线性代数方程组的直接解法
调包消元函数和矩阵求解高斯消去法列主元高斯消去法LU分解法LU分解求矩阵的逆下三角矩阵求逆LU分解求逆演示调包
import numpy as np
消元函数和矩阵求解
#消元def KillYuan(A,i):try:A[i] /= A[i][i]except:print("矩阵奇异")A[i] /= A[i][i]for j in range(i + 1,len(A)):A[j] -= A[j][i] * A[i]return A#矩阵求解def MatrixSolve(A):A = np.copy(A)b = A[:,-1]a = np.delete(A,-1,1)try:np.linalg.solve(a,b)except:print("矩阵奇异")return np.linalg.solve(a,b)
高斯消去法
#高斯消元法def GaussianElimination(A):A = np.copy(A)m = len(A)n = len(A[0])for i in range(m):A = KillYuan(A,i)return MatrixSolve(A),A
列主元高斯消去法
#列主元高斯消元法def MainYuanGaussianElimination(A):A = np.copy(A)m = len(A)n = len(A[0])for i in range(m):maxYuan = 0for j in range(i,m):if abs(A[j][i]) > maxYuan:maxYuan = jA[[i,maxYuan],:] = A[[maxYuan,i],:] A = KillYuan(A,i)return MatrixSolve(A),A
LU分解法
#LU分解法def LU(A):n = len(A[0])L = np.zeros([n, n])U = np.zeros([n, n])for i in range(n):L[i][i] = 1if i == 0:for j in range(n):U[i][j] = A[i][j]L[j][i] = A[j][i] / U[i][i]else:for j in range(i, n): temp = 0for k in range(i):temp = temp + L[i][k] * U[k][j]U[i][j] = A[i][j] - tempfor j in range(i + 1, n): temp = 0for k in range(i):temp = temp + L[j][k] * U[k][i]L[j][i] = (A[j][i] - temp) / U[i][i]return L,U
LU分解求矩阵的逆
下三角矩阵求逆
#下三角矩阵求逆def DownTriangle_Inv(L):n = len(L)Li = np.zeros([n,n])for i in range(n):for j in range(n):if i < j:Li[i][j] = 0if i == j:Li[i][j] = 1 / L[i][j]if i > j:s = 0for k in range(j,i):s += L[i][k] * Li[k][j]Li[i][j] = -1 / L[i][i] * sreturn Li
LU分解求逆
#LU分解法矩阵求逆def A_Inv(A):L, U = LU(A)Li = DownTriangle_Inv(L)Ui = DownTriangle_Inv(U.T).Treturn np.dot(Ui,Li)
演示
Ae2 = np.array([[6, 2, -1, 2], [5, 0, 4, -10], [-9, -4, 2, 0], [6, 8, 2, -10]])print("LU分解法求矩阵的逆：")print(np.around(A_Inv(Ae2), 2))print("numpy验证：")print(np.around(np.linalg.inv(Ae2), 2))
【数值分析实验】线性代数方程组的直接解法：列主元高斯消去法 LU分解法 LU分解法求矩阵的逆（python）
