Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol104.Issue2.1999:
Journal of Computational and Applied Mathematics 104 (1999) 163–171
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
∞
2
f(x)e−[1=(x−
=x)] d x =
−∞
n
XX
(
; )
[f(x)];
f(hn;(
;k;)j )Hn;(
;k;)j + E2n
j=±1 k=1
where the abscissas
and weights Hn;(
;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,
hn;(
;k;)j
(
; )
E2n
[f(x)] =
g(4n) () n! 2n+1
;
(4n)! 2n
for some in (−∞; ∞), provided g(x) := x2n f(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 in nity, of the Laurent–Hermite–Gauss quadrature over the corresponding classical quadrature
c 1999 Elsevier Science B.V. All rights reserved.
for certain integrands.
MSC: primary 65D32; secondary 41A30
Keywords: Quadrature; Strong distribution; Laurent polynomial
1. Introduction
Gauss quadrature is an important tool for approximating de nite 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.
(
c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ - see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 9 ) 0 0 0 5 4 - 0
164
B.A. Hagler / Journal of Computational and Applied Mathematics 104 (1999) 163–171
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.
Assume w(x) is a non-negative weight function on an interval (a; b), −∞6a ¡ b6∞, giving an
inner-product,
(P; Q) :=
Z
b
P(x)Q(x)w(x) d x;
(1.1)
a
for the space of real polynomials. We let {Pn (x)}∞
n=0 denote the orthogonal basis of monic polynomials, ordered by polynomial degree, orthogonal with respect to the above inner-product.
Pn (x) =
Y
(x − x n; k );
(1.2)
k=1; 2; :::; n
for distinct real roots x n; 1 ; x n; 2 ; : : : ; x n; n . For functions f(x) where the integral exists, Gauss quadrature,
as in [7] for example, then gives unique positive real numbers wn; 1 ; wn; 2 ; : : : ; wn; n such that
Z
b
f(x)w(x) d x =
a
X
f(x n; k )wn; k + En [f(x)];
(1.3)
k=1; 2; :::; n
where En (xm ) = 0, for m = 0; 1; : : : ; 2n − 1, or, more generally,
En [f(x)] =
f(2n) ()
(Pn ; Pn );
(2n)!
(1.4)
for f(x) having a continuous (2n)-th derivative and some ∈ (a; b).
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
1
v (x) :=
x−
:
x
Its inverses are
(
; )
vj(
; ) (y) :=
s
(1.5)
4
y + j y2 + 2 ;
2
(1.6)
for j = ±1. In our current context with w(x), a non-negative weight function, results in [4,5] include
an inner-product,
hR; Si :=
XZ
j=±1
vj(
; ) (b)
vj(
; ) (a)
R(x)S(x)w(
; ) (x) d x;
(1.7)
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 {Pn(
; ) (x)}∞
n=0 denote the orthogonal basis of monic Laurent
B.A. Hagler / Journal of Computational and Applied Mathematics 104 (1999) 163–171
165
polynomials, ordered by L-degree, orthogonal with respect to Eq. (1.7). [4,5] give
(
; )
P2n
(x) = n Pn (v(
; ) (x));
(
; )
P2n+1
(x)
= −
n
(1.9)
1
Pn (v(
; ) (x)):
x
(1.10)
It is also shown in [4,5] that
(
; )
(x) = x−n
P2n
Y
(x − xn;(
;k;)j );
Y
(1.11)
j=±1 k=1; 2; :::; n
where the 2n distinct real roots are
xn;(
;k;)j := vj(
; ) (x n; k );
(1.12)
for k = 1; 2; : : : ; n and j = ±1. Another result reported in [4,5] that will be used below is
(
; )
(
; )
hP2n
; P2n
i = 2n+1 (Pn ; Pn ):
(1.13)
Gustafson and Hagler in [2] show that, when the integrals exist, there are unique positive numbers
k = 1; 2; : : : ; n and j = ±1, such that
wn;(
;k;)j ,
XZ
j=±1
vj(
; ) (b)
vj(
; ) (a)
f(x)w(
; ) (x) d x =
X
X
(
; )
f(xn;(
;k;)j )wn;(
;k;)j + E2n
[f(x)];
(1.14)
j=±1 k=1;2;:::; n
(
; ) m
where E2n
(x ) = 0, for m = −2n; −2n + 1; : : : ; 2n − 1, or, more generally,
(
; )
(f) =
E2n
X g(4n) (j ) Z
(4n)!
