GAUSS-LAGUERRE AND GAUSS-HERMITE QUADRATURE ON 64, 96 AND 128 NODES
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1 GAUSS-LAGUERRE AND GAUSS-HERMITE QUADRATURE ON 64, 96 AND 128 NODES RICHARD J. MATHAR Abstract. The manuscript provides tables of abscissae and weights for Gauss- Laguerre integration on 64, 96 and 128 nodes, and abscissae and weights for Gauss-Hermite integration on 96 and 128 nodes. 1. Gauss-Laguerre We tabulate abscissae x i and weights w i for Gauss-Laguerre integration of the form n (1.1) f(x)e x dx w i f(x i ). 0 The objective is to extend the tables provided for up to n = 15 in the Handbook [1, Tabl 25.9][9], up to 32 points by Krylov [4] and for n = 50 in my thesis [6] to higher numbers of nodes. The abscissae are the zeros of Laguerre Polynomials, which are [1, ] (1.2) L n (x) = L (0) n (x) = n i=0 i=1 ( ) n 1 i i! ( x)i. The weights are related to the derivatives at these nodes [8], (1.3) w i = 1 x i [L n (x i ) ] 2. Shao, Chen and Frank provide the Newton iteration to converge on the roots x i of generalized Laguerre Polynomials L (α) n (x) [10, 3]: (1.4) x i x i f(x i) f (x i ) [ and the terminating continued fraction (1.5) L (α) n (x) = x (x) n L (α) n ( 1 α + 1 x i ) ] f(xi ) f (x i ) (n + α)n (n 1 + α)(n 1) (n 2 + α)(n 2) 2n + α 1 x 2n + α 3 x 2n + α 5 x (1 + α) 1 + α x. Date: October 2, Mathematics Subject Classification. Primary 41A55, 65A05; Secondary 65D30. Key words and phrases. Gaussian Integration, Tables, Laguerre Polynomials. 1
2 2 RICHARD J. MATHAR 2. Gauss-Hermite We also tabulate abscissae x i and weights w i for Gauss-Hermite integration of the form n (2.1) f(x)e x2 dx w i f(x i ). The objective is to extend the tables provided by Steen et al. up to n = 15 [12], by Shizgal up to n = 16 [11], by Krylov and in the Handbook [1, Table 25.9][4] up to n = 20, and by Shao, Chen and Frank [10] up to n = 64. The abscissae are the zeros of Hermite Polynomials, which are [1, ] n/2 ( ) i (2.2) H n (x) = n! i!(n 2i)! (2x)n 2i. The weights are related to the derivatives at these nodes [1, ], (2.3) w i = 2n 1 n! π [nh n 1 (x i )] 2 = 2n+1 n! π [H n(x i )] 2. i=0 i=1 The first order Newton iteration to stabilize the roots x i is (2.4) x i x i f(x i) f (x i ), and the supporting terminating continued fraction [13] is (2.5) H n (x) H n(x) = 1 2n H n (x) H n 1 (x) = 1 2n [2x 2(n 1) 2x 2(n 2) 2x 2 2x ]. (3.1) 3. Gauss-Hermite of Moments The w i and x i for integrations for the m-th moment of the form f(x)x m e x2 dx n w i f(x i ) with integer m are computed with the standard theory from roots of a system of orthogonal polynomials p n with norm [2, 5, 14] (3.2) f, g i=1 f(x)g(x)x m e x2 dx and moments µ [1, ] { 0, n + m odd; (3.3) µ n x n+m e x2 dx = Γ( n+m+1 2 ) = π (n+m 1)!!, n + m even. 2 (n+m)/2 The following subsections summarize the known results [10]. m is always assumed to be a positive even integer for odd m the weight function is not always positive over the base interval and the standard theory is not applicable.
