修改龙格库塔bug

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<N:
+        r1 = [b[i] - sum([A[i][j] * x0[j] for j in range(len(A[0]))]) for i in range(len(A))]
+        A_c = [[A[i][j] for j in range(len(A[0]))] for i in range(len(A))]
+        d1 = SovleRowMain(A_c, r1,fake_round_num)[3]
+        x0 = [x0[i] + d1[i] for i in range(len(x0))]
+        print(f"第{count+1}次迭代, r{count+1} = {r1}, d{count+1} = {d1}, x{count+2} = {x0}")
+        err_now = max(abs(r1[i]) for i in range(len(r1)))
+        count += 1
+        if err_now < err:
+            break
+    return x0,count
+
+
+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 = [
+        [51,82],
+        [151/3,81]
+    ]
+    b = [235,232]
+
+    err = 1e-4    # 精度要求
+    N = 1000      # 迭代次数上限
+    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
+
+    x = IterativeMethod(A, b, err, N, fake_round_num)[0]
+    print(f"解为: {x}")
+
+# 先求范数与逆矩阵（条件数）
+