Uploaded March 2019 | Updated September 2026, 2 weeks ago

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Fixed Point Iteration Method to solve Systems of Nonlinear Equations with discussion of Banach Fixed Point Theorem, finding the Jacobian, convergence, and order. Example code on GitHub: github.com/osveliz/numerical-veliz

Chapters:
00:00 Intro
00:25 Systems of Equations
00:33 Solving Nonlinear Systems
01:03 Fixed Point Iteration
01:26 Rewriting Equations
02:03 Example 1
02:23 Visualized Example
03:12 Measuring Distance and Norm
03:45 End Conditions
04:09 Different Combinations of Rewrites
04:45 When Does it Converge?
05:10 Banach Fixed Point Theorem
05:56 The Jacobian
06:48 Contraction Mapping Test
07:24 Contraction Mapping Test Examples
08:20 Notes on the Contraction Mapping Test
09:06 Order of Convergence
09:41 Exact Order
10:31 Summary
10:49 Thank You

See Video Mistakes II: The Sequel youtu.be/YEUbzqkJBf0 for a small correction.

Recommended Viewing:
Fixed Point Iteration youtu.be/OLqdJMjzib8
Fixed Point Iteration Q&A youtu.be/FyCviw2ZA2o
Generalized Aitken-Steffensen youtu.be/-x9_fSNrX3w
Generalized Newton's Method youtu.be/p0SBubUfwiI

Reference Link:
"Elements of Numerical Analysis" by Peter Henrici archive.org/details/ElementsOfNumericalAnalysis/page/n57
#FixedPointIteration #NumericalAnalysis