Shanaz Ansari et al. /International Journal of Engineering and Technology Vol.1(3), 2009, 104-108 104 A Computerizable Iterative-Algorithmic Quadrature Operator Using an Efficient Two-Phase Modification of Bernstein Polynomial Shanaz Ansari Wahid, *Ashok Sahai Department of Mathematics & Computer Science; Faculty of Science & Agriculture St. Augustine Campus; The University of The West Indies. Trinidad & Tobago; West Indies. M. Raghunadh Acharya Department of Statistics and Computer Science, Aurora’s Post Graduate College; Osmania University, Hyderabad (Andhra Pradesh); India. Department of Mathematics & Computer Science The University of the West Indies, Trinidad & Tobago Abstract.: A new quadrature formula has been proposed which uses modified weight functions derived from those of ‘Bernstein Polynomial’ using a ‘Two- Phase Modification’ therein. The quadrature formula has been compared empirically with the simple method of numerical integration using the well-known “Bernstein Operator”. The percentage absolute relative errors for the proposed quadrature formula and that with the “Bernstein Operator” have been computed for certain selected functions, with different number of usual equidistant node-points in the interval of integration~ [0, 1]. It has been observed that both of the proposed modified quadrature formulae, respectively after the ‘Phase-I’ and after the ‘Phases-I & II’ of these modifications, produce significantly better results than that using, simply, the “Bernstein Operator”. Inasmuch as the proposed “Two-Phase Improvement” is available iteratively again-and-again at the end of the current iteration, the proposed improvement algorithm, which is ‘Computerizable’, is an “Iterative-Algorithm”, leading to more-and-more efficient “Quadrature-Operator”, till we are pleased! Keywords: Quadrature Formula; Percentage Absolute Relative Errors; Weight Functions. AMS Classification Number: 41A10; 41A36; 41A55; 65D32; 68W25. I. INTRODUCTION Classically, even the celebrated Bernstein polynomial approximation operator has been used for numerical integration in the interval [0, 1], without any loss of generality (in the sense of change of origin-and-scale). In a rather simple set-up, such similar quadrature methods were of the form Where x i ’s are equally spaced nodes & w i ’s are the respective weights. For example: Trapezoid: Simpson: Similarly, for higher order polynomial Newton- Cotes rules. We note one known thing already from interpolation: equally-spaced nodes result in wiggle. The preceding fact motivated us to ponder-n- wonder as to whether or not we would be able to retain the simplicity of “equidistant points in the interval of integration” in our proposed quadrature formula, and still aspire successfully for its optimality. In fact we were successful. The next section details the ‘How’ part of it. And the section following that numerically illustrates that the potential of this new/proposed ‘two-phase’ modifications of the usual ‘Bernstein Operator’ was very significant. ISSN : 0975-4024