# Project1 - Gram-Schmidt Process for Hermite and Laguerre...

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Gram-Schmidt Process for Hermite and Laguerre Polynomials Brian Cosey Mihai Gheorghe Brian Willhelm Weston Wands Spencer Davidson Ellen Bulka

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For the first problem, we wish to construct the first five orthogonal polynomials from the set {x n }, n = 0, 1, 2,..., using the Gram-Schmidt process. The polynomials are labeled as H n (x), n = 0, 1, 2, 3, 4, and the weight of the function, w (x), equals e -x 2 . The range of the values for x is [-∞,∞]. The coefficient of the highest power of x is 2 n , and the polynomials are normalized to (H n ,H m ) = π ½ 2 n mn . The first five polynomials of this function, known as the Hermite Polynomials, are listed below. H 0 (x) = 1 H 1 (x) = 2x H 2 (x) = 4x 2 - 2 H 3 (x) = 8x 3 – 12 H 4 (x) = 16x 4 – 48x 2 + 12 H 5 (x) = 32x 5 – 160x 3 + 120x These polynomials were normalized using the following equation: H n (x) = (-1) n e x 2 (d n /dx n )e -x 2 = e x 2 /2 (x-d/dx) n e -x 2 /2 The first five Hermite polynomials, listed above, are graphed by color on the attached sheet. Although not all data are included, each polynomial was calculated with integrals of 0.1, from a range of
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