Uploaded February 2025 | Updated September 2026, 1 week ago
Bernoulli's Method for finding zeros of polynomials using only coefficients as well as discussion of the Quotient-Difference Method extension to finding all roots simultaneously. Includes examples, proofs, and visual explanations as well as drawbacks and historical context. Example code hosted on GitHub github.com/osveliz/numerical-veliz
Chapters:
00:00 Intro
00:24 History
01:10 Bernoulli's Method
01:36 Examples
04:06 Why does this work?
05:16 Chage starting value?
05:40 Converge on largest
06:48 Picking starting x values
07:26 Bernoulli Properties
07:50 Finding Smallest Root
09:40 Speed Up Convergence
10:40 Bernoulli with Aitken
11:42 Aitken's Paper
12:15 QD Algorithm w/ Examples
13:34 What's with e and q?
13:55 Properties of QD
14:20 Oscar's Notes
15:04 Outro
Recommended Viewing:
Steffensen's Method with Aitken's Δ² youtu.be/BTYTj0r5PZE
Jenkins-Traub youtu.be/3_yxAnhLiAk
Bairstow's Method youtu.be/iUGEk6kngFw
Graeffe's Method youtu.be/92oh5gYUP7Y
Reference links:
Wikipedia: Daniel Bernoulli en.wikipedia.org/wiki/Daniel_Bernoulli
Introductio in analysin infinitorum, Book I, Chap.XVII, Lausanne by Euler (1748) doi.org/10.1007/978-1-4612-1021-4_17
Traité de la résolution des équations numériques des tous les degrés, Note 6, also Oeuvres by Lagrange, J.L. (1798) books.google.com/books?id=eo0AAAAAMAAJ&ots=SjO4thgbx-&dq=Trait%C3%A9%20de%20la%20r%C3%A9solution%20des%20%C3%A9quations%20num%C3%A9riques%20des%20tous%20les%20degr%C3%A9s%201879.&lr&pg=PA3#v=onepage&q&f=false
XXV.—On Bernoulli's Numerical Solution of Algebraic Equations by A. C. Aitken doi.org/10.1017/S0370164600022070
H. Der Quotienten-Differenzen-Algorithmus by Rutishauser (1954) doi.org/10.1007/BF01600331
Elements of numerical analysis by Henrici (1964) archive.org/details/elements-of-numerical-analysis-by-peter-henrici
Numerical Methods for Roots of Polynomials-Part II by McNamee & Pan doi.org/10.1016/B978-0-444-52730-1.00004-7
Numerical analysis and computation theory and practice by Blum (1972) archive.org/details/numericalanalysi0000blum/page/n3/mode/2up
Finding zeros of a polynomial by the Q-D algorithm by Henrici & Watkins doi.org/10.1145/365559.365619
Further contributions to solution of simultaneous linear equations and determination of eigenvalues, vol. 49.: United States Department of Commerce, National Bureau of Standards, 1958.
babel.hathitrust.org/cgi/pt?id=uiug.30112007252650&seq=441
Background music "Drifting at 432 Hz" by @UnicornHeads
#numericalanalysis #rootfinding #polynomials
Bernoulli's Method for finding zeros of polynomials using only coefficients as well as discussion of the Quotient-Difference Method extension to finding all roots simultaneously. Includes examples, proofs, and visual explanations as well as drawbacks and historical context. Example code hosted on GitHub github.com/osveliz/numerical-veliz
Chapters:
00:00 Intro
00:24 History
01:10 Bernoulli's Method
01:36 Examples
04:06 Why does this work?
05:16 Chage starting value?
05:40 Converge on largest
06:48 Picking starting x values
07:26 Bernoulli Properties
07:50 Finding Smallest Root
09:40 Speed Up Convergence
10:40 Bernoulli with Aitken
11:42 Aitken's Paper
12:15 QD Algorithm w/ Examples
13:34 What's with e and q?
13:55 Properties of QD
14:20 Oscar's Notes
15:04 Outro
Recommended Viewing:
Steffensen's Method with Aitken's Δ² youtu.be/BTYTj0r5PZE
Jenkins-Traub youtu.be/3_yxAnhLiAk
Bairstow's Method youtu.be/iUGEk6kngFw
Graeffe's Method youtu.be/92oh5gYUP7Y
Reference links:
Wikipedia: Daniel Bernoulli en.wikipedia.org/wiki/Daniel_Bernoulli
Introductio in analysin infinitorum, Book I, Chap.XVII, Lausanne by Euler (1748) doi.org/10.1007/978-1-4612-1021-4_17
Traité de la résolution des équations numériques des tous les degrés, Note 6, also Oeuvres by Lagrange, J.L. (1798) books.google.com/books?id=eo0AAAAAMAAJ&ots=SjO4thgbx-&dq=Trait%C3%A9%20de%20la%20r%C3%A9solution%20des%20%C3%A9quations%20num%C3%A9riques%20des%20tous%20les%20degr%C3%A9s%201879.&lr&pg=PA3#v=onepage&q&f=false
XXV.—On Bernoulli's Numerical Solution of Algebraic Equations by A. C. Aitken doi.org/10.1017/S0370164600022070
H. Der Quotienten-Differenzen-Algorithmus by Rutishauser (1954) doi.org/10.1007/BF01600331
Elements of numerical analysis by Henrici (1964) archive.org/details/elements-of-numerical-analysis-by-peter-henrici
Numerical Methods for Roots of Polynomials-Part II by McNamee & Pan doi.org/10.1016/B978-0-444-52730-1.00004-7
Numerical analysis and computation theory and practice by Blum (1972) archive.org/details/numericalanalysi0000blum/page/n3/mode/2up
Finding zeros of a polynomial by the Q-D algorithm by Henrici & Watkins doi.org/10.1145/365559.365619
Further contributions to solution of simultaneous linear equations and determination of eigenvalues, vol. 49.: United States Department of Commerce, National Bureau of Standards, 1958.
babel.hathitrust.org/cgi/pt?id=uiug.30112007252650&seq=441
Background music "Drifting at 432 Hz" by @UnicornHeads
#numericalanalysis #rootfinding #polynomials
