Sources of Error

There are three main sources of errors in numerical computation: rounding, data uncertainty, and truncation.

• Rounding errors, also called arithmetic errors, are an unavoidable consequence of working in finite precision arithmetic. We will deal with these errors in the context of polynomial evaluation and solving linear equations.
• Uncertainty in the data is always a possibility when we are solving practical problems. It may arise in several ways: from errors in measuring physical quantities, from errors in storing the data on the computer (rounding errors), or, if the data is itself the solution to another problem, it may be the result of errors in an earlier computation. The effects of errors in the data are generally easier to understand than the effects of rounding errors committed during a computation, because data errors can be analyzed using perturbation theory for the problem at hand, while intermediate rounding errors require an analysis specific to the given method.
• Truncation or Discretization or Approximation errors are much harder to analyze. We will deal with them in the context of solving differential equations. Many standard numerical methods (for example, the trapezoidal rule for quadrature, Euler's method for differential equations, and Newton's method for nonlinear equations) can be derived by taking finitely many terms of a Taylor series. The terms omitted constitute the truncation error, and for many methods, the size of this error depends on a parameter (often called the stepsize), whose appropriate value is a compromise between obtaining a small error and a fast computation. For many scientific problems, such approximations are necessary.