The projects deal with the application of homotopy methods to solving nonlinear equations describing transistors circuits and other nonlinear systems. Homotopy methods can be used when traditional methods for solving systems of nonlinear equations fail, are difficult to converge, and cannot find all the solutions. These methods are becoming a viable alternative to the existing options in circuit simulators where they can be used to resolve convergence difficulties in ``problem situations.''

The students are developing software implementations of new numerical algorithms for simulating analog electronic circuits such as flip-flops, Schmitt triggers, and oscillators. We are applying new numerical methods (called homotopies), which have been successfully used to solve nonlinear equations describing transistors circuits and other nonlinear systems. These algorithms can be used with the SPICE circuit simulator, or be implemented as stand alone packages with MATLAB.

Fall'97 URO: Willy Horia worsk on developing a MATLAB Toolbox for solving nonlinear equations using homotopies.

Summer'97 CRA Distributed Mentor Project: Andria Dyess from the University of Alabama is working on MATLAB functions for solving transistor circuit equations. Project report.

Summer'97: Ashish Agarwal is solving polynomial equations that describe chemical reactions by using homotopy methods.

Spring'97 URO: William Horia is using MATLAB to solve circuit equations using homotopy routines.

Fall'96 URO: Edward Chan developed a parser that can read SPICE input and create nodal and modified nodal equations suitable for MATLAB.

Recently, there has been growing interest in the area of signal-adaptive systems system, such as quadrature-mirror filter banks. Adaptivity is obtained by designing filter banks that match the statistics of the input signal, by maximizing the coding gain of the system. Methods used in the literature for numerical solution of such problems include quasi-Newton methods, the ring algorithm (a coordinate-descent scheme), and multi-start algorithms.

We investigated the use of new algorithms, known as continuation (homotopy) methods in the design of such filter banks. These methods have been increasingly used for solving a variety of nonlinear problems in fluid dynamics, structural mechanics, systems identification, and circuit theory. They are popular in mathematical programming and are globally convergent provided certain conditions are satisfied by the system's equations. Moreover, they yield all the solutions to the nonlinear system.

Fall'96: Almudena Ordonez used homotopies and MATLAB to find coefficients of digital filter banks.