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, National Bureau of Standards, Washington, D.C., 1966, Edited by M. Abramowitz and I. A. Stengun. MR 34:8607
  • 2. A. I. Babich and A. I. Bobenko, Willmore tori with umbilic lines and minimal surfaces in hyperbolic space, Duke Math. J. 72 (1993), no. 1, 151-185. MR 94j:53070
  • 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 $r^3, s^3, h^3$ in terms of theta-functions, Math. Ann. 290 (1991), 209-245. MR 92h:53072
  • 5. A. I. Bobenko and L. A. Bordag, Periodic multiphase solutions of the Kadomsev-Petviashvili equation, J. Phys. A. 22 (1989), 1259-1274. MR 90i:35228
  • 6. B. Deconinck and M. van Hoeij, Computing Riemann matrices of algebraic curves, Physica D 152-153 (2001), 28-46. MR 2002j:30001
  • 7. B. A. Dubrovin, Theta functions and nonlinear equations, Russian Math. Surveys 36 (1981), no. 2, 11-80. MR 83i:35149
  • 8. B. A. Dubrovin, R. Flickinger, and H. Segur, Three-phase solutions of the Kadomtsev-Petviashvili equation, Stud. Appl. Math. 99 (1997), no. 2, 137-203. MR 98m:35179
  • 9. D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin, 2001. MR 2001k:35004
  • 10. M. Heil, Numerical tools for the study of finite gap solutions of integrable systems, Ph.D. thesis, Technischen Universität Berlin, 1995.
  • 11. J.-I. Igusa, Theta functions, Springer-Verlag, New York, 1972. MR 48:3972
  • 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), 515-534. MR 84a:12002
  • 14. David Mumford, Tata lectures on theta. I, Birkhäuser Boston Inc., Boston, MA, 1983. MR 85h:14026
  • 15. -, Tata lectures on theta. II, Birkhäuser Boston Inc., Boston, MA, 1984. MR 86b:14017
  • 16. -, Tata lectures on theta. III, Birkhäuser Boston Inc., Boston, MA, 1991. MR 93d:14065
  • 17. G. F. B. Riemann, Theorie der Abel'schen functionen, Journal für reine und angewandte Mathematik 54 (1857), 101-155.
  • 18. C. L. Siegel, Topics in complex function theory. Vol. III, John Wiley & Sons, Inc., New York, 1989. MR 57:16317b
  • 19. B. Vallée, A central problem in the algorithmic geometry of numbers: lattice reduction, CWI Quarterly 3 (1990), 95-120. MR 92c:11065
  • 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

American Mathematical Society