Expressing the Algorithm

There is a more systematic way of describing the Gaussian elimination process.

• Easy-to-solve systems
  • A is a lower-triangular matrix

    

    It can be solved as

    

    This is called the forward substitution.

  • A is an upper-triangular matrix

    

    It can be solved as

    

    This is called the back substitution

  • Constraint:
  • # operations for both cases -- .
• Strategy
  If we can write A = LU, with L lower-triangular and U upper-triangular, then what we need to do is
  1. Solve z in Lz = b, then
  2. Solve x in Ux = z