j=±1
vj(
; ) (b)
vj(
; ) (a)
(
; )
[P2n
(x)]2 w(
; ) (x) d x;
(1.15)
for g(x) := x2n f(x) having a continuous (4n)-th derivative and some numbers ±1 in the interval
(
; )
(v−1
(a); v1(
; ) (b)). They also show that the weights in the above quadrature formula satisfy
wn;(
;k;)j =
wn; k
v′ (xn;(
;k;)j )
;
(1.16)
where we have used the abbreviation v′ for the derivative dv(
; ) =d x 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
2
(1.8) is e−[1=(x−
=x)] on (−∞; ∞). A Table of numerical values for the abscissas and weights,
(
; )
an expression for the remainders E2n
[f(x)] and speci c case studies with comparisons to results
yielded by various other quadrature rules are included. However, before narrowing our focus, we rst
present results which describe how the classical quadrature (1.3) is a limiting case of the transformed
quadrature (1.14). An example of this convergence will be provided in Section 4.
166
B.A. Hagler / Journal of Computational and Applied Mathematics 104 (1999) 163–171
2. Resolution to classical quadrature
Theorem 2.1 (Resolution). Let x n;k and wn; k denote; respectively; an abscissas and a corresponding weight in Gauss quadrature; (1:3). Let xn;(
;k;)j and wn;(
;k;)j be given by Eqs. (1:12) and (1:16);
respectively. Taking = 1; the following hold as
→ 0+ :
√
1. If x n; k = 0; then xn;(
;k;1)j = j
→ 0 = x n; k and wn;(
;k;1)j = 12 wn; k ; for j = ±1.
1)
1)
1)
1)
→ x n; k and wn;(
;k;−1
→ wn; k ; but xn;(
;k;1
→ 0 and wn;(
;k;1
→ 0.
2. If x n; k ¡ 0; then xn;(
;k;−1
1)
1)
1)
1)
3. If x n; k ¿ 0; then xn;(
;k;1
→ x n; k and wn;(
;k;1
→ wn; k ; but xn;(
;k;−1
→ 0 and wn;(
;k;−1
→ 0.
q
Proof. xn;(
;k;1)j := 21 (x n; k + j xn;2 k + 4
) → 12 (x n; k + j|x n; k |) as
→ 0+ . Hence, the statements concerning
xn;(
;k;1)j are seen to hold. If x n; k = 0, wn;(
;k;1)j = 21 wn; k by consideration of the de nitions. If x n; k ¡ 0 and
j = −1, or if x n; k ¿ 0 and j = 1, we have that xn;(
;k;1)j → x n; k 6= 0 as
→ 0+ , and it follows that
wn;(
;k;1)j := wn; k =(1 +
=[xn;(
;k;1)j ]2 ) → wn; k in these cases. If x n; k ¡ 0 and j = 1, or if x n; k ¿ 0 and j = −1,
then xn;(
;k;1)j → 0 as
→ 0+ , and it follows by an application of L’Hopital’s rule that
lim
→0+
[xn;(
;k;1)j ]2
[xn;(
;k;1)j ]2 +
= lim+
→0
d (
; 1)
xn; k; j
2xn;(
;k;1)j d
d (
; 1)
xn; k; j + 1
2xn;(
;k;1)j d
= 0:
(2.1)
Hence, in these cases, wn;(
;k;1)j := wn; k =(1 +
=[xn;(
;k;1)j ]2 ) → 0.
Theorem 2.2. If f(x) is continuous at 0 and at x n; k ; k = 1; 2; : : : ; n; then
X
X
j=±1 k=1; 2;:::; n
f(xn;(
;k;1)j )wn;(
;k;1)j →
X
f(x n; k )wn; k ;
k=1; 2; :::; n
as
→ 0+ :
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 :=
Z
∞
2
P(x)Q(x)e−x d x:
(3.1)
−∞
Using the known formula
n!
