Numerical instability
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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.
There are open questions at the frontiers of numerical analysis which try to deal with issues of numerical instability, such as nonlinear instability, but this article is written to be accessible to those who need only understand the simplest definition of a differential equation.
A typical computation will be analyzed algebraically to reveal how the phenomenon arises. A special case of the solution of the equation
y'(x)=f(x,y(x)) with y(0)=1 on the closed interval [0,1]
will be examined. This equation is studied very closely in the subject of Ordinary Differential Equations, but we'll solve it simply using plausible notions found in elementary calculus and then evaluate the numerical stability of the method.
For the simplified (but useful) approach we first form the grid:
GN={x | xn = n*h n=0,1,...,N with h=1/N}. N is a positive integer.
In order to elucidate the main point, other simplifications will be made along the way.
Consider approximate values of y at xn. These approximations of y(xn) will be denoted by yn.