Newton Bisection Hybrid (Newt-Safe) @OscarVeliz

Uploaded December 2018 | Updated September 2026, 2 weeks ago

Newton Bisection Hybrid Method for root finding. Example code available at github.com/osveliz/numerical-veliz

Chapters
0:00 Intro
0:26 Viewer Request
0:49 Numerical Recipes
1:12 Numerical Methods That Work
1:54 Motivation Examples
3:04 Problems with Newton Recap
3:17 Newt-Safe Basic Algorithm
4:00 Newt-Safe Basic Examples
4:53 Additional Conditions
5:39 ΔX Condition Explanation
6:33 ΔX Example
7:18 Newt-Safe without Interval
7:55 Other Hybrid Methods
8:33 Notes and Summary
8:57 Thank You

Correction: The bracket size test at 5:26 used DeltaX which should be DeltaX_old. A video covering this mistake can be found here youtu.be/4jw0cjddmB8

Further Viewing:
Bisection Method youtu.be/MlP_W-obuNg
Newton's Method youtu.be/E24zUEKqgwQ
Newton's Method Interval of Convergence youtu.be/zyXRo8Qjj0A
Newton Fractal youtu.be/MWD2A0Vg2V0
Halley's Method youtu.be/3WiVGSy_084
Householder's Method youtu.be/F9DFewL0mho

Reference Links:
Based of off NewtSafe from Marc Spiegelman https://www.ldeo.columbia.edu/~mspieg/e4300/BlankPDFs/Lecture06_blank.pdf
rtsafe from Numerical Recipes http://numerical.recipes/
Numerical Methods That Work by Forman S. Acton books.google.com/books?id=cGnSMGSE5Y4C&lpg=PA41&pg=PA51#v=onepage&q&f=false

#BisectionMethod #NewtonsMethod #NumericalAnalysis

Approximating the Jacobian: Finite Difference Method for Systems of Nonlinear Equations

$Newton Fractals Using Newtons Method to create Fractals by plotting convergence behavior on the complex plane. Functions used in this video include arctan(z), z^3-1, sin(z), z^8-15z^4+16. Example code and images available at https://github.com/osveliz/numerical-veliz Correction: The derivative of arctan(x) should be 1/(1+x^2). This error is only impacts the slides, not the numerical examples nor fractals which used the correct derivative. A video covering this mistake can be found here https://youtu.be/4jw0cjddmB8 Chapters 0:00 Intro 0:16 Convergence Interval Recap 0:42 Imaginary Numbers 1:05 Newtons Method in Complex Plane 1:29 Basin of Convergence 1:38 Arctangent Fractal 1:50 Newton Fractal 2:10 Why Fractals Emerge 2:54 Example z^3-1 4:07 Example sin(z) 4:33 Example z^8+15z^4-16 4:50 Generalized Newton Fractal 5:05 Generalized Newton Fractal Examples 6:16 Summary 6:42 Thank You Further Viewing: Newtons Method https://youtu.be/E24zUEKqgwQ Newtons Method Interval of Convergence https://youtu.be/zyXRo8Qjj0A Newton Bisection Hybrid (Newt-Safe) https://youtu.be/FD3BPTMGJds Laguerres Method and Fractal https://youtu.be/blOARV4lnIM Halleys Method and Fractal https://youtu.be/3WiVGSy_084 Generalized Newtons Method https://youtu.be/p0SBubUfwiI Further Reading: Numerical Recipes http://numerical.recipes/ Wikipedia https://en.wikipedia.org/wiki/Newton_fractal #Fractal #NewtonsMethod #NumericalAnalysis$