Romberg integration

Richardson extrapolation, applied repeatedly to a table of composite trapezoidal-rule estimates at doubling panel counts, to accelerate convergence dramatically.

Romberg integration builds a triangular table $R(k,j)$ whose first column $R(k,0)$ is the composite trapezoidal rule at $n=2^k$ panels, and whose later columns apply Richardson extrapolation ( $p=2j$ ) to the column before it: $R(k,j)=\dfrac{4^jR(k,j-1)-R(k-1,j-1)}{4^j-1}$ . It is Richardson extrapolation (Chapter 5) applied without modification to the composite trapezoidal rule's own pure-even-power error expansion; the table's diagonal converges to the true integral far faster than its first column alone.

Examples

$R(1,1)$ , built from $R(1,0)$ and $R(0,0)$ (trapezoidal at 2 and 1 panels), already equals Simpson's rule on the same interval -- Romberg's second column reproduces Simpson for free.