# Gauss Three Point Rule (Gauss-legendre Three Point Rule)

**Gauss Three point rule (Gauss-Legendre Three point rule) :** The three point rule is given by

where λ_{0} ≠ 0, λ_{1} ≠ 0, λ_{2} ≠ 0, and x_{0} ≠ x_{1} ≠ x_{2}. The method has six unknowns λ_{0}, x_{0}, λ_{1}, x_{1}, λ_{2}, x_{2}.

Making the formula exact for f(x) = 1, x, x^{2}, x^{3}, x^{4}, x5, we get

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

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

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

....................1.4

........................1.5

.....................1.6

Solving this system as in the two point rule, we obtain

Therefore, the three point Gauss rule (Gauss-Legendre rule) is given by

...................1.7

**Error of approximation :**

The error term is obtained when f(x) = x^{6}. We obtain

The error term is given by

**Note :** Since the error term contains f^{ (6)}(ξ), Gauss three point rule integrates exactly polynomials of degree less than or equal to 5. Further, the error coefficient is very small (1/15750 ≈ 0.00006349). Therefore, the results obtained from this rule are very accurate. We have not derived any Newton-Cotes rule, which can be compared with the Gauss three 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 three point rule can then be applied to each of the integrals