1. Grayson Ishihara Math 480 April 15, 2013

2.  What is Partial Pivoting?  What is the PA=LU Factorization?  What kinds of things can we use these tools for?

3.  Used to solve matrix equations  Compare absolute values of entries in and below pivot position  If |a p1 |≥|a i1 |, switch rows 1 and p  Repeat for each pivot position

4.  Writing a matrix A as the product of a lower triangular matrix L and an upper triangular matrix U

6.  Row reduce A into upper triangular U  Keep track of steps in lower triangular L

7.  Not all matrices allow such a factorization… a AbAb adad acac b = 0, ab = 1 ?

8.  Apply permutation matrix P to A  Find LU Factorization of the resulting matrix  Ensures all matrices have LU Factorization

9.  Solving systems of equations:  Matrix form Ax=b  Multiply both sides by P: PAx=Pb  LUx=Pb  Two-step solution: Ly=Pb, Ux=y  Easy: back substitution