legendreP

Legendre polynomials

Syntax

y = legendreP(n,x)

Description

y = 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.

y = legendreP(3,5.6)

y = 
430.6400

Find the Legendre polynomial of degree 2 at x.

syms x
y = legendreP(2,x)

y = 
$$\frac{3\hspace{0.17em}{x}^2}{2}-\frac{1}{2}$$

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
y = legendreP(n,x)

y = $$\mathrm{legendreP}\left(n,x\right)$$

Find Legendre Polynomial with Vector and Matrix Inputs

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

syms x
y = legendreP([1 2],x)

y = 
$$\left(\begin{array}{cc}x& \frac{3\hspace{0.17em}{x}^2}{2}-\frac{1}{2}\end{array}\right)$$

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];
yM = legendreP(n,xM)

yM = 
$$\left(\begin{array}{cc}\frac{3\hspace{0.17em}{x}^4}{2}-\frac{1}{2}& \frac{2519}{343}\\ -\frac{16}{5}& \frac{3\hspace{0.17em}{x}^2}{2}-\frac{1}{2}\end{array}\right)$$

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 $-\infty$.

syms x
P3 = legendreP(3,x);
P3 = limit(P3,x,-Inf)

P3 = $$-\infty$$

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

P5 = legendreP(5,x);
D3P5 = diff(P5,x,3)

D3P5 = 
$$\frac{945\hspace{0.17em}{x}^2}{2}-\frac{105}{2}$$

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
y = legendreP(2,x)

y = 
$$\frac{3\hspace{0.17em}{x}^2}{2}-\frac{1}{2}$$

t = taylor(y,x)

t = 
$$\frac{3\hspace{0.17em}{x}^2}{2}-\frac{1}{2}$$

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 = 
$$\left(\begin{array}{c}-0.94910791234275852452618968404785\\ -0.74153118559939443986386477328079\\ -0.40584515137739716690660641207696\\ 0\\ 0.40584515137739716690660641207696\\ 0.74153118559939443986386477328079\\ 0.94910791234275852452618968404785\end{array}\right)$$

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.

Output Arguments

y — Legendre polynomial

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

Legendre polynomial, returned 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