Nonlinear Solver Lab
Japanese page

Update log

  • 2024/2/21 : [update]
    The W4 paper is featured on Research Features Magazine!
...
  • 2023/12/11 : [upload][poster]
    How to numerically solve differential equations?
  • 2023/7/10 : [upload][docs]
    on an efficient stopping criteria for the W4 method
  • 2023/4/7 : [upload][docs]
    on the criterion to stop the W4 iteration
  • 2023/3/20 : [update]
    Our paper on a cooling model of rotating White Dwarfs has been accepted in Monthly Notices of the Royal Astronomical Society!
  • 2023/1/23 : [update]
    Our paper on the Lagrangian construction of relativistic rotating stars has been accepted in Monthly Notices of the Royal Astronomical Society!
  • 2022/10/18 : [update]
    Our new paper on the construction of rotating stars has been submitted to arXiv.
  • 2022/9/17 : [update]
    Our paper on the W4 method has been accepted in Applied Numerical Mathematics!
  • 2022/5/9 : [upload][docs] on the W4 matrix decomposition
  • 2022/4/23 : [upload][docs] W4 manual
  • 2022/4/22 : [upload] [demo] W4SV method
  • 2022/2/24 : [update] Contact Information
  • 2022/1/22 : [upload] [demo] Lane-Emden Problem
  • 2022/1/16 : [upload] [demo] Problem Analyzer
  • 2022/1/10 : [upload] [demo] Fujisawa Problem
  • 2022/1/9 : [upload] W4 WEB started.

Overview

  • The aim of this web page is to introduce a by-product of our work, a multi-dimensional root-finder for the system of nonlinear equations.
  • The system of nonlinear equations can be solved by the well-known Newton-Raphson method which finds the root of nonlinear system equations when one starts from an approximate solution close to the solution.
  • Based on the Newton-Raphson method, many extensions have been invented such as quasi-Newton methods, while many problems in physics, astrophysics, and engineerings remain unsolved so far because of the failure of the Newton-type method.
  • We have recently proposed a new method(W4 method) which has the wider convergence property and shown it is an extension of the Newton-Raphson method.

Demonstrations & Downloads

  • To demonstrate how the W4 method works, we show Python notebook below.
Name of Problem (HTML) (ipynb)
Fujisawa Problem by the Newton-Raphson method and the W4 method HTML Notebook
Problem Analyzer by W4 (to check how difficult your problem is) HTML Notebook
Lane-Emden equation by W4 HTML Notebook
Singularity Avoiding Root-Finder (W4SV) HTML Notebook

Documents

  • Getting Started.
  • How important is the matrix decomposition in the W4 method?
  • When should the W4 iteration be stopped? ( I , II, III)
  • How to solve differential equations ( for students and researchers in different fields )
  • At this moment, we have the following papers.
    • "The W4 method: a new multi-dimensional root-finding scheme for nonlinear systems of equations", H.Okawa, K.Fujisawa, Y.Yamamoto, R.Hirai, N.Yasutake, H.Nagakura, S.Yamada, Applied Numerical Mathematics 183, 157-172, 2023, Paper
    • "Effects of rotation and magnetic field on the revival of a stalled shock in supernova explosions", K.Fujisawa, H.Okawa, Y.Yamamoto, S.Yamada, APJ. 872, 155(2019), arXiv page
    • "A novel Lagrangian formulation to construct relativistic rotating stars: Towards its application to their evolution calculations", H.Okawa, K.Fujisawa, N.Yasutake, M.Ogata, Y.Yamamoto, S.Yamada, Monthly Notices of the Royal Astronomical Society, stad075, 2023, Paper
    • "A Lagrangian construction of rotating star models", M.Ogata, H.Okawa, K.Fujisawa, N.Yasutake, Y.Yamamoto, S.Yamada, Monthly Notices of the Royal Astronomical Society, stad647, 2023, Paper
    • "Singularity-Avoiding Multi-Dimensional Root-Finder", H.Okawa, K.Fujisawa, Y.Yamamoto, N.Yasutake, M.Ogata, S.Yamada, arXiv[math.NA],2204.09941, arXiv page
    • "W4 – a multi-dimensional root-finding method for nonlinear equations", H.Okawa, Research Features Magazine, 151, 142-145, article

Acknowledgements

  • This work is in collaboration with K.Fujisawa, R.Hirai, R. Magno, H.Nagakura, M.Ogata, S.Yamada, Y.Yamamoto, N.Yasutake.
  • H.O. would like to thank all supports from Waseda Institute for Advanced Study(WIAS).
  • This work was supported by JSPS KAKENHI Grant Number JP17K18792, JP20K03951, JP20K03953, JP20K14512, JP20H04728, JP20H04742 and Waseda University Grant for Special Research Projects (Project Number:2019C-640).

Contact

  • Email: h.okawa_@_aoni.waseda.jp
  • If you have any question or if you think the W4 method may be useful for your problem, please feel free to contact us since we have already applied it to astrophysical problems in our research field. (Could you change "_@_" to "@" in the above email address?)