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LinearAlgebra[Modular][LUApply] - apply PLU Decomposition to a mod m Matrix or Vector
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Calling Sequence
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LUApply(m, A, pvec, B)
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Parameters
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m
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modulus
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A
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mod m Matrix (from LUDecomposition)
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pvec
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permutation vector (from LUDecomposition)
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B
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mod m Matrix or Vector representing right-hand side of problem
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Description
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The LUApply function applies the permutation pvec and the forward and backward substitutions encoded in A directly to the right-hand side mod m Matrix or Vector B, where pvec and A are the output of the LUDecomposition function.
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B must have the same number of rows as columns in A.
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The function works directly on B, returning the solution in B on successful completion. If the function fails, B can be altered.
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This command is part of the LinearAlgebra[Modular] package, so it can be used in the form LUApply(..) only after executing the command with(LinearAlgebra[Modular]). However, it can always be used in the form LinearAlgebra[Modular][LUApply](..).
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Examples
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Compute LU decomposition of a random 5 x 5 Matrix, and use LUApply to obtain a solution.
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![with(LinearAlgebra[Modular])](/support/helpjp/helpview.aspx?si=4508/file06075/math145.png)
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| (1) |
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![A := Mod(p, Matrix(5, 5, proc (i, j) options operator, arrow; rand() end proc), integer[])](/support/helpjp/helpview.aspx?si=4508/file06075/math156.png)
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![Matrix([[77, 96, 10, 86, 58], [36, 80, 22, 44, 39], [60, 39, 43, 12, 55], [2, 24, 71, 45, 29], [21, 48, 7, 33, 57]])](/support/helpjp/helpview.aspx?si=4508/file06075/math163.png)
| (2) |
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![Matrix([[77, 96, 10, 86, 58], [37, 20, 40, 63, 27], [94, 60, 1, 79, 64], [29, 56, 63, 7, 78], [62, 54, 40, 10, 5]]), Vector[column]([[1], [2], [3], [4]])](/support/helpjp/helpview.aspx?si=4508/file06075/math182.png)
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![B := Mod(p, Matrix(5, 2, proc (i, j) options operator, arrow; rand() end proc), integer[])](/support/helpjp/helpview.aspx?si=4508/file06075/math186.png)
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![B := Matrix([[65, 16], [93, 96], [71, 44], [70, 58], [25, 29]])](/support/helpjp/helpview.aspx?si=4508/file06075/math189.png)
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![Matrix([[11, 37], [12, 20], [57, 11], [83, 68], [1, 11]])](/support/helpjp/helpview.aspx?si=4508/file06075/math204.png)
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Check the solution.
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![Matrix([[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]])](/support/helpjp/helpview.aspx?si=4508/file06075/math213.png)
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Use float[8] with a nontrivial permutation.
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![A := Mod(13, [[0, 0, 12], [12, 0, 3], [1, 1, 1]], float[8])](/support/helpjp/helpview.aspx?si=4508/file06075/math223.png)
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![A := Matrix([[0., 0., 12.], [12., 0., 3.], [1., 1., 1.]])](/support/helpjp/helpview.aspx?si=4508/file06075/math226.png)
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![Matrix([[12., 0., 3.], [12., 1., 4.], [0., 0., 12.]]), Vector[column]([[2], [3]])](/support/helpjp/helpview.aspx?si=4508/file06075/math245.png)
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Now apply to a random vector and check it.
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![B := Mod(p, Vector(3, proc (i) options operator, arrow; rand() end proc), float[8])](/support/helpjp/helpview.aspx?si=4508/file06075/math251.png)
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](/support/helpjp/helpview.aspx?si=4508/file06075/math254.png)
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](/support/helpjp/helpview.aspx?si=4508/file06075/math269.png)
| (10) |
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](/support/helpjp/helpview.aspx?si=4508/file06075/math276.png)
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Download Help Document
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