Romberg Integration

Revised Jan 16, 2001

The general formula are:

$\begin{displaymath} T_{1,k} = {h \over 2} \left[f(a) + f(b) + 2 \sum_{j = 1}^l f(a + j h) \right] \end{displaymath}$

where

$\begin{displaymath} h = {b - a \over 2^{k-1}} \end{displaymath}$

and $l = 2^{k-1} - 1$ .

$\begin{displaymath} T_{l,k} = {1 \over 4^{l-1} - 1} \left(4^{l-1} T_{l-1,k+1} - T_{l-1,k} \right). \end{displaymath}$

For example, for

,

$\begin{displaymath} T_{2,1} = {1 \over 3}\left(4 T_{1,2} - T_{1,1}\right), \end{displaymath}$

and for

,

$\begin{displaymath} T_{3,1} = {1 \over 15}\left(16 T_{2,2} - T_{2,1}\right). \end{displaymath}$

These terms can be arranged in a tableau:

$T_{1,1}$

$T_{1,2}$ $T_{2,1}$

$T_{1,3}$ $T_{2,2}$ $T_{3,1}$

$\vdots$ $\ddots$

$T_{1,l}$ $T_{2,l-1}$ $T_{3,l-2}$ $\cdots$ $T_{l,1}$

Where increasing points in the trapezoid rule goes down the left side, and increasing extrapolation goes along the diagonal.

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The translation was initiated by Tom Quinn on 2001-01-16

Tom Quinn 2001-01-16