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Zeros, Zeros, Zeros: How To Find Them Using Broyden's Method

Sutherland, James

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2017, Master of Mathematical Sciences, Ohio State University, Mathematical Sciences.
Zero-finding methods are used in a wide variety of applications throughout computational mathematics. Newton's method is arguably the most well-known iterative zero-finding method, but its need of the Jacobian matrix in higher dimensions can sometimes lead to serious problems. In 1965, C.G. Broyden developed two n-dimensional secant methods that use the Jacobian only at the initial iterate and perform a rank-one update at all other iterations. Over time these methods became known as Broyden's ``good" method and Broyden's ``bad" method. There have been many studies done with a few carefully selected initial points that conclude the naming of Broyden's methods are accurate. However, rather than focusing on just a few isolated initial points, in this project convergence plots were utilized as a primary tool in the analysis of these methods to facilitate an investigation of several million initial points at once. Using a variety of examples involving nonlinear functions, the results show that each method has its benefits depending on the problem under investigation. The labels ``good" and ``bad" can be misleading and should not influence one method to be used more frequently than the other.
Edward Overman (Advisor)
Ching-Shan Chou (Committee Member)
103 p.

Recommended Citations

Citations

  • Sutherland, J. (2017). Zeros, Zeros, Zeros: How To Find Them Using Broyden's Method [Master's thesis, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu150305892468206

    APA Style (7th edition)

  • Sutherland, James. Zeros, Zeros, Zeros: How To Find Them Using Broyden's Method. 2017. Ohio State University, Master's thesis. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu150305892468206.

    MLA Style (8th edition)

  • Sutherland, James. "Zeros, Zeros, Zeros: How To Find Them Using Broyden's Method." Master's thesis, Ohio State University, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=osu150305892468206

    Chicago Manual of Style (17th edition)