Zeros of the Legendre Polynomials

Legendre functions (Legendre polynomials) are the solutions of the following linear Ordinary Differential Equation (ODE), \[
\frac{\mathrm{d}}{d x}\left[ (1-x^2) \frac{\mathrm{d}}{\mathrm{d}x} P_n(x) \right] + n(n+1)P_n(x) = 0
\]

The Legendre function $P_n(x)$ is a degree $n$ polynomial. It has $n$ distinct roots within (-1,1), $x_{n,k}, k=1,2,\cdots n$, $x_{n,1} < x_{n,2} < \cdots < x_{n,n}$. There is no analytical solution for high order Legendre polynomials. It is useful to know the analytical bounds of the roots.

According to Bruns, Markoff, Stieltjes and Szego [1], the roots satisfy the following inequality, \[
\cos\frac{(n-k+1)\pi}{n+1} < x_{n,k} < \cos\frac{(n-k+\frac{3}{4})\pi}{n+\frac{1}{2}}
 \] for $k = 1, \cdots \left[\frac{n}{2}\right]$. For the other half, recall $x_{n,n-k+1} = -x_{n,k}$.

It may also be useful to give a pair of global bounds, \[
\cos\frac{(n-k+\frac{1}{2})\pi}{n+\frac{1}{2}} < x_{n,k} < \cos\frac{(n-k+1)\pi}{n+\frac{1}{2}}
\] for $k=1,2,\cdots n$, although this is a coarse one.

The roots of the Legendre polynomials also admit asymptotic expansion due to Francesco Tricomi [2]. Let $\theta_{n,k} = \frac{n-k+3/4}{n+1/2}\pi$. Then the $k$-th root (in ascent order) is \[
x_{n,k} = \left\{1 - \frac{1}{8n^2} + \frac{1}{8n^3} - \frac{1}{384n^4}\left( 39 - \frac{28}{\sin^2\theta_{n,k}} \right) + \mathcal{O}(n^{-5})\right\} \cos\theta_{n,k}, \] which can be improve by Newton's algorithm $x_{\nu+1} = x_{\nu} - \frac{1-x_\nu^2}{n} \frac{P_n(x_\nu)}{P_{n-1}(x_\nu) -x_\nu P_n(x_\nu)}$.

Another approximation due to Gatteschi and Pittaluga [3] is \[
x_{n,k} = \cos\left\{ \theta_{n,k} + \frac{1}{16(n+1/2)^2}(\cot\frac{1}{2}\theta_{n,k} - \tan\frac{1}{2}\theta_{n,k}) \right\} + \mathcal{O}(n^{-4}).
\]

Let $x_{n,k}$ be the $k$-th root of $P_n(x)$, $\xi_{n,k}$ be its approximations.

The $x_{n,k} \simeq \cos\theta_{n,k}$ is a pretty good estimator of the location $k$ of Gaussian nodes from $x_{n,k}$.

Last but not least, the zeros are the nodes of the Gaussian quadrature. \[
 \int_{-1}^{+1} \mathrm{d}x f(x) \simeq \sum_{k=1}^n w_{n,k} f(x_{n,k})
\] where $w_{n,k} = \frac{2}{(1-x_{n,k}^2) \left[ P'_n(x_{n,k}) \right]^2}$ are called the Gaussian weights. Gaussian quadrature can be illustrated by the bellow figure ($n = 16$).

References:

[1] Gabriel Szego, Inequalities for the Zeros of Legendre Polynomials and Related Functions, Transactions of the American Mathematical Society, Vol. 39, No. 1 (1936)

[2] F. G. Tricomi, Sugli zeri dei polinomi sferici ed ultrasferici, Ann. Mat. Pura Appl., 31 (1950), pp. 93–97; F.G. Lether and P.R. Wenston, Minimax approximations to the zeros of $P_n(x)$ and Gauss-Legendre quadrature, Journal of Computational and Applied Mathematics Volume 59, Issue 2, 19 May 1995, Pages 245–252

[3] L. Gatteschi and G. Pittaluga, An asymptotic expansion for the zeros of Jacobi polynomials, in Mathematical Analysis. Teubner-Texte Math., 79 (1985), pp. 70–86

Source codes:

John Burkardt (GNU LGPL): http://people.sc.fsu.edu/~jburkardt/cpp_src/quadrule/quadrule.html

pomax.github.io: http://pomax.github.io/bezierinfo/legendre-gauss.html

rosettacode: http://rosettacode.org/wiki/Numerical_integration/Gauss-Legendre_Quadrature

Mathematica:
Needs["NumericalDifferentialEquationAnalysis`"]

GaussianQuadratureWeights[n, a, b, prec]