We know that the linear systems

where the matrix A has distinct eigen values and corresponding eigen vector has a general solution of the form

Let us assume that then the term as we therefore call this term the transient solution, and call the remaining term the steady state solution. Let and be two eigen values of A such that

If our aim is to find numerically the steady state solution , then we must pursue the numerical solution until the slowest decaying exponential in the transient solution, namely is negligible. Thus, the smaller , the longer will be the range of integration. On the other hand, the presence of eigen values of A far out to the left in the complex plane will force us to use excessively small step lengths in order that will lie within the range of absolute stability of the method. The further out such eigen values lie, the more severe is the restriction on step length. A rough measure of this difficulty is the magnitude of . If , we are forced into the highly undesirable computational situation of having to integrate numerically over a long range, using a step length which is everywhere excessively small relative to the interval, this is the problem of stiffness. We can make the following somewhat heuristic definition.