Gaussian integration. In an introductory calculus course, you may have seen

Chapter 5, Problem 31

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Gaussian integration. In an introductory calculus course, you may have seen approximation formulas for integrals of the forinb n f(t)dt % YwiHaj), i=iwhere the a{ are equally spaced points on the interval (ia, b), and the Wi are certain weights (Riemann sums, trapezoidal sums, and Simpsons rule). Gauss has shown that, with the same computational effort, we can get better approximations if we drop the requirement that the ax be equally spaced. Next, we outline his approach. Consider the space Pn with the inner product = j f(t)g(t)dt.Let /o, f\........fn be an orthonormal basis of this space, with degree(fk) = k. (To construct such a basis, apply the Gram-Schmidt process to the standard basis 1, f,..., tn.) It can be shown that / has n distinct roots a i, a2____

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