84 lines
2.6 KiB
Python
84 lines
2.6 KiB
Python
#模 范数
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def Norm(x,v):
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if len(x[0]) == 1:
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if v == 1:
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return sum([abs(i[0]) for i in x])
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elif v == 2:
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return (sum([i[0]**2 for i in x]))**0.5
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elif v == float("inf"):
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return max([abs(i[0]) for i in x])
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else:
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if v == 1:
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return max([sum([abs(x[i][j]) for i in range(len(x))]) for j in range(len(x[0]))])
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elif v == float("inf"):
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return max([sum([abs(i) for i in x[j]]) for j in range(len(x))])
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return None
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# 计算矩阵的行列式
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def Det(A):
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if len(A) == 2:
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return A[0][0] * A[1][1] - A[0][1] * A[1][0]
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det = 0
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for c in range(len(A)):
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sub_matrix = [row[:c] + row[c+1:] for row in A[1:]]
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det += ((-1) ** c) * A[0][c] * Det(sub_matrix)
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return det
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# 计算矩阵的逆矩阵
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def Inverse(A):
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n = len(A)
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# 计算代数余子式矩阵
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B = [[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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minor = [row[:j] + row[j+1:] for row in (A[:i] + A[i+1:])]
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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))
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det = Det(A)
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print("det(A):",det)
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if det == 0:
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print("矩阵不可逆")
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return None
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A_inv = [[B[i][j] / det for j in range(n)] for i in range(n)]
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return A_inv
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def FixInv(A):
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n = len(A)
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if n == 2:
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return [[A[1][1] / Det(A), -A[0][1] / Det(A)],
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[-A[1][0] / Det(A), A[0][0] / Det(A)]]
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B = [[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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t = [row[:j] + row[j+1:] for row in (A[:i] + A[i+1:])]
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B[j][i] = ((-1) ** (i + j)) * Det(t)
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det = Det(A)
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if det == 0:
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print("矩阵不可逆")
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return None
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A_inv = [[B[i][j] / det for j in range(n)] for i in range(n)]
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return A_inv
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# 计算矩阵的条件数
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def Cond(A,v):
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# inv_A = Inverse(A)
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inv_A = FixInv(A)
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print(inv_A,Norm(A, v), Norm(inv_A, v))
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return Norm(A, v) * Norm(inv_A, v)
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#把矩阵换成题干的矩阵#########################
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if __name__ == "__main__":
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A = [
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[1,2],
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[1.001,2.001]
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]
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#把范数的种类数换成题干的要求，inf是无穷范数#########################
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print(f"矩阵A的条件数为: {Cond(A, float('inf')):.5f}")
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#把矩阵换成题干的矩阵#########################
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A = [
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[1,2],
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[3,4]
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]
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#把范数的种类数换成题干的要求，inf是无穷范数########################
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print(f"矩阵A的条件数为: {Cond(A, float('inf')):.5f}")
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