LU-factorization

• Algorithm
  1. If , fail!
  2. If n = 1, return L = 1 and .
  3. Write A as

     

  4. Define . (Schur complement, see CLR p.755)
  5. Recursively compute the LU-factorization of

     

  6. Return

     

• Correctness of the algorithm
  • Show
    1. Base case --
    2. Otherwise --

       

  • Show L is lower-triangular and U is upper-triangular (Exercise)
  • Termination: # recursive calls
• Operation count
  • Computing (Schur complement) costs operations
  • 
  • Solve the recurrence relation. (Exercise)