Solving High-Dimensional Nonlinear Equations with Infinite Solutions by the SVM Visualization Method

Conference paper
Part of the Lecture Notes in Networks and Systems book series (LNNS, volume 283)


There are many standard mathematical methods for solving nonlinear equations. But when it comes to equations with infinite solutions in high dimension, the results from current methods are quite limited. Usually these methods apply to differentiable functions only and have to satisfy some conditions to converge during the iteration. Even if they converge, only one single root is found at a time. However, using the features of SVM, we present a simple fast method which could tell the distribution of these infinite solutions and is capable of finding approximation of the roots with accuracy up to at least \(10^{-12}\). In the same time, we could also have a visual understanding about these solutions.


High-dimensional nonlinear equations Infinite solutions SVM 


  1. 1.
    Radi, B., El Hami, A.: Advanced Numerical Methods with Matlab 2 Resolution of Nonlinear. Differential and Partial Differential Equations. John Wiley & Sons, Incorporated (2018)Google Scholar
  2. 2.
    Wall, D.D.: The order of an iteration formula. Math. Comput. 10(55), 167–168 (1956). Scholar
  3. 3.
    Wegstein, J.H.: Accelerating convergence of iterative processes. Comm. ACM 1(6), 9–13 (1958). Scholar
  4. 4.
    Gutzler, C.H.: An iterative method of Wegstein for solving simultaneous nonlinear equations. (1959)Google Scholar
  5. 5.
    Ezquerro, J.A., Grau, A., Grau-Sánchez, M., Hernández, M.A., Noguera, M.: Analysing the efficiency of some modifications of the secant method. Comput. Math. Appl. 64(6), 2066–2073 (2012).
  6. 6.
    Liu, G., Nie, C., Lei, J.: A novel iterative method for nonlinear equations. IAENG Int. J. Appl. Math. 48, 44–448 (2018)Google Scholar
  7. 7.
    Kumar, M., Singh, A.K., Srivastava, A.: Various newtontype iterative methods for solving nonlinear equations. J. Egypt. Math. Soc. 21(3), 334–339 (2013).
  8. 8.
    Alefeld, G.: On the convergence of Halley’s method. Am. Math. Mon. 88(7), 530 (1981). Scholar
  9. 9.
    George, H.: Brown: on Halley’s variation of newton’s method. Am. Math. Mon. 84(9), 726 (1977). Scholar
  10. 10.
    Levin, Y., Ben-Israel, A.: Directional Newton methods in n variables. Math. Comput. 71(237), 251–263 (2001). Scholar
  11. 11.
    An, H.-B., Bai, Z.-Z.: Directional secant method for nonlinear equations. J. Comput. Appl. Math. 175(2), 291–304 (2005). Scholar
  12. 12.
    An, H.-B., Bai, Z.-Z.: Math. Num. Sin. 26(4), 385–400 (2004)Google Scholar

Copyright information

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022

Authors and Affiliations

  1. 1.National Chengchi UniversityTaipei CityTaiwan R.O.C.

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