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<br /> \text{Roots}, x_i &amp; \text{Coefficients}, c_i \\ \hline<br /> 0.8611363116 &amp; 0.3478548451 \\<br /> 0.339981436 &amp; 0.6521451549 \\<br /> -0.339981436 &amp; 0.6521451549 \\<br /> -0.8611363116 &amp; 0.3478548451 \\ \hline<br /> \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<br /> \text{Roots}, x_i &amp; \text{Coefficients}, c_i \\ \hline<br /> 0.8611363116 &amp; 0.3478548451 \\<br /> 0.339981436 &amp; 0.6521451549 \\<br /> -0.339981436 &amp; 0.6521451549 \\<br /> -0.8611363116 &amp; 0.3478548451 \\ \hline<br /> \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.