# ME205: Numerical Methods for Engineers

Unit 3: Solution of Nonlinear Equations   You have already encountered this topic in* ME101: Introduction to Mechanical Engineering; you may wish to review section 3.4.2 of that course.  In this unit you will examine the topic in more detail and work through several more examples using established techniques.

For motivation, you may wish to refer to the introduction of this course. Consider that you have an equation of the form f(x) = ax2 /[(bx3  + c sin(dx) + f ln(x]). As part of an ongoing effort you need to find the zeros of this equation several times per day for different values of a, b, c, d, and f.  In particular, you need to find the zero near a value of x = m.  The values of a, b, c, d, and f are known but change hourly.

One approach to finding the zero near m, would be to plot the expression as a function of x near m and visually inspect the resulting plot; if necessary, one could zoom in on the area of interest in order to evaluate the zero with more precision. This process is, however, expensive because it requires human intervention. It is more cost effective in many situations to have a machine evaluate the zero near m with user-defined precision.

Note that we have limited the problem here to finding *a zero
near m. To find all zeros is a much more difficult problem and is beyond the scope of this unit. *

The methods in this section are closely related. As you study the resource materials methods, seek to find similarities and differences between the methods. You may wish to refer to Unit 2 on numerical differentiation as you study the secant method. *

Time Advisory: This unit will take you approximately 15 hours to complete.

☐    Subunit 3.1: 5 hours

☐    Subunit 3.2: 10 hours

☐    Sub-subunit 3.2.1: 7 hours

☐    Web Media: 2 hours

☐    Assessment: 1 hour

☐    Sub-subunit 3.2.2: 3 hours

Unit3 Learning Outcomes
Upon successful completion of this unit, the student will be able to:

• Describe situations in which numerical solutions to nonlinear equations are needed.
• Implement the bisection method for solving equations.
• Implement both Newton-Raphson and secant methods.
• Describe the difference between Newton-Raphson and secant methods.
• Demonstrate the relative performance of bisection, Newton-Raphson, and secant methods.

3.1 Bisection Method   - Web Media: University of South Florida: Holistic Numerical Methods Institute’s “Background of Bisection Method,” “Algorithm of Bisection Method,”, “ Example of Bisection Method,” and “Advantages & Drawbacks of Bisection Method” Links:  University of South Florida: Holistic Numerical Methods Institute’s “Background of Bisection Method,” “Algorithm of Bisection Method,” “Example of Bisection Method,” and “Advantages & Drawbacks of Bisection Method”  (YouTube)

Instructions: Please view these four video lectures as an introduction to the topic of bisection method.  You will have a chance to review written descriptions in more detail later on in this subunit.  The total run time for these videos is approximately 35 minutes.

• Reading: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter of Bisection Method” Links:University of South Florida:Holistic Numerical Methods Institute’s “Textbook Chapter of Bisection Method” (PDF)

Instructions: Please read the text with particular attention to the advantages and disadvantages of the method.  You may wish to consider how the method might perform, if there were a discontinuity in the expression of interest. (12 pages)

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

• Assessment: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Bisection Method” Links: University of South Florida:Holistic Numerical Methods Institute’s “Test Your Knowledge of Bisection Method” (PDF)

Instructions: Complete the entire multiple choice quiz.  Refer to “Textbook Chapter of Bisection Method” for any troublesome questions.  You can find the answers here (PDF).

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

• Reading: University of South Florida: Holistic Numerical Methods Institute’s “Mechanical Engineering Example of Bisection Method” Links: University of South Florida:Holistic Numerical Methods Institute’s “Mechanical Engineering Example of Bisection Method” (PDF)

Instructions: Follow the example carefully.  Try to reproduce the calculations.

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

3.2 Newton-Raphson and Secant Methods   3.2.1 Newton-Raphson   - Web Media: University of South Florida: Holistic Numerical Methods Institute’s Lectures on the “Newton-Raphson Method” Link: University of South Florida: Holistic Numerical Methods Institute’s Lectures on the “Newton-Raphson Method” (YouTube)

Instructions: Under “Digital Audiovisual Lectures,” select the YouTube link after each title to view all of the video lectures.  There are 9 lectures total with an approximate run time of 65 minutes.  How do you perform division by the Newton-Raphson method?

• Reading: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter of Newton-Raphson Method” Link: University of South Florida:Holistic Numerical Methods Institute’s “Textbook Chapter of Newton-Raphson Method” (PDF)

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

• Assessment: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Newton-Raphson Method” Links: University of South Florida:Holistic Numerical Methods Institute’s “Test Your Knowledge of Newton-Raphson Method” (PDF)

Instructions: Please attempt all questions on this multiple choice quiz.  If necessary, refer back to the “Textbook Chapter of Newton-Raphson Method” for any questions that pose difficulties.  You can find the answers here (PDF).

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

• Reading: University of South Florida: Holistic Numerical Methods Institute’s “Mechanical Engineering Example on Newton-Raphson Method” Link: University of South Florida: Holistic Numerical Methods Institute’s “Mechanical Engineering Example on Newton-Raphson Method” (PDF)

Instructions: Please follow along with the example, and try to reproduce the calculations.

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

3.2.2 Secant Method   - Web Media: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “Secant Method” Links: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “Secant Method” (YouTube)

Instructions: Under “Digital Audiovisual Lectures,” select the YouTube link to view all four videos: “Derivation of Secant Method: Approach 1 of 2,” “Derivation of Secant Method: Approach 2 of 2,” “Algorithm of Secant Method,” and “Example of Secant Method” (total run time: approximately 27 minutes).

• Reading: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter of Secant Method” Link: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter of Secant Method” (PDF)

Instructions: Please read the entire chapter (6 pages).  How does this method differ from the Newton-Raphson method?

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

• Assessment: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Secant Method” Link: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Secant Method” (PDF, DOC, HTML, FLASH)

Instructions: Under “Multiple Choice Test,” select the appropriate format (PDF, DOC, HTML, or FLASH) to download the quiz.  Please attempt all questions on the multiple choice quiz.  If necessary, refer to “Textbook Chapter of Secant Method” for any questions that pose difficulties.  Please note that the solutions only appear in the Flash version.