This preview shows pages 1–8. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Lecture Note 3: Sept 18  22, 2006 Dr. Jeff ChakFu WONG Department of Mathematics Chinese University of Hong Kong [email protected] MAT 2310 Linear Algebra and Its Applications Fall, 2006 Produced by Jeff ChakFu WONG 1 S OLVING THE L INEAR S YSTEMS Gaussian Elimination and LU Factorization (or Decomposition) 1. Doolittle’s method 2. Crout’s method 3. Choleski’s method SOLVING THE LINEAR SYSTEMS 2 Things to remember Gaussian Elimination and LU Factorization (or Decomposition) • Suppose A is nonsingular • Factor A = LU , where L is lower triangular, U is upper triangular – Computed by Gaussian elimination. – There are many techniques like direct decomposition, Doolittle’s method or Crout’s method executed to find L and U . – Rows and columns often must be reordered to avoid division by zero – pivoting – Given LUfactorization (or decomposition), find x by Solve L z = b by forward substitution Solve U x = z by backward (or back) substitution SOLVING THE LINEAR SYSTEMS 3 The method can be used to solve a system of equations or to find the inverse of a matrix. In this method, the matrix A is decomposed or factorized as the product of a lower triangular matrix L and an upper triangular matrix U . We write the matrix A as A = LU. where L = l 11 ... l 21 l 22 ... l 31 l 32 l 33 ... . . . . . . . . . ... l n 1 l n 2 l l 3 ... l nn , U = u 11 u 12 u 13 ... u 1 n u 22 u 23 ... u 2 n u 33 ... u 3 n . . . . . . . . . ... ... u nn . Multiplying the matrices L and U and comparing the elements of the product matrix with the corresponding elements of A , we get l il u 1 j + l i 2 u 2 j + ··· + l in u nj = a ij , i,j = 1 , 2 , ··· ,n where l ij = 0 , j > i and u ij = 0 , i > j . SOLVING THE LINEAR SYSTEMS 4 To obtain a unique solution, we can choose the values for n elements in either L or U arbitrary. The simplest choices are Doolittle’s method l ii = 1 , i = 1 , 2 , ··· ,n Crout’s method u ii = 1 , i = 1 , 2 , ··· ,n where 1 0 0 * 1 * * 1 , 1 * * 1 * 1 Here * and * are unknowns. The method fails when any of diagonal elements (pivots), l ii , in L or u ii in U becomes zero . SOLVING THE LINEAR SYSTEMS 5 Doolittle’s Method We look for 1 0 0 m 21 1 m 31 m 32 1 • We begin with the augmented matrix, and display ( in the column headed m ) the multipliers required for the transformations. m a 11 a 12 a 13 b 1 a 21 a 22 a 23 b 2 a 31 a 32 a 33 b 3 R 1 R 2 R 3 SOLVING THE LINEAR SYSTEMS 6 Step 1. Eliminate the coefficients a 21 and a 31 , using row R 1 : m 21 = a 21 a 11 m 31 = a 31 a 11 a 11 a 12 a 13 b 1 a 22 a 23 b 2 a 32 a 33 b 3 R 1 R 2 R 3 ( R 2 m 21 × R 1 ) ( R 3 m 32 × R 1 ) Step 2.Step 2....
View
Full
Document
This note was uploaded on 10/15/2009 for the course MATHEMATIC MAT2310B taught by Professor Jeffwong during the Spring '09 term at CUHK.
 Spring '09
 JeffWong
 Linear Algebra, Algebra

Click to edit the document details