Lecture

Bisection Algorithm: Convergence and Continuity

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In course

MATH-101(g): Analysis I

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable. ...

Description

This lecture covers the Bisection Algorithm, focusing on the convergence of sequences and the continuity of functions. It explains the properties of supremum and infimum, the Bolzano-Weierstrass theorem, and the concept of intermediate values. The lecture also discusses the convergence criteria of sequences and the uniqueness of limits, with a detailed demonstration of the algorithm's application. Additionally, it explores the image of intervals under functions, emphasizing the importance of continuity and the relationship between minimum and maximum values.

Instructors (2)

Annalisa Buffa

Teaching & PhD PhD Students Gonzalo Bonilla Moreno, Philipp Christoph Weder, Yannis Dirk Voet, Fabio Marcinno, Ferhat Sindy, Francesco Sala, Markus Renoldner, Mohamed Ben Abdelouahab, Davor Dobrota Past EPFL PhD Students Luca Coradello, Ondine Chanon, Riccardo Puppi Courses Numerical approximation of PDEs MATH-451 The course is about the derivation, theoretical analysis and implementation of the finite element method for the numerical approximation of partial differential equations in on

Lénaïc Chizat

Teaching & PhD PhD Students Javier Esteban Maass Martinez Past EPFL PhD Students Guillaume Yitian Wang Courses Computational optimal transport MATH-528 Computational aspects of measure transportation, from classical optimal transport to modern denoising diffusion models. Topics in machine learning MATH-520 Mathematical analysis of modern supervised machine learning techniques, with an emphasis on the mathematics of artificial neural networks.

Official source

Ontological neighbourhood

Mathematics

Topology: General topology

Related concepts (32)

Limit inferior and limit superior

In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant.

Intermediate value theorem

In mathematical analysis, the intermediate value theorem states that if is a continuous function whose domain contains the interval , then it takes on any given value between and at some point within the interval. This has two important corollaries: If a continuous function has values of opposite sign inside an interval, then it has a root in that interval (Bolzano's theorem). The of a continuous function over an interval is itself an interval.

Infimum and supremum

In mathematics, the infimum (abbreviated inf; plural infima) of a subset of a partially ordered set is the greatest element in that is less than or equal to each element of if such an element exists. In other words, it is the greatest element of that is lower or equal to the lowest element of . Consequently, the term greatest lower bound (abbreviated as ) is also commonly used. The supremum (abbreviated sup; plural suprema) of a subset of a partially ordered set is the least element in that is greater than or equal to each element of if such an element exists.

Extreme value theorem

In calculus, the extreme value theorem states that if a real-valued function is continuous on the closed interval , then must attain a maximum and a minimum, each at least once. That is, there exist numbers and in such that: The extreme value theorem is more specific than the related boundedness theorem, which states merely that a continuous function on the closed interval is bounded on that interval; that is, there exist real numbers and such that: This does not say that and are necessarily the maximum and minimum values of on the interval which is what the extreme value theorem stipulates must also be the case.

Essential infimum and essential supremum

In mathematics, the concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, but adapted to measure theory and functional analysis, where one often deals with statements that are not valid for all elements in a set, but rather almost everywhere, that is, except on a set of measure zero. While the exact definition is not immediately straightforward, intuitively the essential supremum of a function is the smallest value that is greater than or equal to the function values everywhere while ignoring what the function does at a set of points of measure zero.

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