3 GAUSS QUADRATURE ON 64, 96 AND 128 NODES Generalized Hermite Polynomials. The set of orthogonal (monic) polynomials p n (x) is bootstrapped from (3.4) p 0 (x) = 1; p 1 (x) = x; p n+1 (x) = (x a)p n (x) b n p n 1 (x); (3.5) b n = { n/2; n even (m + n)/2. n odd. The vanishing of the moments µ n at odd indices induces that the general term on the right hand side x a reduces to x, a parity p n (x) = ( 1) n p n ( x). The first of the orthogonal monic polynomials are for m = 2 (3.6) (3.7) (3.8) (3.9) (3.10) (3.11) (3.12) p 2 = x ; p 3 = x x; p 4 = x 4 5x ; p 5 = x 5 7x x; p 6 = x x x ; p 7 = x x x x; p 8 = x 8 18x x x , for m = 4 (3.13) (3.14) (3.15) (3.16) (3.17) (3.18) (3.19) p 2 = x ; p 3 = x x; p 4 = x 4 7x ; p 5 = x 5 9x x; p 6 = x x x ; p 7 = x x x x; p 8 = x 8 22x x x ,
4 4 RICHARD J. MATHAR and for m = 6 (3.20) (3.21) (3.22) p 2 = x ; p 3 = x x; p 4 = x 4 9x ; (3.23) (3.24) (3.25) p 5 = x 5 11x x; p 6 = x x x ; p 7 = x x x x; (3.26) p 8 = x 8 26x x x These are conveniently written down separately for even and odd polynomial orders as (3.27) (3.28) p 2n = ( ) n n!l n (m/2 1/2) (x 2 ); p 2n+1 = ( ) n n!xl n (m/2+1/2) (x 2 ) Abscissae and Weights. The standard further steps are normalization of the polynomials such that their norm is unity, (3.29) p n(x) p n(x) pn, p n, computation of all zeros x i of p N (x) at some degree N. computation of the weights w i by (3.30) w i = [xn+1 ]p N+1 [x N ]p N 1 p N+1 (x i)p N (x i ) = p N, p N p N+1 (x i )p N (x i ), where [x N+1 ]p N+1 and [xn ]p N are the leading coefficients of the two polynomials after normalization, and where the prime at p denotes the derivative with respect to x. The final table shows in each line a pair (x i, w i ) for N fixed at 96 or 128. Because the p n are even or odd in our case, the x i emerge in pairs of the same weight; only the non-negative values need to be shown. 4. Results First the x i and then the associated w i are shown for the Laguerre polynomials. For Hermite polynomials, the duplicates with the opposite sign are not reproduced. The originating Maple program is shown in the appendix. Laguerre n= e e e e e e e e e e-01
5 GAUSS QUADRATURE ON 64, 96 AND 128 NODES e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-48
6 6 RICHARD J. MATHAR e e e e e e e e e e e e e e e e e e e e e e e e e e e e-101 Laguerre n= e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-10
7 GAUSS QUADRATURE ON 64, 96 AND 128 NODES e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-68
8 8 RICHARD J. MATHAR e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-155 Laguerre n= e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-04
9 GAUSS QUADRATURE ON 64, 96 AND 128 NODES e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-36
10 10 RICHARD J. MATHAR e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-119
11 GAUSS QUADRATURE ON 64, 96 AND 128 NODES e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-210 Hermite n=96 m= e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-13
12 12 RICHARD J. MATHAR e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-75 Hermite n=96 m= e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-07
13 GAUSS QUADRATURE ON 64, 96 AND 128 NODES e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-74 Hermite n=96 m= e e e e e e e e e e e e e e e e e e e e e e e e e e-03
14 14 RICHARD J. MATHAR e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-73 Hermite n=96 m= e e e e e e e e e e e e e e e e-01
15 GAUSS QUADRATURE ON 64, 96 AND 128 NODES e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-71 Hermite n=96 m= e e e e e e-02
16 16 RICHARD J. MATHAR e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-63
17 GAUSS QUADRATURE ON 64, 96 AND 128 NODES e e-70 Hermite n=128 m= e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-33
18 18 RICHARD J. MATHAR e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-102 Hermite n=128 m= e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-07
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