§21.10 Methods of Computation

Permalink:: http://dlmf.nist.gov/21.10

§21.10(i) General Riemann Theta Functions
§21.10(ii) Riemann Theta Functions Associated with a Riemann Surface

§21.10(i) General Riemann Theta Functions

Notes:: See Deconinck et al. (2004).
Keywords:: Riemann theta functions, scaled Riemann theta functions
Permalink:: http://dlmf.nist.gov/21.10.i

Although the defining Fourier series (21.2.1) is uniformly convergent on compact sets, its evaluation is cumbersome when one or more of the eigenvalues of $\imagpart{(\boldsymbol{{\Omega}})}$ is near zero. Furthermore, for fixed $\boldsymbol{{\Omega}}$ different terms of the Fourier series dominate for different values of $\mathbf{z}$ .

To overcome these obstacles, we compute instead the scaled function $\mathop{\hat{\theta}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ (§21.2(i)) from the expansion

21.10.1 $\mathop{\hat{\theta}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=\sum _{{\mathbf{n}\in S(\epsilon)}}e^{{\pi i\left[\mathbf{n}-[\mathbf{Y}^{{-1}}\mathbf{y}]\right]\cdot\mathbf{X}\cdot\left[\mathbf{n}-[\mathbf{Y}^{{-1}}\mathbf{y}]\right]}}\* e^{{2\pi i\left[\mathbf{n}-[\mathbf{Y}^{{-1}}\mathbf{y}]\right]\cdot\mathbf{x}}}\* e^{{-\pi\left[\mathbf{n}+[\mathbf{Y}^{{-1}}\mathbf{y}]\right]\cdot\mathbf{Y}\cdot\left[\mathbf{n}+[\mathbf{Y}^{{-1}}\mathbf{y}]\right]}},$

Symbols:: $\mathop{\hat{\theta}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : scaled Riemann theta function, $\in$ : element of, $e$ : base of exponential function, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\epsilon$ : maximum error and $S(\epsilon)$ : set of vectors
Referenced by:: §21.10(i)
Permalink:: http://dlmf.nist.gov/21.10.E1
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where $\epsilon$ is the tolerated maximum absolute error for $\mathop{\hat{\theta}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ . Here $\mathbf{X}=\realpart{(\boldsymbol{{\Omega}})}$ , $\mathbf{Y}=\imagpart{(\boldsymbol{{\Omega}})}$ , $\mathbf{x}=\realpart{(\mathbf{z})}$ , $\mathbf{y}=\imagpart{(\mathbf{z})}$ , and

21.10.2 $S(\epsilon)=\left\{\mathbf{m}\in\Integer^{g}\Big|\pi\left[\mathbf{m}+[\mathbf{Y}^{{-1}}\mathbf{y}]\right]\cdot\mathbf{Y}\cdot\left[\mathbf{m}+[\mathbf{Y}^{{-1}}\mathbf{y}]\right]\leq R(\epsilon)\right\}.$

Symbols:: $\in$ : element of, $\Integer$ : set of all integers, $g$ : positive integer, $\epsilon$ : maximum error and $S(\epsilon)$ : set of vectors
Referenced by:: §21.10(i)
Permalink:: http://dlmf.nist.gov/21.10.E2
Encodings:: TeX, pMML, png

Thus $S(\epsilon)$ is the set of all integer vectors that are contained in an ellipsoid centered at the fractional part of $\mathbf{Y}^{{-1}}\mathbf{y}$ , and whose size is determined by the allowed absolute error. The value of $R(\epsilon)$ is determined as follows. Let $r$ be the length of the shortest vector of the lattice $\Lambda=\left\{\sqrt{\pi}\,\mathbf{T}\mathbf{m}|\mathbf{m}\in\Integer^{g}\right\}$ , and $\mathbf{T}^{{\mathrm{T}}}\mathbf{T}=\mathbf{Y}$ be the Cholesky decomposition of $\mathbf{Y}$ (Atkinson (1989, p. 254)). Then $R(\epsilon)$ is the greater of $\sqrt{g/2}+r$ and the smallest positive root of the equation

21.10.3 $\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}g,R^{2}\right)/(2gr^{g})=\epsilon.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $g$ : positive integer, $\epsilon$ : maximum error and $r$ : length
Permalink:: http://dlmf.nist.gov/21.10.E3
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For the incomplete gamma function $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ , see §8.2(i).

The construction (21.10.2) amounts to determining all integer vectors in a $g$ -dimensional ellipsoid. For this purpose it is convenient to have the ellipsoid as spherical as possible (Siegel (1973, pp. 144–159), Heil (1995)).

Usually, (21.10.1) can also be used for the efficient evaluation of $\mathop{\hat{\theta}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ for fixed $\boldsymbol{{\Omega}}$ and varying $\mathbf{z}$ , by addition of a few vectors to the set $S(\epsilon)$ .

§21.10(ii) Riemann Theta Functions Associated with a Riemann Surface

Keywords:: Hurwitz system, Riemann matrix, Riemann surface, Schottky group, algebraic curves
Permalink:: http://dlmf.nist.gov/21.10.ii

In addition to evaluating the Fourier series, the main problem here is to compute a Riemann matrix originating from a Riemann surface. Various approaches are considered in the following references:

Belokolos et al. (1994, Chapter 5) and references therein. Here the Riemann surface is represented by the action of a Schottky group on a region of the complex plane. The same representation is used in Gianni et al. (1998).
Tretkoff and Tretkoff (1984). Here a Hurwitz system is chosen to represent the Riemann surface.
Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.

§21.10 Methods of Computation

Contents

§21.10(i) General Riemann Theta Functions

§21.10(ii) Riemann Theta Functions Associated with a Riemann Surface