;
2n
we write the quadrature (1.3) in the compact form
(Hn ; Hn )H =
Z
∞
−∞
2
f(x)e−x d x =
X
k=1; 2;:::; n
f(hn; k )Hn; k +
f(2n) () n!
;
(2n)! 2n
(3.2)
(3.3)
B.A. Hagler / Journal of Computational and Applied Mathematics 104 (1999) 163–171
167
for f(x) having a continuous (2n)-th derivative, and some ∈ (−∞; ∞). We have thus denoted
the zeros of Hn (x) by hn; k and the corresponding quadrature weights by Hn; k , for k = 1; 2; : : : ; n. The
formula
√
2n−1 n!
(3.4)
Hn; k = 2
n [Hn−1 (hn; k )]2
is also well known in the literature.
Since
2
lim xm e−[1=(x−
=x)] = 0;
(3.5)
x→0
for m = 0; ±1; ±2; : : :, we can then write Eq. (1.7), in this case, as
hR; SiH (
; ) :=
Z
∞
2
R(x)S(x)e−[1=(x−
=x)] d x;
(3.6)
−∞
as an inner-product for the space of real Laurent polynomials.
Denote by {Hn(
; ) (x)}∞
n=0 the sequence of monic orthogonal Laurent polynomials with respect to
Eq. (3.6). By Eq. (1.12), the zeros of Hn(
; ) (x) are given by
)
h(
;
n; k; j
s
4
= hn; k + j h2n; k + 2
2
(3.7)
for k =1; 2; : : : ; n and j =±1. The corresponding weight factors in Eq. (1.14), considering Eq. (1.16),
we can write as
Hn;(
;k;)j
=
) 2
(h(
;
n; k; j )
) 2
(h(
;
n; k; j ) +
(3.8)
Hn; k ;
or, by Eq. (3.4),
Hn;(
;k;)j
=
√
) 2
2n−1 n! (h(
;
n; k; j )
) 2
2
n2 [(h(
;
n; k; j ) +
][Hn−1 (hn; k )]
;
(3.9)
for k = 1; 2; : : : ; n and j = ±1.
The following theorem introduces the (2n)-point Laurent–Hermite–Gauss quadrature formula of
parameters
; ¿ 0.
)
(
; )
Theorem 3.1. Let
; ¿ 0; and let n be a positive integer. With h(
;
n; k; j and Hn; k; j given by (3:7)
and (3:9); respectively;
Z
∞
2
f(x)e−[1=(x−
=x)] d x =
−∞
n
XX
)
(
; )
(
; )
f(h(
;
n; k; j )Hn; k; j + E2n [f(x)]
(3.10)
j=±1 k=1
with
(
; )
E2n
[f(x)] :=
g(4n) () n! 2n+1
;
(4n)! 2n
for some in (−∞; ∞); provided g(x) := x2n f(x) has a continuous (4n)-th derivative.
