Improved Convergence for Product Integration

• K. S. Thomas
Chapter
Part of the ISNM 57: International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique book series (ISNM, volume 57)

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 fN(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). fN(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 ϕi(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)
.

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