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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Fast evaluation of modular functions using Newton iterations and the AGM
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by Régis Dupont PDF
Math. Comp. 80 (2011), 1823-1847


We present an asymptotically fast algorithm for the numerical evaluation of modular functions such as the elliptic modular function $j$. Our algorithm makes use of the natural connection between the arithmetic-geometric mean (AGM) of complex numbers and modular functions. Through a detailed complexity analysis, we prove that for a given $\tau$, evaluating $N$ significative bits of $j(\tau )$ can be done in time $O(\mathcal {M}(N)\log N)$, where $\mathcal {M}(N)$ is the time complexity for the multiplication of two $N$-bit integers. However, this is only true for a fixed $\tau$ and the time complexity of this first algorithm greatly increases as $\mathrm {Im}(\tau )$ does. We then describe a second algorithm that achieves the same time complexity independently of the value of $\tau$ in the classical fundamental domain $\mathcal {F}$. We also show how our method can be used to evaluate other modular forms, such as the Dedekind $\eta$ function, with the same time complexity.
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Additional Information
  • Régis Dupont
  • Affiliation: INRIA Futurs, Projet TANC, Laboratoire LIX, École polytechnique, 91128 Palaiseau, France
  • Email:
  • Received by editor(s): May 27, 2005
  • Published electronically: March 4, 2011
  • © Copyright 2011 Régis Dupont
  • Journal: Math. Comp. 80 (2011), 1823-1847
  • MSC (2000): Primary 65D20; Secondary 33E05, 11Y16
  • DOI:
  • MathSciNet review: 2785482