lapack_plu():
  Demonstrate the use of tril() and triu() functions.
  Reconstruct P, L, U factors from compressed LAPACK output.

  Given the following matrix A:
[[1 2 3]
 [4 5 6]
 [7 8 0]]
  Calling LAPACK for a P' * L * U factorization
    call dgetrf ( a, n, n, lda, ipiv, info )
  returns the following compressed information:

    compressed LU info:
[[7.       8.       0.      ]
 [0.142857 0.857143 3.      ]
 [0.571429 0.5      4.5     ]]

    compressed pivot info P:
[3 3 3]

  U = np.triu ( ALU, 0 )
[[7.       8.       0.      ]
 [0.       0.857143 3.      ]
 [0.       0.       4.5     ]]

  L = np.tril ( ALU, -1 ) + np.identity ( n )
[[1.       0.       0.      ]
 [0.142857 1.       0.      ]
 [0.571429 0.5      1.      ]]

  P is a permutation, reconstructed from p
[[0. 0. 1.]
 [1. 0. 0.]
 [0. 1. 0.]]

  Check PLU = P' * L * U = A?
[[0.999999  1.999999  3.       ]
 [4.000003  5.0000035 6.       ]
 [7.        8.        0.       ]]