Improved Convergence for Product Integration

Abstract

Product integration is a well established technique for evaluating the definite integral

$$\int \begin{gathered}1 \hfill \\0 \hfill \\\end{gathered} g(t)f(t)dt$$

((1.1))

The function f(t) is approximated by f_N(t) and the resulting integral

$$\int \begin{gathered}1 \hfill \\0 \hfill \\\end{gathered} g(t){f_N}(t)dt$$

((1.2))

is taken as our approximation to the integral in (1.1). f_N(t) is usually constructed by piecewise-polynomial interpolation. A simple example is the product trapezoidal rule which uses piecewise-linear interpolation, namely

$${f_N}(t) = \sum\limits_{i = 0}^N {{\phi _i}(t)f(i/N)} $$

((1.3))

. In (1.3), the ϕ_i(t) I = 0,...,N are the usual “hat” functions. Then we have

$$\int \begin{gathered}1 \hfill \\0 \hfill \\ \end{gathered} g(t)f(t)dt \simeq \sum\limits_{i = 0}^N {{w_i}f(i/N)} $$

((1.4))

where

$${w_i} = \int \begin{gathered}1 \hfill \\0 \hfill \\\end{gathered} g(t){\phi _i}(t)dt$$

((1.5))

.