IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v8y2020i11p2017-d443874.html
   My bibliography  Save this article

Fast Switch and Spline Function Inversion Algorithm with Multistep Optimization and k-Vector Search for Solving Kepler’s Equation in Celestial Mechanics

Author

Listed:
  • Daniele Tommasini

    (Applied Physics Department, School of Aeronautic and Space Engineering, Universidade de Vigo, As Lagoas s/n, 32004 Ourense, Spain)

  • David N. Olivieri

    (Computer Science Department, School of Informatics (ESEI), Universidade de Vigo, As Lagoas s/n, 32004 Ourense, Spain
    Centro de Intelixencia Artificial, La Molinera, s/n, 32004 Ourense, Spain)

Abstract

Obtaining the inverse of a nonlinear monotonic function f ( x ) over a given interval is a common problem in pure and applied mathematics, the most famous example being Kepler’s description of orbital motion in the two-body approximation. In traditional numerical approaches, this problem is reduced to solving the nonlinear equation f ( x ) − y = 0 in each point y of the co-domain. However, modern applications of orbital mechanics for Kepler’s equation, especially in many-body problems, require highly optimized numerical performance. Ongoing efforts continually attempt to improve such performance. Recently, we introduced a novel method for computing the inverse of a one-dimensional function, called the fast switch and spline inversion (FSSI) algorithm. It works by obtaining an accurate interpolation of the inverse function f − 1 ( y ) over an entire interval with a very small generation time. Here, we describe two significant improvements with respect to the performance of the original algorithm. First, the indices of the intervals for building the spline are obtained by k-vector search combined with bisection, thereby making the generation time even smaller. Second, in the case of Kepler’s equation, a multistep method for the optimized calculation of the breakpoints of the spline polynomial was designed and implemented in Cython. We demonstrate results that accurately solve Kepler’s equation for any value of the eccentricity e ∈ [ 0 , 1 − ϵ ] , with ϵ = 2.22 × 10 − 16 , which is the limiting error in double precision. Even with modest current hardware, the CPU generation time for obtaining the solution with high accuracy in a large number of points of the co-domain can be kept to around a few nanoseconds per point.

Suggested Citation

  • Daniele Tommasini & David N. Olivieri, 2020. "Fast Switch and Spline Function Inversion Algorithm with Multistep Optimization and k-Vector Search for Solving Kepler’s Equation in Celestial Mechanics," Mathematics, MDPI, vol. 8(11), pages 1-18, November.
  • Handle: RePEc:gam:jmathe:v:8:y:2020:i:11:p:2017-:d:443874
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/8/11/2017/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-7390/8/11/2017/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Arnas, David & Mortari, Daniele, 2018. "Nonlinear function inversion using k-vector," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 754-768.
    2. Sharma, Janak Raj & Arora, Himani, 2016. "A new family of optimal eighth order methods with dynamics for nonlinear equations," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 924-933.
    3. Kansal, Munish & Kanwar, V. & Bhatia, Saurabh, 2015. "New modifications of Hansen–Patrick’s family with optimal fourth and eighth orders of convergence," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 507-519.
    4. Sharifi, Somayeh & Salimi, Mehdi & Siegmund, Stefan & Lotfi, Taher, 2016. "A new class of optimal four-point methods with convergence order 16 for solving nonlinear equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 119(C), pages 69-90.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Ramandeep Behl & Arwa Jeza Alsolami & Bruno Antonio Pansera & Waleed M. Al-Hamdan & Mehdi Salimi & Massimiliano Ferrara, 2019. "A New Optimal Family of Schröder’s Method for Multiple Zeros," Mathematics, MDPI, vol. 7(11), pages 1-14, November.
    2. Arnas, David & Leake, Carl & Mortari, Daniele, 2020. "The n-dimensional k-vector and its application to orthogonal range searching," Applied Mathematics and Computation, Elsevier, vol. 372(C).
    3. Zhanlav, T. & Chuluunbaatar, O. & Ulziibayar, V., 2017. "Generating function method for constructing new iterations," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 414-423.
    4. Moin-ud-Din Junjua & Fiza Zafar & Nusrat Yasmin, 2019. "Optimal Derivative-Free Root Finding Methods Based on Inverse Interpolation," Mathematics, MDPI, vol. 7(2), pages 1-10, February.
    5. Vinay Kanwar & Puneet Sharma & Ioannis K. Argyros & Ramandeep Behl & Christopher Argyros & Ali Ahmadian & Mehdi Salimi, 2021. "Geometrically Constructed Family of the Simple Fixed Point Iteration Method," Mathematics, MDPI, vol. 9(6), pages 1-13, March.
    6. Min-Young Lee & Young Ik Kim & Beny Neta, 2019. "A Generic Family of Optimal Sixteenth-Order Multiple-Root Finders and Their Dynamics Underlying Purely Imaginary Extraneous Fixed Points," Mathematics, MDPI, vol. 7(6), pages 1-26, June.
    7. Young Hee Geum & Young Ik Kim & Beny Neta, 2018. "Developing an Optimal Class of Generic Sixteenth-Order Simple-Root Finders and Investigating Their Dynamics," Mathematics, MDPI, vol. 7(1), pages 1-32, December.
    8. Janak Raj Sharma & Ioannis K. Argyros & Sunil Kumar, 2019. "Convergence Analysis of Weighted-Newton Methods of Optimal Eighth Order in Banach Spaces," Mathematics, MDPI, vol. 7(2), pages 1-14, February.
    9. Syahmi Afandi Sariman & Ishak Hashim & Faieza Samat & Mohammed Alshbool, 2021. "Modification of Newton-Househölder Method for Determining Multiple Roots of Unknown Multiplicity of Nonlinear Equations," Mathematics, MDPI, vol. 9(9), pages 1-12, April.
    10. Ramandeep Behl & Munish Kansal & Mehdi Salimi, 2020. "Modified King’s Family for Multiple Zeros of Scalar Nonlinear Functions," Mathematics, MDPI, vol. 8(5), pages 1-17, May.
    11. Yanlin Tao & Kalyanasundaram Madhu, 2019. "Optimal Fourth, Eighth and Sixteenth Order Methods by Using Divided Difference Techniques and Their Basins of Attraction and Its Application," Mathematics, MDPI, vol. 7(4), pages 1-22, March.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:8:y:2020:i:11:p:2017-:d:443874. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.