legendreP

Legendre polynomials

Syntax

legendreP(n,x)

Description

legendreP(n,x) returns the nth degree Legendre polynomial at x.

Examples

Find Legendre Polynomials for Numeric and Symbolic Inputs

Find the Legendre polynomial of degree 3 at 5.6.

legendreP(3,5.6)

ans =
  430.6400

Find the Legendre polynomial of degree 2 at x.

syms x
legendreP(2,x)

ans =
(3*x^2)/2 - 1/2

If you do not specify a numerical value for the degree n, the legendreP function cannot find the explicit form of the polynomial and returns the function call.

syms n
legendreP(n,x)

ans =
legendreP(n, x)

Find Legendre Polynomial with Vector and Matrix Inputs

Find the Legendre polynomials of degrees 1 and 2 by setting n = [1 2].

syms x
legendreP([1 2],x)

ans =
[ x, (3*x^2)/2 - 1/2]

legendreP acts element-wise on n to return a vector with two elements.

If multiple inputs are specified as a vector, matrix, or multidimensional array, the inputs must be the same size. Find the Legendre polynomials where input arguments n and x are matrices.

n = [2 3; 1 2];
xM = [x^2 11/7; -3.2 -x];
legendreP(n,xM)

ans =
[ (3*x^4)/2 - 1/2,        2519/343]
[           -16/5, (3*x^2)/2 - 1/2]

legendreP acts element-wise on n and x to return a matrix of the same size as n and x.

Differentiate and Find Limits of Legendre Polynomials

Use limit to find the limit of a Legendre polynomial of degree 3 as x tends to -∞.

syms x
expr = legendreP(4,x);
limit(expr,x,-Inf)

ans =
Inf

Use diff to find the third derivative of the Legendre polynomial of degree 5.

syms n
expr = legendreP(5,x);
diff(expr,x,3)

ans =
(945*x^2)/2 - 105/2

Find Taylor Series Expansion of Legendre Polynomial

Use taylor to find the Taylor series expansion of the Legendre polynomial of degree 2 at x = 0.

syms x
expr = legendreP(2,x);
taylor(expr,x)

ans =
(3*x^2)/2 - 1/2

Plot Legendre Polynomials

Plot Legendre polynomials of orders 1 through 4.

syms x y
fplot(legendreP(1:4, x))
axis([-1.5 1.5 -1 1])
grid on

ylabel('P_n(x)')
title('Legendre polynomials of degrees 1 through 4')
legend('1','2','3','4','Location','best')

Find Roots of Legendre Polynomial

Use vpasolve to find the roots of the Legendre polynomial of degree 7.

syms x
roots = vpasolve(legendreP(7,x) == 0)

roots =
 -0.94910791234275852452618968404785
 -0.74153118559939443986386477328079
 -0.40584515137739716690660641207696
                                   0
  0.40584515137739716690660641207696
  0.74153118559939443986386477328079
  0.94910791234275852452618968404785

Input Arguments

n — Degree of polynomial

nonnegative number | vector | matrix | multidimensional array | symbolic number | symbolic vector | symbolic matrix | symbolic function | symbolic multidimensional array

Degree of polynomial, specified as a nonnegative number, vector, matrix, multidimensional array, or a symbolic number, vector, matrix, function, or multidimensional array. All elements of nonscalar inputs should be nonnegative integers or symbols.

x — Input

number | vector | matrix | multidimensional array | symbolic number | symbolic vector | symbolic matrix | symbolic function | symbolic multidimensional array

Input, specified as a number, vector, matrix, multidimensional array, or a symbolic number, vector, matrix, function, or multidimensional array.

More About

Legendre Polynomial

• The Legendre polynomials are defined as

  $$P\left(n,x\right)=\frac{1}{2^{n}n!}\frac{d^n}{d{x}^n}{\left(x^2-1\right)}^n.$$

• The Legendre polynomials satisfy the recursion formula

  $$\begin{array}{l}P\left(n,x\right)=\frac{2n-1}{n}xP\left(n-1,x\right)-\frac{n-1}{n}P\left(n-2,x\right),\\ \text{where}\\ P\left(0,x\right)=1\\ P\left(1,x\right)=x.\end{array}$$

• The Legendre polynomials are orthogonal on the interval [-1,1] with respect to the weight function w(x) = 1, where

  $${\displaystyle \underset{x=-1}{\overset{x=1}{\int }}P(n,x)P(m,x)dx}=\{\begin{array}{cc}0& \text{if }n\ne m\\ \frac{1}{n+1/2}& \text{if }n=m.\end{array}$$

• The relation with Gegenbauer polynomials G(n,a,x) is

  $$P\left(n,x\right)=G\left(n,\frac{1}{2},x\right).$$

• The relation with Jacobi polynomials P(n,a,b,x) is

  $$P\left(n,x\right)=P\left(n,0,0,x\right).$$

Version History

Introduced in R2014b