Formal Definition
Let f(x) and g(x) be two functions defined on some subset of the real numbers. One writes
if and only if there is a positive constant M such that for all sufficiently large values of x, f(x) is at most M multiplied by g(x) in absolute value. That is, f(x) = O(g(x)) if and only if there exists a positive real number M and a real number x0 such that
In many contexts, the assumption that we are interested in the growth rate as the variable x goes to infinity is left unstated, and one writes more simply that f(x) = O(g(x)). The notation can also be used to describe the behavior of f near some real number a (often, a = 0): we say
if and only if there exist positive numbers δ and M such that
If g(x) is non-zero for values of x sufficiently close to a, both of these definitions can be unified using the limit superior:
if and only if
Read more about this topic: Big O Notation
Famous quotes containing the words formal and/or definition:
“There must be a profound recognition that parents are the first teachers and that education begins before formal schooling and is deeply rooted in the values, traditions, and norms of family and culture.”
—Sara Lawrence Lightfoot (20th century)
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)