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.
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 Grm-Schmidt 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).
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 Grm-Schmidt 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: 0 + 1 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 = 0.555555 c 2 = 0.555555 c 3 = 0.888888 x 1 = 0.774597 x 2 = 0.774597 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).
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 1. 1. 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.
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 Grm-Schmidt 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
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!