also do we have theta Notation also in Sympy?
On Tue, Feb 28, 2012 at 7:33 PM, prateek papriwal <papriwalprat...@gmail.com
> wrote:
> 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>`_
>>
>>
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>>
>
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