  We consider the problem of proving termination for triangular weakly
non-linear loops (twn-loops) over some ring $\mathcal{S}$ like $\mathbb{Z}$,
$\mathbb{Q}$, or $\mathbb{R}$. The guard of such a loop is an arbitrary
quantifier-free Boolean formula over (possibly non-linear) polynomial
inequations, and the body is a single assignment of the form $(x_1, \ldots,
x_d) \longleftarrow (c_1 \cdot x_1 + p_1, \ldots, c_d \cdot x_d + p_d)$ where
each $x_i$ is a variable, $c_i \in \mathcal{S}$, and each $p_i$ is a (possibly
non-linear) polynomial over $\mathcal{S}$ and the variables
$x_{i+1},\ldots,x_{d}$.
  We show that the question of termination can be reduced to the existential
fragment of the first-order theory of $\mathcal{S}$. For loops over
$\mathbb{R}$, our reduction implies decidability of termination. For loops over
$\mathbb{Z}$ and $\mathbb{Q}$, it proves semi-decidability of non-termination.
  Furthermore, we present a transformation to convert certain non-twn-loops
into twn-form. Then the original loop terminates iff the transformed loop
terminates over a specific subset of $\mathbb{R}$, which can also be checked
via our reduction. Moreover, we formalize a technique to linearize (the updates
of) twn-loops in our setting and analyze its complexity. Based on these
results, we prove complexity bounds for the termination problem of twn-loops as
well as tight bounds for two important classes of loops which can always be
transformed into twn-loops.
  Finally, we show that there is an important class of linear loops where our
decision procedure results in an efficient procedure for termination analysis,
i.e., where the parameterized complexity of deciding termination is polynomial.
