We describe a few relevant aspects of the new plasma physics stability code, TERPSICHORE, which breaks new ground both in plasma theory and in its algorithmic implementation. We chose a highly accurate discrete mathematical representation which led to the choice of algorithms that matched the vector-parallel architecture of the Cray Y-MP and Cray-2 supercomputers quite well. The resulting code runs at Gflop/s speeds, which translates into a very high efficiency. This means we can perform very extensive studies within our limited computer budgets.