Welcome back to a new post at thoughts-on-cpp.com. In this post, I would like to discuss the Gauss integration algorithm, more precisely the Gauss-Legendre integration algorithm. The Gauss-Legendre integration is the most known form of the Gauss integrations. Others are

- Gauss-Tschebyschow
- Gauss-Hermite
- Gauss-Laguerre
- Gauss-Lobatto
- Gauss-Kronrod

The idea of the Gauss integration algorithm is to approximate, similar to the Simpson Rule, the function *f(x)* by

While *w(x)* is a weighting function, is a polynomial function (Legendre-Polynomials) with defined nodes which can be exactly integrated. A general form for a range of *a-b* looks like the following.

The Legendre-Polynomials are defined by the general formula and its derivative

The following image is showing the 3rd until the 7th Legendre Polynomials, the 1st and 2nd polynomials are just *1* and *x* and therefore not necessary to show.

Let’s have a closer look at the source code:

The integral is done by the `gaussLegendreIntegral`

(line 69) function which is initializing the `LegendrePolynomial`

class and afterward solving the integral (line 77 – 80). Something very interesting to note: We need to calculate the Legendre-Polynomials only once and can use them for any function of order *n* in the range *a-b*. The Gauss-Legendre integration is therefore extremely fast for all subsequent integrations.

The method `calculatePolynomialValueAndDerivative`

is calculating the value (line 50) at a certain node and its derivative (line 51). Both results are used at method `calculateWeightAndRoot`

to calculate the the node by the Newton-Raphson method (line 33 – 37).

The weight *w(x)* will be calculated (line 40) by

As we can see in the screen capture below, the resulting approximation of

is very accurate. We end up with an error of only . Gauss-Legendre integration works very good for integrating smooth functions and result in higher accuracy with the same number of nodes compared to Newton-Cotes Integration. A drawback of Gauss-Legendre integration might be the performance in case of dynamic integration where the number of nodes are changing.

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