Let x1, x2, . . . , xn be the roots of the Legendre polynomial Pn. If the Ai s are
Chapter 5, Problem 16(choose chapter or problem)
Let x1, x2, . . . , xn be the roots of the Legendre polynomial Pn. If the Ai s are defined as in Exercise 15, then the quadrature formula _ 1 1 f (x)dx = A1 f (x1)+A2 f (x2)+ +An f (xn) will be exact for all polynomials of degree less than 2n. (a) Show that if 1 j < 2n, then Pj (x1)A1 + + Pj (xn)An = _1, Pj_ = 0 (b) Use the results from part (a) and from Exercise 15 to set up a nonhomogeneous n n linear system for determining the quadrature coefficients A1, A2, . . . , An. 1
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