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.

B must have the same number of rows as columns in A.

The function works directly on B, returning the solution in B on successful completion. If the function fails, B can be altered.

LUApply is simply the application of Permute, ForwardSubstitute, and then BackwardSubstitute.

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](..).

$\mathrm{with}\left(\mathrm{LinearAlgebra}\left[\mathrm{Modular}\right]\right)\:$

$p≔97$

$p≔97$

$A≔\mathrm{Mod}\left(p\,\mathrm{Matrix}\left(5\,5\,\left(i\,j\right)\mapsto \mathrm{rand}\left(\right)\right)\,\mathrm{integer}\left[\right]\right)\:$

$A$

$\left[\begin{array}{ccccc}77& 96& 10& 86& 58\\ 36& 80& 22& 44& 39\\ 60& 39& 43& 12& 55\\ 2& 24& 71& 45& 29\\ 21& 48& 7& 33& 57\end{array}\right]$

$\mathrm{A2}≔\mathrm{Copy}\left(p\,A\right)\:$

$\mathrm{pv}≔\mathrm{Vector}\left(4\right)\:$

$\mathrm{LUDecomposition}\left(p\,\mathrm{A2}\,\mathrm{pv}\,0\right)\:$

$\mathrm{A2},\mathrm{pv}$

$\left[\begin{array}{ccccc}77& 96& 10& 86& 58\\ 37& 20& 40& 63& 27\\ 94& 60& 1& 79& 64\\ 29& 56& 63& 7& 78\\ 62& 54& 40& 10& 5\end{array}\right],\left[\begin{array}{c}1\\ 2\\ 3\\ 4\end{array}\right]$

$B≔\mathrm{Mod}\left(p\,\mathrm{Matrix}\left(5\,2\,\left(i\,j\right)\mapsto \mathrm{rand}\left(\right)\right)\,\mathrm{integer}\left[\right]\right)$

$B≔\left[\begin{array}{cc}65& 16\\ 93& 96\\ 71& 44\\ 70& 58\\ 25& 29\end{array}\right]$

$X≔\mathrm{Copy}\left(p\,B\right)\:$

$\mathrm{LUApply}\left(p\,\mathrm{A2}\,\mathrm{pv}\,X\right)\:$

$X$

$\left[\begin{array}{cc}11& 37\\ 12& 20\\ 57& 11\\ 83& 68\\ 1& 11\end{array}\right]$

$\mathrm{Multiply}\left(p\,A\,X\right)-B$

$\left[\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& 0\end{array}\right]$

$p≔13\:$

$A≔\mathrm{Mod}\left(13\,\left[\left[0\,0\,12\right]\,\left[12\,0\,3\right]\,\left[1\,1\,1\right]\right]\,\mathrm{float}\left[8\right]\right)$

$A≔\left[\begin{array}{ccc}0.& 0.& 12.\\ 12.& 0.& 3.\\ 1.& 1.& 1.\end{array}\right]$

$\mathrm{A2}≔\mathrm{Copy}\left(p\,A\right)\:$

$\mathrm{pv}≔\mathrm{Vector}\left(2\right)\:$

$\mathrm{LUDecomposition}\left(p\,\mathrm{A2}\,\mathrm{pv}\,0\right)\:$

$\mathrm{A2},\mathrm{pv}$

$\left[\begin{array}{ccc}12.& 0.& 3.\\ 12.& 1.& 4.\\ 0.& 0.& 12.\end{array}\right],\left[\begin{array}{c}2\\ 3\end{array}\right]$

$B≔\mathrm{Mod}\left(p\,\mathrm{Vector}\left(3\,i\mapsto \mathrm{rand}\left(\right)\right)\,\mathrm{float}\left[8\right]\right)$

$B≔\left[\begin{array}{c}5.\\ 3.\\ 5.\end{array}\right]$

$X≔\mathrm{Copy}\left(p\,B\right)\:$

$\mathrm{LUApply}\left(p\,\mathrm{A2}\,\mathrm{pv}\,X\right)\:$

$X$

$\left[\begin{array}{c}8.\\ 2.\\ 8.\end{array}\right]$

$\mathrm{Multiply}\left(p\,A\,X\right)-B$

$\left[\begin{array}{c}0.\\ 0.\\ 0.\end{array}\right]$