gai

按方法整理/矩阵-G-S得G.py

@@ -0,0 +1,91 @@
+#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
+
+# 计算矩阵的点积
+def Dot(A,B):
+    if len(A[0]) != len(B):
+        return None
+    return [[sum([A[i][j] * B[j][k] for j in range(len(A[0]))]) for k in range(len(B[0]))] for i in range(len(A))]
+
+# 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
+
+# 计算矩阵的逆矩阵
+def Inverse(A):
+    n = len(A)
+    # 计算代数余子式矩阵
+    B = [[0 for i in range(n)] for j in range(n)]
+    for i in range(n):
+        for j in range(n):
+            minor = [row[:j] + row[j+1:] for row in (A[:i] + A[i+1:])]
+            B[j][i] = ((-1) ** (i + j)) * sum(minor[k][l] * (-1) ** (k + l) for k in range(n - 1) for l in range(n - 1))
+    det = Det(A)
+    print("det(A):",det)
+    if det == 0:
+        print("矩阵不可逆")
+        return None
+    A_inv = [[B[i][j] / det for j in range(n)] for i in range(n)]
+    return A_inv
+
+def FixInv(A):
+    n = len(A)
+    B = [[0 for i in range(n)] for j in range(n)]
+    for i in range(n):
+        for j in range(n):
+            t = [row[:j] + row[j+1:] for row in (A[:i] + A[i+1:])]
+            B[j][i] = ((-1) ** (i + j)) * Det(t)
+    det = Det(A)
+    if det == 0:
+        print("矩阵不可逆")
+        return None
+    A_inv = [[B[i][j] / det for j in range(n)] for i in range(n)]
+    return A_inv
+
+
+
+if __name__ == "__main__":
+    ##########################################################################
+    # 把矩阵换成题干的矩阵 #########################
+    A = [
+        [1,2,-2],
+        [1,1,1],
+        [2,2,1]
+    ]
+
+    DL = [[0 for _ in range(len(A))] for __ in range(len(A))]
+    U = [[0 for _ in range(len(A))] for __ in range(len(A))]
+    for i in range(len(A)):
+        for j in range(len(A[0])):
+            if i >= j:
+                DL[i][j] = A[i][j]
+            else:
+                U[i][j] = -A[i][j]
+                
+
+    
+    print(f"LU分解的D-L矩阵为: {DL}")
+    print(f"LU分解的U矩阵的逆矩阵为: {U}")
+    ID = FixInv(DL)
+    G = Dot(ID, U)
+    print(f"LU分解的G矩阵为: {G}")
+    

按方法整理/矩阵-逆矩阵-条件数.py

@@ -42,9 +42,27 @@ def Inverse(A):
     A_inv = [[B[i][j] / det for j in range(n)] for i in range(n)]
     return A_inv
 
+def FixInv(A):
+    n = len(A)
+    if n == 2:
+        return [[A[1][1] / Det(A), -A[0][1] / Det(A)],
+                [-A[1][0] / Det(A), A[0][0] / Det(A)]]
+    B = [[0 for i in range(n)] for j in range(n)]
+    for i in range(n):
+        for j in range(n):
+            t = [row[:j] + row[j+1:] for row in (A[:i] + A[i+1:])]
+            B[j][i] = ((-1) ** (i + j)) * Det(t)
+    det = Det(A)
+    if det == 0:
+        print("矩阵不可逆")
+        return None
+    A_inv = [[B[i][j] / det for j in range(n)] for i in range(n)]
+    return A_inv
+
 # 计算矩阵的条件数
 def Cond(A,v):
-    inv_A = Inverse(A)
+    # inv_A = Inverse(A)
+    inv_A = FixInv(A)
     print(f"inv_A: {inv_A}, Norm(A, v): {Norm(A, v)}, Norm(inv_A, v): {Norm(inv_A, v)}")
     # print(inv_A,Norm(A, v), Norm(inv_A, v))
     return Norm(A, v) * Norm(inv_A, v)