# # Ill Conditioning

In the vast majority of cases the solver will give a correct solution to the problem. However, some problems are by nature ill-conditioned in which case small changes in the input data can lead to more significant changes in the results.

Taking a simple example to look at ill-conditioning; consider a simple
two spring system, where the springs are connected in series. The
stiffness of the first spring is

In this case the equations describing the system is

As in a solver based on a Gaussian elimination technique, we use these
equations to arrive at a relationship between

which when substituted in the other equation gives:

or

With exact arithmetic the term

We have then a system as shown below where the error is like adding a
third spring, which acts in parallel with

The expected reaction is

Thus the reaction is in error by a factor