A New Quadrature Rule Derived from Spline Interpolation with Error Analysis

We present a new quadrature rule based on the spline interpolation along with the error analysis. Moreover, some error estimates for the reminder when the integrand is either a Lipschitzian function, a function of bounded variation or a function whose derivative belongs to Lp are given. We also give some examples to show that, practically, the spline rule is better than the trapezoidal rule.

Authors:

• Hadi Taghvafard

Keywords:

• Quadrature
• Spline interpolation
• Trapezoidal rule
• Numericalintegration
• Error analysis.

References:

[1] S. S. Dragomir, On Simpson-s quadrature formula for Lipschitzian
mappings and applications, Soochow J. Mathematics, 25(2), 175-180,
1999.
[2] S. S. Dragomir, On Simpson-s quadrature formula for mappings of
bounded variation and applications, Tamkang J. Mathematics, 30(1), 53-
58, 1999.
[3] S. S. Dragomir, On Simpson-s quadrature formula for differentiable
mappings whose derivatines belong to Lp spaces and applications, J.
KSIAM, 2(2), 57-65, 1998.
[4] J. Stoer, R. Bulrisch, Introduction to numerical analysis, Second edition,
Springer-Verlag, 1993.
[5] M. B. Allen, I. E. Isaacson, Numerical analysis for applied science, John
Wiley & Sons, 1998.
[6] R. L. Burden, J. D. Faires, Numerical analysis, Seventh edition,
Brooks/Cole, 2001.