Definition: The linear systems is said to be stiff if (i) and (ii) , where are the eigen values of A. The ratio

is called the stiffness ratio.

Non linear systems exhibit stiffness if the eigen values of the Jacobian behave in a similar fashion. The eigen values are no longer constant but depend on the solution, and therefore vary with . Accordingly we say that the system is stiff in an interval of if, for the eigen values of satisfy (i) and (ii) above.

Note that if the partial derivatives appearing in the Jacobian are continuous and bounded in an appropriate region, then the Lipschitz constant of the system may be taken to be . For any matrix , where is the spectral radius, is defined to be being the eigen values of . If it follows that . Thus stiff systems are occasionally referred to as systems with large Lipschitz constants'.