PLU-factorization

PLU is LU with pivoting (although ``P'' stands for permutation).

• Permutation matrices :
  • Each row or each column has exactly one 1 and 0 for the rest.
  • Example:
  • Property:
• What if happens in LU-factorization?
  • Want to find a permutation matrix P, such that A = PLU.
• Algorithm
  1. If n = 1, return .
  2. Find a non-zero element in the first row of A. (If not, then the system is not solvable.)
  3. Define a permutation matrix such that
     • ,
     • for ,
     • All the other elements of Q are 0.
  4. Let B = QA. Notice that .
  5. Write B as

     

  6. Define . Recursively compute
  7. Return

     

• Correctness of the algorithm (HW)
• Operation count (HW)