Code to calculate the partial theta function

Question

The version of the partial theta function I want to compute is

$O(z) = \sum\limits^\infty_{n=0}\exp(-(z+n\pi)^2)$

Does anyone have or know of code to efficiently compute $O(z)$? There are two papers dealing with this computation, but can't find any code.

Hiary, G. (2008). A nearly-optimal method to compute the truncated theta fcn, its derivatives and integrals. Annals of Mathematics, Second Series, Vol 174, No. 2, pp 859-889.

Kuznetsov, A (2015). Computing the truncated theta function via Mordel-Integral. Math comp 84, pp 2911-2926

Answer 1

θ[z_?NumericQ] := NSum[Exp[-(z + n Pi)^2], {n, 0, Infinity}]

Using a partial sum

θp[z_] := Sum[Exp[-(z + n Pi)^2], {n, 0, 100}]

Comparing the plots

Plot[{θ[z], θp[z]}, {z, -2.5, 2.5},
 PlotStyle -> {{AbsoluteThickness[4], Green}, Red},
 PlotLegends -> Placed["Expressions", {.2, .75}]]

Answer 2

The sum $O(z) = \sum\limits^\infty_{n=0}\exp(-(z+n\pi)^2)$ is extremely rapidly convergent, unlike the sums in the references cited in the question.

Pick an accuracy eps. Then one can sum to that accuracy as follows:

(* solve for limits of summation *)
n /. Simplify[Solve[Exp[-(z + n Pi)^2] == eps, n, Reals], 0 < eps < 1]
(*  {-((z + Sqrt[-Log[eps]])/π), (-z + Sqrt[-Log[eps]])/π}  *)

ClearAll[sum];
sum[z_?NumericQ, accgoal_: Automatic] := Module[{n, eps},
   eps = 10^-(accgoal /. Automatic -> $MachinePrecision);
   n = Range @@  (* range of terms to sum *)
     N@Clip[
      Floor@{-((z + Sqrt[-Log[eps]])/π), (-z + Sqrt[-Log[eps]])/π} + {0, 1}, 
      {0, Infinity}];
   Total[Exp[-(z + n Pi)^2]] (* vectorized sum for speed *)
   ];

Plot[sum[z], {z, -30, 10}]

The number of terms in the sum for machine precision accuracy is about 4 or 5:

2*Sqrt[-Log[$MachineEpsilon]]/Pi
(*3.82203  *)