but then what does O(a,b,c) means
On Tue, Feb 28, 2012 at 7:30 PM, Ronan Lamy <ronan.l...@gmail.com> wrote:
> Le mardi 28 février 2012 à 19:25 +0530, prateek papriwal a écrit :
> > what is O notation ?
>
> Typing "O?" in isympy:
>
> Type: WithAssumptions
> Base Class: <class 'sympy.core.assumptions.WithAssumptions'>
> String Form:<class 'sympy.series.order.Order'>
> Namespace: Interactive
> File: /home/ronan/dev/sympy/sympy/series/order.py
> Docstring:
> Represents the limiting behavior of some function
>
> The order of a function characterizes the function based on the limiting
> behavior of the function as it goes to some limit. Only taking the limit
> point to be 0 is currently supported. This is expressed in big O
> notation
> [1]_.
>
> The formal definition for the order of a function `g(x)` about a point
> `a`
> is such that `g(x) = O(f(x))` as `x \rightarrow a` if and only if for
> any
> `\delta > 0` there exists a `M > 0` such that `|g(x)| \leq M|f(x)|` for
> `|x-a| < \delta`. This is equivalent to `\lim_{x \rightarrow a}
> |g(x)/f(x)| < \infty`.
>
> Let's illustrate it on the following example by taking the expansion of
> `\sin(x)` about 0:
>
> .. math ::
> \sin(x) = x - x^3/3! + O(x^5)
>
> where in this case `O(x^5) = x^5/5! - x^7/7! + \cdots`. By the
> definition
> of `O`, for any `\delta > 0` there is an `M` such that:
>
> .. math ::
> |x^5/5! - x^7/7! + ....| <= M|x^5| \text{ for } |x| < \delta
>
> or by the alternate definition:
>
> .. math ::
> \lim_{x \rightarrow 0} | (x^5/5! - x^7/7! + ....) / x^5| < \infty
>
> which surely is true, because
>
> .. math ::
> \lim_{x \rightarrow 0} | (x^5/5! - x^7/7! + ....) / x^5| = 1/5!
>
>
> As it is usually used, the order of a function can be intuitively
> thought
> of representing all terms of powers greater than the one specified. For
> example, `O(x^3)` corresponds to any terms proportional to `x^3,
> x^4,\ldots` and any higher power. For a polynomial, this leaves terms
> proportional to `x^2`, `x` and constants.
>
> Examples
> ========
>
> >>> from sympy import O
> >>> from sympy.abc import x
> >>> O(x)
> O(x)
> >>> O(x)*x
> O(x**2)
> >>> O(x)-O(x)
> O(x)
>
> References
> ==========
>
> .. [1] `Big O notation <http://en.wikipedia.org/wiki/Big_O_notation>`_
>
>
> --
> You received this message because you are subscribed to the Google Groups
> "sympy" group.
> To post to this group, send email to sympy@googlegroups.com.
> To unsubscribe from this group, send email to
> sympy+unsubscr...@googlegroups.com.
> For more options, visit this group at
> http://groups.google.com/group/sympy?hl=en.
>
>
--
You received this message because you are subscribed to the Google Groups
"sympy" group.
To post to this group, send email to sympy@googlegroups.com.
To unsubscribe from this group, send email to
sympy+unsubscr...@googlegroups.com.
For more options, visit this group at
http://groups.google.com/group/sympy?hl=en.
