Chapter 2System 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 = [1345.45677, 1643.56346]

Computing Matrix Condition Number

Computing Matrix Condition Number

Residual

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