Quadrature problems in one and several dimensions

Abstract

In the thesis we have considered the numerical approximation of five types of integrals: newlineone dimensional real, complex and Riemann-Stieltjes integrals, multi-dimensional newlinecomplex integrals and two dimensional real CPV integrals. The mixed type rules which newlineinclude both function values and the values of derivatives in [-1, 1] have been generated newlineso that the degree of precisions of the rules are as high as possible. This thesis includes newlinethe formulation of four point rule, five point rule and a set of six point rules with newlinederivatives and their extrapolatory rules. Extrapolation is applied to the obtained newlineextrapolatory rules second time as well as third time, which makes the degree of newlineprecision higher and higher. In case of singular integrals, subtracting out singularity newlinemethod is applied with corrective factor for better approximation. The derivative based newlineGauss-Legendre two-point rule for the Riemann-Stieltjes integral is derived. For newlinenumerical evaluation of one dimensional contour integrals, rules with five points, nine newlinepoints are generated and some new modified rules of higher degree are derived newlineestablishing different relation among the coefficients and error functions. Extrapolation newlineof Gauss-Legendre two point, three point rules with two point, three point interpolatory newlinerules is being carried out to have rules of higher degree of precision. Seven point degree newlinefive, eight point degree five, sixteen point degree seven, eleven point degree five, newlinethirteen point degree seven non-product rules are formulated for numerical newlineapproximation of two dimensional complex integrals and thirty five point degree seven newlinerule is derived for three dimensional complex integrals of analytic functions inside a newlinecontour. Twelve point rule of degree five and twenty point rule of degree seven are newlinegenerated by the product of four point interpolatory rule with three point Gauss- newlineLegendre rule, five point interpolatory rule with four point Gauss-Legendre rule for two newlinedimensional complex integrals. Some cubature rules for numerical evaluation of