EPODE

Stiffness

In order to classify the problem using the stiff ratio, we must establish the domain ranges for each problem class.

In EpODE, the following classification is proposed. The problem is

nonstiff, if S(J,t₀,t₁,e)<10
limit stiff, if 10<= S(J,t₀,t₁,e)<10²
middling stiff, if 10²<= S(J,t₀,t₁,e)<10³
strongly stiff, if 10³<= S(J,t₀,t₁,e)<10⁶
extremely stiff, if 10⁶<= S(J,t₀,t₁,e)<10⁹
pathologically stiff, if 10⁹<= S(J,t₀,t₁,e)

If the S>= 10¹⁰ the stiff ratio is declared 'Infinity'. If the smallest eigenvalue of the Jacobian matrix is zero, the stiff ratio will also be declared 'Infinity' and the problem classification, 'Pathologically stiff' (since an oldest definition of the stiff ratio is S=max_{i=1, ... ,n}|l_i| / min_{i=1, ... ,n}|l_i|).

Example for the system B1.

In order to find the maximum eigenvalue of a matrix A, we can adopt the {\tt Rayley}'s method (this method works well if A is a symmetric matrix). Consider the iterative process

y_n+1=Ay_n /||y_n||, y₀=(1, ... ,1)^T

If y_n converges to y when n goes to infinity, the maximum eigenvalue can be approximated by the scalar product of y with Ay. If the iterative process is not convergent we can set maximum eigenvalue to a big value and we apply the Lehmer-Schur algorithm. The smallest eigenvalue is the largest eigenvalue of A^-1.