algorithms that work on infinite inputs by means of operating on finite approximations and are proven correct in the…