Orthogonal Polynomials


 Harry Barry Leonard
 2 years ago
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1 Mth 4401 Gussin Qudrture Pge 1 Orthogonl Polynomils Orthogonl polynomils rise from series solutions to differentil equtions, lthough they cn be rrived t in vriety of different mnners. Orthogonl polynomils re well studied, nd their properties re generlly well understood, so they re useful tool, especilly when used s bsis set. The set of functions {φ 0 (x),..., φ n (x)} is linerly independent on [, b] if whenever c 0 φ 0 (x) + c 1 φ 1 (x) + + c n φ n (x) = c i φ i (x) = 0 for ll x [, b] then c 0 = c 1 = = c n = 0. Otherwise, the set is linerly dependent. An integrble function w is clled weight function on [, b] if w(x) 0 for x [, b] but w(x) 0 on ny subintervl of [, b]. The weight function ssigns vrying degrees of importnce to portions of the intervl [, b]. The set of functions {φ 0 (x),..., φ n (x)} is orthogonl on [, b] with respect to the weight function w if w(x)φ i (x)φ j (x) dx = α i δ ij where δ ij = 1 if i = j nd is zero otherwise. The constnt α i > 0. If α i = 1 for ll i, then the set of functions is orthonorml. Exmple For given positive integer n, the set of functions {φ 0 (x),..., φ 2n (x)} φ 0 (x) = 1 2π, φ k (x) = 1 π cos kx, where k = 1, 2,..., n, φ n+k (x) = 1 π sin kx, where k = 1, 2,..., n 1, is n orthonorml set on intervl [ π, π] with respect to weight function w(x) = 1. This set of functions cn be used s bsis set to crete lest squres pproximtion to ny function f in the intervl [ π, π] where f(x) S n (x) = k = π π 2n k=0 f(x)φ k (x) dx. k φ k (x), The function lim n S n (x) is clled the Fourier series of f. The bsic ide is tht for given weight function nd intervl we cn crete set of orthogonl polynomils using Grm Schmidt orthogonliztion.
2 Mth 4401 Gussin Qudrture Pge 2 Here is tble of common orthogonl polynomils. Polynomil Intervl weight Normliztion φ n (x) (, b) w(x) w(x)φ2 n(x) dx Legendre P n (x) (, 1) 1 2/(2n + 1) Lguerre L n (x) (0, ) e x 1 Hermite H n (x) (, ) e x2 2 n n! π Chebyshev 1st Kind T n (x) (, 1) (1 x 2 ) /2 π(n = 0) or π/2(n 0) Chebyshev 2nd Kind U n (x) (, 1) (1 x 2 ) 1/2 π/2 Note we could include the endpoints on the intervls, (, 1) or [, 1], etc. The GrmSchmidt Process to Construct Orthogonl Polynomils The set of polynomils {φ 0 (x),..., φ n (x)} defined in the following wy is orthogonl on [, b] with respect to the weight function w. φ 0 (x) = 1, B 1 = φ 1 (x) = x B 1, nd then for k 2 use x w(x) [φ 0 (x)] 2 dx, w(x) [φ 0 (x)] 2 dx φ k (x) = (x B k )φ k (x) C k φ k 2 (x), B k = C k = x w(x) [φ k (x)] 2 dx, w(x) [φ k (x)] 2 dx x w(x) φ k (x)φ k 2 (x) dx. w(x) [φ k 2 (x)] 2 dx You cn prove this using induction. The Mthemtic file shows how this cn be used to construct the Legendre polynomils mentioned erlier. Theorem If the set of orthogonl polynomils {φ 0 (x),..., φ k (x),..., φ n (x)} is defined on [, b] with weight function w, nd φ k (x) is polynomil of degree k then φ k (x) hs k distinct roots (when k 1) in the intervl (, b).
3 Mth 4401 Gussin Qudrture Pge 3 Gussin Qudrture: Initil Thoughts 1. Assume we cn pproximte the integrl s the following sum: w(x)f(x) dx c i f(x i ), where we will hve nodes x 1,..., x n [, b]. 2. We wnt to find these nodes in n optiml wy, rther thn just hving them eqully spced. 3. Our erlier theorem tells us the orthogonl polynomil φ n (x) we define through the GrmSchmidt process will hve n roots in the intervl. 4. We hve 2n prmeters to determine: c i nd x i for i = 1,..., n. 5. A polynomil of degree 2n 1 hs 2n prmeters: x + + 2n x 2n = 2n i=0 i x i Key Point: If f(x) is polynomil of degree 2n 1 or less, then we should get n exct result for the integrl since we hve enough prmeters to fit the curve exctly! A qudrture rule with degree of precision k mens the qudrture rule gives the exct result for ny polynomil of degree less thn or equl to k Exmple If n = 3 nd w(x) = 1 nd [, b] = [, 1], our pproximtion should be exct if f(x) = 1, x, x 2, x 3, x 4, x 5. Choose c i nd x i such tht 3 x k dx = c i x k i, k = 0, 1,..., 5. This is set of 6 nonliner equtions in 6 unknowns, which we will use Mthemtic to solve. We get c 1 = c 2 = c 3 = x 1 = x 2 = x 3 = 0 Ech c i is the weight ssocited with the node x i. Now tht we hve these prmeters, we cn use the formul f(x) dx c i f(x i ), for ny other function f we like. In this cse, since w(x) = 1 nd [, b] = [, 1] nd n = 3, the nodes re the roots of the Legendre polynomil P 3 (x).
