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(1) This problem will allow you to experience the Legendre polynomials, which we used to define Gauss-Legendre and Gauss-Legendre-Lobatto quadrature. A good discussion for this problem will involve the word "orthogonal." (a) Starting with Lo(x) = 1 and L₁(x) = x, generate L2(x), L3(x), L₁(x) and L5(x) by the 3-term recurrence on page 125 of the textbook. (b) Compute (you may use Wolfram alpha or some other symbolic calcu- lator if you like) 1 L L5(x) · Lj(x)dx = 0 for j = 0,..., 4. [(L₁(x))²dx for j = 0,...,5. (c) Compute

(1) This problem will allow you to experience the Legendre polynomials, which we used to define Gauss-Legendre and Gauss-Legendre-Lobatto quadrature. A good discussion for this problem will involve the word "orthogonal." (a) Starting with Lo(x) = 1 and L₁(x) = x, generate L2(x), L3(x), L₁(x) and L5(x) by the 3-term recurrence on page 125 of the textbook. (b) Compute (you may use Wolfram alpha or some other symbolic calcu- lator if you like) 1 L L5(x) · Lj(x)dx = 0 for j = 0,..., 4. [(L₁(x))²dx for j = 0,...,5. (c) Compute

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The maximum value that m can take is n + 1 and is achieved pro- vided wn+1 is proportional to the so-called Legendre polynomial of degree n+1, Ln+1(x). The Legendre polynomials can be computed recursively, through the following three-term relation Lo(x) = 1, Lk+1(x) = = L₁(x) = x, 2k + 1 k+1 k k + 1 -xLk (x) -Lk-1(x), k = 1,2,... For every n = 0, 1,..., every polynomial pn € Pn can be obtained by a linear combination of the polynomials Lo, L₁, ..., Ln. Moreover, Ln+1 is orthogonal to all the Legendre polynomials of degree less than or equal to n, i.e., S²₁ Ln+1(x)L; (x) dx 0 for all j = 0, . . . , n. This explains why (4.26) is true with m less than or equal to n + 1. = 1
Transcribed Image Text:The maximum value that m can take is n + 1 and is achieved pro- vided wn+1 is proportional to the so-called Legendre polynomial of degree n+1, Ln+1(x). The Legendre polynomials can be computed recursively, through the following three-term relation Lo(x) = 1, Lk+1(x) = = L₁(x) = x, 2k + 1 k+1 k k + 1 -xLk (x) -Lk-1(x), k = 1,2,... For every n = 0, 1,..., every polynomial pn € Pn can be obtained by a linear combination of the polynomials Lo, L₁, ..., Ln. Moreover, Ln+1 is orthogonal to all the Legendre polynomials of degree less than or equal to n, i.e., S²₁ Ln+1(x)L; (x) dx 0 for all j = 0, . . . , n. This explains why (4.26) is true with m less than or equal to n + 1. = 1
(1) This problem will allow you to experience the Legendre polynomials, which we used to define Gauss-Legendre and Gauss-Legendre-Lobatto quadrature. A good discussion for this problem will involve the word "orthogonal." (a) Starting with L₁(x) 1 and L₁(x) = x, generate L₂(x), L3(x), L4(x) and L5(x) by the 3-term recurrence on page 125 of the textbook. (b) Compute (you may use Wolfram alpha or some other symbolic calcu- lator if you like) = 1 [ Lo(x) (c) Compute L5(x) Lj(x)dx=0 for j = 0,...,4. . [(L₁(x))²dx for j=0,..., 5.
Transcribed Image Text:(1) This problem will allow you to experience the Legendre polynomials, which we used to define Gauss-Legendre and Gauss-Legendre-Lobatto quadrature. A good discussion for this problem will involve the word "orthogonal." (a) Starting with L₁(x) 1 and L₁(x) = x, generate L₂(x), L3(x), L4(x) and L5(x) by the 3-term recurrence on page 125 of the textbook. (b) Compute (you may use Wolfram alpha or some other symbolic calcu- lator if you like) = 1 [ Lo(x) (c) Compute L5(x) Lj(x)dx=0 for j = 0,...,4. . [(L₁(x))²dx for j=0,..., 5.
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