# LUDecomposition

Usage

LUDecomposition[m] 产生一个矩阵 m的 LU 分解表示。

Notes

LUDecomposition 返回三个元素的一个列表。第一个元素是上三角和下三角矩阵的组合，第二个元素是一个指定用于绕轴旋转的行向量，对近似数值矩阵 ，第三个元素是 条件数的一个估计。 • 由LUDecomposition产生的数据是为了在LUBackSubstitution中使用。
• 参见 Mathematica 全书: 3.7.8 and 节 3.7.10.
• 实现注释: 参见节 A.9.4.
Further Examples

 In[1]:=
 Out[1]=
 In[2]:=
 Out[2]=
 In[3]:=
 Out[3]=
 In[4]:=

LUDecomposition satisfies the following relation: Given a matrix M, determine a unit lower triangular matrix L, an upper triangular matrix U and a permutation vector P such that Part[M, P] L. U . Part[M, P] rearranges the rows of M. The permutation ensures numerical stability during the decomposition phase; after that, L and U are determined using Gaussian elimination. Solving the system M.x b is equivalent to solving the system L.U.x b but it is easier because it amounts to solving two triangular systems in a row.

For reasons of efficiency, the matrices L and U are returned as a single matrix, where the ones on the main diagonal of L are left out. The functions Lower and Upper recover the matrices by multiplying pairwise (not dot multiplying!) with appropriate matrices of zeros and ones.

 In[5]:=
 In[6]:=
 In[7]:=
 In[8]:=
 Out[8]=
 In[9]:=
 Out[9]//MatrixForm=
 In[10]:=
 Out[10]//MatrixForm=
 In[11]:=
 Out[11]//MatrixForm=
Further Examples
 In[1]:=
 Out[1]=
 In[2]:=
 Out[2]=
 In[3]:=
 Out[3]=
 In[4]:=

LUDecomposition satisfies the following relation: Given a matrix M, determine a unit lower triangular matrix L, an upper triangular matrix U and a permutation vector P such that Part[M, P] L. U . Part[M, P] rearranges the rows of M. The permutation ensures numerical stability during the decomposition phase; after that, L and U are determined using Gaussian elimination. Solving the system M.x b is equivalent to solving the system L.U.x b but it is easier because it amounts to solving two triangular systems in a row.

For reasons of efficiency, the matrices L and U are returned as a single matrix, where the ones on the main diagonal of L are left out. The functions Lower and Upper recover the matrices by multiplying pairwise (not dot multiplying!) with appropriate matrices of zeros and ones.

 In[5]:=
 In[6]:=
 In[7]:=
 In[8]:=
 Out[8]=
 In[9]:=
 Out[9]//MatrixForm=
 In[10]:=
 Out[10]//MatrixForm=
 In[11]:=
 Out[11]//MatrixForm=