ETNA
Kent State University and
Johann Radon Institute (RICAM)
Electronic Transactions on Numerical Analysis.
Volume 51, pp. 219–239, 2019.
Copyright
c
2019, Kent State University.
ISSN 1068–9613.
DOI: 10.1553/etna_vol51s219
EFFICIENT CUBATURE RULES
∗
JAMES R. VAN ZANDT
†
Abstract.
67 new cubature rules are found for three standard multi-dimensional integrals with spherically
symmetric regions and weight functions using direct search with a numerical zero-finder. 63 of the new rules have
fewer integration points than known rules of the same degree, and 20 are within three points of Möller’s lower bound.
Most have all positive coefficients, and most have some symmetry, including some supported by one or two concentric
spheres. They include degree-7 formulas for the integration over the sphere and Gaussian-weighted integrals over the
entire space, each in 6 and 7 dimensions, with 127 and 183 points, respectively.
Key words.
multiple integrals, Gaussian weight, cubature formula, integration rule, numerical integration,
regular simplex
AMS subject classifications. 65D30, 65D32, 41A55, 41A63
1. Introduction.
We are concerned with estimating multi-dimensional integrals of the
form
(1.1)
Z
Ω
w(x)f(x) dx,
where
x = [x
1
x
2
···x
n
]
T
, for the integration regions
Ω
and weighting functions
w(x)
given
in Table 1.1. Applications of
(1.1)
include the evaluation of quantum-mechanical matrix
elements with Gaussian wave functions in atomic physics [33], nuclear physics [18], and
particle physics [19]. For applications in statistics, particularly Bayesian inference, see [11].
For applications in target tracking, see [2, 21].
We approximate these integrals using cubature formulas or integration rules of the form
(1.2)
N
X
i=1
W
i
f(x
i
),
where the weights W
i
and nodes or points x
i
are independent of the function f.
In the following, we use the notation
G
n
,
E
r
2
n
,
E
r
n
, and
S
n
for the integrals defined
in Table 1.1. The first two integrals in the table are of course closely related. Given an
approximation of
G
n
of the form
(1.2)
, we can construct an equivalent approximation
E
r
2
n
≈
P
N
i=1
B
i
f(b
i
)
, where
b
i
= x
i
/
√
2
and
B
i
= π
n/2
W
i
. In this paper we address
E
r
2
n
following the numerical analysis convention. However, in the supplemental material we quote
the parameters for the corresponding
G
n
formulas for the convenience of researchers using
another commonly used convention.
If an integration rule is exact for all polynomials up to and including degree
d
but not for
some polynomial of degree
d + 1
, then we say the rule has algebraic degree of exactness (or
simply degree) d.
One can construct cubature formulas being exact for a space of polynomials by solving
the large system of polynomial equations associated with it. In describing this method, Cools
∗
Received August 6, 2018. Accepted March 22, 2019. Published online on July 24, 2019. Recommended by
Sotirios Notaris.
†
The MITRE Corporation, 202 Burlington Rd, Bedford, MA, 01730, USA (
jrv@mitre.org
). The author’s
affiliation with The MITRE Corporation is provided for identification purposes only and is not intended to convey
or imply MITRE’s concurrence with, or support for, the positions, opinions or viewpoints expressed by the author.
Approved for Public Release; Distribution Unlimited. MITRE Case Number 17-4134.
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