How Newton's method solves multiple equations at once

This video explains how Newton's method (also called the Newton-Raphson method) can solve more than one equation simultaneously.
MDO Lab: mdolab.engin.u...
Engineering Design Optimization: mdobook.github...
Animations done using Manim: docs.manim.com...
Video by Eytan Adler
#some2

[@johnjasa]

Eytan, amazing work! This type of visualization and explanation is so valuable to the community. Thank you for it!

[@monsieur910]

This video is great. It just came up on my feed but I was smiling through the whole video. I used this method extensively during my phd :) Anyway I subscribe to your channel.

[@alexkleb1941]

What do you do when the two lines created by the tangent planes are parallel and either always intersect or never intersect?

[@MDOLab]

Great question Alex! That represents an unsolvable (i.e., singular) linear system. Newton's method would not be able to continue, though you could imagine workarounds where the current states are perturbed slightly with the hope of finding a solvable point nearby. When applying Newton's method to numerical codes in practice, finding a point that is exactly singular is unlikely. However, it can come close in which case Newton's method would return a bad step, but to somewhere that is hopefully better conditioned.

[@isaiahfleischer8173]

Great video Eytan!