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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

A new type of Chebyshev quadrature


Authors: R. E. Barnhill, J. E. Dennis and G. M. Nielson
Journal: Math. Comp. 23 (1969), 437-441
MSC: Primary 65.55
MathSciNet review: 0242367
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Abstract: A Chebyshev quadrature is of the form

$\displaystyle \int_{ - 1}^1 {w(x)f(x)dx \simeq } c\sum\limits_{k = 1}^n {f({x_k})} $

It is usually desirable that the nodes $ {x_k}$ be in the interval of integration and that the quadrature be exact for as many monomials as possible (i.e., the first $ n + 1$ monomials). For $ n = 1, \cdot \cdot \cdot ,7$ and $ 9$, such a choice of nodes is possible, but for $ n = 8$ and $ n > 9$, the nodes are complex. In this note, the idea used is that the $ {l^2}$-norm of the deviations of the first $ n + 1$ monomials from their moments be a minimum. Numerical calculations are carried out for $ n = 8,10$, and $ 11$ and one interesting feature of the numerical results is that a ``multiple'' node at the origin is required. The above idea is then generalized to a minimization of the $ {l^2}$-norm of the deviations of the first $ k$ monomials, $ k \geqq n + 1$, including $ k = \infty $, and corresponding numerical results are presented.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1969-0242367-4
PII: S 0025-5718(1969)0242367-4
Article copyright: © Copyright 1969 American Mathematical Society