Chapter 3 Solution of Nonlinear Equations Using Root Finding Techniques_V2

Chapter 3 Solution of Nonlinear Equations Using Root Finding Techniques_V2

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- 1 - Chapter 3 Nonlinear Equations and Root Finding Methods Contents 3.1 Nonlinear Equations in Science and Engineering . .................................................... - 2 - 3.2 Nonlinear Equations and Root Finding Functions . ................................................... - 2 - 3.3 Converting the Nonlinear Equation to a Root Finding Problem. .............................. - 3 - 3.4 Choosing Better Root Finding Functions . ................................................................ - 4 - 3.5 Newton's method for Finding Roots . ........................................................................ - 4 - Example 1 An Important Application of Newton's method . .......................................... - 8 - 3.6 Specifying a Good Initial Guess for Newton's method . .......................................... - 15 - 3.7 Equations with No Solution . ................................................................................... - 16 - 3.8 Solving for Multiple Roots Using Newton's method . ............................................. - 18 - 3.9 Root Finding Methods that Use the Secant Line Rather than the Tangent Line . ... - 24 - 3.11 The Secant Method and the False Position Method. ............................................. - 26 - 3.12 Secant Method Summary . ..................................................................................... - 27 - 3.13 Comparing Newton's method to the Secant Method . ........................................... - 27 - 3.14 Summary . .............................................................................................................. - 30 - Exercises . ...................................................................................................................... - 31 -
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- 2 - Chapter 3 Solution of Nonlinear Equations Using Root Finding Techniques 3.1 Nonlinear Equations in Science and Engineering Scientist and engineers can often find an equation g(G) = 0 of one variable that characterizes some physical problem which they want to solve for ‘x’. However, they are not able to find an analytic solution for x such that g(G) = 0 . This problem is often encountered by engineers when solving engineering design problems. Once the engineer has studied the design problem, and is able to determine the number of significant figures of accuracy it requires, the problem g(G) = 0 can then be solved for G using a numerical method. It can then be solved to the number of significant figures of accuracy that the design requires. The first step in solving a nonlinear equation is to convert it into a root finding problem. The second step is to find the roots. In this chapter we will present two effective root finding functions: Newton's method and the Secant Method. First, however, let’s exam how root finding functions are related to nonlinear equations. 3.2 Nonlinear Equations and Root Finding Functions
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This note was uploaded on 01/26/2011 for the course CH E 2112 taught by Professor Dr.harwell during the Spring '10 term at The University of Oklahoma.

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Chapter 3 Solution of Nonlinear Equations Using Root Finding Techniques_V2

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