103 lines
3.3 KiB
Python
103 lines
3.3 KiB
Python
import math
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#列主元高斯消元法
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def SovleRowMain(A,b,round_num=15):
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ks = 0.00000001
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n = len(A)
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if len(A[0]) != n:
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print("A要为方阵")
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return None, None, None, None
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if len(b) != n:
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print("b与A的行数不匹配")
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return None, None, None, None
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p = list(range(n))
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for i in range(n):
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row_max = abs(A[i][i])
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row_max_index = i
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for j in range(i + 1, n):
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if abs(A[j][i]) > row_max:
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row_max = abs(A[j][i])
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row_max_index = j
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A[i], A[row_max_index] = A[row_max_index], A[i]
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b[i], b[row_max_index] = b[row_max_index], b[i]
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p[i], p[row_max_index] = p[row_max_index], p[i]
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if abs(A[i][i]) < ks:
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print("A矩阵奇异，无法进行高斯消元")
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return None, None, None, None
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for j in range(i + 1, n):
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m = round(A[j][i] / A[i][i],round_num)
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A[j][i] = m
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for k in range(i + 1, n):
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A[j][k] -= round(m * A[i][k],round_num)
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b[j] -= round(m * b[i],round_num)
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if abs(A[n - 1][n - 1]) < ks:
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print("A矩阵奇异，无法进行高斯消元")
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return None, None, None, None
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# 回代求解
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b[n - 1] = round(b[n - 1]/A[n - 1][n - 1],round_num)
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for i in range(n - 2, -1, -1):
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for j in range(i + 1, n):
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b[i] -= round(A[i][j] * b[j],round_num)
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b[i] /= round(A[i][i])
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b = [round(b[i], round_num) for i in range(n)]
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# 得到L,U和P矩阵
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L = [[0 for i in range(n)] for j in range(n)]
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U = [[0 for i in range(n)] for j in range(n)]
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P = [[0 for i in range(n)] for j in range(n)]
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for i in range(n):
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for j in range(n):
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if i == j:
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L[i][j] = 1
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U[i][j] = A[i][j]
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elif i < j:
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U[i][j] = A[i][j]
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else:
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L[i][j] = A[i][j]
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P[i][p[i]] = 1
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return P,L,U,b
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#迭代改善法
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def IterativeMethod(A, b, err, N, fake_round_num=15):
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b_c = [b[i] for i in range(len(b))]
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A_c = [[A[i][j] for j in range(len(A[0]))] for i in range(len(A))]
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P,L,U,x0 = SovleRowMain(A_c, b_c,fake_round_num)
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print(L)
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print(U)
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print(f"初始解为: {x0}")
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count = 0
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while count<N:
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r1 = [b[i] - sum([A[i][j] * x0[j] for j in range(len(A[0]))]) for i in range(len(A))]
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A_c = [[A[i][j] for j in range(len(A[0]))] for i in range(len(A))]
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d1 = SovleRowMain(A_c, r1,fake_round_num)[3]
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x0 = [x0[i] + d1[i] for i in range(len(x0))]
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print(f"第{count+1}次迭代, r{count+1} = {r1}, d{count+1} = {d1}, x{count+2} = {x0}")
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err_now = max(abs(r1[i]) for i in range(len(r1)))
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count += 1
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if err_now < err:
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break
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return x0,count
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if __name__ == "__main__":
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##########################################################################
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#把矩阵改成题干的矩阵，b改成题干结果，err精度要求修改##########################
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A = [
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[51,82],
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[151/3,81]
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]
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b = [235,232]
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err = 1e-4 # 精度要求
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N = 1000 # 迭代次数上限
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fake_round_num = 4 # 模拟的四舍五入精度，根据题目情况或者凑过程修改
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x = IterativeMethod(A, b, err, N, fake_round_num)[0]
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print(f"解为: {x}")
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# 先求范数与逆矩阵（条件数）
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