4 Mth 4401 Gussin Qudrture Pge 4 Proof of Gussin Qudrture Technique for [, b] = [, 1] nd w(x) = 1 cse Now tht we hve n ide of wht to expect, let s work out the detils for generl n, nd show tht Gussin qudrture hs degree of precision 2n We gin begin with the Lgrnge interpolting polynomil. f(x) = f(x i ) (x i x k ) + 1 n! f (n) (c(x)). (1) k=1 We will use the following qudrture rule, which is found by integrting Eq. (1). Since the error in Eq. (1) involves f (n), the qudrture rule will be hve degree of precision t lest n 1. f(x) dx = where c i = c i f(x i ), (x i x k ) dx. (2) 2. Suppose W is ny polynomil with degree k 2n 1 (W is not necessrily Legendre polynomil). Let s show the qudrture rule Eq. (2) is exct for f = W. () Dividing W by the nth degree Lengendre polynomil gives W (x) = Q(x)P n (x) + R(x), (3) where both Q nd R re polynomils of degree less thn n. (b) Since Q is degree less thn n, Q cn be written in terms of the Legendre polynomils s: n Q(x) = d i P i (x) i=0 since the Legendre polynomils {P 0 (x),..., P n (x)} form bsis set for polynomils of degree t most n 1. (c) Using the orthogonlity property of the Legendre polynomils, we hve n Q(x)P n (x) dx = d i P i (x)p n (x) dx = 0, i=0 nd, upon integrting Eq. (3), W (x) dx = 0 + R(x) dx = c i R(x i ), (4) where the lst equlity follows by using the qudrture rule (2) (with degree of precision n 1) on the polynomil R(x) which hs degree t most n 1.
5 Mth 4401 Gussin Qudrture Pge 5 (d) If we choose x i, i = 1, 2,..., n, to be the roots of the Lengendre polynomil P n (x), then Eq. (3) tells us tht W (x i ) = Q(x i ) 0 P n (x i ) + R(x i ) = R(x i ), nd therefore, we hve from (4) W (x) dx = c i W (x i ), so the qudrture rule Eq. (2) is exct for the ny polynomil W of degree 2n 1. This nlysis shows us why the nodes x i should be chosen s the roots of the Legendre polynomil, nd lso gives us formul to compute the weights, c i = (x i x k ) dx. The nice thing is tht for vriety of n the nodes nd weights hve lredy been clculted, so you don t hve to work them out yourself! The mzing connection to orthogonl polynomils is tht the nodes re the roots of the orthogonl polynomil φ n (x) we define through the GrmSchmidt process on intervl [, b] with weight function w(x)! Notice the text defined Gussin qudrture only for [, b] = [, 1] nd w(x) = 1, but these ides pply to the other orthogonl polynomils s well. Gussin Qudrture Technique for [, b] [, 1] nd w(x) = 1 If the finite intervl [, b] is not [, 1], use the liner trnsformtion t = (2x b )/(b ): w(x)f(x) dx = ( ) ( ) (b )t + b + (b )t + b + w f 2 2 b 2 dt
6 Mth 4401 Gussin Qudrture Pge 6 To Perform Gusssin Qudrture to Evlute w(x)f(x) dx L ni = (x i x k ) I prefer to think of how it is built, so I get the weight in the correct plce. f(x) = w(x)f(x) dx = w i = f(x i )L ni, w i f(x i ), w(x)l ni (x) dx = w(x) (x i x k ) dx You will sometimes see tbles tht reports weights bsed on the following, so be sure to red your tbles closely! f(x) = f(x) dx = w i = f(x i )L in, w i f(x i ), L ni (x) dx = (x i x k ) dx You cn use the formuls when the pproprite weight function is not present, but the results will not be s good. Exmple 0 e x cos x dx Identify the intervl [0, ) nd w(x) = e x s the weight function ssocited with Lguerre polynomils, L n (x). We choose the nodes to be roots of L n (x) nd the weights cn either be computed or looked up in tble. The function is f(x) = cos x. 0 cos(x) = e x cos x dx = w i = 5 cos(x i )L 5i, 5 w i cos(x i ), 5 e x 0 (x i x k ) dx The rel power of the method is tht the weights/nodes hve lredy been tbulted nd you cn just plug them in!
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