Newton's Method for Systems of Nonlinear Equations

Generalized Newton's method for systems of nonlinear equations. Lesson goes over numerically solving multivariable nonlinear equations step-by-step with visual examples and explanation of the Jacobian, the backslash operator, and the inverse Jacobian. Example code in MATLAB / GNU Octave on GitHub: http://github.com/osveliz/numerical-veliz

Chapters
0:00 Intro
0:12 Prerequisites
0:32 Background
0:58 Setup
1:54 Jacobian
2:55 Historical Context
5:11 Newton's Method Example Step-by-Step
6:57 End Condition
7:12 Numerical Example in Table
7:33 Newton's Method with Backslash
7:53 Newton's Method with Inverse Jacobian
8:16 MATLAB / GNU Octave
9:01 Newton Fractals
10:51 3D Fractal
11:37 Historical Optimization Newton's Method
12:18 Oscar's Notes
12:59 Thank You

Recommended Viewing:
Newton's Method https://youtu.be/E24zUEKqgwQ
Newton Fractals https://youtu.be/MWD2A0Vg2V0
Newton-Bisection Hybrid https://youtu.be/FD3BPTMGJds
Video Mistakes https://youtu.be/4jw0cjddmB8
Fixed Point Iteration Systems of Equations https://youtu.be/xa2vUsYJD-c
Generalized Aitken-Steffensen Method https://youtu.be/-x9_fSNrX3w
Secant Method for Systems of Nonlinear Equations https://youtu.be/p2OPlnHJPNI
Generalized Finite Difference https://youtu.be/G25sGOWkOuQ
Broyden's Method https://youtu.be/UlLGzW-KjHs
Global Newton Method https://youtu.be/BGZfHxzZ-7c

@3Blue1Brown 's series on @Khan Academy:
Jacobian Prerequisite https://www.youtube.com/watch?v=VmfTXVG9S0U
The Jacobian Matrix https://www.youtube.com/watch?v=bohL918kXQk
Computing a Jacobian Matrix https://www.youtube.com/watch?v=CGbBbH1e7Yw
Local Linearity for a Multivariable Function https://www.youtube.com/watch?v=Vnga_psnCAo

References:
"Historical Development of the Newton–Raphson Method" by Ypma https://doi.org/10.1137/1037125
"Thomas Simpson and ‘Newton's method of approximation’: an enduring myth" by Kollerstrom https://doi.org/10.1017/S0007087400029150
"Essays on Several Curious and Useful Subjects, in Speculative and Mix'd Mathematicks" by Simpson https://books.google.com/books?id=3HpYAAAAcAAJ&dq
"A New Treatise of Fluxions" by Simpson https://books.google.com/books?id=GGZbUevBqskC

Background music "The Golden Present" by @Jesse Gallagher
#NewtonsMethod #NumericalAnalysis #NonlinearSystem

Видео Newton's Method for Systems of Nonlinear Equations канала Oscar Veliz