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

**Gauss Two point rule (Gauss-Legendre Two point rule) :
**

The two point rule is given by

where λ_{0} ≠ 0, λ_{1 }≠ 0 and x_{0} ≠ x1. The method has four unknowns λ_{0}, x_{0}, λ_{1}, x_{1}. Making the

formula exact for f(x) = 1, x, x^{2}, x^{3}

^{We get,}

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

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

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

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

Eliminating λ0 from (1.2) and (1.4), we get

Now, λ_{1 }≠ 0 and x_{0} ≠ x_{1}. Hence, x_{1} = 0, or x_{1} = – x_{0}. If x_{1} = 0, (1.2) gives x_{0} = 0, which is not possible. Therefore, x_{1} = – x_{0}.

Substituting in (1.2), we get

λ_{0} – λ_{1} = 0, or λ_{0} = λ_{1}.

Substituting in (1.3), we get

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

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

**Error of approximation**

The error term is obtained when f(x) = x^{4}. We obtain

The error term is given by

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