Steffensen's Method
The video covering Steffensen's Method and Aitken's Δ² Method can be found here https://youtu.be/BTYTj0r5PZE. Example code is given in Steffensen.sml file, written in SML/NJ it has Aitken's Δ² Method and both versions of Steffensen's Method. You can run the program online from CodingGround. To run the program locally have SML/NJ installed (smlnj.org) then type sml Steffensen.smlin the terminal in the directory where Steffensen.sml is saved. If you'd rather not use the terminal use an IDE instead like Geany to run the program (change the execute option to sml %e.sml).
Reference links:
- Aitken's paper https://www.cambridge.org/core/journals/proceedings-of-the-royal-society-of-edinburgh/article/xxvon-bernoullis-numerical-solution-of-algebraic-equations/64D4A7C56F1EFEC696AF68D7870DB451
- Steffensen's paper https://www.tandfonline.com/doi/abs/10.1080/03461238.1933.10419209
- Version 2.0 https://www.sciencedirect.com/science/article/pii/S1110256X1300028X
The video showing how to derive generalized Aitken's Δ² Method and generalized Steffensen's Method can be found here. The code and documentation are given along with Fixed Point Iteration for Nonlinear Systems of Equations.
Reference Links:
- Peter Henrici "Elements of Numerical Analysis" https://archive.org/details/ElementsOfNumericalAnalysis/page/n65
- Tatsuo Noda "The Steffensen Iteration Method for Systems of Nonlinear Equations" https://projecteuclid.org/euclid.pja/1195515278 https://projecteuclid.org/euclid.pja/1195513725 https://projecteuclid.org/euclid.pja/1195514312
- Yves Nievergelt "Aitken's and Steffensen's Accelerations in Several Variables" https://link.springer.com/article/10.1007/BF01385782