Numerical Methods with Programming

Section 15.3 Laguerre polynomials

In an earlier homework we found that finding integrals of the form \(\int_0^\infty f(x)\,dx\) is hard: we had to pick some large upper limit \(b\) (mostly by guessing) to replace the infinite upper limit. Otherwise, there was no way to choose evaluation points by placing them at equal distances throughout the interval.

But now we have a way of choosing evaluation points differently, as roots of orthogonal polynomials. So we can try to integrate on an infinite interval by using orthogonal polynomials on such an interval.

However, on an infinite interval we cannot define the inner product of polynomials in the sense of the integral \(\int_0^\infty fg\text{,}\) the integral will diverge. Instead we define dot product as \(\int_0^\infty f(x)g(x)e^{-x}\,dx\text{,}\) introducing a weight \(e^{-x}\) which allows the integral to converge. This way we can build orthogonal polynomials: for example, \(1\) and \(1-x\) are orthogonal (check).

The Laguerre polynomial of degree \(n\text{,}\) denoted \(L_n\text{,}\) is the unique \(n\)-th degree polynomial such that \(\int_0^\infty x^k L_n(x)e^{-x}\,dx = 0\) for \(k =0, 1, \dots, n-1\text{,}\) and \(L_n(0)=1\text{.}\) Note that \(\int_0^\infty Q(x)P_n(x)e^{-x}\,dx = 0\) for every polynomial \(Q\) with degree \(\deg Q< n\text{.}\)

Similarly to Legendre polynomials, the Laguerre polynomials have a recursive formula:

which can be used to compute them starting with \(L_0(x)=1\) and \(L_1(x)=1-x\text{.}\)