Lecture_10_Optimization_-_Newtons_Method

Lecture_10_Optimization_-_Newtons_Method - EGR 102...

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EGR 102 Lecture 10 1 EGR 102 Introduction to Engineering Modeling Optimization - Newton’s Method Chapter 7.1-7.2 Figures from: “Applied Numerical Methods with MATLAB,” Steven Chapra, McGraw Hill
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Objectives Understand why and where optimization occurs in engineering and scientific problem solving. Recognize the difference between one- dimensional and multi-dimensional optimization. Know the difference between global and local optima. Know how to recast a maximization problem so that it can be solved with a minimizing algorithm. EGR 102 2 Lecture 10
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EGR 102 Lecture 10 3 Root Finding Methods ± Graphical ± Gauss Elimination ± Bisection ± Newton-Raphson
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EGR 102 Lecture 10 4 Graphical Method
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EGR 102 Lecture 10 5 Naïve Gauss Elimination
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EGR 102 Lecture 10 6 Bisection Method ± Two initial guesses for the root are required. These guesses must “bracket” or be on either side of the root. ± If one root of a real and continuous function, f(x)=0, is bounded by values x= x l , x = x u then f( x l ) · f( x u ) <0. ( The function changes sign on opposite sides of the root )
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EGR 102 Lecture 10 7 ± A tangent to the function of x i , [ƒ’( x i )], is extrapolated down to the x-axis to provide an estimate of the root at x i+1 Newton-Raphson Method ) ( ' ) ( 1 i i i i x f x f x x = +
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Optimization Optimization is the process of creating something that is as effective as possible. From a mathematical perspective, optimization
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Lecture_10_Optimization_-_Newtons_Method - EGR 102...

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