In this paper, quadrature formulas on the real line with the highest degree of accuracy, with positive weights, and with one or two prescribed nodes anywhere on the interval of integration are ...characterized. As an application, the same kind of rules but with one or both (finite) endpoints being fixed nodes and one or two more prescribed nodes inside the interval of integration are derived. An efficient computation of such quadrature formulas is analyzed by considering certain modified Jacobi matrices. Some numerical experiments are finally presented.
In this paper, we present the Gauss-type quadrature formula as a rigorous method for statistical moment estimation involving arbitrary input distributions and further extend its use to robust design ...optimization. The mathematical background of the Gauss-type quadrature formula is introduced and its relation with other methods such as design of experiments (DOE) and point estimate method (PEM) is discussed. Methods for constructing one dimensional Gauss-type quadrature formula are summarized and the insights are provided. To improve the efficiency of using it for robust design optimization, a semi-analytic design sensitivity analysis with respect to the statistical moments is proposed for two different multi-dimensional integration methods, the tensor product quadrature (TPQ) formula and the univariate dimension reduction (UDR) method. Through several examples, it is shown that the Gauss-type quadrature formula can be effectively used in robust design involving various non-normal distributions. The proposed design sensitivity analysis significantly reduces the number of function calls of robust optimization using the TPQ formulae, while using the UDR method, the savings of function calls are observed only in limited situations.
Let
H
n
be the
nth Hermite polynomial, i.e., the
nth orthogonal on
R
polynomial with respect to the weight
w(
x)=exp(−
x
2). We prove the following: If
f is an arbitrary polynomial of degree at most
...n, such that |
f|⩽|
H
n
| at the zeros of
H
n+1
, then for
k=1,…,
n we have ‖
f
(
k)
‖⩽‖
H
n
(
k)
‖, where ‖·‖ is the
L
2(w;
R)
norm. This result can be viewed as an inequality of the Duffin and Schaeffer type. As corollaries, we obtain a Markov-type inequality in the
L
2(w;
R)
norm, and estimates for the expansion coefficients in the basis of Hermite polynomials.
We show that the Christoffel function
λ
n
associated with the Hermite weight function
w
H
(
x)=exp(−
x
2) is bell-shaped. As a consequence, we describe completely how the weights in a Gauss-type ...quadrature formula associated with
w
H
(
x) are arranged in magnitude.
Given a symmetric rule for the approximate numerical integration of
we choose criteria for optimising the parameters in that rule. The method is extended to find integration rules in higher ...dimensions.
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