Laurent–Hermite–Gauss Quadrature
(Brian A. Hagler
Department of Mathematics, University of Colorado, Boulder, CO 80309-0395, USA
Received 6 October 1998
Abstract
This paper exends the results presented in Gustafson and Hagler (in press) by explicating the (2n)-point Laurent– Hermite–Gauss quadrature formula of parameters ; ¿0:
Z ∞
−∞
f(x)e−[1=(x−=x)]2dx=X j=±1
n X
k=1
f(h(n; k; j; ))H
(; )
n; k; j +E
(; ) 2n [f(x)]; where the abscissash(n; k; j; ) and weightsH
(; )
n; k; j are given in terms of the abscissas and weights associated with the classical Hermite–Gauss Quadrature, as prescribed in Gustafson and Hagler (J. Comput. Appl. Math. 105 (1999) to appear). By standard numerical methods, it is shown in the present work that, for xed; ¿0,
E2(; n )[f(x)] = g(4n)()
(4n)! n! 2n
2n+1
;
for somein (−∞;∞), providedg(x) :=x2nf(x) has a continuous (4n)-th derivative. The resolution as→0+, with=1, of the transformed quadratures introduced in Gustafson and Hagler (in press) to the corresponding classical quadratures is presented here for the rst time, with the (2n)-point Laurent–Hermite–Gauss quadrature providing an example, displayed graphically in a gure. Error comparisons displayed in another gure indicate the advantage in speed of convergence, as the number of nodes tends to innity, of the Laurent–Hermite–Gauss quadrature over the corresponding classical quadrature for certain integrands. c1999 Elsevier Science B.V. All rights reserved.
MSC:primary 65D32; secondary 41A30
Keywords:Quadrature; Strong distribution; Laurent polynomial
1. Introduction
Gauss quadrature is an important tool for approximating denite integrals, and scholarly works related to this technique continue to abound in the literature. In particular [1–3,6 –10] were consulted
E-mail address: [email protected] (B.A. Hagler).
(
Supported in part by NSF Grant DMS-9701028.
in the preparation of this article. The developing theories of orthogonal functions have yielded new such quadrature formulas, and the theory of orthogonal Laurent polynomials is no exception, as we shall now indicate.
n=0 denote the orthogonal basis of monic
polyno-mials, ordered by polynomial degree, orthogonal with respect to the above inner-product.
Pn(x) =
Hagler, in [4], and Hagler et al. in [5], presented a two parameter family of transformations taking systems of orthogonal polynomials to systems of orthogonal Laurent polynomials. Their work
is based on a Laurent polynomial called the doubling transformation of parameters ; ¿0, which
we denote here by
for j=±1. In our current context withw(x), a non-negative weight function, results in [4,5] include
an inner-product,
for the space of real Laurent polynomials, where
w(; )(x) :=w(v(; )(x)): (1.8)
With the terms L-degree and monic implied by the ordered basis {1, x−1, x, x−2, x2; : : :} for the
space of Laurent polynomials, we let {P(; )
n (x)}
∞
polynomials, ordered by L-degree, orthogonal with respect to Eq. (1.7). [4,5] give
It is also shown in [4,5] that
P2(; n )(x) =x−n Y
j=±1 Y
k=1;2; :::; n
(x−xn; k; j(; )); (1.11)
where the 2n distinct real roots are
xn; k; j(; ):=vj(; )(xn; k); (1.12)
for k= 1;2; : : : ; n and j=±1. Another result reported in [4,5] that will be used below is
hP2(; n ); P(2; n )i=2n+1(P
n; Pn): (1.13)
Gustafson and Hagler in [2] show that, when the integrals exist, there are unique positive numbers
w(n; k; j; ), k= 1;2; : : : ; n and j=±1, such that
1 (b)). They also show that the weights in the above quadrature formula satisfy
w(n; k; j; )= wn; k
v′(x(; )
n; k; j)
; (1.16)
where we have used the abbreviation v′ for the derivative dv(; )=dx of v(; )(x).
Cursory numerical examples of the transformed quadrature (1.14) are included in [2]. In the present paper, we concentrate on quadrature in the Hermite case, where the strong weight function (1.8) is e−[1=(x−=x)]2
on (−∞;∞). A Table of numerical values for the abscissas and weights,
an expression for the remainders E2(; n )[f(x)] and specic case studies with comparisons to results
2. Resolution to classical quadrature
Theorem 2.1 (Resolution). Let xn;k and wn; k denote; respectively; an abscissas and a correspond-ing weight in Gauss quadrature; (1:3). Let x(n; k; j; ) and wn; k; j(; ) be given by Eqs. (1:12) and (1:16);
+. Hence, the statements concerning
xn; k; j(;1) are seen to hold. If xn; k= 0, wn; k; j(;1)=
Proof. Apply the previous theorem.
