In this thesis a nodal equation extension technique to solve ill-conditioned circuits on a short word-length computer is presented. Ill-conditioned circuits are characterised by widely different conductance values which cause 'catastrophic cancellations' during Gaussian elimination on the nodal admittance matrix. Two algorithms, called the MNA1 and the MNA2, which circumvent catastrophic cancellations by extending the nodal equation sets, are presented. Basically, these algorithms use branch-currents (MNA1) or algebraic combinations or branch currents (MNA2) as additional variables to extend the circuitequation set. Cancellations during the forward and backward substitution parts of the Gaussian elimination are avoided by an accurate floating point summation algorithm. A new data structure which allows partial pivoting on sparse matrices is described and compared with the Azar-Nichols data structure. Whereas the latter partitions the nodal admittance matrix into sparse and non-sparse parts, the new data structure has only a sparse partition. A pivoting technique due to Stewart, which employs modification of unsuitable pivots, rather than interchanges of rows or columns, to preserve matrix sparsity is also investigated. Computer programs developed to implement the algorithms under the program suite SUICIDES are described and results compared with other standard analysis algorithms. The nodal extension algorithms, namely the MNA1 and the MNA2, are found to produce accurate results for ill-conditioned circuits, the MNA2 being generally slower than the MNA1. For well-conditioned circuits these methods are computationally more expensive than the standard Nodal Analysis (NA). As a compromise between speed and accuracy, we also present an algorithm which normally uses the computationally inexpensive NA algorithm but switches to the MNA1 algorithm when ill-conditioning is detected. Simulation of a number of test circuits shows that for well-conditioned circuits, the switching algorithm is as fast as the NA algorithm and for ill-conditioned circuits is only marginally slower than the MNA1. The overhead for implementation of the algorithm in a general purpose circuit analysis program suite is reasonable.