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

 

A characterization of positive quadrature formulae


Author: Yuan Xu
Journal: Math. Comp. 62 (1994), 703-718
MSC: Primary 41A55; Secondary 65D32
MathSciNet review: 1223234
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Abstract: A positive quadrature formula with n nodes which is exact for polynomials of degree $ 2n - r - 1,0 \leq r \leq n$, is based on the zeros of certain quasi-orthogonal polynomials of degree n. We show that the quasi-orthogonal polynomials that lead to the positive quadrature formulae can all be expressed as characteristic polynomials of a symmetric tridiagonal matrix with positive subdiagonal entries. As a consequence, for a fixed n, every positive quadrature formula is a Gaussian quadrature formula for some nonnegative measure.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1994-1223234-0
PII: S 0025-5718(1994)1223234-0
Keywords: Quadrature formula, quasi-orthogonal polynomial, tridiagonal matrix
Article copyright: © Copyright 1994 American Mathematical Society