Preconditioned Iterative Methods for Sparse Linear Algebra Problems Arising in Circuit Simulation

The DC operating point of a circuit may be computed by tracking the zero curve of an associated artificial-parameter homotopy. Homotopy algorithms exist that are globally convergent with probability one for the DC operating point problem. These algorithms require computing the one-dimensional kernel of the Jacobian matrix of the homotopy mapping at each step along the zero curve, and hence the solution of a linear system of equations at each step. These linear systems are typically large, highly sparse, non-symmetric and indefinite. Several iterative methods which are applicable to such problems, including Craig's method, GMRES(k), BiCGSTAB, QMR, KACZ, and LSQR, are applied to a suite of test problems derived from simulations of actual bipolar circuits. Preconditioning techniques considered include incomplete LU factorization (ILU), sparse submatrix ILU, and ILU allowing restricted fill in bands or blocks. Timings and convergence statistics are given for each iterative method and preconditioner.