Orthogonal polynomials | Polynomials | Special hypergeometric functions
In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation: which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer. Sometimes the name Laguerre polynomials is used for solutions of where n is still a non-negative integer.Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonine polynomials, after their inventor Nikolay Yakovlevich Sonin). More generally, a Laguerre function is a solution when n is not necessarily a non-negative integer. The Laguerre polynomials are also used for Gaussian quadrature to numerically compute integrals of the form These polynomials, usually denoted L0, L1, …, are a polynomial sequence which may be defined by the Rodrigues formula, reducing to the closed form of a following section. They are orthogonal polynomials with respect to an inner product The sequence of Laguerre polynomials n! Ln is a Sheffer sequence, The rook polynomials in combinatorics are more or less the same as Laguerre polynomials, up to elementary changes of variables. Further see the Tricomi–Carlitz polynomials. The Laguerre polynomials arise in quantum mechanics, in the radial part of the solution of the Schrödinger equation for a one-electron atom. They also describe the static Wigner functions of oscillator systems in quantum mechanics in phase space. They further enter in the quantum mechanics of the Morse potential and of the 3D isotropic harmonic oscillator. Physicists sometimes use a definition for the Laguerre polynomials that is larger by a factor of n! than the definition used here. (Likewise, some physicists may use somewhat different definitions of the so-called associated Laguerre polynomials.) (Wikipedia).
Laguerre's method for finding real and complex roots of polynomials. Includes history, derivation, examples, and discussion of the order of convergence as well as visualizations of convergence behavior. Example code available on github https://www.github.com/osveliz/numerical-veliz Chapte
From playlist Root Finding
Lagrange Polynomials for function approximation including simple examples. Chapters 0:00 Intro 0:08 Lagrange Polynomials 0:51 Visualizing L2 1:00 Numeric Example 1:11 Example Visualized 1:27 Why Lagrange Works 1:47 Lagrange Accuracy 2:12 Error 2:59 Error Visualized 3:20 Error Bounds 4:08
From playlist Numerical Methods
Moving on from Lagrange's equation, I show you how to derive Hamilton's equation.
From playlist Physics ONE
Ch04n2: Integrals over Infinite Intervals, Gauss Laguerre, Gauss Hermite
Integrals over Infinite Intervals. Gauss Laguerre, Gauss Hermite Numerical Computation, chapter 4, additional video no 2. To be viewed after the video ch04n1. Wen Shen, Penn State University, 2018.
From playlist CMPSC/MATH 451 Videos. Wen Shen, Penn State University
Calculus BC - Unit 5 Lesson 2: Lagrange Error Bound
Calculus BC - Taylor's Remainder Theorem and the Lagrange Error Bound
From playlist AP Calculus BC
Number Theory | Lagrange's Theorem of Polynomials
We prove Lagrange's Theorem of Polynomials which is related to the number of solutions to polynomial congruences modulo a prime.
From playlist Number Theory
The Aberth-Ehrlich Method for solving all roots of a polynomial simultaneously including history, methodology, examples, and order as well as comparison to Durand-Kerner. Example code github: http://github.com/osveliz/numerical-veliz Chapters 0:00 Intro 0:19 History 0:41 Methodology 0:59
From playlist Root Finding
Lagrange Multipliers Maximum of f(x, y, z) = xyz subject to x + y + z - 3 = 0
Lagrange Multipliers Maximum of f(x, y, z) = xyz subject to x + y + z - 3 = 0
From playlist Calculus 3
Halley's Method (the method of tangent hyperbolas) for finding roots including history, derivation, examples, and fractals. Also discusses Taylor's Theorem relating to Halley's Method as well as Halley's Comet. Sample code and images available on GitHub https://www.github.com/osveliz/numer
From playlist Root Finding
Multivariable Calculus | Lagrange multipliers
We give a description of the method of Lagrange multipliers and provide some examples -- including the arithmetic/geometric mean inequality. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist Multivariable Calculus
Lagrange Multipiers: Find the Max and Min of a Function of Two Variables
This video explains how to use Lagrange Multipliers to maximum and minimum a function under a given constraint. The results are shown in using level curves. http://mathispower4u.com
From playlist Lagrange Multipliers
Energy levels and diagram for hydrogen
MIT 8.04 Quantum Physics I, Spring 2016 View the complete course: http://ocw.mit.edu/8-04S16 Instructor: Barton Zwiebach License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT 8.04 Quantum Physics I, Spring 2016
Horner's Method (Ruffini-Horner Scheme) for evaluating polynomials including a brief history, examples, Ruffini's Rule with derivatives, and root finding using Newton-Horner. Example code on GitHub https://github.com/osveliz/numerical-veliz Chapters 0:00 Intro 0:11 - History 1:33 - TLDR 1
From playlist Root Finding
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
Householder's Method for finding roots of equations including history, derivation, examples, and fractals. Example code is available on GitHub https://github.com/osveliz/numerical-veliz Chapters 0:00 Intro 0:25 Derivation 1:58 History 2:34 Householder's Method 4:07 Householder's Method Ex
From playlist Root Finding
Exact solution of a left-permeable open ASEP by Arvind Ayyer
Indian Statistical Physics Community Meeting 2018 DATE:16 February 2018 to 18 February 2018 VENUE:Ramanujan Lecture Hall, ICTS Bangalore This is an annual discussion meeting of the Indian statistical physics community which is attended by scientists, postdoctoral fellows, and graduate s
From playlist Indian Statistical Physics Community Meeting 2018
Download the free PDF from http://tinyurl.com/EngMathYT This video shows how to apply the method of Lagrange multipliers to a max/min problem. Such ideas are seen in university mathematics.
From playlist Lagrange multipliers
Maximize a Function of Two Variable Under a Constraint Using Lagrange Multipliers
This video explains how to use Lagrange Multipliers to maximize a function under a given constraint. The results are shown in 3D.
From playlist Lagrange Multipliers
Sparsification of graphs and matrices - Daniel Spielman
Daniel Spielman Yale University November 3, 2014 Random graphs and expander graphs can be viewed as sparse approximations of complete graphs, with Ramanujan expanders providing the best possible approximations. We formalize this notion of approximation and ask how well an arbitrary graph
From playlist Mathematics