Termination criterion and error analysis of a mixed rule using an anti-gaussian rule in whole interval and adaptive algorithm

A mixed quadrature rule of higher precision for approximate evaluation of real definite integrals has been constructed using an anti-Gaussian rule. The analytical convergence of the rule has been studied. The error bounds have been determined asymptotically. In adaptive quadrature routines not before mixed quadrature rules basing on anti-Gaussian quadrature rule have been used for fixing termination criterion. Adaptive quadrature routines being recursive by nature, a termination criterion is formed taking in to account a mixed quadrature rule. The algorithm presented in this paper and successfully tested on different integrals by C program. The relative efficiency of the mixed quadrature rule is reflected in the table at the end.