Bairstow's Method

Bairstow's Method for finding the roots of polynomials including complex roots. Discussion of method derivation, relation to synthetic division of two variables, stopping condition, selection of initial values, fractals, and historical context. Submission for Summer of Math Exposition 2 contest by @3blue1brown. Example code hosted on GitHub https://github.com/osveliz/numerical-veliz Chapters: 00:00 Intro 00:36 Factoring with monomial 01:13 Factoring with quadratic 01:40 Synthetic Division 1 variable 02:16 Synthetic Division 2 variables 03:22 Solving for roots 03:50 Different u and v 04:24 Describing Notation 05:22 Solving Nonlinear System 07:01 Another Synthetic Division 07:45 Updating u & v 08:27 Bairstow Iteration Example 08:56 Bairstow Iteration Example 2 09:22 Note on Quadratic Equation 09:42 Bairstow Full Algorithm 10:51 Complex Roots Example 11:33 Bairstow Fractals 12:23 Picking u & v 12:46 Henrici Starting Values 13:08 Bairstow's Original Problem 14:40 Oscar's Notes 15:24 Outro Recommended Viewing: Horner's Method https://youtu.be/zEvfkSuPqWk Graeffe's Method https://youtu.be/92oh5gYUP7Y Generalized Newton's Method https://youtu.be/p0SBubUfwiI Reference links: "Applied Aerodynamics" by Bairstow https://books.google.com/books?id=GIUQAQAAMAAJ "Elements of Numerical Analysis" by Henrici https://archive.org/details/elements-of-numerical-analysis-by-peter-henrici "A Modified Bairstow Method for Multiple Zeros of a Polynomial" by F. M. Carrano https://doi.org/10.1090/S0025-5718-1973-0334492-5 Background music "Drifting at 432 Hz" by @UnicornHeads #SoME2 #NumericalAnalysis #BairstowsMethod