def Newton_Cotes (f, a, b, n):
    In=0
    Coef = [0,[1./2,1./2], [1./6,4./6,1./6], [1./8,3./8,3./8,1./8], [7./90,32./90,12./90,32./90,7./90], [19./288,75./288,50./288,50./288,75./288,19./288], [41./840,216./840,27./840,272./840,27./840,216./840,41./840]]
    pas= (b-a) / n
    Wn= Coef[n]
    for k in range (n+1):
        In += Wn[k]*f(a+k*pas)
    In= In * (b-a)
    return In

def NewtC_Comp (f, a, b, n, r):
    if n > 6:
        print ("n trop grand")
    else:
        Itot=0
        pas= (b-a)/r
        X = [a+k*pas for k in range (r+1)]
        for k in range (r):
            Itot += Newton_Cotes(f,X[k],X[k+1],n)
        return Itot

f(x)= 1/(1+x**2)
n= Newton_Cotes(f,-10,10,6)
Iex=numerical_approx(arctan(10)-arctan(-10))
Iex

def NewtC_Comp1(f,a,b,r):
    Itot=0
    pas= (b-a)/r
    X = [a+k*pas for k in range (r+1)]
    for k in range (r):
        Itot += Newton_Cotes(f,X[k],X[k+1],1)
    return Itot

NewtC_Comp(f,-10,10,2, 10)

Pts=[[] for i in range (6)]
x=""
for j in range (1,7):
    for k in range (1,21):
        Iapprox=NewtC_Comp(f,-10,10,j,k)
        err=abs(numerical_approx(Iex-Iapprox))
        Pts[j-1].append((k,err))
line(Pts[0], color='red')+line(Pts[1],color='yellow')+line(Pts[2])+line(Pts[3])+line(Pts[4])+line(Pts[5],color='black')

def Legendre(n):
    L=[1,x]
    for k in range (2,n):
        Pk = ((2*k-1)*x*L[k-1] - (k-1)*L[k-2]) / (k)
        L.append(Pk)
    return L

def Racines(L,n):
    Racine=[]
    X = srange(-1,1.1,0.1)
    Intervalle=[]
    for i in range (len(X)-1):
        if (L[n](X[i]) * L[n](X[i+1])) < 0:
            Intervalle.append([X[i],X[i+1]])

    '''for j in range Intervalle:
        Racine.append(Racine_newt(L[n],i[0],10))'''


    for j in range (len(Intervalle)):
        Racine.append(Racine_Newt(L[n], (Intervalle[j][0]+Intervalle[j][1])/2, 10))
    return Racine

def Racine_Newt (fa, x0, m):
    dfa=diff(fa,x)
    phi(x)=x-fa(x)/dfa(x)
    xn=x0
    while abs(xn-phi(xn)) > 10**(-m):
        xn=phi(xn)
    return phi(xn)

n=4
L=Legendre(n)
XX=[]
for k in range(2,n):
    XX.append(Racines(L,k))
XX

def Poids(A):
    W=[]
    for n in range(4):
        Pn(x)=L[n+1]
        dPn=diff(Pn,x)
        W.append(1/(1-xk**2)*dPn(xk)**2)
    return W

def Gauss_L (f,a,b,n,W):
    for i in range(2,n):
        P += W[i]*f(xk)
    P= 2*P