# 1.7 Stiffness and Implicit Methods

## 1.7.2 Spectral Condition Number

Stiffness can also arise in linear or linearized systems when eigenvalues exist with significantly different magnitudes. For example,

 $u_ t = Au, \qquad A = \left(\begin{array}{cc} -1 & 1 \\ 0 & -1000 \end{array} \right).$ (1.120)

The eigenvalues of $$A$$ are $$\lambda = -1$$ and $$\lambda = -1000$$. Since the timestep must be set so that both eigenvalues are stable, the larger eigenvalue will dominate the timestep. The spectral condition number is the ratio of the largest to smallest eigenvalue magnitudes,

 $\mbox{Spectral Condition Number} = \frac{\max |\lambda _ j|}{\min |\lambda _ j|}$ (1.121)

When the spectral condition number is greater than around 1000, problems are starting to become stiff and implicit methods are likely to be more efficient than explicit methods.

Exercise What is the spectral condition number of the matrix $$A = \left[\begin{array}{cc} 1032 & 18 \\ 54 & -200 \end{array}\right]$$. Please answer with at least 3 significant digits.

Exercise 1

Answer: The eigenvalues are $$\lambda _1=1032.788$$ and $$\lambda _2=-200.788$$. The ratio of their absolute values gives the correct answer.