Révisions Centrale Maths II

Voici des propositions pour les exercices de préparation à cette mystérieuse épreuve de Maths II de Centrale...

Python
# Exo 1
import math
import numpy as np
import scipy as sp
import scipy.integrate as integ
import scipy.optimize as resol
 
def approche(a,n):
    f = lambda x: integ.quad(lambda t: t**n*np.exp(-t)/math.factorial(n), 0, x)[0] - a
    borne = 0
    while f(borne) < 0:
        borne +=  1
    gauche, droite = borne - 1, borne
    while (droite - gauche) > 0.01:
        milieu = gauche + (droite - gauche)*0.5
        (gauche, droite) = (gauche, milieu) if f(milieu) > 0 else (milieu, droite)
    return (gauche, droite)
 
 
# ou bien en utilisant fsolve :
def approche2(a,n):
    f = lambda x: integ.quad(lambda t: t**n*np.exp(-t)/math.factorial(n), 0, x)[0] - a
    return resol.fsolve(f,0)
 
#Exo 2
import scipy.integrate as integ
 
def p(x,n):
    poly_tmp = 1
    for k in range(n):
        poly_tmp *= (x - k) / (k + 1)
    return poly_tmp
 
 
def a(i):
    return integ.quad(lambda x: p(x,i), 0, 1)[0]
 
#Exo 3
import numpy as np
import numpy.linalg as nl
from math import sqrt
 
S = np.matrix([[-1,0],[0,1]])
T = np.matrix([[-1,1],[1,1]]) / sqrt(2)
R = S*T
 
L = [R**(k % 8)*S**(k // 8) for k in range(16)] 
 
def indice(N):
    i = 0
    M = S*N
    for i in range(16):
        if (abs(M - L[i]) < 1e-15).all():
            return i
    
def signature(G):
    P = 1
    for i in range(16):
        p = 1
        Gi = indice(G[i])
        for j in range(i+1,16):
            p *= (indice(G[j]) - Gi) / (j - i)
        P *= p
    return P
 
#Exo 4
import matplotlib.pyplot as plt
import numpy as np
import math
 
def pu(n):
    p = 1
    X, Y = [],[]
    for k in range(1,n+1):
        p *= 1 + 1j/k**2
        X.append(p.real)
        Y.append(p.imag)
    plt.plot(X,Y)
    
def u(n):
    p = 1
    for k in range(1,n+1):
        p *= 1 + 1j/k**2
    return abs(p), math.atan(p.imag / p.real)
    
 
def pv(n):
    p = 1
    X, Y = [],[]
    for k in range(1,n+1):
        p *= 1 + 2*1j/k
        X.append(p.real)
        Y.append(p.imag)
    plt.plot(X,Y)
    
def v(n):
    p = 1
    for k in range(1,n+1):
        p *= 1 + 2*1j/k
    return abs(p), math.atan(p.imag / p.real)
 
#Exo 5
import numpy.random as rnd
from collections import Counter
 
def tirage():
    return rnd.randint(1,15)
    
def lancers():
    return sum([0 if tirage() <= 9 else 1 for k in range(5)])
    
def tableau(n):
    return Counter([lancers() for k in range(n)])    
 
#Exo 6
import numpy.random as rnd
from collections import Counter
 
def course():
    lancer     = rnd.randint(1,7)
    nb_lancers = 1
    while (lancer < 6) and (nb_lancers < 5):
         lancer      = rnd.randint(1,7)
         nb_lancers += 1
    return 1 if (lancer < 6) and (nb_lancers == 5) else 0
    
def simulTortue(n):
    return Counter([course() for k in range(n)])
 
#Exo 8
import matplotlib.pyplot as plt
import numpy as np
 
 
def escargot(f,u0,xi,xf,n):
    X = np.linspace(xi,xf)
    plt.plot(X,X)
    plt.plot(X,f(X))
    Ordo = []
    u = u0
    for k in range(n):
        Ordo += [u,f(u)]
        u = f(u)
    plt.plot([u0] + Ordo, Ordo + [u])
 
#Exo 9
import numpy as np
import numpy.linalg as nl
from math import sqrt
 
A = np.matrix(np.array([20,12,-4,12,-4,-3,9,-5,-4,1,5,-5,-8,-10,6,-2]).reshape(4,4))
B = np.matrix(np.array([-12,-16,-8,-4,4,13,1,-1,4,5,9,-1,8,10,2,6]).reshape(4,4))
 
val_A, vec_A = nl.eig(A)
 
val_B, vec_B = nl.eig(B)
 
DA = nl.inv(vec_A) * A * vec_A
 
DB = nl.inv(vec_B) * B * vec_B

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