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Assume that we have this equation

$$PA=LU$$

Where $A \in \Re^{mxn}$, $L \in \Re^{mxn}$ is a lower triangular matrix and $U \in \Re^{nxn}$ is an upper triangular matrix. $P \in \Re^{mxm}$ is the partial pivoting matrix.

In this case, $A,U,L$ are known. How can I find $P$?

Can I take

$$P = LUA^{\dagger}$$

?

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    $\begingroup$ Every algorithm for computing LU decomposition with partial pivoting gives you also $P$ in some way. Why would you do this? $\endgroup$ – Algebraic Pavel Feb 24 '19 at 0:37
  • $\begingroup$ Because I using Lapack subroutine dgetrf_ , and the P-matrix is not a matrix, it's more like a vector that describe how the P-matrix should be. That's not a problem. The real problem is that the P-vector is one element to short. I think it's a bug inside dgetrf_. Here is an example: github.com/DanielMartensson/EmbeddedLapack/blob/master/… $\endgroup$ – Daniel Mårtensson Feb 24 '19 at 1:17
  • $\begingroup$ @AlgebraicPavel If you want to contribute with some C-code for solving the pivot matrix, it would be very helpfull for this free library. $\endgroup$ – Daniel Mårtensson Feb 24 '19 at 1:18
  • $\begingroup$ What do you mean by one element too short? $\endgroup$ – Algebraic Pavel Feb 24 '19 at 12:11
  • $\begingroup$ @AlgebraicPavel It seems that Lapack solve $A=PLU$ and MATLAB solve $PA=LU$. I have now made so Lapack solve $PA=LU$ :) Just look at the lu.c file from github.com/DanielMartensson/EmbeddedLapack $\endgroup$ – Daniel Mårtensson Feb 24 '19 at 17:02
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Yes, this is correct.

Your problem seems to be that LAPACK doesn't return the pivoting matrix, but a permutation vector. This thread might help: http://icl.cs.utk.edu/lapack-forum/viewtopic.php?f=2&t=1747

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