A procedure for decomposing an matrix into a product of a lower triangular matrix and an upper triangular matrix ,
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LU decomposition is implemented in the Wolfram Language as LUDecomposition[m].
Written explicitly for a matrix, the decomposition is
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This gives three types of equations
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This gives equations for unknowns (the decomposition is not unique), and can be solved using Crout's method. To solve the matrix equation
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first solve for . This can be done by forward substitution
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for , ..., . Then solve for . This can be done by back substitution
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for , ..., 1.