Use Python SciPy to compute the Rodrigues formula P_n(x) (Legendre polynomials)

Question

I'm trying to use Python to calculate the Rodrigues formula, P_n(x).

http://en.wikipedia.org/wiki/Rodrigues%27_formula

That is, I would like a function which takes into two input parameters, n and x, and returns the output of this formula.

However, I don't think SciPy has this function yet. SpiPy does offer a Legendre module:

http://docs.scipy.org/doc/numpy/reference/routines.polynomials.legendre.html

I don't think any of these is the Rodrigues formula. Am I wrong?

Is there a standard way SciPy offers to do this?

EDIT: I would like the input parameters to be arrays, not just single input values.

Answer 1

If you simply want P_n(x), then you can create a suitable object representing the P_n polynomial using scipy.special.legendre and call it with your values of x :

In [1]: from scipy.special import legendre
In [2]: n = 3
In [3]: Pn = legendre(n)
In [4]: Pn(2.5)
Out[4]: 35.3125        # P_3(2.5)

The object Pn is, in a sense, the "output" of the Rodrigues formula: it is a polynomial of the required order, which can be evaluated at a provided value of x . If you want a single function that takes n and x , you can use eval_legendre :

In [5]: from scipy.special import eval_legendre
In [6]: eval_legendre(3, 2.5)
Out[6]: 35.3125

As noted in the docs , this is the recommended way to do it for large-ish n (eg n > 20 ), instead of creating a polynomial object with all the coefficients which does not handle rounding errors and numerical stability as well.

EDIT: Both approaches work with arrays (at least for the x argument). For example:

In [7]: x = np.array([0, 1, 2, 5, 10])
In [8]: Pn(x)
Out[8]: 
array([  0.00000000e+00,   1.00000000e+00,   1.70000000e+01,
     3.05000000e+02,   2.48500000e+03])