[6] ['Brute force' calculation of Gaussian quadrature weights and nodes] Let (3) w₁ƒ(x1) +w2ƒ(x2) be a quadrature formula to approximate the integral ſ¹₁ ƒ(x). Determine the weights {w₁, w₂} and nodes {x₁, x2} such that (3) is exact (no error) for f(x) = 1, x, x², and x³. Then show that the formula is not exact for f(x) = x4 and conclude that the degree of exactness is N = 3. =
[6] ['Brute force' calculation of Gaussian quadrature weights and nodes] Let (3) w₁ƒ(x1) +w2ƒ(x2) be a quadrature formula to approximate the integral ſ¹₁ ƒ(x). Determine the weights {w₁, w₂} and nodes {x₁, x2} such that (3) is exact (no error) for f(x) = 1, x, x², and x³. Then show that the formula is not exact for f(x) = x4 and conclude that the degree of exactness is N = 3. =
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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Gaussian quadrature
![[6] ['Brute force' calculation of Gaussian quadrature weights and nodes]
Let
w₁f(x₁) +w₂f (x₂)
(3)
be a quadrature formula to approximate the integral ₁ f(x).
Determine the weights {w₁, w2} and nodes {x₁, x2} such that (3) is exact
(no error) for f(x) = 1, x, x², and x³. Then show that the formula is not
exact for f(x) = x4 and conclude that the degree of exactness is N = 3.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fd47556-f3ce-4f39-818d-be563d9523c8%2F644dc99b-94c8-4974-96af-445bd3d4b2d8%2Foipss97_processed.png&w=3840&q=75)
Transcribed Image Text:[6] ['Brute force' calculation of Gaussian quadrature weights and nodes]
Let
w₁f(x₁) +w₂f (x₂)
(3)
be a quadrature formula to approximate the integral ₁ f(x).
Determine the weights {w₁, w2} and nodes {x₁, x2} such that (3) is exact
(no error) for f(x) = 1, x, x², and x³. Then show that the formula is not
exact for f(x) = x4 and conclude that the degree of exactness is N = 3.
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