Gauss One Point Rule (Gauss-legendre One Point Rule)

Gauss one point rule (Gauss-Legendre one point rule) :

The one point rule is given by

......................1.1

where λ0 ≠ 0. The method has two unknowns λ₀, x₀. Making the formula exact for f(x) = 1, x

we get,

Since, λ₀ ≠ 0, we get x₀ = 0.

Therefore, the one point Gauss formula is given by

...................1.2

Error of approximation

The error term is obtained when f(x) = x². We obtain

The error term is given by.

...................1.3

Note : Since the error term contains f ′′(ξ) , Gauss one point rule integrates exactly polynomials of degree less than or equal to 1. Therefore, the results obtained from this rule are comparable with the results obtained from the trapezium rule. However, we require two function evaluations in the trapezium rule whereas we need only one function evaluation in the Gauss one point rule. If better accuracy is required, then the original interval [a, b] can be subdivided and the limits of each subinterval can be transformed to [– 1, 1]. Gauss one point rule can then be applied to each of the integrals.