Numerical integration (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 xi and weights wi for i = 1, …, n. The modern formulation using orthogonal polynomials was developed by Carl Gustav Jacobi in 1826. The most common domain of integration for such a rule is taken as [−1, 1], so the rule is stated as which is exact for polynomials of degree 2n − 1 or less. This exact rule is known as the Gauss-Legendre quadrature rule. The quadrature rule will only be an accurate approximation to the integral above if f (x) is well-approximated by a polynomial of degree 2n − 1 or less on [−1, 1]. The Gauss-Legendre quadrature rule is not typically used for integrable functions with endpoint singularities. Instead, if the integrand can be written as where g(x) is well-approximated by a low-degree polynomial, then alternative nodes xi' and weights wi' will usually give more accurate quadrature rules. These are known as Gauss-Jacobi quadrature rules, i.e., Common weights include (Chebyshev–Gauss) and . One may also want to integrate over semi-infinite (Gauss-Laguerre quadrature) and infinite intervals (Gauss–Hermite quadrature). It can be shown (see Press, et al., or Stoer and Bulirsch) that the quadrature nodes xi are the roots of a polynomial belonging to a class of orthogonal polynomials (the class orthogonal with respect to a weighted inner-product). This is a key observation for computing Gauss quadrature nodes and weights. (Wikipedia).
Gaussian Quadrature | Lecture 40 | Numerical Methods for Engineers
An explanation of Gaussian quadrature. An example of how to calculate the weights and nodes for two-point Legendre-Gauss quadrature. Join me on Coursera: https://www.coursera.org/learn/numerical-methods-engineers Lecture notes at http://www.math.ust.hk/~machas/numerical-methods-for-engi
From playlist Numerical Methods for Engineers
Jesús María Sanz-Serna: Gauss's Gaussian quadrature
HYBRID EVENT Recorded during the meeting "1Numerical Methods and Scientific Computing" the November 9, 2021 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on
From playlist Numerical Analysis and Scientific Computing
PUSHING A GAUSSIAN TO THE LIMIT
Integrating a gaussian is everyones favorite party trick. But it can be used to describe something else. Link to gaussian integral: https://www.youtube.com/watch?v=mcar5MDMd_A Link to my Skype Tutoring site: dotsontutoring.simplybook.me or email dotsontutoring@gmail.com if you have ques
From playlist Math/Derivation Videos
Gaussian Quadrature 3: The Explanation of the Technique
https://bit.ly/PavelPatreon https://lem.ma/LA - Linear Algebra on Lemma http://bit.ly/ITCYTNew - Dr. Grinfeld's Tensor Calculus textbook https://lem.ma/prep - Complete SAT Math Prep
From playlist Part 4 Linear Algebra: Inner Products
(PP 6.3) Gaussian coordinates does not imply (multivariate) Gaussian
An example illustrating the fact that a vector of Gaussian random variables is not necessarily (multivariate) Gaussian.
From playlist Probability Theory
(PP 6.1) Multivariate Gaussian - definition
Introduction to the multivariate Gaussian (or multivariate Normal) distribution.
From playlist Probability Theory
Here I evaluate the Gaussian Integral in under a minute. This is a must-see for calculus lovers! Gaussian Integral: https://youtu.be/kpmRS4s6ZR4 Gaussian Integral Playlist: https://www.youtube.com/playlist?list=PLJb1qAQIrmmCgLyHWMXGZnioRHLqOk2bW Subscribe to my channel: https://youtube.co
From playlist Gaussian Integral
Joe Neeman: Gaussian isoperimetry and related topics II
The Gaussian isoperimetric inequality gives a sharp lower bound on the Gaussian surface area of any set in terms of its Gaussian measure. Its dimension-independent nature makes it a powerful tool for proving concentration inequalities in high dimensions. We will explore several consequence
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Slides and more information: https://mml-book.github.io/slopes-expectations.html
From playlist There and Back Again: A Tale of Slopes and Expectations (NeurIPS-2020 Tutorial)
From playlist COMP0168 (2020/21)
Preview: The Magic of Gaussian Quadrature - A Billion Times Better than the Next Best Thing
https://bit.ly/PavelPatreon https://lem.ma/LA - Linear Algebra on Lemma http://bit.ly/ITCYTNew - Dr. Grinfeld's Tensor Calculus textbook https://lem.ma/prep - Complete SAT Math Prep
From playlist Part 1 Linear Algebra: An In-Depth Introduction with a Focus on Applications
Engineering CEE 20: Engineering Problem Solving. Lecture 23
UCI CIvil & Environmental Engineering 20 Engineering Problem Solving (Spring 2013) Lec 23. Engineering Problem Solving View the complete course: http://ocw.uci.edu/courses/cee_20_introduction_to_computational_engineering_problem_solving.html Instructor: Jasper Alexander Vrugt, Ph.D. Licen
From playlist Engineering CEE 20: Engineering Problem Solving
Mod-01 Lec-14 Convergence of Gaussian Integration
Elementary Numerical Analysis by Prof. Rekha P. Kulkarni,Department of Mathematics,IIT Bombay.For more details on NPTEL visit http://nptel.ac.in
From playlist NPTEL: Elementary Numerical Analysis | CosmoLearning Mathematics
Gaussian Integral 7 Wallis Way
Welcome to the awesome 12-part series on the Gaussian integral. In this series of videos, I calculate the Gaussian integral in 12 different ways. Which method is the best? Watch and find out! In this video, I calculate the Gaussian integral by using a technique that is very similar to the
From playlist Gaussian Integral
Engineering CEE 20: Engineering Problem Solving. Lecture 24
UCI CIvil & Environmental Engineering 20 Engineering Problem Solving (Spring 2013) Lec 24. Engineering Problem Solving View the complete course: http://ocw.uci.edu/courses/cee_20_introduction_to_computational_engineering_problem_solving.html Instructor: Jasper Alexander Vrugt, Ph.D. Licen
From playlist Engineering CEE 20: Engineering Problem Solving
Scott Field - Gravitational Wave Parameter Estimation with Compressed Likelihood Evaluations
Recorded 17 November 2021. Scott Field of the University of Massachusetts Dartmouth presents "Gravitational Wave Parameter Estimation with Compressed Likelihood Evaluations" at IPAM's Workshop III: Source inference and parameter estimation in Gravitational Wave Astronomy. Abstract: One of
From playlist Workshop: Source inference and parameter estimation in Gravitational Wave Astronomy
ch4 C: Gaussian quadrature, part 2. Wen Shen
Wen Shen, Penn State University. Lectures are based on my book: "An Introduction to Numerical Computation", published by World Scientific, 2016. See promo video: https://youtu.be/MgS33HcgA_I
From playlist CMPSC/MATH 451 Videos. Wen Shen, Penn State University
Gaussian Quadrature 1: Summary of Legendre Polynomials
https://bit.ly/PavelPatreon https://lem.ma/LA - Linear Algebra on Lemma http://bit.ly/ITCYTNew - Dr. Grinfeld's Tensor Calculus textbook https://lem.ma/prep - Complete SAT Math Prep
From playlist Part 4 Linear Algebra: Inner Products