(3.11)
168
B.A. Hagler / Journal of Computational and Applied Mathematics 104 (1999) 163–171
Table 1
Abscissas and weights for Laurent–Hermite–Gauss quadrature of paramaters
; ¿ 0
Z
∞
f(x) e−[1=
(x−
=x)]2
dx ≈
−∞
X X
f(hn;(
;k;)j )Hn;(
;k;)j
j=±1 k=1;:::; n
1
= [hn; k + j h2n; k + (4
=2 )]
Abscissas :
2
Weights : Hn;(
;k;)j = Hn; k (hn;(
;k;)j )2 =[(hn;(
;k;)j )2 +
]
q
hn;(
;k;)j
n
±hn; k
Hn; k
2
0.70710 67811 86548
8.86226 92545 28 (−1)
3
0.00000 00000 00000
1.22474 48713 91589
1.18163 58006 04 (0)
2.95408 97515 09 (−1)
4
0.52464 76232 75290
1.65068 01238 85785
8.04919 09000 55 (−1)
8.13128 35447 25 (−2)
5
0.00000 00000 00000
0.95857 24646 13819
2.02018 28704 56086
9.45308 72048 29 (−1)
3.93619 32315 22 (−1)
1.99532 42059 05 (−2)
6
0.43607 74119 27617
1.33584 90760 13697
2.35060 49736 74492
7.24629 59522 44 (−1)
1.57067 32032 29 (−1)
4.53000 99055 09 (−3)
Proof. The result follows by standard methods, using Hermite’s interpolation formula for
g(x) := x2n f(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
f(x) d x =
−∞
X f(xi )
i=1;:::; n
w(xi )
wi + en (f):
Fig. 1 shows several gross features of the approximation
Z
∞
−∞
f(x) d x ≈ Q
= Truncated
X X
j=±1 k=1;:::; 8
1)
f(h(
;
8; k; j )
e
1)
1) 2
−[h(
;
−
=h(
;
]
8; k; j
8; k; j
H8;(
;k;1)j ;
(4.1)
B.A. Hagler / Journal of Computational and Applied Mathematics 104 (1999) 163–171
169
Fig. 1. Six abscissas and weights plots. Q
is the truncated value of the quadrature in Eq. (4.1).
2
for f(x) = x−10 e−(x−1=x) , as
is varied. Scaling is uniform for the six plots, and h8;(
;k;1)j and H8;(
;k;1)j 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
(1; 1)
e2n (fp ) in the formulas
Z
∞
−∞
fp (x) d x =
X fp (hn; k )
i=1; :::; n
e−(hn; k )2
Hn; k + en (fp );
(4.2)
170
B.A. Hagler / Journal of Computational and Applied Mathematics 104 (1999) 163–171
(1;1)
(fp ) and constants cp as in Eqs. (4.2) – (4.4).
Fig. 2. Six error comparison graphs with en (fp ), e2n
Z
1)
fp (h(1;
n; k; j )
−∞
fp (x) d x =
∞
X
X
j=±1 k=1;2; :::; n
e
1)
1) 2
−(h(1;
−1=h(1;
)
n; k; j
n; k; j
2
Hn;(1;k;1)j + e(1;1)
2n (fp );
(4.3)
2
for the integrands fp (x) = cp x−2p e−x −1=x , corresponding to p = 3; 4; 5; 6; 7; 8. The constants cp are
chosen so that
Z
∞
fp (x) d x = 1:
(4.4)
−∞
Scaling is uniform for the six plots. Once again, hn;(
;k;1)j and Hn;(
;k;1)j were computed using Eqs. (3.7) and
(3.8) and numerical values for hn; k and Hn; k given in [10]. In each graph, the points (n; |en (fp )|), for
(1;1)
(fp )|), for n=2; 3; : : : ; 10,
n=2; 3; : : : ; 10, have been joined by a dashed line, while the points (n; |e2n
have been joined by a solid line.
It can be seen from Eq. (3.11), and re
ected in the six graphs of Fig. 2, that
(1;1)
e2n
(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,
converges to 0 as n tends to in nity.
B.A. Hagler / Journal of Computational and Applied Mathematics 104 (1999) 163–171
171
References
[1] W. Gautschi, A survay of Gauss–Christo el quadrature formulae, in: P.L. Butzer, F. Feher (Eds.), Christo el
Memorial Volume, Birkhauser, Basel, 1981, pp. 72–147.
[2] P.E. Gustafson, B.A. Hagler, Gaussian quadrature rules and numerical examples for strong extensions of classical
mass distributions, J. Comput. Appl. Math. 105 (1999) to appear.
[3] S. Haber, A note on some quadrature formulas for the interval (−∞; ∞), Math. Comput. 18 (1962) 313–314.
[4] B.A. Hagler, A transformation of orthogonal polynomial sequences into orthogonal Laurent polynomial sequences,
Ph.D. thesis, University of Colorado, 1997.
