associated legendre functions matlab

In the function legendre(1,-0.7071), the value corresponding to P11(-0.7071) is coming wrong when checked with standard solutions. Matlab is giving the solution as -0.7071. whereas, the actual solution is +0.7071. Please have a look at it. Or please suggest me how to correct it.

One can verify using online calculator in the link. https://keisan.casio.com/exec/system/1287453184

[Rik]

The documentation states that the first order is P(x)=x. Do you have any mathematical proof this is incorrect?

[chaitanya acharya]

Yes. Analytical solution for P11(cos(theta))=-sin(theta). But, the solution in matlab is different.

[Rik]

Since the documentation states the first degree returns the input as output, that obviously doesn't match your analysis. If you show your analytical solution we might to see if there is a flaw in your reasoning, or if there is a flaw in the analysis by Mathworks engineers.

[chaitanya acharya]

For the condition when degree=1 order=0, P10(x)=x as can be seen in wikipedia. For this case, the matlab code works fine. The problem starts when order is Odd.

For example,

Lets consider the order=1 degree=1 associated legendre polynomial.

In that case, the solutions is given as P11(x)=-sqrt(1-x^2); as can be seen in wikipedia.

Wikipedia link : https://en.wikipedia.org/wiki/Associated_Legendre_polynomials

if x= cos(theta), P11(x)=-sin(theta).

But, matlab gives wrong solution in third and fourth quadrant i.e., when theta is varied between pi to 2*pi.

I tried the same with mathematica. There i found the same problem.

[Tomy Duby]

The issue is caused by two different definitions of associated Legendre polynomials:

• with non-negative integers.
• is the definition DLMF (Digital Library of Mathematical Functions) and Matlab are using.

The relation between the two definitions for real x is:

This relation is in the printed edition of Abramowitz and Stegun. I could not find it in DLMF.

I hope this helps.

TD

[David Goodmanson]

Hi Tony,

That's yet another reason why Abramowitz and Stegun is a better book than DLMF. There is a cornocopia of useful equations in A&S, and when they did DLMF you would think they would have supplemented those to make it even better. Instead they threw out a bunch of them and refer you to reference book blah blah blah if you want to find what you need. No excuse for that.

Answer 1

[David Goodmanson]

Hi chaitanya,

It's apples and oranges. When the domain of the argument is -1 <= x <= 1, the function is -sqrt(1-x^2). That's what Matlab is doing, and that's what it says it is doing. When the domain is opened up, 0 <= theta < 2pi with x = cos(theta), then the function can become -sin(theta). Both results are in Wikipedia.

[chaitanya acharya]

Run this code: you will understand where the problem is occuring.

theta=linspace(0,2*pi,9);
phi=0;

n=1; m=1;

% legendre calculated from matlab
leg1=legendre(1,cos(theta));
leg_matlab=leg1(m+1,:);

% legendre analytical solution for cos(theta) at n=1 and m=1 is -sin(theta)
leg_analytical=-sin(theta);

figure(1)
plot(leg_matlab)
hold on
plot(leg_analytical)