Use case

It is used to improve the accuracy of a numerical method to a higher order by combining results from different step sizes. It does this by eliminating the lower-order error terms from the asymptotic error expansion.

Mathematically

For a numerical method with approximation of the exact value, where the error has convergence order :

Here,

  • is the convergence order of the original method
  • The new approximation has convergence order (or for methods with only even error terms)

Note: Each extrapolation step increases the convergence order. For integration methods like trapezoid rule, the order increases by 2 in each iteration as the expansion contains only even powers of .

Examples:

  • Trapezoid rule ():
  • Simpson’s rule ():

center

Quadrature MethodOrder of Convergence
Left Hand Rule
Midpoint Rule
Trapezoidal Rule
Simpson’s Rule

Derivation of the Romberg Extrapolation

Consider a numerical method with error expansion:

The dominant error term is (lowest power, largest contribution). Elimination Process With two approximations:

Step 1: Multiply the second equation by :

Step 2: Subtract the first equation:

Step 3: Solve for :

Result

The term cancels out completely, leaving only higher-order error terms . This is what we mean by “eliminating” the lower-order error term.

Example For trapezoid rule with :

  • Original error:
  • After extrapolation: (the term is eliminated)

The approximation becomes much more accurate because we’ve removed the largest source of error.