4. Romberg Integration Romberg integration uses the composite ...

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4. NUMERICAL INTEGRATION AND DIFFERENTIATION

4. Romberg Integration Romberg integration uses the composite trapezoidal rule for preliminary approximations and then uses Richardson extrapolation for improvements. Richardson’s Extrapolation Richardson’s Extrapolation is used to generate high-accuracy results while using low-order formulas such as the trapezoidal rule. It can be applied whenever it is known that an approximation technique has an error term with a predictable form, one that depends on a parameter, usually a step size h. Suppose for all h 6= 0, (#1)

M=

N1(h) | {z }

1st approximation

i.e.

2 +K + K3h3 + · ·}·, | 1h + K2h {z error of O(h)

N1(h) ⇡ K1(h)

M

h unless there is a large variation among the constants. Using instead of h, we 2 get ⇣h⌘ ⇣h⌘ ⇣ h ⌘2 ⇣ h ⌘3 (#2) M = N1 + K1 + K2 + K3 + ··· . 2 2 2 2 Then 2(#2) (#1) gives h ⇣h⌘ ⇣ ⇣h⌘ ⌘i ⇣ h2 ⌘ ⇣ h3 ⌘ 2 3 M = N1 + N1 N1(h) + K2 h + K3 h + ··· 2 2 2 4 | {z } | {z } N2 (h)

(#3) M =

N2(h) | {z }

2nd approximation

⇣K

⌘ 3K3 3 h + h + ··· . 2 4 | {z } 2 2

error of O(h2 )

error of O(h2 )