Scientific Computing

This module illustrates Romberg integration, which is a numerical
quadrature technique based on repeated
Richardson extrapolation of values obtained using the composite
trapezoid rule with successively halved mesh spacing. For
*k* ≥ 0*T*_{k,
0}*f*(*x*)*a*, *b*]*h* =
(*b* − *a*) ⁄ 2^{k}*j* = 1,…,*k**T*_{k, j}*T*_{k−1, j−1}*T*_{k, j−1}*T*_{k, j}*k* increases, the approximations to the integral
become more accurate because the composite trapezoid rule is applied
with decreasing mesh spacing. Accuracy also increases with *j*,
because the order of accuracy of *T*_{k,
j}*j* +1)

The user begins by selecting an integrand function from the menu
provided. The integrand function is displayed in the left panel and
the triangular array of values *T*_{k,
j}*T*_{0, 0})*T*_{k,
j}*T*_{k, j}*j* = 0*j* >
0*T*_{k, j}*I* of the integral is shown below the
selected function for comparison with the values in the triangular
array, and is also plotted as a black point (which may be obscured by
the red point) in the right panel.

**Reference:** Michael T. Heath, *Scientific Computing,
An Introductory Survey*, 2nd edition, McGraw-Hill, New York,
2002. See Section 8.7, especially Example 8.9 and Figure 8.6.

**Developers:** Evan VanderZee and Michael Heath