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  的  条件数的一个估计。
• 由 LUDecomposition产生的数据是为了在 LUBackSubstitution中使用。
 Further Examples
We start with the decomposition of a simple matrix.
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We solve the system m.v = {5,8}.
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This checks the result.
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Recovering the lower and upper triangular matrices from the LUDecomposition 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.
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Consider this inexact matrix.
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This decomposes the matrix into lower and upper diagonal matrices, gives a permutation vector and an estimate of the condition number.
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This is the lower triangular matrix L.
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This is the upper triangular matrix U.
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This checks the result.
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Further Examples
We start with the decomposition of a simple matrix.
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We solve the system m.v = {5,8}.
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This checks the result.
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Out[3]=
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Recovering the lower and upper triangular matrices from the LUDecomposition 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.
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Consider this inexact matrix.
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This decomposes the matrix into lower and upper diagonal matrices, gives a permutation vector and an estimate of the condition number.
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This is the lower triangular matrix L.
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Out[9]//MatrixForm=
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This is the upper triangular matrix U.
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Out[10]//MatrixForm=
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This checks the result.
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