Next: Floating Point Number
Up: No Title
Previous: Relative Error and
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.
Dinesh Manocha / Stephen Pizer
Wed Jan 8 00:43:08 EST 1997 / Jan 19 1999