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*]

## No comments:

## Post a Comment