acb_theta_ql_jet_fd, acb_theta_ql_jet

The finite difference method evaluates $\theta_{a,b}(z + h_n, \tau)$ at $h_n = (\varepsilon \zeta^{n_0},\ldots, \varepsilon \zeta^{n_{g-1}})$ for a primitive $m$-th root of unity $\zeta$, $m = \mathit{ord} + 1$, and all $n \in \{0,\ldots, m - 1\}^g$; a discrete Fourier transform gives $m^g \sum_{k \equiv p \bmod m} a_k \varepsilon^{|k|}$ for each $p$, where $a_k$ are the Taylor coefficients. By the Cauchy formula with the bound of acb_theta_ql_local_bound, the error is at most $2cg\varepsilon^{|p|+m}/\rho^{m}$ before division by $\varepsilon^{|p|}$. The function chooses $\varepsilon$, strips $(z, \tau)$ of their radii, raises the precision to compensate for the division by $\varepsilon^{\mathit{ord}} (\mathit{ord}+1)^g$, and restores certified radii with acb_theta_ql_jet_error and derivatives to order $\mathit{ord} + 2$ computed at low precision; see Derivatives.