Informatique/SPE/IPT/TP0 METHODES NUMERIQUES/main.py

import numpy as np

class Question:
    def __init__(self,i=1, l=0, n=''):
        self.name=n
        self.level=l
        self.number=i
    def __enter__(self):
        print('\n' + (self.level*2)*' ' + f"-> {self.number}. : {self.name} -- Début")
        return self
    def __exit__(self, exc_type, exc_value, exc_traceback):
        print((self.level*2)*' ' + f"<- {self.number}. : {self.name} -- Fin\n")

if False:
    with Question(1):
        with Question(1,1):
            def dichotomie(f, n, a=0, b=2):
                iterations = 0
                while abs(b-a) >= 2*10**(-n):
                    iterations += 1
                    c = (a+b) / 2
                    if f(c) == 0:
                        return c
                    elif f(a)*f(c) > 0:
                        a = c
                    else:
                        b = c
                return c, iterations
            f = lambda x: x**3 - 3*x**2 + 1
            print(dichotomie(f,5,0,1.5))
            print(dichotomie(np.sin, 3, 3, 4))
            print(dichotomie(np.sin, 10, 3, 4))


        with Question(2,1):
            def newton(f, fp, x0, nbiter, epsilon=1e-10):
                for i in range(nbiter):
                    x1 = x0 - f(x0) / fp(x0)
                    if abs(x1 - x0) <= epsilon:
                        break
                    x0 = x1
                return x1
            print(newton(lambda x: x**3 - 3*x**2 + 1, lambda x: 3*x**2 - 6*x, 1.5, 200, epsilon=1e-100))
            print(newton(np.sin, np.cos, 3, 100))
            print(abs(newton(np.sin, np.cos, 3, 100) - dichotomie(np.sin, 10, 3, 4)[0]))


    with Question(2):
        from math import cos, exp, sin, log
        from scipy.integrate import quad
        from time import time
        with Question(1,2):
            def rectangles(f, a, b, n):
                res = 0
                for k in range(n):
                    res += f(a + k*(b - a)/n)
                return res*(b - a)/n
            def f1(x):
                return x
            def f2(x):
                return x**10
            f3 = cos
            f4 = exp
            print(rectangles(f1, 0, 1, 1000), quad(f1, 0, 1)[0])
            print(rectangles(f2, 0, 1, 1000), quad(f2, 0, 1)[0])
            print(rectangles(f3, 0, np.pi/2, 1000), quad(f3, 0, np.pi/2)[0])
            print(rectangles(f4, -3, 3, 1000), quad(f4, -3, 3)[0])


        with Question(2,2):
            def trapezes(f, a, b, n):
                res = 0
                for k in range(n):
                    res += f(a + k*(b - a)/n) + f(a + (k+1)*(b - a)/n)
                return res*(b - a)/(2*n)
            print(trapezes(f1, 0, 1, 1000), quad(f1, 0, 1)[0])
            print(trapezes(f2, 0, 1, 1000), quad(f2, 0, 1)[0])
            print(trapezes(f3, 0, np.pi/2, 1000), quad(f3, 0, np.pi/2)[0])
            print(trapezes(f4, -3, 3, 1000), quad(f4, -3, 3)[0])


        with Question(3, 2):
            def simpson(f, a, b, n):
                res = 0
                for k in range(n):
                    res += f(a + k*(b - a)/n) + 4*f(((a+k*(b - a)/n) + (a + (k+1)*(b - a)/n))/2) + f(a + (k+1)*(b - a)/n)
                return res*(b - a)/(6*n)
            print(simpson(f1, 0, 1, 1000), quad(f1, 0, 1)[0])
            print(simpson(f2, 0, 1, 1000), quad(f2, 0, 1)[0])
            print(simpson(f3, 0, np.pi/2, 1000), quad(f3, 0, np.pi/2)[0])
            print(simpson(f4, -3, 3, 1000), quad(f4, -3, 3)[0])


