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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 g1858(g1876) = 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 g1858(g1876) = 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 g1858(g1876) = 0 can then be solved for g1876 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 Many of the most important equations in science and engineering are nonlinear equations. The van der Waals equation of state is an important example of a nonlinear equation: 2 a (P + )(V - b) = RT V (3.1) The van der Waals equation describes how the pressure ( P ) and molar volume ( V ) of a real gas vary with the product of the absolute temperature ( T ) and the universal gas law constant ( R ). To describe the behavior of a gas like carbon dioxide with this equation, we have to specify the value of the constants a and b . While this equation can be solved explicitly for the pressure, it cannot be solved explicitly for the volume, which is often the principal variable of interest. Another example of a nonlinear equation is provided by surfactants, man made chemical compounds with properties very similar to the phospholipids that constitute the main structural components of the membranes of living cells. Many technologies depend on applications of surfactants, from cleaning compounds to ore processing, including novel environmental remediation technologies, enhanced oil recovery, and the processing of carbon nanotubes. Surfactants form aggregates in aqueous solution called micelles. The
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