System of Linear Equations Chapter 2 System of Linear Equations Solving Linear Systems (2.4) January 26, 2010 New TA (2 TAs) Nejhum Shahed Office Hours: 2-5pm Wed, E309

Elementary Elimination Matrices

Example: Gaussian Elimination

Example: Gaussian Elimination

Example: Gaussian Elimination

LU Factorization

LU Factorization

Example: LU Factorization

Example: LU Factorization M1A=[1 2 2;0 -4 -6;0 -2 -4]; M2=[1 0 0;0 1 0; 0 -0.5 1] M2M1A=[ 1 2 2;0 -4 -6; 0 0 -1]; L=[1 0 0;4 1 0; 4 0.5 1]; What are M1, M1 A, M2 M2M1 A A = LU, what are L and U ?

Row Interchanges

Partial Pivoting

LU Factorization with Partial Pivoting

Complete Pivoting

Example: Pivoting

Example: Small Pivots

Example, cont’d

Pivoting, cont’d

Residual

Residual

Example

Example

Uniqueness of LU Factorization

Complexity of Solving Linear Systems

Inversion vs. Factorization