Perturbation of Linear Equations

This chapter will discuss the behaviour of the system of linear simultaneous equations $$ Ax = b $$ when the matrix A and the vector b are subjected to small order perturbations ∆A and ∆b respectively. The problem then becomes $$ \left( {A + \Delta A} \right)\left( {x + \Delta x} \right) = b + \Delta b $$ and we are mainly concerned with studying the deviation ∆x of the solution with the perturbation. Such an exercise is called sensitivity analysis, for the extent of the deviation ∆x relative to ∆A and ∆b defines the sensitivity of the system. A highly sensitive system is roughly one where a relatively large deviation ∆x is incurred by small perturbations ∆A and ∆b. As we shall see, highly sensitive systems are generally to be avoided in practice. They are referred to as ill-conditioned, for a highly sensitive system would yield large variations in the results for only small uncertainties in the data. To clarify this fact, let us study the behaviour of the system of equations $$ x + y = 2 $$$$ 0.49x + 0.51y = 1 $$ — representing a pair of straight lines intersecting at x = 1, y = 1 — when a small term ε is added to the equations. Surprisingly enough, the set of equations obtained, namely $$ x + y = 2 + \varepsilon $$$$ 0.49x + 0.51y = 1 $$ represents a pair of straight lines meeting at a point (x, y) given by x = 1 + 25.5ε; y = 1 − 24.5ε, and rather distant from x = 1, y = 1. Here, a small change of order ε in the equations has produced a change of 25ε in the solution. This system has definitely a high sensitivity to perturbations in its matrix A and vector b.