Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Computing Riemann theta functions

Authors: Bernard Deconinck, Matthias Heil, Alexander Bobenko, Mark van Hoeij and Marcus Schmies
Translated by:
Journal: Math. Comp. 73 (2004), 1417-1442
MSC (2000): Primary 14K25, 30E10, 33F05, 65D20
Published electronically: December 19, 2003
MathSciNet review: 2047094
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Riemann theta function is a complex-valued function of $g$ complex variables. It appears in the construction of many (quasi-)periodic solutions of various equations of mathematical physics. In this paper, algorithms for its computation are given. First, a formula is derived allowing the pointwise approximation of Riemann theta functions, with arbitrary, user-specified precision. This formula is used to construct a uniform approximation formula, again with arbitrary precision.

References [Enhancements On Off] (What's this?)

  • 1. Handbook of mathematical functions, with formulas, graphs and mathematical tables, Edited by Milton Abramowitz and Irene A. Stegun. Fifth printing, with corrections. National Bureau of Standards Applied Mathematics Series, Vol. 55, National Bureau of Standards, Washington, D.C., (for sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 20402), 1966. MR 0208798
  • 2. M. Babich and A. Bobenko, Willmore tori with umbilic lines and minimal surfaces in hyperbolic space, Duke Math. J. 72 (1993), no. 1, 151–185. MR 1242883,
  • 3. E. D. Belokolos, A. I. Bobenko, V. Z. Enol'skii, A. R. Its, and V. B. Matveev, Algebro-geometric approach to nonlinear integrable problems, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1994.
  • 4. A. I. Bobenko, All constant mean curvature tori in 𝑅³,𝑆³,𝐻³ in terms of theta-functions, Math. Ann. 290 (1991), no. 2, 209–245. MR 1109632,
  • 5. A. I. Bobenko and L. A. Bordag, Periodic multiphase solutions of the Kadomsev-Petviashvili equation, J. Phys. A 22 (1989), no. 9, 1259–1274. MR 994366
  • 6. Bernard Deconinck and Mark van Hoeij, Computing Riemann matrices of algebraic curves, Phys. D 152/153 (2001), 28–46. Advances in nonlinear mathematics and science. MR 1837895,
  • 7. B. A. Dubrovin, Theta-functions and nonlinear equations, Uspekhi Mat. Nauk 36 (1981), no. 2(218), 11–80 (Russian). With an appendix by I. M. Krichever. MR 616797
  • 8. B. A. Dubrovin, Ron Flickinger, and Harvey Segur, Three-phase solutions of the Kadomtsev-Petviashvili equation, Stud. Appl. Math. 99 (1997), no. 2, 137–203. MR 1458597,
  • 9. David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364
  • 10. M. Heil, Numerical tools for the study of finite gap solutions of integrable systems, Ph.D. thesis, Technischen Universität Berlin, 1995.
  • 11. Jun-ichi Igusa, Theta functions, Springer-Verlag, New York-Heidelberg, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 194. MR 0325625
  • 12. C. G. J Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum, Königsberg (1829).
  • 13. A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), no. 4, 515–534. MR 682664,
  • 14. David Mumford, Tata lectures on theta. I, Progress in Mathematics, vol. 28, Birkhäuser Boston, Inc., Boston, MA, 1983. With the assistance of C. Musili, M. Nori, E. Previato and M. Stillman. MR 688651
  • 15. David Mumford, Tata lectures on theta. II, Progress in Mathematics, vol. 43, Birkhäuser Boston, Inc., Boston, MA, 1984. Jacobian theta functions and differential equations; With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura. MR 742776
  • 16. David Mumford, Tata lectures on theta. III, Progress in Mathematics, vol. 97, Birkhäuser Boston, Inc., Boston, MA, 1991. With the collaboration of Madhav Nori and Peter Norman. MR 1116553
  • 17. G. F. B. Riemann, Theorie der Abel'schen functionen, Journal für reine und angewandte Mathematik 54 (1857), 101-155.
  • 18. C. L. Siegel, Vorlesungen über ausgewählte Kapitel der Funktionentheorie. Teil III, Mathematisches Institut der Universität, Göttingen, 1966 (German). MR 0476761
  • 19. Brigitte Vallée, A central problem in the algorithmic geometry of numbers: lattice reduction, CWI Quarterly 3 (1990), no. 2, 95–120. MR 1076479
  • 20. E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge University Press, Cambridge, 1902.
  • 21. W. Wirtinger, Untersuchungen über thetafunctionen, B. G. Teubner, Leipzig, 1895.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 14K25, 30E10, 33F05, 65D20

Retrieve articles in all journals with MSC (2000): 14K25, 30E10, 33F05, 65D20

Additional Information

Bernard Deconinck
Affiliation: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523-1874

Matthias Heil
Affiliation: Fachbereich Mathematik, Technische Universität Berlin, Strasse des 17.Juni 136, 10623 Berlin, Germany

Alexander Bobenko
Affiliation: Fachbereich Mathematik, Technische Universität Berlin, Strass des 17.Juni 136, 10623 Berlin, Germany

Mark van Hoeij
Affiliation: Department of Mathematics, Florida State University, Tallahassee, Florida 32306

Marcus Schmies
Affiliation: Fachbereich Mathematik, Technische Universität Berlin, Strass des 17.Juni 136, 10623 Berlin, Germany

Keywords: Riemann theta function, pointwise approximation, uniform approximation
Received by editor(s): June 7, 2002
Published electronically: December 19, 2003
Article copyright: © Copyright 2003 American Mathematical Society