Broyden's Method
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- Опубліковано 30 чер 2024
- 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-...
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 • Newton's Method for Sy...
Secant Method for Systems of Nonlinear Equations • Secant Method for Syst...
Fixed Point Iteration for Systems of Nonlinear Equations • Approximating the Jaco...
Global Newton Method • Global Newton's Method...
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 www.ams.org/mcom/1971-25-114/...
"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 www.jstor.org/stable/2236561
Background music "The Golden Present" by @JesseGallagher
#BroydensMethod #NumericalAnalysis #NonlinearSystem
Wow! Insane and clear explanation. I tried presenting a weekly research presentation for a course a couple years ago and almost had no clue what this method was about. This would have saved my life back then lol Much better understanding now, thanks for making this!
Thank you so much for this! Love this. This demystifies Broyden's method for me as it is one of the solver methods for SciPy's root function.
Great work! Keep making these. You're helping people all around the world!
I love numerical analysis!
Cool, video, thanks!!
And I bet there is likely a generalization of Halley's method for systems of nonlinear equations? Or even Householder's method?
There is for Halley. Don't know about Householder but I don't see why not.
Good subject to work
@@OscarVeliz nice! Would like to see that.
Great video! I've read Broyden's original paper and it says "The functions that require zeroing are real functions of real variables". Are you aware of any generalisations to complex-valued functions? Thanks a lot!
Rewatching the vid, I thought that the method 3 at 5:30 seems familiar and I was right. Method 3 actually is the David-Fletcher-Powell (DFP) method for unconstrained optimization. It's the earliest quasi-newton method for approximating the hessian which is symmetric so the update should be symmetric. I think there may be a typo because in Wikipedia, the middle and last terms in the inverse Jacobian update are negative and positive respectively while here in the video, it's the opposite.
I've been dabbling with optimization lately and I discovered that both Broyden's good and bad methods perform very badly in optimization problems. In the Rosenbrock function, the bad method simply blows up with nan results while the good method takes hundreds of iterations just to converge at the minimum at (1,1).
I probably did make a typo there and nice catch! I did not notice this relationship at all even after looking at DFP.
Optimization dabbling sounds like a lot of fun. Always a good time when you learn new methods.
A new video has been in the works for a long while but real life keeps getting in the way. Hoping I can get a lot done in the next few weeks.
hi, first of all thank you, I have a question I am using broyden 's method for solving indirect shooting method(optimal control), which is basically based on ode's , but using this method is actually diverging rather than converging.....
The normal Broyden method doesn't guarantee convergence. He does make the case for a variation in his original paper that can induce convergence but it isn't generally used and I didn't discuss it in the video. There are other methods that I have lessons for on this channel that do guarantee convergence like the Global Newton Method.
@@OscarVeliz ok yea actually I had to use damped newton method and it was all good
how to find norm square of matrix In Jn kindly guide me
Sir kindly guide at in vidioe at 3:57
Last step I cannot understand.
When we calculate j1 then how we calculate ||∆x||^2 in denomirator
You find the difference in the step size, then compute the L-2 norm (square root of the sum of squares), and square it. Broyden wrote it as the step size vector transposed multiplied against the step size vector. Same result.
are you 3b1b?
He and I have never been seen in the same room at the same time.
@@OscarVeliz your voice and accent are very similar to his