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Course# MATH-251(b): Numerical analysis

Summary

The students will learn key numerical techniques for solving standard mathematical problems in science and engineering. The underlying mathematical theory and properties are discussed.

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Instructors (2)

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Un MOOC francophone d'algèbre linéaire accessible à tous, enseigné de manière rigoureuse et ne nécessitant aucun prérequis.

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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.

Numerical methods for ordinary differential equations

Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation.

Matrix decomposition

In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems. In numerical analysis, different decompositions are used to implement efficient matrix algorithms. For instance, when solving a system of linear equations , the matrix A can be decomposed via the LU decomposition.

LU decomposition

In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix decomposition). The product sometimes includes a permutation matrix as well. LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix.

Data

In common usage and statistics, data (USˈdætə; UKˈdeɪtə) is a collection of discrete or continuous values that convey information, describing the quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted formally. A datum is an individual value in a collection of data. Data is usually organized into structures such as tables that provide additional context and meaning, and which may themselves be used as data in larger structures.

Lectures in this course (13)

Interpolation by Intervals: Lagrange InterpolationMATH-251(b): Numerical analysis

Covers Lagrange interpolation using intervals to find accurate polynomial approximations.

Finite Differences Formulas: Error AnalysisMATH-251(b): Numerical analysis

Explores finite differences formulas, error analysis, and numerical experiments in computational mathematics.

Numerical Quadrature FormulasMATH-251(b): Numerical analysis

Explains numerical quadrature formulas, with a focus on Simpson's rule application.

Linear Systems: Chapters 4, 5, 6MATH-251(b): Numerical analysis

Explores the link between linear systems and optimization through elimination and LU decomposition.

Differential Equations: Approximation Methods and AnalysisMATH-251(b): Numerical analysis

Covers methods for approximating solutions to differential equations and their real-world applications.