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Unformatted text preview: Lecture 11: Intro to Numerical Methods Daniel Frances c 2012 Contents 1 Introduction 2 2 Matrix Multiplication 3 2.1 O n analysis of alternate orderings . . . . . . . . . . . . . . . . . . . . . . . . 3 3 Solving Linear Equations and Matrix Inversion 4 3.1 Analytical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.1.1 O n analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1 1 Introduction Now that we have completed dealing with some really ”hard” problems (NPhard) in this second and last part of the course we are going to lower our vision and deal with some mundane polynomial algorithms needed to solve optimization problems: • Linear Optimization (LP) – Matrix multiplication – Solving linear equations and Matrix Inversion and – Revised Simplex • Nonlinear Optimization (NLP) – Solving Nonlinear equations – Nonlinear Unconstrained Optimization (NLP) What makes these problems worthy of study, is that they allow us to solve larger and larger problems. As you may have seen in the lab, it takes a perceptibly long time to solve some ”large” LP problems. This is not only true for the KleeMinty cases, but is also true for some ”real” cases. For example infinite horizon MDP (Markov Decision Process) problems can be formulated as LPs  as those who took MIE365 know. BUT what you may not know is that the dimensions of these LPs are so large  in the millions of variables and constraints that present LP codes cannot solve these MDP problems, and alternate methods such as...
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This note was uploaded on 03/31/2012 for the course MIE 335 taught by Professor Frances during the Spring '12 term at University of Toronto.
 Spring '12
 Frances

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