An extremely efficient breakpoint-hopping algorithm is presented for tracing the driving-point and transfer characteristics of any nonlinear circuit made of linear (possibly multi-terminal) resistors, dc independent sources, linear controlled sources (all 4 types) and 2-terminal nonlinear resistors described by piecewise-linearupsilon - icharacteristics. Most resistive nonlinear electronic circuits can be realistically modeled by such circuits. The algorithm can trace not only violently nonlinear (with sharp turning points) and multivalued characteristics, but also characteristics composed of several disconnected branches, provided one point in each branch is given. The remarkable computational efficiency of the breakpoint-hopping algorithm is due to two key properties built into the algorithm: 1) the circuit equation is formulated into a special form; namely, a canonical piecewise-linear equation with a lattice structure. 2) the algorithm finds only the breakpoints and possibly one point on each end (unbounded) segment via explicit formulas (hence no convergence problem). These data points represent the minimal amount of information needed to specify a piecewise-linear characteristic uniquely.