diff --git a/279-3.py b/279-3.py index f3b5fa6..4ddefb9 100644 --- a/279-3.py +++ b/279-3.py @@ -7,7 +7,7 @@ def ClassicRK(x0,y0,h,xk,fxy): k2 = fxy(x0+h/2,y0+h*k1/2) k3 = fxy(x0+h/2,y0+h*k2/2) k4 = fxy(x0+h,y0+h*k3) - y0 += h*(k1+4*k2+k3)/6 + y0 += h*(k1+2*k2+2*k3+k4)/6 x0 += h result.append((x0,y0)) return result diff --git a/按方法整理/常微分方程-经典龙格-库塔格式.py b/按方法整理/常微分方程-经典龙格-库塔格式.py new file mode 100644 index 0000000..fa3deef --- /dev/null +++ b/按方法整理/常微分方程-经典龙格-库塔格式.py @@ -0,0 +1,29 @@ +import math + +#经典R-K，龙格-库塔法 +def ClassicRK(x0,y0,h,xk,fxy): + k1=k2=k3=k4=0 + result = [(x0,y0)] + while x0<=xk: + k1 = fxy(x0,y0) + k2 = fxy(x0+h/2,y0+h*k1/2) + k3 = fxy(x0+h/2,y0+h*k2/2) + k4 = fxy(x0+h,y0+h*k3) + y0 += h*(k1+2*k2+2*k3+k4)/6 + x0 += h + result.append((x0,y0)) + return result + + +if __name__=="__main__": + ########################################################################## + x0 = 0 # x的初始值（左边界）换成题干里面的################# + y0 = -1 # y的初始值换成题干里面的################# + fxy = lambda x,y: x + y #f(x,y)换成题干里面的################# + real_fx = lambda x: -x-1 #真实函数换成题干里面的################# + h = 0.1 #步长换成题干里面的################# + xk = 2 #x的右边界换成题干里面的################# + result = ClassicRK(x0, y0, h, xk, fxy) + print("x\ty\t\treal_y\t\t\t误差") + for x, y in result: + print(f"{x:.2f}\t{y:.10f}\t\t{real_fx(x):.10f}\t\t\t{abs(y - real_fx(x)):.5f}") \ No newline at end of file diff --git a/按方法整理/矩阵-SOR逐次超松弛迭代法.py b/按方法整理/矩阵-SOR逐次超松弛迭代法.py new file mode 100644 index 0000000..80b59e9 --- /dev/null +++ b/按方法整理/矩阵-SOR逐次超松弛迭代法.py @@ -0,0 +1,62 @@ +import math + +#模 范数 +def Norm(x,v): + if len(x[0]) == 1: + if v == 1: + return sum([abs(i[0]) for i in x]) + elif v == 2: + return (sum([i[0]**2 for i in x]))**0.5 + elif v == float("inf"): + return max([abs(i[0]) for i in x]) + else: + if v == 1: + return max([sum([abs(x[i][j]) for i in range(len(x))]) for j in range(len(x[0]))]) + elif v == float("inf"): + return max([sum([abs(i) for i in x[j]]) for j in range(len(x))]) + return None + +#SOR方法 逐次超松弛迭代 +def SOR(A,b,x,w,err,N): + count = 0 + n = len(A) + while True: + count += 1 + for i in range(n): + sum1 = sum(A[i][j] * x[j] for j in range(i)) + sum2 = sum(A[i][j] * x[j] for j in range(i, n)) + x[i] += w*(b[i] - sum1 - sum2) / A[i][i] + r = [[b[i] - sum(A[i][j] * x[j] for j in range(len(A[0])))] for i in range(n)] + err_now = Norm(r, float("inf")) + x_t = [round(i,5) for i in x] + print(f"第{count}次迭代, 误差 = {err_now:.5}, x = {x_t}") + if err_now < err: + return x, count, 1 + + if count > N: + return None,count, 0 + + +if __name__ == "__main__": + ########################################################################## + #把矩阵改成题干的矩阵，b改成题干结果，err精度要求修改########################## + A = [ + [4,-1,0,-1,0,0], + [-1,4,-1,0,-1,0], + [0,-1,4,0,0,-1], + [-1,0,0,4,-1,0], + [0,-1,0,-1,4,-1], + [0,0,-1,0,-1,4] + ] + b = [2,3,2,2,1,2] + + x = [0,0, 0, 0, 0, 0] # 初始解 + err = 1e-5 # 精度要求 + w = 1 # 松弛因子,题干要求 P201 + x1,k,sta = SOR(A, b, x, w, err, 100) + print(f"w = {w}, 解为: {x1}, 迭代次数: {k}, 状态: {'收敛' if sta == 1 else '未收敛'}") + + w = 1.1 + x = [0, 0, 0, 0, 0, 0] + x2,k,sta = SOR(A, b, x, w, err, 100) + print(f"w = {w}, 解为: {x2}, 迭代次数: {k}, 状态: {'收敛' if sta == 1 else '未收敛'}") diff --git a/按方法整理/矩阵-模范数-点积.py b/按方法整理/矩阵-模数-范数-点积.py similarity index 100% rename from 按方法整理/矩阵-模范数-点积.py rename to 按方法整理/矩阵-模数-范数-点积.py diff --git a/按方法整理/矩阵-特征值-谱半径.py b/按方法整理/矩阵-特征值-谱半径.py new file mode 100644 index 0000000..322a7c3 --- /dev/null +++ b/按方法整理/矩阵-特征值-谱半径.py @@ -0,0 +1,94 @@ +import math + +#抛物线法解方程 +def MullerSolve(fx,x0,x1,x2,err1,err2,N): + count = 0 + f0 = fx(x0) + f1 = fx(x1) + f2 = fx(x2) + q = (x2 - x1) / (x1 - x0) + p = 0 + a = 0 + b = 0 + c = 0 + while True: + p = (x2 - x0) / (x1 - x0) + + a = q**2 * f0 - q*p*f1 + q*f2 + b = q**2 *f0 - p**2 *f1 + (p + q)*f2 + c = p*f2 + h1 = 0 + + if b.real < 0: + h1 = -2 * c / (b - (b**2 - 4*a*c)**0.5) + else: + h1 = -2 * c / (b + (b**2 - 4*a*c)**0.5) + x3 = x2 + h1 * (x2 - x1) + f3 = fx(x3) + k = err1 + 1 + if abs(f3) < 1: + k = abs(x3 - x2) + else: + k = abs(x3 - x2) / abs(f3) + + if abs(f3) < err2 or k < err1: + return x3, 1 + count += 1 + if count > N: + return None, 0 + x0 = x1 + x1 = x2 + x2 = x3 + f0 = f1 + f1 = f2 + f2 = f3 + q = h1 +#计算矩阵的行列式 +def Det(A): + if len(A) == 2: + return A[0][0] * A[1][1] - A[0][1] * A[1][0] + det = 0 + for c in range(len(A)): + sub_matrix = [row[:c] + row[c+1:] for row in A[1:]] + det += ((-1) ** c) * A[0][c] * Det(sub_matrix) + return det + + +if __name__ == "__main__": + ########################################################################## + #把矩阵换成题干的矩阵，通常建议自己再验算一遍，这个能不能算不一定######################### + A =[ + [1,0,1], + [2,2,1], + [-1,0,0] + ] + + lam = [] + + count = 0 + k = -100 + fx = lambda x: Det([[A[i][j] - x * (1 if i == j else 0) for j in range(len(A))] for i in range(len(A))]) + k = 10 + while len(lam) < len(A): + for i in range(-k,k): + re,sta = MullerSolve(fx, i, i + 1, i + 2, 1e-10, 1e-10, 100) + if sta == 1: + a = round(re.real, 9) + b = round(re.imag, 9) + re_t = complex(a, b) + if re_t not in lam: + if re_t.imag != 0: + lam.append(re_t) + lam.append(re_t.conjugate()) + else: + lam.append(a) + if len(lam) == len(A): + break + k *= 10 + p = abs(lam[0]) + for i in range(len(lam)): + print(f"λ{i+1} = {lam[i]}") + if abs(lam[i]) > p: + p = abs(lam[i]) + + print(f"谱半径 = {p:.3f}") \ No newline at end of file diff --git a/按方法整理/矩阵-迭代改善法.py b/按方法整理/矩阵-迭代改善法.py new file mode 100644 index 0000000..8a8737f --- /dev/null +++ b/按方法整理/矩阵-迭代改善法.py @@ -0,0 +1,102 @@ +import math + +#列主元高斯消元法 +def SovleRowMain(A,b,round_num=15): + ks = 0.00000001 + n = len(A) + if len(A[0]) != n: + print("A要为方阵") + return None, None, None, None + if len(b) != n: + print("b与A的行数不匹配") + return None, None, None, None + p = list(range(n)) + for i in range(n): + row_max = abs(A[i][i]) + row_max_index = i + for j in range(i + 1, n): + if abs(A[j][i]) > row_max: + row_max = abs(A[j][i]) + row_max_index = j + A[i], A[row_max_index] = A[row_max_index], A[i] + b[i], b[row_max_index] = b[row_max_index], b[i] + p[i], p[row_max_index] = p[row_max_index], p[i] + + if abs(A[i][i]) < ks: + print("A矩阵奇异，无法进行高斯消元") + return None, None, None, None + for j in range(i + 1, n): + m = round(A[j][i] / A[i][i],round_num) + A[j][i] = m + for k in range(i + 1, n): + A[j][k] -= round(m * A[i][k],round_num) + b[j] -= round(m * b[i],round_num) + + if abs(A[n - 1][n - 1]) < ks: + print("A矩阵奇异，无法进行高斯消元") + return None, None, None, None + + # 回代求解 + b[n - 1] = round(b[n - 1]/A[n - 1][n - 1],round_num) + for i in range(n - 2, -1, -1): + for j in range(i + 1, n): + b[i] -= round(A[i][j] * b[j],round_num) + b[i] /= round(A[i][i]) + b = [round(b[i], round_num) for i in range(n)] + # 得到L,U和P矩阵 + L = [[0 for i in range(n)] for j in range(n)] + U = [[0 for i in range(n)] for j in range(n)] + P = [[0 for i in range(n)] for j in range(n)] + for i in range(n): + for j in range(n): + if i == j: + L[i][j] = 1 + U[i][j] = A[i][j] + elif i < j: + U[i][j] = A[i][j] + else: + L[i][j] = A[i][j] + + P[i][p[i]] = 1 + return P,L,U,b + +#迭代改善法 +def IterativeMethod(A, b, err, N, fake_round_num=15): + b_c = [b[i] for i in range(len(b))] + A_c = [[A[i][j] for j in range(len(A[0]))] for i in range(len(A))] + P,L,U,x0 = SovleRowMain(A_c, b_c,fake_round_num) + print(L) + print(U) + print(f"初始解为: {x0}") + count = 0 + while count