Computing theta function

Let $f: {\Bbb R}^n \longrightarrow {\Bbb R}$ be a positive definite quadratic form and let $y \in {\Bbb R}^n$ be a point. We present a fully polynomial randomized approximation scheme (FPRAS) for computing $\sum_{x \in {\Bbb Z}^n} e^{-f(x)}$, provided the eigenvalues of $f$ lie in the interval roughly between $s$ and $e^{s}$ and for computing $\sum_{x \in {\Bbb Z}^n} e^{-f(x-y)}$, provided the eigenvalues of $f$ lie in the interval roughly between $e^{-s}$ and $s^{-1}$ for some $s \geq 3$. To compute the first sum, we represent it as the integral of an explicit log-concave function on ${\Bbb R}^n$, and to compute the second sum, we use the reciprocity relation for theta functions. Choosing $s \sim \log n$, we apply the results to test the existence of sufficiently many short integer vectors in a given subspace $L \subset {\Bbb R}^n$ or in the vicinity of $L$.