Legendre Polynomials Library

This library includes:

Gauss-Legendre Quadrature
Numerically stable, accurate and efficient calculation of an order n Legendre Polynomial at values -1<=x<=1
Numerically stable calculation of the derivative of an order n Legendre Polynomial at x.
Calculation of the zeros of Legendre polynomials.
Calculation of the Legendre polynomial coefficients.

Download

You can download the library here:

The library was created using Analytica 6.0, but doesn't use any recently introduced features, so is expected to work fine in any post 5.0 build (perhaps even earlier).

Gauss-Legendre Quadrature

Also called Gaussian Quadrature, is commonly used to compute definite integrals of a function F(x) with high accuracy using a small number of points where F(x) is evaluated. The method using n points is exact when F(x) is a polynomial of degree 2*n-1 or less, and is very accurate when F(x) can be accurately approximated by a polynomial of degree n (i.e., by its Taylor series with n terms).

Note that an example of Quadrature is included with the library.

Function Gauss_Quadrature_Pts(K, lb, ub)

The function used to compute a Gauss-Legendre quadrature. To use, supply and index, K:=1..n. The length of K is used for n, the number of points used. The function has two return values, both indexed by K:

The points where you should evaluate your function.
The weights

The bounds for the integration, «lb» and «ub» default to -1 to 1 if not specified. Both bounds much be finite. The following pattern illustrates how this function is used to integrate F(x) from a to b using n points:

Index K := 1..n;

Local (xi,wi) := Gauss_Quadrature_Pts(K, a, b) Do Sum( wi * F(xi), K )

An example is included with the library. In the example, Gaussian quadrature achieves 6 digits of precision with 12 points (i.e., evaluates of F(x)) and 12 digits with 17 points, whereas the trapezoidal rule has almost 4 digits of precision with 100 terms and the trend suggests it would take thousands of points to get to 6 digits of precision. Here is the function to be integrated in the example and the accuracy achieved as a function of the number of points used:

Evaluation of Legendre Polynomials

With the exception of very low degree, n, the direct evaluation of a Legendre polynomial [math]\displaystyle{ P_n(x) }[/math] using the polynomial coefficients is numerically unstable and results is low accuracy. Hence, when you need to evaluate a Legendre Polynomial, you should use the numerically stable, efficient and accurate evaluation function, LegendreP. For the first derivative, use LegendreP_deriv.

Function LegendreP(n,x)

Computes [math]\displaystyle{ P_n(x) }[/math], which is the nth order Legendre polynomial evaluated at x, where [math]\displaystyle{ -1\le x \le 1 }[/math] and n>=0.

Function LegendreP_deriv(n,x)

Computes [math]\displaystyle{ {{d P_n(x)}\over{d x}} }[/math], which is the derivative of the nth order Legendre polynomial evaluated at x, where [math]\displaystyle{ -1\le x \le 1 }[/math] and n>=0.

The Roots (zeros) of the Legendre Polynomial

The roots or zeros of a Legendre polynomial are the points where [math]\displaystyle{ P_n(x)=0 }[/math]. All polynomials for n>1 have n roots between -1 and 1.

Function Legendre_kth_root(n,k)

Computes the kth root of [math]\displaystyle{ P_n }[/math], where [math]\displaystyle{ 1\le k\le n }[/math].

Legendre polynomial coefficients

A Legendre polynomial of degree n is a polynomial of the form:

[math]\displaystyle{ P_n(x) = \sum_{k=0}^n c_k x^k }[/math]

The coefficients [math]\displaystyle{ c_k }[/math] are the coefficients.

Be aware that evaluation of polynomials using the coefficients for large n often has a lot of numeric round-off. It is better to use the LegendreP function to evaluate the polynomial at a point, especially for large n.

Function LegendreP_Coefs(K, n)

To obtain the numeric coefficients for the nth Legendre polynomial (n>=0), supply an index, K:=0..n. The n+1 coefficients are return indexed by K. The first item in the returned array is the constant. The 2nd item is the coefficient for [math]\displaystyle{ x }[/math]. The 3rd is the coefficient for [math]\displaystyle{ x^2 }[/math], etc. The values of the K index do not effect the returned result.

Legendre Library

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