3. Laurent–Hermite–Gauss quadrature
Let {Hn(x)}∞n=0 be the sequence of monic Hermite polynomials, orthogonal with respect to
(P; Q)H:=
we write the quadrature (1.3) in the compact form
for f(x) having a continuous (2n)-th derivative, and some ∈ (−∞;∞). We have thus denoted
is also well known in the literature. Since
as an inner-product for the space of real Laurent polynomials.
Denote by {H(; )
n (x)}
∞
n=0 the sequence of monic orthogonal Laurent polynomials with respect to
Eq. (3.6). By Eq. (1.12), the zeros of H(; )
The following theorem introduces the (2n)-point Laurent–Hermite–Gauss quadrature formula of
Table 1
Abscissas and weights for Laurent–Hermite–Gauss quadrature of paramaters; ¿0 Z ∞
2 0.70710 67811 86548 8.86226 92545 28 (−1)
3 0.00000 00000 00000 1.18163 58006 04 (0)
1.22474 48713 91589 2.95408 97515 09 (−1)
4 0.52464 76232 75290 8.04919 09000 55 (−1)
1.65068 01238 85785 8.13128 35447 25 (−2)
5 0.00000 00000 00000 9.45308 72048 29 (−1)
0.95857 24646 13819 3.93619 32315 22 (−1) 2.02018 28704 56086 1.99532 42059 05 (−2)
6 0.43607 74119 27617 7.24629 59522 44 (−1)
1.33584 90760 13697 1.57067 32032 29 (−1) 2.35060 49736 74492 4.53000 99055 09 (−3)
Proof. The result follows by standard methods, using Hermite’s interpolation formula for
g(x) :=x2nf(x), considering (1.13) and (3.2).
Table 1 shows how the formulas (3.7) and (3.8) incorporate numerical values for hn; k and Hn; k
in (3.3) to give h(n; k; j; ) and Hn; k; j(; ) in Eq. (3.10). The abscissas hn; k and weight factors Hn; k, for
n= 2;3;4;5;6, were taken from the values tabulated in [10].
4. Examples and comparisons
For convenience, we will henceforth consider formulas written in the form
Z ∞
Fig. 1 shows several gross features of the approximation
Fig. 1. Six abscissas and weights plots. Q is the truncated value of the quadrature in Eq. (4.1).
for f(x) =x−10e−(x−1=x)2
, as is varied. Scaling is uniform for the six plots, and h8(;; k; j1) andH8(;; k; j1) used
in the computation of Q, for each of the values of , were derived using Eqs. (3.7) and (3.8) and
numerical values for h8; k and H8; k given in [10]. The 16-point Laurent–Hermite–Gauss quadrature
for this integral, represented by the sum on the right-hand side of Eq. (4:1), is theoretically exact
for = 1, and the value 8.8604 of Q1 given in the gure is correct in each of its digits. = 0 gives
the classical Hermite–Gauss quadrature.
The Resolution Theorem shows the convergence, as tends to 0, with = 1, of the quadrature
associated with the Laurent polynomial (1.5) to the corresponding classical quadrature. The sequence of graphs in Fig. 1 demonstrates this convergence. The plots also give an indication of why we call
(1.5) the doubling transformation, especially when comparing the structures of the = 10 and = 0
graphs.
Since Q varies so greatly, perhaps the rst question that Fig. 1 suggests is this: For a given
integrand, which values of the parameters will give the best approximation of the integral? Such problems are beyond the goals of this paper.
Fig. 2 shows six large scale graphical comparisons of the magnitude of the error terms en(fp) and
e(12n;1)(fp) in the formulas
Z ∞
−∞
fp(x) dx=
X
i=1; :::; n
fp(hn; k)
Fig. 2. Six error comparison graphs with en(fp), e(12n;1)(fp) and constantscp as in Eqs. (4.2) – (4.4).
Z −∞
∞
fp(x) dx=
X
j=±1 X
k=1;2; :::; n
fp(h(1n; k; j;1)) e−(h(1n; k; j;1)−1=h
(1;1)
n; k; j)2
Hn; k; j(1;1)+ e(12n;1)(fp); (4.3)
for the integrands fp(x) =cpx−2p e−x
2−
1=x2, corresponding to p= 3;4;5;6;7;8. The constants c
p are chosen so that
Z ∞
−∞
fp(x) dx= 1: (4.4)
Scaling is uniform for the six plots. Once again, h(n; k; j;1) andHn; k; j(;1) were computed using Eqs. (3.7) and
(3.8) and numerical values for hn; k andHn; k given in [10]. In each graph, the points (n;|en(fp)|), for
n=2;3; : : : ;10, have been joined by a dashed line, while the points (n;|e2(1n;1)(fp)|), forn=2;3; : : : ;10, have been joined by a solid line.
It can be seen from Eq. (3.11), and reected in the six graphs of Fig. 2, that
e(12n;1)(fp) = 0; n¿p: (4.5)
In dramatic contrast, it is not at all clear from the numerical evidence represented in Fig. 2 that the
error en(fp), associated with the classical Hermite–Gauss quadrature, for any of the six values of p,
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