Uploaded August 2026 | Updated September 2026, 2 weeks ago
Wilf's Method Global Bisection for finding roots of polynomials in the complex plane. Example code hosted on GitHub github.com/osveliz/numerical-veliz
Clarification and Correction:
The V(x) test example on x^2-x-1 is used on the Sturm Sequence of the given polynomial f_n = {x^2-x-1, 2x-1, 5/4}. Therefore, the four test points result in the series:
f_n(-1) = {1, -3, 5/4}, f_n(0) = {-1, -1, 5/4}, f_n(1) = {-1, 1, 5/4}, & f_n(2) = {1, 3, 5/4}
Applying the V(x) function looks at the sign changes in the sequence so - → + counts as does + → - but not - → - nor + → +. Going from non-zero to zero and vice versa also counts. Giving these results:
V(-1) = {+,-,+} = 2, V(0) = {-, -, +} = 1, V(1) = {-, +, +} = 1, & V(2) = {+, +, +} = 0
Some of these values did not appear in the final rendering of the video.
Chapters:
0:00-0:36 Intro
0:37-0:58 Lehmer-Schur video
0:59-2:37 Wilf Search
2:38-4:49 Sturm Sequence
4:50-5:50 Fourier's Approach
5:51-6:44 Historical Context
6:45-9:01 Adapting for Complex Plane
9:02-10:02 Oscar's Thoughts
10:03-10:18 Outro
Recommended Viewing:
Lehmer-Schur youtu.be/hT0EY2rxLlQ
Bernoulli's Method youtu.be/zB_pQJWnhTc
Jenkins-Traub youtu.be/3_yxAnhLiAk
Bairstow's Method youtu.be/iUGEk6kngFw
Graeffe's Method youtu.be/92oh5gYUP7Y
Aberth-Ehrlich youtu.be/XIzCzfMDSzk
Durand-Kerner youtu.be/5JcpOj2KtWc
Horner's Method youtu.be/zEvfkSuPqWk
Generalized Bisection for Systems of Nonlinear Equations youtu.be/rMg61nfkZ3M
Bisection Method youtu.be/MlP_W-obuNg
References:
Wilf, Herbert S. "A global bisection algorithm for computing the zeros of polynomials in the complex plane." Journal of the ACM (JACM) 25.3 (1978): 415-420.
Sturm Biography mathshistory.st-andrews.ac.uk/Biographies/Sturm
Heindel, Lee E. "Integer arithmetic algorithms for polynomial real zero determination." Proceedings of the second ACM symposium on Symbolic and algebraic manipulation. 1971.
Pinkert, James R. "An exact method for finding the roots of a complex polynomial." ACM Transactions on Mathematical Software (TOMS) 2.4 (1976): 351-363.
Background music "Drifting at 432 Hz" by @UnicornHeads
#numericalanalysis #rootfinding #bisectionmethod
Wilf's Method Global Bisection for finding roots of polynomials in the complex plane. Example code hosted on GitHub github.com/osveliz/numerical-veliz
Clarification and Correction:
The V(x) test example on x^2-x-1 is used on the Sturm Sequence of the given polynomial f_n = {x^2-x-1, 2x-1, 5/4}. Therefore, the four test points result in the series:
f_n(-1) = {1, -3, 5/4}, f_n(0) = {-1, -1, 5/4}, f_n(1) = {-1, 1, 5/4}, & f_n(2) = {1, 3, 5/4}
Applying the V(x) function looks at the sign changes in the sequence so - → + counts as does + → - but not - → - nor + → +. Going from non-zero to zero and vice versa also counts. Giving these results:
V(-1) = {+,-,+} = 2, V(0) = {-, -, +} = 1, V(1) = {-, +, +} = 1, & V(2) = {+, +, +} = 0
Some of these values did not appear in the final rendering of the video.
Chapters:
0:00-0:36 Intro
0:37-0:58 Lehmer-Schur video
0:59-2:37 Wilf Search
2:38-4:49 Sturm Sequence
4:50-5:50 Fourier's Approach
5:51-6:44 Historical Context
6:45-9:01 Adapting for Complex Plane
9:02-10:02 Oscar's Thoughts
10:03-10:18 Outro
Recommended Viewing:
Lehmer-Schur youtu.be/hT0EY2rxLlQ
Bernoulli's Method youtu.be/zB_pQJWnhTc
Jenkins-Traub youtu.be/3_yxAnhLiAk
Bairstow's Method youtu.be/iUGEk6kngFw
Graeffe's Method youtu.be/92oh5gYUP7Y
Aberth-Ehrlich youtu.be/XIzCzfMDSzk
Durand-Kerner youtu.be/5JcpOj2KtWc
Horner's Method youtu.be/zEvfkSuPqWk
Generalized Bisection for Systems of Nonlinear Equations youtu.be/rMg61nfkZ3M
Bisection Method youtu.be/MlP_W-obuNg
References:
Wilf, Herbert S. "A global bisection algorithm for computing the zeros of polynomials in the complex plane." Journal of the ACM (JACM) 25.3 (1978): 415-420.
Sturm Biography mathshistory.st-andrews.ac.uk/Biographies/Sturm
Heindel, Lee E. "Integer arithmetic algorithms for polynomial real zero determination." Proceedings of the second ACM symposium on Symbolic and algebraic manipulation. 1971.
Pinkert, James R. "An exact method for finding the roots of a complex polynomial." ACM Transactions on Mathematical Software (TOMS) 2.4 (1976): 351-363.
Background music "Drifting at 432 Hz" by @UnicornHeads
#numericalanalysis #rootfinding #bisectionmethod
