Numerical instability

Numerical Instability deals with an unexpected phenomenon, whose provenence is very obscure, which arises from the numerical solution of differential equations. The usual notion of instability includes concepts such as fickleness, unreliability, and oversensitivity. If a computation is numerically unstable it gives inaccurate results. A major source of numerical instability results from a cumulative magnification effect of small errors on the numerical results.

As an example, there is the special case of the solution of the equation

      y'(x)=f(x,y(x))  with y(0)=1 on the closed interval [0,1] 

This equation is studied very closely in the subject of Ordinary Differential Equations, but simply using plausible notions found in elementary calculus the numerical stability of the method can be examined.

For the simplified (but useful) approach this grid may be used:

      GN={x | xn = n*h   n=0,1,...,N with h=1/N}. N is a positive integer.

Other simplifications can be made as we go along, in order to elucidate the main point. For example, consider approximate values of y at x_n. These approximations of y(x_n) will be denoted by y_n.