Theta.jl

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Theta.jl is a Julia package for computing the Riemann theta function and its derivatives.

For more information, refer to our preprint.

Download Julia 1.4. Start Julia and run

import Pkg
Pkg.add("Theta")

First load the package in Julia.

using Theta

We start with a matrix M in the Siegel upper-half space.

M = [0.794612+1.9986im 0.815524+1.95836im 0.190195+1.21249im 0.647434+1.66208im 0.820857+1.68942im; 
0.0948191+1.95836im 0.808422+2.66492im 0.857778+1.14274im 0.754323+1.72747im 0.74972+1.95821im; 
0.177874+1.21249im 0.420423+1.14274im 0.445617+1.44248im 0.732018+0.966489im 0.564779+1.57559im; 
0.440969+1.66208im 0.562332+1.72747im 0.292166+0.966489im 0.433763+1.91571im 0.805161+1.46982im; 
0.471487+1.68942im 0.0946854+1.95821im 0.837648+1.57559im 0.311332+1.46982im 0.521253+2.29221im];      

We construct a RiemannMatrix using M.

R = RiemannMatrix(M);

We can then compute the theta function on inputs z and M as follows.

z = [0.30657351+0.34017115im; 0.71945631+0.87045964im; 0.19963849+0.71709398im; 0.64390182+0.97413482im; 0.02747232+0.59071266im];
theta(z, R)

We can also compute first derivatives of theta functions by specifying the direction using the optional argument derivs. The following code computes the partial derivative of the theta function with respect to the first coordinate of z.

theta(z, R, derivs=[[1,0,0,0,0]])

We specify higher order derivatives by adding more elements into the input to derivs, where each element specifies the direction of the derivative. For instance, to compute the partial derivative of the theta function with respect to the first, second and fifth coordinates of z, we run

theta(z, R, derivs=[[1,0,0,0,0], [0,1,0,0,0], [0,0,0,0,1]])

We can compute theta functions with characteristics using the optional argument char.

theta(z, R, char=[[0,1,0,1,1],[0,1,1,0,0]])

We can also compute derivatives of theta functions with characteristics.

theta(z, R, derivs=[[1,0,0,0,0]], char=[[0,1,0,1,1],[0,1,1,0,0]])