Chapter 2System of Linear Equations

• Solving Linear Systems (2.4)

January 26, 2010

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Elementary Elimination Matrices

Example: Gaussian Elimination

Example: Gaussian Elimination

Example: Gaussian Elimination

LU Factorization

LU Factorization

Example: LU Factorization

Example: LU Factorization

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