Post on 04-Jan-2016
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