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Doolittle方法（即LU分解）及Python实现
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目录 一、LU分解原理 二、LU分解过程 三、Python实现完整代码 四、矩阵的一些补充概念 一、LU分解原理1. Gauss elimination takes more computational time for higher-order matrices. To reduce time consumption a matrix can be decomposed using LU factoring methods. A system of linear equations can be written in a matrix form Ax = b where |A| ≠0 即系数矩阵A需要是可逆矩阵。 2. Using LU factorization A can be decomposed as A = LU where L is lower triangular matrix and U is an upper triangular matrix. 由可逆矩阵的性质，若A是可逆矩阵，L、U均为方阵，则L、U也是可逆矩阵。 这里L为下三角矩阵，U为上三角矩阵。 另外，并不是每个矩阵都有LU分解。但是这些矩阵可借由排列其各行顺序来解决，所以最终会得到一个PLU 分解。 3. Hence we have If we let then we have 1. Decompose matrix A as 2. Determine all values of unknown entries in L and U 这一步解并没有它看起来那么麻烦。 Alternatively we can also find L and U by using elementary row operation (addition of 2 rows) 我们也可以用仅进行列变换的方法来得到L，用仅进行行变换的方法来得到U 3. Solve using forward substitution using backward substitution 三、Python实现完整代码 #LU分解求Ax=b #过程：A=LU,LZ=b,Ux=Z import numpy as np import pandas as pd from scipy import linalg np.random.seed(2) def LU_decomposition(A,b): n=len(A[0]) L = np.zeros([n,n]) U = np.zeros([n, n]) for i in range(n): L[i][i]=1 if i==0: U[0][0] = A[0][0] for j in range(1,n): U[0][j]=A[0][j] L[j][0]=A[j][0]/U[0][0] else: for j in range(i, n):#U temp=0 for k in range(0, i): temp = temp+L[i][k] * U[k][j] U[i][j]=A[i][j]-temp for j in range(i+1, n):#L temp = 0 for k in range(0, i ): temp = temp + L[j][k] * U[k][i] L[j][i] = (A[j][i] - temp)/U[i][i] Z=linalg.solve(L, b) x=linalg.solve(U, Z) print("L=",L) print("U=",U) print("Z=",Z) print("Exact solution: x=",x) return #定义A,b的值 A=[[-3,-5,10,13],[2,0,-4,6],[8,-2,1,-3],[5,-1,1,15]] b=[-14,8,7,0] LU_decomposition(A,b) 四、矩阵的一些补充概念pivot: Find the entry in the left column with the largest absolute value. This entry is called the pivot. 即首元，取当前列的最大元素 naive Gaussian elimination: meaning no row exchanges allowed gaussian elimination with partial pivoting gaussian elimination with full pivoting gaussian elimination with scaled partial pivoting Crout method Cholesky method |
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