MECN 3500 Inter - Bayamon Lecture 8 Numerical Methods for Engineering MECN 3500 Professor: Dr. Omar E. Meza Castillo Department of Mechanical Engineering Inter American University of Puerto Rico Bayamon Campus

Lecture 8 MECN 3500 Inter - Bayamon 2 Tentative Lectures Schedule TopicLecture Mathematical Modeling and Engineering Problem Solving 1 Introduction to Matlab 2 Numerical Error 3 Root Finding System of Linear Equations 7 Least Square Curve Fitting Polynomial Interpolation Numerical Integration Ordinary Differential Equations

Lecture 8 MECN 3500 Inter - Bayamon Elimination Method LU Decomposition and Matrix Inversion 3

Lecture 8 MECN 3500 Inter - Bayamon  To solve linear algebraic equations using the technique LU Decomposition in order to provides an efficient means to compute the matrix inverse. 4 Course Objectives

Lecture 8 MECN 3500 Inter - Bayamon Provides an efficient way to compute matrix inverse by separating the time consuming elimination of the Matrix [A] from manipulations of the right-hand side {B}. Provides an efficient way to compute matrix inverse by separating the time consuming elimination of the Matrix [A] from manipulations of the right-hand side {B}. Gauss elimination, in which the forward elimination comprises the bulk of the computational effort, can be implemented as an LU decomposition Gauss elimination, in which the forward elimination comprises the bulk of the computational effort, can be implemented as an LU decomposition 5 Introduction

Lecture 8 MECN 3500 Inter - Bayamon 6 LU Decomposition If L- lower triangular matrix U- upper triangular matrix Then, [A]{X} = {B} or [A]{X} - {B} = 0 [L]{[U]{X}-{D}} = [A]{X} - {B}

Lecture 8 MECN 3500 Inter - Bayamon LU Decomposition [L][U]=[A] and [L]{D}={B} A two-step strategy 1.-LU Decomposition step: [A] can be decomposed into two matrices [L] and [U] such that [L][U]=[A] 2.-Substitution step: Similar to first phase of Gauss elimination, consider [U]{X}={D}[L]{D}={B} [L]{D}={B} is used to generate an intermediate vector {D} by forward substitution Then, [U]{X}={D} is used to get {X} by back substitution.

Lecture 8 MECN 3500 Inter - Bayamon LU Decomposition

Lecture 8 MECN 3500 Inter - Bayamon LU Decomposition Requires the same total FLOPS as for Gauss elimination. Requires the same total FLOPS as for Gauss elimination. Saves computing time by separating time- consuming elimination step from the manipulations of the right hand side. Saves computing time by separating time- consuming elimination step from the manipulations of the right hand side. Provides efficient means to compute the matrix inverse Provides efficient means to compute the matrix inverse

Lecture 8 MECN 3500 Inter - Bayamon NGaussElimwLUDecomp.m

Lecture 8 MECN 3500 Inter - Bayamon >>Enter Matrix A > [ ; ; ] A = >>Enter Solution Vector B > [ ] B = S = Bc = >>Enter Matrix A > [ ; ; ] A = >>Enter Solution Vector B > [ ] B = S = Bc = Example 10.2

Lecture 8 MECN 3500 Inter - Bayamon L = U = Ac = Lc = Uc = L = U = Ac = Lc = Uc =

Lecture 8 MECN 3500 Inter - Bayamon Homework6  Omar E. Meza Castillo Ph.D. 13