Application to Vector Equations

A little more care is required when $\bi{y}$ and $\bi{f}$ are vectors. In this case $\delta\bi{y}$ is an arbitrary infinitesimal vector and the derivative $\partial\bi{f}/\partial\bi{y}$ is a matrix $\bss{F}$ with components

$$F_{ij} = {\partial f_i\over \partial y_j}$$ (1.25)

in which $f_i$ and $y_j$ represent the components of $\bi{f}$ and $\bi{y}$ respectively. Hence (1.18) takes the form

$$\delta y^{(n+1)}_i = \delta y^{(n)}_i - \delta t\sum_j \left.{\partial f_i\over \partial y_j}\right\vert _n \delta y^{(n)}_j$$ (1.26)

$$\delta\bi{y}_{n+1} =\left[\bss{I} - \delta t \bss{F}\right]\delta\bi{y}_n= \bss{G}\delta\bi{y}_n.$$ (1.27)

This leads directly to the stability condition that all the eigenvalues of $\bss{G}$ must have modulus less than unity (see problem 6). In general any of the stability conditions derived in this course for scalar equations can be re-expressed in a form suitable for vector equations by applying it to all the eigenvalues of an appropriate matrix.