Chapter 5. Root Solving and Optimization Methods

Chapter 5. Root Solving and Optimization Methods - 5 Root...

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0-8493-????-?/00/$0.00+$.50 © 2000 by CRC Press LLC © 2001 by CRC Press LLC 5 Root Solving and Optimization Methods In this chapter, we frst learn some elementary numerical techniques and the use oF the fsolve and fzero commands From the MATLAB library to obtain the real roots (or zeros) oF an arbitrary Function. Then, we discuss the use oF the MATLAB command roots For fnding all roots oF a polynomial. ±ollowing this, we consider the Golden Section method and the fmin and fmins MATLAB commands For optimizing (fnding the minimum or maxi- mum value oF a Function) over an interval. Our discussions pertain exclu- sively to problems with one and two variables (input) and do not include the important problem oF optimization with constraints. 5.1 Finding the Real Roots of a Function This section explores the diFFerent categories oF techniques For fnding the real roots (zeros) oF an arbitrary Function. We outline the required steps For com- puting the zeros using the graphical commands, the numerical techniques known as the Direct Iterative and the Newton-Raphson methods, and the built-in fsolve and fzero Functions oF MATLAB. 5.1.1 Graphical Method In the graphical method, we fnd the zeros oF a single variable Function by implementing the Following steps: 1. Plot the particular Function over a suitable domain. 2. IdentiFy the neighborhoods where the curve crosses the x -axis (there may be more than one point); and at each such point, the Following steps should be independently implemented. 3. Zoom in on the neighborhood oF each intersection point by repeated application oF the MATLAB axis or zoom commands.
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© 2001 by CRC Press LLC 4. Use the crosshair of the ginput command to read the coordinates of the intersection. In problems where we desire to Fnd the zeros of a function that depends on two input variables, we follow (conceptually) the same steps above, but use 3-D graphics. In-Class Exercises Pb. 5.1 ±ind graphically the two points in the x-y plane where the two sur- faces, given below, intersect: ( Hint: Use the techniques of surface and contour renderings, detailed in Chapter 1, to plot the zero height contours for both surfaces; then read off the intersections of the resulting curves.) Pb. 5.2 Verify your graphical answer to Pb. 5.1 with that you would obtain analytically. 5.1.2 Numerical Methods This chapter subsection brie²y discusses two techniques for Fnding the zeros of a function in one variable, namely the Direct Iterative and the Newton- Raphson techniques. We do not concern ourselves too much, at this point, with an optimization of the routine execution time, nor with the inherent lim- its of each of the methods, except in the most general way. ±urthermore, to avoid the inherent limits of these techniques in some pathological cases, we assume that we plot each function under consideration, verify that it crosses the x -axis, and satisfy ourselves in an empirical way that there does not seem to be any pathology around the intersection point before we embark on the application of the following algorithms. These statements will be made more rigorous to you in future courses in numerical analysis.
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This note was uploaded on 10/19/2010 for the course ENGINEERIN ELEC121 taught by Professor Tang during the Spring '10 term at University of Liverpool.

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Chapter 5. Root Solving and Optimization Methods - 5 Root...

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