Gregory type quadrature based on quadratic nodal spline interpolation |
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Authors: | SA De Swardt JM De Villiers |
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Institution: | (1) Department of Mathematics, University of Stellenbosch, Stellenbosch 7600, South Africa; Fax : (021) 8083828; e-mail:jmdv@sunvax.sun.ac.za , ZA |
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Abstract: | Summary. Using a method based on quadratic nodal spline interpolation, we define a quadrature rule with respect to arbitrary nodes,
and which in the case of uniformly spaced nodes corresponds to the Gregory rule of order two, i.e. the Lacroix rule, which
is an important example of a trapezoidal rule with endpoint corrections. The resulting weights are explicitly calculated,
and Peano kernel techniques are then employed to establish error bounds in which the associated error constants are shown
to grow at most linearly with respect to the mesh ratio parameter. Specializing these error estimates to the case of uniform
nodes, we deduce non-optimal order error constants for the Lacroix rule, which are significantly smaller than those calculated
by cruder methods in previous work, and which are shown here to compare favourably with the corresponding error constants
for the Simpson rule.
Received July 27, 1998/ Revised version received February 22, 1999 / Published online January 27, 2000 |
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Keywords: | Mathematics Subject Classification (1991):41A55 41A15 41A05 65D32 65D30 65D07 65D05 |
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