MATLAB: How does matlab compute legendreP(n,x)

interpolation

A bad way to compute P_n(x) for large n, with P_n the nth Legendre polynomial, is to compute all the coefficients of P_n and build up the polynomial. My question is, how does matlab's legendreP(n,x) function get round this? What is the algorithm it is using?

Best Answer

At the MATLAB level, command

mupad

When that window comes up, type into it

expose(orthpoly::legendre)

and you will be shown the source code for legendreP

I think it might be using the recurrence method mentioned on the legendreP page.

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MATLAB: Extracting the coefficient of a polynomials matrix

You could differentiate and evaluate P at x=0, so

P0 = subs(P, x, 0);
P1 = subs(diff(P), x, 0);
P2 = subs(diff(P,2)/2, x, 0);
P3 = subs(diff(P,3)/factorial(3), x, 0);
P4 = subs(diff(P,4)/factorial(4), x, 0);

would give you your matrices.

In general the nth polynomial matrix of P can be found with,

Pn = subs(diff(P,n)/factorial(n), x, 0);

MATLAB: Math Legendre problem

That's almost right, but wherever you had p(n) etc, you shouldn't adjust n. You should only have subtracted 1 from the instances of n that are on their own. So change all your p(??) bits back to their original values and it should work.

Best Answer

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