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Mathematics of Computation

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The exact degree of precision of generalized Gauss-Kronrod integration rules

Author: Philip Rabinowitz
Journal: Math. Comp. 35 (1980), 1275-1283
MSC: Primary 65D30
MathSciNet review: 583504
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Abstract: It is shown that the Kronrod extension to the n-point Gauss integration rule, with respect to the weight function $ {(1 - {x^2})^{\mu - {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}}$, $ 0 < \mu \leqslant 2$, $ \mu \ne 1$, is of exact precision $ 3n + 1$ for n even and $ 3n + 2$ for n odd. Similarly, for the $ (n + 1)$-point Lobatto rule, with $ - {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}} < \mu \leqslant 1$, $ \mu \ne 0$, the exact precision is 3n for n odd and $ 3n + 1$ for n even.

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Keywords: Kronrod rule, Gauss integration rule, Lobatto integration rule, Gegenbauer polynomials, Szegö polynomials, Fourier coefficients
Article copyright: © Copyright 1980 American Mathematical Society