Gauss' Quadrature (Integration) – Formal

Covered chapter 19.3 from Numerical Methods for Scientists and Engineers, Hamming.

If having followed the previous post, which introduced the formulation by way of sample points as parameters technique, one will now have some understanding with respect to the ideas presented on this next topic. The ideas are really the same except generalized a bit more. Prerequisites here are also the same. There will be use of series summations included in this topic. As a refresher to series summation, I'd present the following series here and define a sequence alongside this.

Firstly, let's consider a simple sequence like  

a = {1, 2, 3, 4 , 5, 6, 7}

Here the series of a is written

or we can also express sequence of a as

where the sequence {a_i}is written

Now introducing Gauss quadrature integration. Let N points exist such that

  

where both the weights and the sample points are regarded as parameters.

For all w_k and x_k we have a total of N w_k x_k terms which means there are N+N = 2N parameters in total.

The defining equations in this case like the previous example posting are generalized with 2N defining equations:

 As in the previous posting we use a polynomial only it is composed of N factors as opposed to 2 meaning this is a polynomial of Nth degree

Remember before when we had a polynomial of 2nd degree, we had 2 coefficients c₀ and c₁, in this case we have N coefficients c₀, c₁, …, c_N-1. Notice the sequence case pattern here?

Procedurally we'd do much the same as we did in the previous postings which were to multiply each of the N coefficients c₀, c₁, …, c_N-1 respectively times each of the defining equations above, or in the generalized form, we multiply the j th equations by c_j and sum these equations which takes the form

noting here that x_k is a factor (or zero) in the polynomial π (x) and zeros the polynomial.

In the previous posting it were more or less demonstrated that the added c_j multiplied m_jth equations plus m_N th equations would lead to grouping of terms leading to a given w_k π(x_k) series. See the previous posting example for the case N = 2. As it turns out the same as generally true for all cases of N.

Procedurally like the previous posting, we shift the multipliers c_j down one line and repeat the process to get

Thus we have 2 equations of this form so far. We need a total of N equations of this form so we repeat the process of shifting our multipliers down to the k th case so that which results in N total equations of this form. Where the final equation in such equation generating sequence is

  

At the moment, will for go providing an example here, but it suffices to say we can use the same methods found solving the set of N linear equations for all coefficients c₀, c₁, …, c_N-1. You could apply solution techniques found in linear algebra to accomplish this.  Once having coefficients for the given N the degree polynomial one would have to determine factorization here.  Fortunately, there are methods for doing this, but this extends beyond typical high school and college algebra.  Here is a link to some factorization techniques however.  Probably wise to have a computer or computational aids to do much of this since much of this may require number crunching and extends beyond what one might want to do by hand in this case of higher degree polynomials.  Having this, we then reapply the same techniques to solve the the N generated equations for our given weights w₀, w₁, …, w_N-1.

Also we can apply this form of integration with respect to any function f(x).  Here we did so where f(x) = 1.