  We present a new method for numerically computing generic multi-loop Feynman
integrals. The method relies on an iterative application of Feynman's trick for
combining two propagators. Each application of Feynman's trick introduces a
simplified Feynman integral topology which depends on a Feynman parameter that
should be integrated over. For each integral family, we set up a system of
differential equations which we solve in terms of a piecewise collection of
generalized series expansions in the Feynman parameter. These generalized
series expansions can be efficiently integrated term by term, and segment by
segment. This approach leads to a fully algorithmic method for computing
Feynman integrals from differential equations, which does not require the
manual determination of boundary conditions. Furthermore, the most complicated
topology that appears in the method often has less master integrals than the
original one. We illustrate the strength of our method with a five-point
two-loop integral family.
