Compute L and U for the symmetric matrix A:

A = |a a a a|
    |a b b b|
    |a b c c|
    |a b c d|

Find four conditions on a, b, c, d to get A = LU with four pivots.

== Solution ==
|a a a a|    |a   a   a   a|    |a   a   a   a|    |a   a   a   a|
|a b b b| => |  b-a b-a b-a| => |  b-a b-a b-a| => |  b-a b-a b-a| = U
|a b c c|    |  b-a c-a c-a|    |      c-b c-b|    |      c-b c-b| 
|a b c d|    |  b-a c-a d-a|    |      c-b d-b|    |          d-c|
This means:
    |1      |
L = |1 1    |
    |1 1 1  |
    |1 1 1 1|
The four conditions are, obviously,
a != 0
a != b
b != c
c != d