Previous Page Next Page Contents

O -- the domain of order terms (Landau symbols)

Introduction

O(f, x = x0) represents the Landau symbol O(f, x -> x0).

Call(s)

O(f <, x = x0, y = y0...>)

Parameters

f - an arithmetical expression representing a function in x, y etc.
x, y... - the variables: identifiers
x0, y0... - the limit points: arithmetical expressions

Returns

an element of the domain O.

Related Functions

asympt, limit, series, taylor

Details

Example 1

For polynomial expressions, certain simplifications occur:

>> O(x^4 + 2*x^2), O(7*x^3), O(x, x = 1)
                            2      3
                         O(x ), O(x ), O(1, x = 1)

A zero limit point is not printed on the screen:

>> O(1), O(1, x = 1), O(x^2/(y + 1), x = 0, y = -1, z = PI)
                                   /   2                   \
                                   |  x                    |
               O(1), O(1, x = 1), O| -----, z = PI, y = -1 |
                                   \ y + 1                 /

The arithmetical operations are overloaded for order terms:

>> 7*O(x), O(x^2) + O(x^13), O(x^3) - O(x^3), O(x^2)^2 + O(x^4)
                                  2      3      4
                         O(x), O(x ), O(x ), O(x )

Example 2

For multivariate polynomial expression, higher order terms are discarded if they are divisible by lower order terms:

>> O(15*x*y^2 + 3*x^2*y + x^2*y^2)
                                    2    2
                             O(5 x y  + x  y)
>> O(x + x^2*y) = O(x)*O(1 + x*y)
                                O(x) = O(x)

Example 3

We demonstrate how to access the variables and the limit points of an order term:

>> a := O(x^2*y^2)
                                    2  2
                                 O(x  y )
>> indets(a) = O::indets(a), O::points(a)
                      {x, y} = {x, y}, {x = 0, y = 0}
>> delete a:

Changes




Do you have questions or comments?


Copyright © SciFace Software GmbH & Co. KG 2000