Python: piecewise function integration

Question

I want to integrate a piecewise defined function that is multiplied by Legendre polynomials. Unfortunately, I cannot find how to use the nth Legendre polynomial x in the documentation . I want to integrate every Legendre polynomial from x when n = 1,..., 50, so I set n = np.arange(1, 51, 1).

import numpy as np
import pylab
from scipy import integrate

n = np.arange(1, 51, 1)                                                   


def f(x):
    if 0 <= x <= 1:
        return 1
    if -1 <= x <= 0:
        return -1

I suppose I need to define another function, say u(x).

c = []


def u(x):
    c.append((2. * n + 1) / 2. * integrate.quad(f(x) * insert Legendre polynomials here, -1., 1.)) 
    return sum(c * Legendre poly, for nn in range(1, 51)) 

So, I would then return some u(x)with the first 50 members extending my piecewise function to Legendre polynomials.

Change 1:

If this is not done, I could use the Rodrigas Formula to calculate the nth Legendre polynomial. However, I could not find anything useful when I was looking for calculations of the nth derivative in Python.

P_n(x) = \frac{1}{2^n n!}\frac{d^n}{dx^n}(x^2 - 1)^n

, , - , ​​ Python.

2:

, :

import numpy as np
from scipy.integrate import quad

def f(x, coef):
    global p
    p = np.polynomial.legendre.Legendre(coef=coef)
    if 0 <= x <= 1:
        return 1*p(x)
    if -1 <= x <= 0:
        return -1*p(x)

c = []
for n in range(1, 51):
    c.append((2. * n + 1.) / 2. * quad(f, -1, 1, args=range(1,n+1))[0])

def g(x)
    return sum(c * p(x) for n in range(1, 51))

, c, . 1.5, 0, -7/8, 0, ...

, g, x = np.linspace(-1, 1, 500000), , c - 50. ?

Answer 1

, f (x) * Ln (x), f (x) - , python. , .

, legval .

import numpy as np
import matplotlib

x = np.linspace(-1, 1, 201)

L = np.polynomial.legendre.legval(x, np.identity(50))

plt.plot(x, L.T)

. , legendre Ln (x), n .

import numpy as np    
from numpy.polynomial.legendre import leggauss, legval

def f(x):
    if 0 <= x <= 1:
        return 1
    if -1 <= x <= 0:
        return -1

# of course you could write a vectorized version of
# this particular f(x), but I assume you have a more
# general piecewise function
f = np.vectorize(f)

deg = 100
x, w = leggauss(deg) # len(x) == 100

L = np.polynomial.legendre.legval(x, np.identity(deg))
# Sum L(xi)*f(xi)*wi
integral = (L*(f(x)*w)[None,:]).sum(axis=1)

c = (np.arange(1,51) + 0.5) * integral[1:51]

x_fine = np.linspace(-1, 1, 2001) # 2001 points
Lfine = np.polynomial.legendre.legval(x_fine, np.identity(51))

# sum_1_50 of c(n) * Ln(x_fine)
cLn_sum = (c[:,None] * Lfine[1:51,:]).sum(axis=0)

c = 1.5, 0, -8.75e-1, 0,... , , , .

Answer 2

, :

import numpy as np
from scipy.integrate import quad

def f(x, coef):
    n = coef[-1]
    p = (2*n+1)/2.*np.polynomial.legendre.Legendre(coef=coef)
    if 0 <= x <= 1:
        return 1*p(x)
    if -1 <= x <= 0:
        return -1*p(x)

c = []
for n in range(1, 51):
    c.append(quad(f, -1, 1, args=range(1,n+1))[0])

:

print c
#[0.0, 5.0, 6.9999999999999991, 4.5, ... , 62.975635570615466, 64.274102283412574, 77.143785770271251]