Julia versions	master build	Coverage

Julia implementation of the Riemann Theta function. This package is mostly a port from Python of the same function in the Sage library Abelfunction (https://github.com/abelfunctions/abelfunctions). Beyond a given problem size (number of z in zs, dimension of z's, number of integration points), the functions switch to a different algorithm using matrix operations resulting in very competitive timings (at the cost of memory usage).

The Sage library is itself an implementation of :

[CRTF] B. Deconinck, M. Heil, A. Bobenko, M. van Hoeij and M. Schmies, Computing Riemann Theta Functions, Mathematics of Computation, 73, (2004), 1417-1442.

Exported function are :

     riemanntheta(zs::Vector{Vector{Complex128}},
                  Ω::Matrix{Complex128};
                  eps::Float64=1e-8,
                  derivs::Vector{Vector{Complex128}}=Vector{Complex128}[],
                  accuracy_radius::Float64=5.)::Vector{Complex128}

Return the value of the Riemann theta function for Ω and all z in zs if derivs is empty, or the derivatives at all z in zs for the given directional derivatives in derivs.

Parameters :

• zs : A vector of complex vectors at which to evaluate the Riemann theta function.
• Omega : A Riemann matrix.
• eps : (Default: 1e-8) The desired numerical accuracy.
• derivs : A vector of complex vectors giving a directional derivative.
• accuracy_radius : (Default: 5.) The radius from the g-dimensional origin where the requested accuracy of the Riemann theta is guaranteed when computing derivatives. Not used if no derivatives of theta are requested.

     oscillatory_part(zs::Vector{Vector{Complex128}},
                      Ω::Matrix{Complex128};
                      eps::Float64=1e-8,
                      derivs::Vector{Vector{Complex128}}=Vector{Complex128}[],
                      accuracy_radius::Float64=5.)::Vector{Complex128}

Return the value of the oscillatory part of the Riemann theta function for Ω and all z in zs if derivs is empty, or the derivatives at all z in zs for the given directional derivatives in derivs.

Parameters :

• zs : A vector of complex vectors at which to evaluate the Riemann theta function.
• Omega : A Riemann matrix.
• eps : (Default: 1e-8) The desired numerical accuracy.
• derivs : A vector of complex vectors giving a directional derivative.
• accuracy_radius : (Default: 5.) The radius from the g-dimensional origin where the requested accuracy of the Riemann theta is guaranteed when computing derivatives. Not used if no derivatives of theta are requested.

And :

     exponential_part(zs::Vector{Vector{Complex128}},
                      Ω::Matrix{Complex128})::Vector{Float64}

Return the value of the exponential part of the Riemann theta function for Ω and all z in zs.

Parameters :

• zs : A vector of complex vectors at which to evaluate the Riemann theta function.
• Omega : A Riemann matrix.