na_lec_5

na_lec_5 - A Then 29 A i i l x = n i 2 1 = and 2 1 2 i i u...

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1 10. Numerical integration 10.1. Gauss Quadrature (Due to Golub, Welsch, Math. Comp. 23 , pp. 221-230) Suppose ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 - - - - - - - - = 0 1 1 4 1 1 3 4 3 0 1 2 4 2 1 2 4 2 0 1 1 4 1 1 1 4 1 0 2 2 2 2 2 2 n n O O O A and let T U U A Λ = be the spectral decomposition of A , where ( 29 ( 29 A i diag l = Λ is the spectral matrix and ] [ ij u = U is the eigenvector matrix of
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Unformatted text preview: A . Then ( 29 A i i l x = , n i ,..., 2 , 1 = and 2 1 2 i i u w = , n i ,..., 2 , 1 = are the gauss quadratures and their associate weights, respectively, i.e., ( 29 ( 29 ∫ ∑-= = 1 1 1 n i i k i k P w dx x P x for all polynomials of degree 1 2 ,..., 2 , 1 ,-= n k ....
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