Designers of RF circuits such as power amplifiers, mixers, and filters make extensive use of simulation tools which perform periodic steady-state analysis and its extensions, but until the mid 1990s, the computational costs of these simulation tools restricted designers from simulating the behavior of complete RF subsystems. The introduction of fast matrix-implicit iterative algorithms completely changed this situation, and extensions of these fast methods are providing tools which can perform periodic, quasi-periodic, and periodic noise analysis of circuits with thousands of devices. Even though there are a number of research groups continuing to develop extensions of matrix-implicit methods, there is still no compact characterization which introduces the novice researcher to the fundamental issues. In this paper, we examine the basic periodic steady-state problem and provide both examples and linear algebra abstractions to demonstrate connections between seemingly dissimilar methods and to try to provide a more general framework for fast methods than the standard time-versus-frequency domain characterization of finite-difference, basis-collocation, and shooting methods

In this paper we present a preconditioned recycled Krylov-subspace method to accelerate a recently developed approach for AC and noise analysis of linear periodically-varying communication circuits. Examples are given to show that the combined method can be used to analyze switching filter frequency response, mixer 1/f noise frequency translation, and amplifier intermodulation distortion. In addition, it is shown that for large circuits the pre-conditioned recycled Krylov-subspace method is up to forty times faster than the standard optimized direct methods.

Gaussian-elimination based shooting-Newton methods, a commonly used approach for computing steady-state solutions, grow in computational complexity like N 3 , where N is the number of circuit equations.Just using iterative methods to solve the shooting-Newton equations results in an algorithm which is still order N 2 because of the cost of calculating the dense sensitivity matrix.Below, a matrix-free Krylov-subspace approach is presented, and the method is shown to reduce shooting-Newton computational complexity to that of ordinary transient analysis.Results from several examples are given to demonstrate that the matrix-free approach is more than ten times faster than using iterative methods alone for circuits with as few as 400 equations.2 Shooting Methods Finding the periodic steady-state solution of a circuit involves nding the initial condition for the circuit's associated system of dierential equations such that the solution at the end of the period matches the initial condition.More precisely, nding the steady-state solution means nding a particular solution to the circuit equations, as

The factorization of sparse matrices is used in the inner loop of many engineering algorithms. including circuit simulation. This time consuming operation can be speeded up by utilizing multiprocessor architectures. Distributed memory architectures can overcome the memory bottleneck normally associated with shared memory machines but require a careful distribution of matrix data to the processors. The authors present partitioning schemes that distribute the rows of the matrix to processors to allow a better utilization of the processing resources. They also present bounds on the speedup achieveable with partitioned matrices. They demonstrate that a good partitioning strategy coupled with more sophisticated scheduling algorithms improves the overall processor utilization.>

The problem of sparse matrix decomposition using distributed memory multiprocessors is addressed. The data partitioning scheme is simple and is based on equalizing the load among the processors. A new O(n log n) task scheduling heuristic with provably deadlock-free properties is presented. The key idea is the ordering of nodes in a task graph that represents the matrix decomposition steps in a levelized manner, based on a new measure, delta the remaining completion time. The method tends to minimize the idle time of processors by revising the overall decomposition schedule by permitting the execution of tasks within these idle periods. For large sparse matrices, the analysis and simulation results show that a multiprocessor with even a small number of processors will exceed the performance of a supercomputer like the Cray X-MP.>