Tutorial: Fundamentals of ill-conditioning

System identification techniques using modal analysis can often lead to ill-conditioned equations since the inverse problem may not be unique. An ill-conditioned system of linear algebraic equations is defined as a set for which difficulty is encountered when trying to obtain a numerical solution. It may be that there are fewer equations than unknowns or that the equations are not sufficiently linearly independent. Thus, the analyst will require additional independent information, or will have to perform very precise numerical computations in order to obtain a unique solution. The usual manifestation of ill-conditioning is that the coefficient matrix will be difficult to decompose and/or small perturbations in the right hand side will produce large changes in the solution. This tutorial paper presents a mathematical insight into how and why such problems occur and new material on how such problems may be resolved by means of algorithms which are accurate and stable. Accuracy and computational efficiency comparisons with existing methods are also presented.