[5] B.A. Hagler, W.B. Jones, W.J. Thron, Orthogonal Laurent polynomials of Jacobi, Hermite and Laguerre Types,
Orthogonal Functions, Moment Theory, and Continued Fractions: Theory and Applications (William B. Jones and
A. Sri Ranga, eds.), Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, 1998, 187–208.
[6] W.H. Harper, Quadrature formulas for in nite integrals, Math. Comput. 16 (1962) 170–175.
[7] F.B. Hildebrand, Introduction to Numerical Analysis, McGraw-Hill, New York, 1956.
[8] W.B. Jones, W.J. Thron, Orthogonal Laurent polynomials and Gaussian quadrature, in: K.E. Gustafson,
W.P. Reinhardt (Eds.), Quantum Mechanics in Mathematics, Chemistry, and Physics, Plenum Press, New York,
1981, pp. 449 – 455.
[9] O. Nja stad, W.J. Thron, The theory of sequences of orthogonal L-polynomials, Pade approximants and continued
fractions, Det Kong. Norske Vid. Selsk. Skr. 1 (1983) 54–91.
[10] H.E. Salzer, R. Zucker, R. Capuano, Table of zeros and weight factors of the rst twenty hermite polynomials, J.
Res. NBS 48 (1952) 111–116.
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
∞
2
f(x)e−[1=(x−
=x)] d x =
−∞
n
XX
(
; )
[f(x)];
f(hn;(
;k;)j )Hn;(
;k;)j + E2n
j=±1 k=1
where the abscissas
and weights Hn;(
;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,
hn;(
;k;)j
(
; )
E2n
[f(x)] =
g(4n) () n! 2n+1
;
(4n)! 2n
for some in (−∞; ∞), provided g(x) := x2n f(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 in nity, of the Laurent–Hermite–Gauss quadrature over the corresponding classical quadrature
c 1999 Elsevier Science B.V. All rights reserved.
for certain integrands.
MSC: primary 65D32; secondary 41A30
Keywords: Quadrature; Strong distribution; Laurent polynomial
1. Introduction
Gauss quadrature is an important tool for approximating de nite 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.
(
c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ - see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 9 ) 0 0 0 5 4 - 0
164
B.A. Hagler / Journal of Computational and Applied Mathematics 104 (1999) 163–171
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.
Assume w(x) is a non-negative weight function on an interval (a; b), −∞6a ¡ b6∞, giving an
inner-product,
(P; Q) :=
Z
b
P(x)Q(x)w(x) d x;
(1.1)
a
for the space of real polynomials. We let {Pn (x)}∞
n=0 denote the orthogonal basis of monic polynomials, ordered by polynomial degree, orthogonal with respect to the above inner-product.
Pn (x) =
Y
(x − x n; k );
(1.2)
k=1; 2; :::; n
for distinct real roots x n; 1 ; x n; 2 ; : : : ; x n; n . For functions f(x) where the integral exists, Gauss quadrature,
as in [7] for example, then gives unique positive real numbers wn; 1 ; wn; 2 ; : : : ; wn; n such that
Z
b
f(x)w(x) d x =
a
X
f(x n; k )wn; k + En [f(x)];
(1.3)
k=1; 2; :::; n
where En (xm ) = 0, for m = 0; 1; : : : ; 2n − 1, or, more generally,
En [f(x)] =
f(2n) ()
(Pn ; Pn );
(2n)!
(1.4)
for f(x) having a continuous (2n)-th derivative and some ∈ (a; b).
