Skip to main content

Connections Between Power Series Methods and Automatic Differentiation

  • Conference paper
  • First Online:
Recent Advances in Algorithmic Differentiation

Abstract

There is a large overlap in the work of the Automatic Differentiation community and those whose use Power Series Methods. Automatic Differentiation is predominately applied to problems involving differentiation, and Power series began as a tool in the ODE setting. Three examples are presented to highlight this overlap, and several interesting results are presented.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Apostol, T.: Calculating higher derivatives of inverses. Amer. Math. Monthly 107(8), 738–741 (2000)

    Google Scholar 

  2. Carothers, D., Parker, G.E., Sochacki, J., Warne, P.G.: Some properties of solutions to polynomial systems of differential equations. Electronic Journal of Differential Equations 2005, 1–18 (2005)

    Google Scholar 

  3. Fehlberg, E.: Numerical integration of differential equations by power series expansions, illustrated by physical examples. Tech. Rep. NASA-TN-D-2356, NASA (1964)

    Google Scholar 

  4. Gofen, A.: The ordinary differential equations and automatic differentiation unified. Complex Variables and Elliptic Equations 54, 825–854 (2009)

    Google Scholar 

  5. Liu, J., Parker, G.E., Sochacki, J., Knutsen, A.: Approximation methods for integrodifferential equations. Proceedings of the International Conference on Dynamical Systems and Applications, III pp. 383–390 (2001)

    Google Scholar 

  6. Liu, J., Sochacki, J., Dostert, P.: Chapter 16: Singular perturbations and approximations for integrodifferential equations. In: S. Aizicovici, N.H. Pavel (eds.) Differential Equations and Control Theory. CRC press (2001). ISBN: 978-0-8247-0681-4

    Google Scholar 

  7. Neidinger, R.: Introduction to automatic differentiation and matlab object-oriented programming. SIAM Review 52(3), 545–563 (2010)

    Google Scholar 

  8. Parker, G.E., Sochacki, J.: Implementing the Picard iteration. Neural, Parallel Sci. Comput. 4(1), 97–112 (1996)

    Google Scholar 

  9. Parker, G.E., Sochacki, J.: A Picard-Maclaurin theorem for initial value PDE’s. Abstract Analysis and its Applications 5, 47–63 (2000)

    Google Scholar 

  10. Perlin, I., Reed, C.: The application of a numerical integation procedure developed by Erwin Fehlberg to the restricted problem of three bodies. Tech. Rep. NAS8-11129, NASA (1964)

    Google Scholar 

  11. Sochacki, J.: Polynomial ordinary differential equations – examples, solutions, properties. Neural Parallel & Scientific Computations 18(3-4), 441–450 (2010)

    Google Scholar 

  12. Warne, P.G., Warne, D.P., Sochacki, J.S., Parker, G.E., Carothers, D.C.: Explicit a-priori error bounds and adaptive error control for approximation of nonlinear initial value differential systems. Comput. Math. Appl. 52(12), 1695–1710 (2006). DOI http://dx.doi.org/10.1016/ j.camwa.2005.12.004

Download references

Acknowledgements

The authors would like to thank the reviewers and the editor for improving this paper with their comments and suggestions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Roger J. Thelwell .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Carothers, D.C. et al. (2012). Connections Between Power Series Methods and Automatic Differentiation. In: Forth, S., Hovland, P., Phipps, E., Utke, J., Walther, A. (eds) Recent Advances in Algorithmic Differentiation. Lecture Notes in Computational Science and Engineering, vol 87. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30023-3_16

Download citation

Publish with us

Policies and ethics