9.1
Application
to Differential Equations
9.2
Geometry of
Linear Operators on R2
9.3
Least Squares
Fitting to Data
9.4
Approximation
Problems; Fourier Series
9.5
Quadratic
Forms
9.6
Diagonalizing Quadratic Forms; Conic Sections
Hwk: 1b,
5b, 6, 7abcd, 9, 11, 15ac
9.7
Quadric
Surfaces
9.8
Comparison of
Procedures for Solving Linear Systems
9.9
LU-Decompositions
(Appropriate for computer solutions)
Task: solve a linear system
technique is based on factoring the coefficient matrix into lower and upper triangular matrices
Summary of technique to solve Ax = b:
Ax = b Let A = LU, substitute Technique: below
LUx = b Define Ux = y, substitute
Ly = b
y = ? Solve Ly = b by "forward-substitution" (working from the top down)
x = ? Solve Ux = y by back-substitution
Def: LU-decomposition (or triangular decomposition) of a square matrix A is a factorization A=LU, where L is lower triangular and U is upper triangular.
Thm 1: If A is a square matrix that can be reduced to row-echelon form U without row swaps, an LU-decomposition is possible.
Fact: LU-decompositions are not unique
Fact: If an LU-decomposition is not possible, it is still possible to factor A=PLU, where P is a matrix obtained by interchanging the rows of In appropriately (See Exercise 17)
Easier: to solve a system of equations, rearrange the equations first so that no row swaps are necessary.
Steps to find LU-decomposition: See pg 483 and Example 3.
Return to: Merced College; Don Power Updated 05/08/07 by Don Power