Numerical integration - Gauss Lobatto

[Sofie RK]

Homework Statement

I need calculate the points ($x_i$) and weights ($w_i$) with Gauss Lobatto seven points on the interval [a,b]. With the points and the weights I am going to approximate any integral at this interval.

Homework Equations

I have found the relevant points and weights at the interval [-1,1] using tables etc (https://www.math.ntnu.no/emner/TMA4125/2019v/notater/tabell_gauss.pdf)

The Attempt at a Solution

If I could adjust $x_i$ and $w_i$ from the interval [-1,1] to [a,b] with a linear transformation, I think the problem would be solved. But I have only found the values, and can't find a general formula for $x_i$ and $w_i$.

Thanks

 

[DrClaude]

The abscissae and weights are based on the Legendre polynomial. See 3.5 Gauss–Lobatto rules.

 

[Ray Vickson]

Homework Statement

I need calculate the points ($x_i$) and weights ($w_i$) with Gauss Lobatto seven points on the interval [a,b]. With the points and the weights I am going to approximate any integral at this interval.

Homework Equations

I have found the relevant points and weights at the interval [-1,1] using tables etc (https://www.math.ntnu.no/emner/TMA4125/2019v/notater/tabell_gauss.pdf)

The Attempt at a Solution

If I could adjust $x_i$ and $w_i$ from the interval [-1,1] to [a,b] with a linear transformation, I think the problem would be solved. But I have only found the values, and can't find a general formula for $x_i$ and $w_i$.

Thanks

Convert your integral $\int_a^b f(x) \, dx$ into $\int_{-1}^1 c f(r+cy) \, dy$. Apply Gauss-Lobato to the latter, then invert the transformation to get the correct formula for your original problem.