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
1
v (x) :=
x−
:
x
Its inverses are
(
; )
vj(
; ) (y) :=
s
(1.5)
4
y + j y2 + 2 ;
2
(1.6)
for j = ±1. In our current context with w(x), a non-negative weight function, results in [4,5] include
an inner-product,
hR; Si :=
XZ
j=±1
vj(
; ) (b)
vj(
; ) (a)
R(x)S(x)w(
; ) (x) d x;
(1.7)
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 {Pn(
; ) (x)}∞
n=0 denote the orthogonal basis of monic Laurent
B.A. Hagler / Journal of Computational and Applied Mathematics 104 (1999) 163–171
165
polynomials, ordered by L-degree, orthogonal with respect to Eq. (1.7). [4,5] give
(
; )
P2n
(x) = n Pn (v(
; ) (x));
(
; )
P2n+1
(x)
= −
n
(1.9)
1
Pn (v(
; ) (x)):
x
(1.10)
It is also shown in [4,5] that
(
; )
(x) = x−n
P2n
Y
(x − xn;(
;k;)j );
Y
(1.11)
j=±1 k=1; 2; :::; n
where the 2n distinct real roots are
xn;(
;k;)j := vj(
; ) (x n; k );
(1.12)
for k = 1; 2; : : : ; n and j = ±1. Another result reported in [4,5] that will be used below is
(
; )
(
; )
hP2n
; P2n
i = 2n+1 (Pn ; Pn ):
(1.13)
Gustafson and Hagler in [2] show that, when the integrals exist, there are unique positive numbers
k = 1; 2; : : : ; n and j = ±1, such that
wn;(
;k;)j ,
XZ
j=±1
vj(
; ) (b)
vj(
; ) (a)
f(x)w(
; ) (x) d x =
X
X
(
; )
f(xn;(
;k;)j )wn;(
;k;)j + E2n
[f(x)];
(1.14)
j=±1 k=1;2;:::; n
(
; ) m
where E2n
(x ) = 0, for m = −2n; −2n + 1; : : : ; 2n − 1, or, more generally,
(
; )
(f) =
E2n
X g(4n) (j ) Z
(4n)!
j=±1
vj(
; ) (b)
vj(
; ) (a)
(
; )
[P2n
(x)]2 w(
; ) (x) d x;
(1.15)
for g(x) := x2n f(x) having a continuous (4n)-th derivative and some numbers ±1 in the interval
(
; )
(v−1
(a); v1(
; ) (b)). They also show that the weights in the above quadrature formula satisfy
wn;(
;k;)j =
wn; k
v′ (xn;(
;k;)j )
;
(1.16)
where we have used the abbreviation v′ for the derivative dv(
; ) =d x 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
2
(1.8) is e−[1=(x−
=x)] on (−∞; ∞). A Table of numerical values for the abscissas and weights,
(
; )
an expression for the remainders E2n
[f(x)] and speci c case studies with comparisons to results
yielded by various other quadrature rules are included. However, before narrowing our focus, we rst
present results which describe how the classical quadrature (1.3) is a limiting case of the transformed
quadrature (1.14). An example of this convergence will be provided in Section 4.
166
B.A. Hagler / Journal of Computational and Applied Mathematics 104 (1999) 163–171
2. Resolution to classical quadrature
Theorem 2.1 (Resolution). Let x n;k and wn; k denote; respectively; an abscissas and a corresponding weight in Gauss quadrature; (1:3). Let xn;(
;k;)j and wn;(
;k;)j be given by Eqs. (1:12) and (1:16);
respectively. Taking = 1; the following hold as
→ 0+ :
√
1. If x n; k = 0; then xn;(
;k;1)j = j
→ 0 = x n; k and wn;(
;k;1)j = 12 wn; k ; for j = ±1.
1)
1)
1)
1)
→ x n; k and wn;(
;k;−1
→ wn; k ; but xn;(
;k;1
→ 0 and wn;(
;k;1
→ 0.
2. If x n; k ¡ 0; then xn;(
;k;−1
1)
1)
1)
1)
3. If x n; k ¿ 0; then xn;(
;k;1
→ x n; k and wn;(
;k;1
→ wn; k ; but xn;(
;k;−1
→ 0 and wn;(
;k;−1
→ 0.
