Tracing solution curves of non-linear equations with sharp turning points

Non-linear equations having multiple Solutions are usually associated with sharp turning points in the associated solution curves. Existing solution methods often fail or become extremely slow in the neighbourhood of such turning points. This paper presents an efficient algorithm based on the backward-differentiation formula (BDF) which overcomes this problem. For the special subclass of non-linear series-parallel circuits, this algorithm is generalized to allow all branches of driving-point and transfer characteristics to be traced efficiently. For any fixed value of the driving-point voltage or current source, this algorithm guarantees that all solutions are found.