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How can I (succesfully) unpack the information module rmatrixlu returns me in the first argument? I.e. i need the matrices L and U explicitely.

Lower part of the result contains L, upper one contains U. They should be easy to unpack.

Sure, if this were it. What about the Permutation Matrix (Pivots) and the Decomposition A=P L U? I don't understand the function of the returned pivots in order to construct that Decomposition.

Here is extract from documentation:

It seems we are talking past each other. Please take a look at this example (mycode)

LU Decomposition A = L U.
A:
2,000000 -1,000000 0,000000
1,000000 0,000000 -1,000000
3,000000 7,000000 -1,000000

The Decomposition of A is
L:
0,666667 1,000000 0,000000
0,333333 0,411765 1,000000
1,000000 0,000000 0,000000
U:
3,000000 7,000000 -1,000000
0,000000 -5,666667 0,666667
0,000000 0,000000 -0,941176

Now please consider the output of alglib routine rmatrixlux

---> LUA:
3,0000 7,0000 -1,0000
0,6667 -5,6667 0,6667
0,3333 0,4118 -0,9412
---> PIVOTS:
2 2 2

Matrix U can easily be extracted. But the construction of L is what is making me a headache when using alglib's rmatrixlu.

L is unitriangular, which means that its diagonal elements are equal to 1.0. In your case L is equal to
1,0000 0,0000 0,0000
0,6667 1,0000 0,0000
0,3333 0,4118 1,0000

Which in this case would mean, that LU != A. And, what about the Pivots? I know i am intrusive, and i am really sorry about that, but i hope to solve this (little) problem with your very appreciated help.