Chapter 2 System of Linear Equations Sensitivity and Conditioning (2.3) Solving Linear Systems (2.4) January 19, 2010

Vector Norms

Example: Vector Norm

Properties of Vector Norms

Matrix Norm Since A is linear, we can evaluate the max for ||x||=1

Illustration

Matrix Norms

Properties of Matrix Norms

Example

Condition Number

Properties of Condition Number

Error Bounds

Error Bounds, cont’d For example, if ε = 10 -6, cond(A) = 100 X = [ , ]

Computing Matrix Condition Number

Residual

Solving Linear System (2.4) What should the form be for MA ?

Triangular Linear System

Triangular Matrices

Back-Substitution

Forward-Substitution

Example: Permutations

Example: Diagonal Scaling