Four-point Gaussian quadrature rule

[Heimisson]

I need to use the four-point Gaussian quadrature rule to do some intense numerical calculations. Could anyone link to this page where it's written out explicitly over an [a,b] interval. I haven't been able to find it, I'm trying to derive it now but it's crucial that I'm 100% correct. I haven't used a Gaussian quadrature before so seeing clearly what it should look like would make me feel a lot better.

This isn't really homework but a part of a much larger project I'm working on for school so you will not be doing my homework for my or anything like that.

thanks

 

[Kreizhn]

This is for single-integral calculations? Also, are you looking for the theoretical derivation or the pseudo-code?

 

[Kreizhn]

This book gives a decent derivation:

Numerical Analysis, 8th Edition, Burden and Faires, Brooks/Cole 2005, 978-0534382162

This book has a list of the coefficients and roots tabulated for general n-point quadratures:

Gaussian Quadrature Formulas, Stroud and Secrest, Prentice Hall 1966,

 

[Heimisson]

This is for single-integral calculations? Also, are you looking for the theoretical derivation or the pseudo-code?

I need to use this on a integral of two dimensions x and x'. But I figured that I would just use it first on x and then look at that result as a function of x' and use it again. I don't really care for a theoretical derivation I just want the formula. I'm trying to calculate the kernel of a integro-differential equation. This means heavy numerical calculations so my professor recommended this rule because it uses the fewest points with best accuracy. But I' very new to it.

 

[Kreizhn]

Suppose that $x_1,\ldots, x_n$ are the roots of the nth Legendre polynomial, and that for each $i=1,2,\ldots, n$ we define $c_i$ by
$$c_i = \int_{-1}^1 \prod_{j=1, j\neq i}^n \frac{ x- x_j}{x_i-x_j} dx$$
If P(x) is any polynomial of degree less than 2n then
$$\int_{-1}^1 P(x) dx = \sum_{i=1}^n c_i P(x_i)$$.

This gives a (2n)th order approximation to the integral of any function. You can look up the roots of the nth Legendre polynomial (or calculate them computationally if you like). The c_i can then easily be calculated, but again, there are extensive tables with this information already computed. Hence

$$\int_{-1}^1 f(x) dx \approx \sum_{i=1}^n c_i f(x_i)$$

To make the algorithm work between [a,b] rather than just [-1,1], apply the transformation
$$t = \frac{2x-a-b}{b-a}, \qquad x = \frac12[(b-a)t + a + b]$$
and calculate
$$\int_a^b f(x) dx = \int_{-1}^1 f\left( \frac{(b-a)t + (b+a) }{2} \right) \frac{b-a}2 dt$$

In the particular case when you want a 4 point approximation, the following is the table of the values (you can get better values by computing them using the formula I've given above)

$$\begin{array}{|r|r|} \hline \text{Roots}, x_i & \text{Coefficients}, c_i \\ \hline 0.8611363116 & 0.3478548451 \\ 0.339981436 & 0.6521451549 \\ -0.339981436 & 0.6521451549 \\ -0.8611363116 & 0.3478548451 \\ \hline \end{array}$$

 

[Heimisson]

Suppose that $x_1,\ldots, x_n$ are the roots of the nth Legendre polynomial, and that for each $i=1,2,\ldots, n$ we define $c_i$ by
$$c_i = \int_{-1}^1 \prod_{j=1, j\neq i}^n \frac{ x- x_j}{x_i-x_j} dx$$
If P(x) is any polynomial of degree less than 2n then
$$\int_{-1}^1 P(x) dx = \sum_{i=1}^n c_i P(x_i)$$.

This gives a (2n)th order approximation to the integral of any function. You can look up the roots of the nth Legendre polynomial (or calculate them computationally if you like). The c_i can then easily be calculated, but again, there are extensive tables with this information already computed. Hence

$$\int_{-1}^1 f(x) dx \approx \sum_{i=1}^n c_i f(x_i)$$

To make the algorithm work between [a,b] rather than just [-1,1], apply the transformation
$$t = \frac{2x-a-b}{b-a}, \qquad x = \frac12[(b-a)t + a + b]$$
and calculate
$$\int_a^b f(x) dx = \int_{-1}^1 f\left( \frac{(b-a)t + (b+a) }{2} \right) \frac{b-a}2 dt$$

In the particular case when you want a 4 point approximation, the following is the table of the values (you can get better values by computing them using the formula I've given above)

$$\begin{array}{|r|r|} \hline \text{Roots}, x_i & \text{Coefficients}, c_i \\ \hline 0.8611363116 & 0.3478548451 \\ 0.339981436 & 0.6521451549 \\ -0.339981436 & 0.6521451549 \\ -0.8611363116 & 0.3478548451 \\ \hline \end{array}$$

Thank you so much!

This is a cool method of doing numerical integrations, I wonder why they didn't teach this in my numerical analysis class.