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Student[LinearAlgebra][LUDecomposition] - compute the PLU or PLU1R decomposition of a Matrix
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Calling Sequence
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LUDecomposition(A, out, options)
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Parameters
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A
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Matrix
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out
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(optional) equation of the form output = obj where obj is one of 'P', 'L', 'U', 'U1', or 'R', or a list consisting of one or more of these names; selects result objects to compute
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options
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(optional) parameters; for a complete list, see LinearAlgebra[LUDecomposition]
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Description
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The LUDecomposition command computes a PLU decomposition or a modified PLU1R decomposition of the Matrix A.
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Depending on what is included in the output option (out), an expression sequence containing one or more of the factors P, L, U, U1, and R can be returned. The objects are returned in the same order as specified in the output list.
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Note: Either U or the pair U1 and R may be returned, but not both.
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The LUDecomposition(A) calling sequence is equivalent to LUDecomposition(A, output=['P','L','U']). This LU decomposition generates a square unit lower triangular L factor and an upper triangular factor U with the same dimensions as A so that  . The Matrix P is a permutation Matrix.
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The PLU1R decomposition is achieved by using LUDecomposition(A, output=['P','L','U1','R']). This further factors U into  where U1 is square upper triangular factor and R is the unique reduced row echelon form of the Matrix A. In this case,  .
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The output option (out) determines the content of the returned expression sequence.
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Examples
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![with(Student[LinearAlgebra])](/support/helpjp/helpview.aspx?si=4294/file05003/math153.png)
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![A := Matrix([[0, 1, 1, -3], [-2, 3, 1, 4], [0, 0, 0, 1], [3, 1, 0, 0]])](/support/helpjp/helpview.aspx?si=4294/file05003/math160.png)
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![p, l, u := Matrix([[0, 1, 0, 0], [1, 0, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0]]), Matrix([[1, 0, 0, 0], [0, 1, 0, 0], [-3/2, 11/2, 1, 0], [0, 0, 0, 1]]), Matrix([[-2, 3, 1, 4], [0, 1, 1, -3], [0, 0, -4, 45/2], [0, 0, 0, 1]])](/support/helpjp/helpview.aspx?si=4294/file05003/math167.png)
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![Matrix([[0, 1, 1, -3], [-2, 3, 1, 4], [0, 0, 0, 1], [3, 1, 0, 0]])](/support/helpjp/helpview.aspx?si=4294/file05003/math174.png)
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To reduce a Matrix using Gaussian elimination, specify the 'U' object:
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![Matrix([[-2, 3, 1, 4], [0, 1, 1, -3], [0, 0, -4, 45/2], [0, 0, 0, 1]])](/support/helpjp/helpview.aspx?si=4294/file05003/math189.png)
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To reduce a Matrix using Gauss-Jordan elimination, specify the 'R' object:
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![Matrix([[1, 0, 0, -1], [0, 1, 0, 4/3], [0, 0, 1, -1/3]])](/support/helpjp/helpview.aspx?si=4294/file05003/math200.png)
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References
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For more information on the P*L*U1*R decomposition see:
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Corless, Robert M., and Jeffrey, David J. "The Turing Factorization of a Rectangular Matrix." Sigsam Bulletin, Vol. 31 no. 3. (September 1997): 20-28.
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This paper names the U1 factor U.
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