    with Question(3):
        import matplotlib.pyplot as plt
        with Question(1, 2):
            def euler(F, t0, y0, t1, n):
                res = [y0]*(n+1)
                for i in range(1,n+1):
                    res[i] = res[i-1] + (t1-t0)/n * F(t0 + i*(t1-t0)/n, res[i-1])
                return res
            def expo_euler(t, yt):
                return yt
            ne = 100
            n = 10
            plt.plot(np.linspace(0, 1, num=n+1), euler(expo_euler, 0, 1, 1, n), 'bo')
            plt.plot(np.linspace(0, 1, num=ne+1), np.exp(np.linspace(0, 1, num=ne+1)), 'r') 
            plt.show()


print("--- Calcul Matriciel ---\n\n")
with Question(1):
    with Question(2, 2):
        with Question(1,3):
            def matrice_nulle(n, m):
                return [[0 for _ in range(m)] for _ in range(n)]
        with Question(2,3):
            def dimension(A):
                if len(A):
                    return len(A),len(A[0])
                else:
                    return 0
        with Question(3,3):
            def addition(A, B):
                n,m = dimension(A)
                return [[A[i][j] + B[i][j] for j in range(m)] for i in range(n)]
            A3=[[1,2,3],
                [0,1,0],
                [-1,2,-6]]
            B3=[[-1,5,0],
                [1,0,-2],
                [1,2,3]]
            print(addition(A3, B3), '\n', np.array(A3)+np.array(B3))
            print("Le résultat pour cette exemple est le même.")
        with Question(4,3):
            def transposee(A):
                n,m = dimension(A)
                return [[A[j][i]for j in range(n)] for i in range(m)]
            print(transposee([[1,2,3],
                              [4,5,6]]))
            print(list(map(list,zip(*[[1,2,3],[4,5,6]])))) # Parce que c'est amusant de chercher à faire une version courte.
        with Question(5,3):
            def multiple(A, coef):
                n,m = dimension(A)
                return [[coef * A[i][j] for j in range(m)] for i in range(n)]
            print(multiple([[1,2],
                            [3,4]], 2))
        with Question(6,3):
            def multiplication(A, B):
                n,p = dimension(A)
                _,m = dimension(B)
                return [[sum(A[i][k] * B[k][j] for k in range(p))for j in range(m)] for i in range(n)]
            A6 = [[1,2,3],
                  [4,5,6]]
            B6 = [[3,6],
                  [2,5],
                  [1,4]]
            print(multiplication(A6,B6))
        with Question(7,3):
            def puissance(A, n):
                _,m = dimension(A)
                res = [[1 if j==i else 0 for j in range(m)] for i in range(m)]
                for _ in range(0, n):
                    print(n)
                    res = multiplication(res, A)
                return res
            print(puissance([[0,1],
                             [0,0]], 2))
    with Question(3, 2):
            with Question(2,3):
                with Question(1,4):
                    def echange_ligne(A, i, j):
                        A[i], A[j] = A[j], A[i]
                    A1 = [[1,2,3],
                          [4,5,6]]
                    echange_ligne(A1, 0, 1)
                    print(A1)
                with Question(2,4):
                    def transvection(A, i, j, mu):
                        A[i] = addition([A[i]], multiple([A[j]], mu))[0]
                    transvection(A1, 0, 1, -1)
                    print(A1)
                with Question(3, 4):
                    def pivot_partiel(A, j0):
                        n, m = dimension(A)
                        if n<j0 or m<j0:
                            raise IndexError("j0 trop grand")
                        maxi = A[j0][j0]
                        for i in range(j0, n):
                            maxi = max(A[i][j0], maxi)
                        return maxi
                    print(pivot_partiel([[1,2,3],
                                         [4,5,6],
                                         [7,3,9]], 1))
                with Question(4,4):
                    def devient_triangle(A):
                        pass