Complex Numerical Analysis

Namyong Lee, Minnesota State University-Mankato, Mathematics & Statistics
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Initial Publication Date: October 6, 2026
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Summary

This Group Activity is designed for experimenting the advantages of Complex Numerical Analysis. The submitted module is for numerical differentiation. Through this Group Work students may understand the advantage of using complex variables in numerical algorithms. Also, students may understand the substrative cancellation issue in finite precision computing.

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Learning Goals

Students will learn issues in finite difference scheme and how to avoid the problem. Matlab is mainly used as an experimental tool to discover the contents by students. Students will experience vibe coding to peer coding, and understanding computational obstacles by experimenting. If schedule allowed, each group can be presented in class.

Context for Use

This module is part of "Experimental Mathematics". A series of computational experiments and discovery through the Group Work in the Lab setting.
Matlab is used but can be replaced by any other scientific programing language.
Both vibe coding and peer coding are used for the Group Work.

Description and Teaching Materials

We attached the Group Work (Complex Numerical Analysis I), a note for instructors (for the Group Work), two Matlab sample files and a key reference.
Student Handout for "Complex Numerical Analysis I" (Acrobat (PDF) 30kB Sep30 26)
Instructor Note (Acrobat (PDF) 359kB Sep30 26)
Forward Finite Difference Scheme (Matlab File 2kB Sep30 26)
Complex Step Differentiation (Matlab File 1kB Sep30 26)
Numerical Differentiation of Analytic Functions (Acrobat (PDF) 499kB Sep30 26)

Teaching Notes and Tips

Posted in the file "Note for the instructors.pdf"

Assessment

Each group suppose to submit a group report for the class activities. We grade them with mastery based grading. If schedule permit, students can also present their learning in class.

References and Resources

Numerical Differentiation - Wikipedia
https://en.wikipedia.org/wiki/Numerical_differentiation

Lyness, James N., and Moler, Cleve B., Numerical Differentiation of Analytic Functions, SIAM J. Numerical Analysis 4, 1967, pp. 202-210. https://epubs.siam.org/doi/abs/10.1137/0704019

Squire, William, and Trapp, George, Using complex variables to estimate derivatives of real functions, SIAM Review 40, 1998, pp. 110-112.
https://epubs.siam.org/doi/abs/10.1137/S003614459631241X