Lecture 10 - 10 Gaussian Quadrature So far we have...

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10 Gaussian Quadrature So far we have encountered the Newton-Cotes formulas b a f ( x ) dx n i =0 A i f ( x i ) , A i = b a i ( x ) dx, which are exact if f is a polynomial of degree at most n . It is important to note that in the derivation of the Newton-Cotes formulas we assumed that the nodes x i were equally spaced and fixed. The main idea for obtaining more accurate quadrature rules is to treat the nodes as additional degrees of freedom, and then hope to find “good” locations that ensure higher accuracy. Therefore, we now have n + 1 nodes x i in addition to n + 1 polynomial coefficients for a total of 2 n + 2 degrees of freedom. This should be enough to derive a quadrature rule that is exact for polynomials of degree up to 2 n + 1. Gaussian quadrature, indeed accomplishes this: Theorem 10.1 Let q be a nonzero polynomial of degree n + 1 and w a positive weight function such that b a x k q ( x ) w ( x ) dx = 0 , k = 0 , . . . , n. (95) If the nodes x i , i = 0 , . . . , n , are the zeros of q , then b a f ( x ) w ( x ) dx n i =0 A i f ( x i ) (96) with A i = b a i ( x ) w ( x ) dx,
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