This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 1 LECTURE 3.1A Dr. Vyas Overview: (Covers section 3.3) • Upper Triangular Linear Systems • Back Substitution • Examples/Demos UPPER TRIANGULAR LINEAR SYSTEMS 1. Definition: An N N × matrix [ ] i j N N a × = A is called upper triangular provided that the elements satisfy 0, i j a for all i j = . The N N × matrix [ ] i j N N a × = A is called lower triangular provided that the elements satisfy 0, i j a for all i j = < . 2. If A is an upper triangular matrix, then = A X B is said to be an upper triangular system of linear equations and has the form, 11 12 13 1 1 1 1 1 22 23 2 1 2 2 2 3 3 33 3 1 3 1 1 1 1 1 N N N N N N N N N N N N N N NN a a a a a x b a a a a x b x b a a a x b a a x b a = KK KK KK M M M M MKKKMKKKM KK KKK 3. The lower triangular system will have nonzero terms below the diagonal of the matrix while zeroes above the diagonal, unlike the case above. 4. Theorem 3.5 : Suppose that = A X B is an upper triangular system with the form given above. If kk a ≠ for 1, 2, ....., k N = , then there exists a unique solution to the upper triangular system. The unique solution is found by first determining / N N NN x b a = and then using a backward substitution algorithm given as, 1 1, 2, ........,1 N k k j j j k k kk b a x x for k N N a...
View
Full
Document
This note was uploaded on 04/14/2008 for the course MATH 353 taught by Professor Vyas during the Spring '08 term at University of Delaware.
 Spring '08
 Vyas
 Math, Linear Systems

Click to edit the document details