# lect_12 - Introduction to Numerical Analysis for Engineers...

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Numerical Methods for Engineers 13.002 Lecture 12 Introduction to Numerical Analysis for Engineers • Minimization Problems 5 • Least Square Approximation – Normal Equation – Parameter Estimation – Curve fitting • Optimization Methods – Simulated Annealing • Traveling salesman problem – Genetic Algorithms Mathews

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Numerical Methods for Engineers 13.002 Lecture 12 Minimization Problems Data Modeling – Curve Fitting Linear Model Minimimize Overall Error Non-linear Model Objective: Find c that minimizes error
Numerical Methods for Engineers 13.002 Lecture 12 Least Square Approximation m measurements n unknowns m > n m n n model parameters m measurements Linear Measurement Model Overdetermined System Least Square Solution Minimize Residual Norm

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Numerical Methods for Engineers 13.002 Lecture 12 Least Square Approximation A Theorem ) Proof Normal Equation Symmetric n x n matrix. Non- singular if columns of A are linearly independent q.e.d
Numerical Methods for Engineers 13.002 Lecture 12 Least Square Approximation Parameter estimation Example Island Survey A B C E D F Normal Equation Residual Vector Measured Altitude Differences Points D, E, and F at sea level. Find altitude of inland points A, B, and C. A=[ [1 0 0 -1 0 -1]' [0 1 0 1 -1 0]' [0 0 1 0 1 1]']

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## This note was uploaded on 02/24/2012 for the course MECHANICAL 2.993J taught by Professor Henrikschmidt during the Spring '05 term at MIT.

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lect_12 - Introduction to Numerical Analysis for Engineers...

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