Construction of Gauss Quadrature Rules

Transcription

1 Jim Lmbers MAT 772 Fll Semester Lecture 15 Notes These notes correspond to Sections 10.2 nd 10.3 in the text. Construction of Guss Qudrture Rules Previously, we lerned tht Newton-Cotes qudrture rule with n nodes hs degree t most n. Therefore, it is nturl to sk whether it is possible to select the nodes nd weights of n n-point qudrture rule so tht the rule hs degree greter thn n. Gussin qudrture rules hve the surprising property tht they cn be used to integrte polynomils of degree 2n 1 exctly using only n nodes. Gussin qudrture rules cn be constructed using technique known s moment mtching, or direct construction. For ny nonnegtive integer k, the k th moment is defined to be μ k x k dx. For given n, our gol is to select weights nd nodes so tht the first 2n moments re computed exctly; i.e., μ k w i x k i, k 0, 1,..., 2n 1. Since we hve 2n free prmeters, it is resonble to think tht pproprite nodes nd weights cn be found. Unfortuntely, this system of equtions is nonliner, so it cn be quite difficult to solve. Suppose g(x) is polynomil of degree 2n 1. For convenience, we will write g P 2,, for ny nturl number k, P k denotes the spce of polynomils of degree t most k. We shll show tht there exist weights {w i } nd nodes {x i} Furthermore, for more generl functions, G(x), such tht g(x) dx w i g(x i ). G(x) dx w i G(x i ) + E[G] 1

2 1. x i re rel, distinct, nd < x i < b for i 0, 1,..., n The weights {w i } stisfy w i > 0 for i 0, 1,..., n The error E[G] stisfies E[G] G(2n) (ξ) (2n)! (x x i) 2 dx. Notice tht this method is exct for polynomils of degree 2n 1 since the error functionl E[G] depends on the (2n) th derivtive of G. To prove this, we shll construct n orthonorml fmily of polynomils {q i (x)} n so tht { 0 r s, q r (x)q s (x) dx 1 r s. This cn be ccomplished using the fct tht such fmily of polynomils stisfies three-term recurrence reltion β j q j (x) (x α j )q j 1 (x) β j 1 q j 2 (x), q 0 (x) (b ) 1/2, q 1 (x) 0, α j xq j 1 (x) 2 dx, β 2 j xq j (x)q j 1 (x) dx, j 1, β 0 1. We choose the nodes {x i } to be the roots of the n th -degree polynomil in this fmily, which re rel, distinct nd lie within (, b), s proved erlier. Next, we construct the interpolnt of degree n 1, denoted p (x), of g(x) through the nodes: L (x) g(x i )L,i (x),, for i 0,..., n 1, L,i (x) is the ith Lgrnge polynomil for the points x 0,..., x. We shll now look t the interpoltion error function e(x) g(x) p (x). Clerly, since g P 2, e P 2. Since e(x) hs roots t ech of the roots of q n (x), we cn fctor e so tht e(x) q n (x)r(x), r P. It follows from the fct tht q n (x) is orthogonl to ny polynomil in P tht the integrl of g cn then be written s I(g) p (x) dx + q n (x)r(x) dx 2

3 w i p (x) dx g(x i )L,i (x) dx g(x i ) g(x i )w i L,i (x) dx L,i (x) dx, i 0, 1,..., n 1. For more generl function G(x), the error functionl E[G] cn be obtined from the expression for the interpoltion error presented erlier. Exmple We will use Gussin qudrture to pproximte the integrl 0 e x2 dx. The prticulr Gussin qudrture rule tht we will use consists of 5 nodes x 1, x 2, x 3, x 4 nd x 5, nd 5 weights w 1, w 2, w 3, w 4 nd w 5. To determine the proper nodes nd weights, we use the fct tht the nodes nd weights of 5-point Gussin rule for integrting over the intervl [ 1, 1] re given by i Nodes r 5,i Weights c 5,i To obtin the corresponding nodes nd weights for integrting over [ 1, 1], we cn use the fct tht in generl, ( ) b + b b f(x) dx f t + dt, s cn be shown using the chnge of vrible x [(b )/2]t+(+b)/2 tht mps [, b] into [ 1, 1]. We then hve ( ) b + b b f(x) dx f t + dt

4 5 f i1 ( b 2 r 5,i + 5 f(x i )w i, i1 ) + b b 2 2 c 5,i b x i 2 r + b 5,i + 2, w b i 2 c 5,i, i 1,..., 5. In this exmple, 0 nd b 1, so the nodes nd weights for 5-point Gussin qudrture rule for integrting over [0, 1] re given by which yields It follows tht 0 e x2 dx x i 1 2 r 5,i + 1 2, w i 1 2 c 5,i, i 1,..., 5, 5 i1 i Nodes x i Weights w i e x2 i wi e e e e e Since the exct vlue is , the bsolute error is , which is remrkbly ccurte considering tht only fives nodes re used. The high degree of ccurcy of Gussin qudrture rules mke them the most commonly used rules in prctice. However, they re not without their drwbcks: They re not progressive, so the nodes must be recomputed whenever dditionl degrees of ccurcy re desired. An lterntive is to use Guss-Kronrod rules. A (2n + 1)-point Guss-Kronrod rule uses the nodes of the n-point Gussin rule. For this reson, prcticl qudrture procedures use both the Gussin rule nd the corresponding Guss-Kronrod rule to estimte ccurcy. 4

5 Becuse the nodes re the roots of polynomil, they must be computed using trditionl root-finding methods, which re not lwys ccurte. Errors in the computed nodes led to lost degrees of ccurcy in the pproximte integrl. In prctice, however, this does not normlly cuse significnt difficulty. 5