Uploaded September 2025 | Updated September 2026, 2 weeks ago

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Lehmer-Schur method for finding zeros of polynomials in the complex plane, as well as discussion of the Schur-Cohn test for determining if a polynomial contains a root in a circle given a radius and center. Includes examples and visual explanations as well as suggestions regarding hybrid algorithms. Example code hosted on GitHub github.com/osveliz/numerical-veliz

Chapters:
00:00 Intro
00:30 Quadratic Equation
01:37 Issue with Solving Directly
01:54 Solving Numerically
02:30 Bisection
03:16 Finding Complex Roots
03:50 Lehmer-Schur Intro
04:07 Find the First Ring
06:03 Search the Ring
08:42 Circle Test
12:15 A Fresh Start
12:30 Hybrid Approach
12:48 Bisection Support
13:05 Oscar's Notes
13:53 Outro

Recommended Viewing:
Horner's Method youtu.be/zEvfkSuPqWk
Jenkins-Traub youtu.be/3_yxAnhLiAk
Bairstow's Method youtu.be/iUGEk6kngFw
Graeffe's Method youtu.be/92oh5gYUP7Y
Bernoulli's Method youtu.be/zB_pQJWnhTc
Newton-Bisection Hybrid youtu.be/FD3BPTMGJds

Reference links:
"A machine method for solving polynomial equations" by D. H. Lehmer doi.org/10.1145/321062.321064
"Search procedures for polynomial equation solving." by D. H. Lehmer https://apps.dtic.mil/sti/tr/pdf/AD0698798.pdf
"Improvements on the Lehmer-Schur Root Detection Method" by Dan Loewenthal doi.org/10.1006/jcph.1993.1209

Background music "Drifting at 432 Hz" by @UnicornHeads

#numericalanalysis #rootfinding #polynomials #some4 #some