q
Proof. xn;(
;k;1)j := 21 (x n; k + j xn;2 k + 4
) → 12 (x n; k + j|x n; k |) as
→ 0+ . Hence, the statements concerning
xn;(
;k;1)j are seen to hold. If x n; k = 0, wn;(
;k;1)j = 21 wn; k by consideration of the de nitions. If x n; k ¡ 0 and
j = −1, or if x n; k ¿ 0 and j = 1, we have that xn;(
;k;1)j → x n; k 6= 0 as
→ 0+ , and it follows that
wn;(
;k;1)j := wn; k =(1 +
=[xn;(
;k;1)j ]2 ) → wn; k in these cases. If x n; k ¡ 0 and j = 1, or if x n; k ¿ 0 and j = −1,
then xn;(
;k;1)j → 0 as
→ 0+ , and it follows by an application of L’Hopital’s rule that
lim
→0+
[xn;(
;k;1)j ]2
[xn;(
;k;1)j ]2 +
= lim+
→0
d (
; 1)
xn; k; j
2xn;(
;k;1)j d
d (
; 1)
xn; k; j + 1
2xn;(
;k;1)j d
= 0:
(2.1)
Hence, in these cases, wn;(
;k;1)j := wn; k =(1 +
=[xn;(
;k;1)j ]2 ) → 0.
Theorem 2.2. If f(x) is continuous at 0 and at x n; k ; k = 1; 2; : : : ; n; then
X
X
j=±1 k=1; 2;:::; n
f(xn;(
;k;1)j )wn;(
;k;1)j →
X
f(x n; k )wn; k ;
k=1; 2; :::; n
as
→ 0+ :
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 :=
Z
∞
2
P(x)Q(x)e−x d x:
(3.1)
−∞
Using the known formula
n!
;
2n
we write the quadrature (1.3) in the compact form
(Hn ; Hn )H =
Z
∞
−∞
2
f(x)e−x d x =
X
k=1; 2;:::; n
f(hn; k )Hn; k +
f(2n) () n!
;
(2n)! 2n
(3.2)
(3.3)
B.A. Hagler / Journal of Computational and Applied Mathematics 104 (1999) 163–171
167
for f(x) having a continuous (2n)-th derivative, and some ∈ (−∞; ∞). We have thus denoted
the zeros of Hn (x) by hn; k and the corresponding quadrature weights by Hn; k , for k = 1; 2; : : : ; n. The
formula
√
2n−1 n!
(3.4)
Hn; k = 2
n [Hn−1 (hn; k )]2
is also well known in the literature.
Since
2
lim xm e−[1=(x−
=x)] = 0;
(3.5)
x→0
for m = 0; ±1; ±2; : : :, we can then write Eq. (1.7), in this case, as
hR; SiH (
; ) :=
Z
∞
2
R(x)S(x)e−[1=(x−
=x)] d x;
(3.6)
−∞
as an inner-product for the space of real Laurent polynomials.
Denote by {Hn(
; ) (x)}∞
n=0 the sequence of monic orthogonal Laurent polynomials with respect to
Eq. (3.6). By Eq. (1.12), the zeros of Hn(
; ) (x) are given by
)
h(
;
n; k; j
s
4
= hn; k + j h2n; k + 2
2
(3.7)
for k =1; 2; : : : ; n and j =±1. The corresponding weight factors in Eq. (1.14), considering Eq. (1.16),
we can write as
Hn;(
;k;)j
=
) 2
(h(
;
n; k; j )
) 2
(h(
;
n; k; j ) +
(3.8)
Hn; k ;
or, by Eq. (3.4),
Hn;(
;k;)j
=
√
) 2
2n−1 n! (h(
;
n; k; j )
) 2
2
n2 [(h(
;
n; k; j ) +
][Hn−1 (hn; k )]
;
(3.9)
for k = 1; 2; : : : ; n and j = ±1.
The following theorem introduces the (2n)-point Laurent–Hermite–Gauss quadrature formula of
parameters
; ¿ 0.
)
(
; )
Theorem 3.1. Let
; ¿ 0; and let n be a positive integer. With h(
;
n; k; j and Hn; k; j given by (3:7)
and (3:9); respectively;
Z
∞
2
f(x)e−[1=(x−
=x)] d x =
−∞
n
XX
)
(
; )
(
; )
f(h(
;
n; k; j )Hn; k; j + E2n [f(x)]
(3.10)
j=±1 k=1
with
(
; )
E2n
[f(x)] :=
g(4n) () n! 2n+1
;
(4n)! 2n
for some in (−∞; ∞); provided g(x) := x2n f(x) has a continuous (4n)-th derivative.
