On the convergence of block relaxation methods for circuit simulation

The authors developed sufficient conditions for the convergence of several block relaxation methods. They first consider time-point relaxation methods, namely the block Gauss-Seidel-Newton (G-S-N) and the block Newton-Gauss-Seidel (N-G-S) algorithms. The previously known sufficient condition for convergence of the G-S-N and the N-G-S algorithms requires: (1) a capacitor connected between every node in the circuit and the reference ground node: and (2) the choice of a sufficiently small time step for the implicit integration formula used to discretize (in time) the circuit equations. The authors derive a sufficient condition that is less restrictive than (1) above. For a given partitioning of a circuit, they define a set (possibly empty) of feedback nodes that capture the topology of the partitioned circuit to a certain extent. They then show that the G-S-N and the N-G-S algorithms converge