Uploaded June 2021 | Updated September 2026, 2 weeks ago
Broyden's Method for solving systems of nonlinear equations. Lesson covers motivation, history, examples, discussion, and order of this Quasi-Newton Method. It also explains the "Good" and "Bad", as well as the third version of the method. Example code hosted on GitHub github.com/osveliz/numerical-veliz
Chapters:
0:00 Intro
0:22 Newton's Method According to Broyden
1:08 Nonlinear System Example
1:18 Newton's Method Example
1:28 Analyzing Newton's Behavior
2:07 Solving for J
2:51 Broyden's Approach
3:25 Almost Broyden's Method
4:13 Solving for the Inverse
4:46 Broyden's "Good" Method
5:11 More Options
5:35 Broyden's "Bad" Method
6:07 Generalizing Root-Finding
6:32 The Case for Secant Method
6:52 The Case for Newton's Method
7:17 Broyden's Take
8:11 The question of Order
8:31 Oscar's Notes
9:03 Outro
Recommended Viewing:
Newton's Method for Systems of Nonlinear Equations youtu.be/p0SBubUfwiI
Secant Method for Systems of Nonlinear Equations youtu.be/p2OPlnHJPNI
Fixed Point Iteration for Systems of Nonlinear Equations youtu.be/G25sGOWkOuQ
Global Newton Method youtu.be/BGZfHxzZ-7c
Reference links:
"A class of methods for solving nonlinear simultaneous equations" by Charles G. Broyden doi.org/10.2307/2003941
"A faster Broyden method" by Kvaalen doi.org/10.1007/BF01931297
"On Some Methods Based on Broyden's Secant Approximation to the Hessian" by Dennis hdl.handle.net/1813/5946
"On the discovery of the 'good Broyden' method" by C. G. Broyden doi.org/10.1007/s101070050111
"The convergence of an algorithm for solving sparse nonlinear systems" by C. G. Broyden ams.org/mcom/1971-25-114/S0025-5718-1971-0297122-5/S0025-5718-1971-0297122-5.pdf
"On the Local and Superlinear Convergence of Quasi-Newton Methods" by Broyden, Dennis, and Moré doi.org/10.1093/imamat/12.3.223
"Some Convergence Properties of Broyden's Method" by Gay doi.org/10.1137/0716047
"Adjustment of an Inverse Matrix Corresponding to a Change in One Element of a Given Matrix" by Sherman and Morrison jstor.org/stable/2236561
Background music "The Golden Present" by @JesseGallagher
#BroydensMethod #NumericalAnalysis #NonlinearSystem
Broyden's Method for solving systems of nonlinear equations. Lesson covers motivation, history, examples, discussion, and order of this Quasi-Newton Method. It also explains the "Good" and "Bad", as well as the third version of the method. Example code hosted on GitHub github.com/osveliz/numerical-veliz
Chapters:
0:00 Intro
0:22 Newton's Method According to Broyden
1:08 Nonlinear System Example
1:18 Newton's Method Example
1:28 Analyzing Newton's Behavior
2:07 Solving for J
2:51 Broyden's Approach
3:25 Almost Broyden's Method
4:13 Solving for the Inverse
4:46 Broyden's "Good" Method
5:11 More Options
5:35 Broyden's "Bad" Method
6:07 Generalizing Root-Finding
6:32 The Case for Secant Method
6:52 The Case for Newton's Method
7:17 Broyden's Take
8:11 The question of Order
8:31 Oscar's Notes
9:03 Outro
Recommended Viewing:
Newton's Method for Systems of Nonlinear Equations youtu.be/p0SBubUfwiI
Secant Method for Systems of Nonlinear Equations youtu.be/p2OPlnHJPNI
Fixed Point Iteration for Systems of Nonlinear Equations youtu.be/G25sGOWkOuQ
Global Newton Method youtu.be/BGZfHxzZ-7c
Reference links:
"A class of methods for solving nonlinear simultaneous equations" by Charles G. Broyden doi.org/10.2307/2003941
"A faster Broyden method" by Kvaalen doi.org/10.1007/BF01931297
"On Some Methods Based on Broyden's Secant Approximation to the Hessian" by Dennis hdl.handle.net/1813/5946
"On the discovery of the 'good Broyden' method" by C. G. Broyden doi.org/10.1007/s101070050111
"The convergence of an algorithm for solving sparse nonlinear systems" by C. G. Broyden ams.org/mcom/1971-25-114/S0025-5718-1971-0297122-5/S0025-5718-1971-0297122-5.pdf
"On the Local and Superlinear Convergence of Quasi-Newton Methods" by Broyden, Dennis, and Moré doi.org/10.1093/imamat/12.3.223
"Some Convergence Properties of Broyden's Method" by Gay doi.org/10.1137/0716047
"Adjustment of an Inverse Matrix Corresponding to a Change in One Element of a Given Matrix" by Sherman and Morrison jstor.org/stable/2236561
Background music "The Golden Present" by @JesseGallagher
#BroydensMethod #NumericalAnalysis #NonlinearSystem