(3.11)
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B.A. Hagler / Journal of Computational and Applied Mathematics 104 (1999) 163–171
Table 1
Abscissas and weights for Laurent–Hermite–Gauss quadrature of paramaters
; ¿ 0
Z
∞
f(x) e−[1=
(x−
=x)]2
dx ≈
−∞
X X
f(hn;(
;k;)j )Hn;(
;k;)j
j=±1 k=1;:::; n
1
= [hn; k + j h2n; k + (4
=2 )]
Abscissas :
2
Weights : Hn;(
;k;)j = Hn; k (hn;(
;k;)j )2 =[(hn;(
;k;)j )2 +
]
q
hn;(
;k;)j
n
±hn; k
Hn; k
2
0.70710 67811 86548
8.86226 92545 28 (−1)
3
0.00000 00000 00000
1.22474 48713 91589
1.18163 58006 04 (0)
2.95408 97515 09 (−1)
4
0.52464 76232 75290
1.65068 01238 85785
8.04919 09000 55 (−1)
8.13128 35447 25 (−2)
5
0.00000 00000 00000
0.95857 24646 13819
2.02018 28704 56086
9.45308 72048 29 (−1)
3.93619 32315 22 (−1)
1.99532 42059 05 (−2)
6
0.43607 74119 27617
1.33584 90760 13697
2.35060 49736 74492
7.24629 59522 44 (−1)
1.57067 32032 29 (−1)
4.53000 99055 09 (−3)
Proof. The result follows by standard methods, using Hermite’s interpolation formula for
g(x) := x2n f(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
f(x) d x =
−∞
X f(xi )
i=1;:::; n
w(xi )
wi + en (f):
Fig. 1 shows several gross features of the approximation
Z
∞
−∞
f(x) d x ≈ Q
= Truncated
X X
j=±1 k=1;:::; 8
1)
f(h(
;
8; k; j )
e
1)
1) 2
−[h(
;
−
=h(
;
]
8; k; j
8; k; j
H8;(
;k;1)j ;
(4.1)
B.A. Hagler / Journal of Computational and Applied Mathematics 104 (1999) 163–171
169
Fig. 1. Six abscissas and weights plots. Q
is the truncated value of the quadrature in Eq. (4.1).
2
for f(x) = x−10 e−(x−1=x) , as
is varied. Scaling is uniform for the six plots, and h8;(
;k;1)j and H8;(
;k;1)j 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
(1; 1)
e2n (fp ) in the formulas
Z
∞
−∞
fp (x) d x =
X fp (hn; k )
i=1; :::; n
e−(hn; k )2
Hn; k + en (fp );
(4.2)
170
B.A. Hagler / Journal of Computational and Applied Mathematics 104 (1999) 163–171
(1;1)
(fp ) and constants cp as in Eqs. (4.2) – (4.4).
Fig. 2. Six error comparison graphs with en (fp ), e2n
Z
1)
fp (h(1;
n; k; j )
−∞
fp (x) d x =
∞
X
X
j=±1 k=1;2; :::; n
e
1)
1) 2
−(h(1;
−1=h(1;
)
n; k; j
n; k; j
2
Hn;(1;k;1)j + e(1;1)
2n (fp );
(4.3)
2
for the integrands fp (x) = cp x−2p e−x −1=x , corresponding to p = 3; 4; 5; 6; 7; 8. The constants cp are
chosen so that
Z
∞
fp (x) d x = 1:
(4.4)
−∞
Scaling is uniform for the six plots. Once again, hn;(
;k;1)j and Hn;(
;k;1)j were computed using Eqs. (3.7) and
(3.8) and numerical values for hn; k and Hn; k given in [10]. In each graph, the points (n; |en (fp )|), for
(1;1)
(fp )|), for n=2; 3; : : : ; 10,
n=2; 3; : : : ; 10, have been joined by a dashed line, while the points (n; |e2n
have been joined by a solid line.
It can be seen from Eq. (3.11), and re
ected in the six graphs of Fig. 2, that
(1;1)
e2n
(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,
converges to 0 as n tends to in nity.
B.A. Hagler / Journal of Computational and Applied Mathematics 104 (1999) 163–171
171
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