An Algebraic Study of the Levin Transformation in Integration

Carino, R.L., Robinson, I., & DeDoncker, E. (1992). An Algebraic Study of the Levin Transformation in Integration. In T.O. Espelid and A. Genz (Eds.), Numerical Integration: Recent Developments, Software, and Applications. Springer. 175-186.

The asymptotic error expansion for the m^N-copy quadrature rule approximation over the hypercube H N is known for some integrand functions. In this paper, we use Macsyma to investigate the outcome of applying the Levin transformation to a sequence of quadrature approximations Q^(m) f, m = n + 1, n + 2, … for which these known forms of error expansion are valid. The resulting expansions suggest that in certain cases, iterated extrapolation by a low-order Levin transformation is capable of obtaining approximations more accurate than those obtained by the straightforward use of the transformation. Numerical results are presented to illustrate the effectiveness of the iterated transformation in these cases.