Numerical integration of the syms type in MATLAB

Question

I am using a generating function of Legendre polynomials (no matter what math entity it is) and for that I need to use "syms" type and then take the derivative by "diff" function.

function p=Plm(l,m)
syms x    
p = diff((x^2-1)^l,m);
end

Then the output of this function, "p", should be used to take a definite integral as

fun = @(x) Plm(l,m).*cos(x).^2
integral(fun,-1,1)

where "l" and "m" could be replaced by any positive integer. Matlab gives the error that it cannot integrate over symbolic x that comes from "p". What is the way to get around this?

I am sure, this is because p is of type sym. Also, you differentiate p with respect to m that gives 0 because, p is not function of m. — brainkz
@AminR. brainkx is rigth. p is not dependant of m, thus it should be zero (it is, if you try) — Ander Biguri

Ander Biguri Ander Biguri · Accepted Answer · 2016-02-03T16:12:13

The problem is your definition of fun, as it doesn't return a numerical output. then you have a symbolic variable x and a numeric substitution of x (not the same) in the anonymous function. This makes things complicated, as the symbolic output of Plm returns a function that depends on a symbol that is not defined anymore, as it was created in the function workspace. This is a very bad programming practice, as in general, you wont be able to mix it with more symbolic x outside the function declaration

My suggestions would be either

include the cos inside Plm
Define Plm to accept a symbol defined outside the function scope, function p=Plm(l,m,x) being x the said symbol.

Else the following will work. The anonymous function must return a numerical value, so you should add the function subs and double to it as:

fun = @(xin)double(subs(Plm(l,m).*cos(x).^2,'x',xin))

The problem here is that the x in cos(x) is not a symbol, and if you substitute it by xin it will behave weirdly due to the behaviour of subs. However, we could hack the thing a bit, making a symbolic cos() using the variable that Plm depends on, by using symbar. This will force the anonymous function to evaluate Plm twice (which is not good).

In short, this works :

fun = @(xin)double(subs(Plm(l,m).*cos(symvar(Plm(l,m))).^2,xin));
integral(fun,-1,1);

Numerical integration of the syms type in MATLAB

1 Answers