Numerical Methods

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 λ0 ≠ 0, λ1 ≠ 0 and x0 ≠ x1. The method has four unknowns λ0, x0, λ1, x1. Making the
formula exact for f(x) = 1, x, x2, x3

We get,

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

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

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

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

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

 

Now, λ1 ≠ 0 and x0 ≠ x1. Hence, x1 = 0, or x1 = – x0. If x1 = 0, (1.2) gives x0 = 0, which is not possible. Therefore, x1 = – x0.

Substituting in (1.2), we get 

λ0 – λ1 = 0, or λ0 = λ1.

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) = x4. We obtain

The error term is given by

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