We present theoretical results on the convergence of iterative methods for the solution of linear differential-algebraic equations arising form circuit simulation. The iterative methods considered include the continuous-time and discretetime waveform relaxation methods and the Krylov subspace methods in function space. The waveform generalized minimal residual method for solving linear differential-algebraic equations in function space is developed, which is one of the waveform Krylov subspace methods. Some new criteria for convergence of these iterative methods are derived. Examples are given to verify the convergence conditions.

In this letter we present a new way for computing generalized eigenvalue problems in engineering applications. To transform a generalized eigenvalue problem into an associated problem for solving nonlinear dynamic equations by using optimization techniques, we can determine all eigenvalues and their eigenvectors for general complex matrices. Numerical examples are given to verify the formula of dynamic equations.

In this paper we derive the expressions of the spectra of waveform relaxation operators for linear differential-algebraic equations which stem from circuit simulation. These expressions suggest ways to split the matrices of the circuit equations such that waveform relaxation will converge. Numerical experimental results are given.

In this paper we study general complex eigenvalue problems in engineering fields. The eigenvalue problems can be transformed into the associated problems for solving large systems of nonlinear ordinary differential equations (dynamic equations) by optimization techniques. The known waveform relaxation method in circuit simulation can then be successfully applied to compute the resulting dynamic equations. The approach reported here, which is implemented on a message passing multi-processor system, can determine all eigenvalues and their eigenvectors for general complex matrices without any restriction on the matrices. The numerical results are provided to confirm the theoretical analysis.