Lecture

Numerical Analysis: Quadrature Formulas

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In course

MATH-250: Advanced numerical analysis I

Construction and analysis of numerical methods for the solution of problems from linear algebra, integration, approximation, and differentiation. ...

Description

This lecture covers the theory behind quadrature formulas, including the choice of nodes and weights, the order of accuracy, and the construction of polynomials to approximate functions. The instructor demonstrates the application of these formulas through examples and proofs.

Instructors (3)

Simone Deparis

Expertise Simone Deparis is a professor at EPFL specializing in applied mathematics, numerical analysis, and the didactics of university-level mathematics. His research spans computational modeling for biomedical applications and innovations in mathematics education, with a focus on equity, self-efficacy, and assessment reform in large STEM cohorts. Mission Our group conducts research at the intersection of applied mathematics and learning sciences. In applied mathematics, we focus on modeling and simulatio

Daniel Kressner

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

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Related concepts (30)

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts.

Gaussian quadrature

In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. (See numerical integration for more on quadrature rules.) An n-point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the nodes x_i and weights w_i for i = 1, ..., n.

Numerical stability

In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues.

Mathematical proof

A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation".

Numerical integration

In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of definite integrals. The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for numerical integration, especially as applied to one-dimensional integrals.

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