Is There a Computer Algebra System That Can Handle Big O Error Terms?

[CRGreathouse]

I was wondering if there was a computer algebra system (like Math'ca, Maple, or Maxima) that could handle manipulations with big O error terms.

Mathematica has an O function, but it doesn't work properly:

> x^2 Log[x] + x Log[x]^2 + x / Log[x] + O[x]

O[x^1]

(it should be "x^2 Log[x] + x Log[x]^2 + O[x]")

But I'm interested in more than just cutting off small terms. I'd like to be able to do division

> (x^2 + 3)/(x^2 + O(x))

1 + O(1/x)

and harder functions like inverses and integrals

> inverse(x^2 + O(x), x)

±sqrt(x) + O(1)

(Note: the Os in my examples are about +infinity; a system handling other limits would be nice.)

Is there any system that can do this? If not, is there a reason? Is it harder than it seems?

 

[Evalesco]

I'm not aware of any computer algebra systems that can do this, but I'm sure they exist. It may be difficult to implement this kind of functionality, as the underlying algorithms must be able to accurately track and manipulate Big O error terms. However, it is likely possible to implement some basic manipulation with Big O error terms in a computer algebra system.

 

[oswaler]

I can understand why you would be interested in a computer algebra system that can handle manipulations with big O error terms. This type of functionality can be useful in many fields of science, particularly in areas where approximations and estimations are common.

Unfortunately, I am not aware of any existing computer algebra system that specifically focuses on handling error terms. While Mathematica does have an O function, as you mentioned, it seems to have limitations and may not be suitable for more complex manipulations.

One possible reason for the lack of a dedicated system for error term handling could be the complexity and variability of error terms. Error terms can come in different forms and depend on various factors, making it challenging to create a universal system that can handle all cases.

Additionally, the demand for such a system may not be high enough to justify the development and maintenance costs. Most existing computer algebra systems already have a wide range of capabilities and may not see a need to add error term handling as a separate feature.

However, there may be other ways to achieve the desired functionality. For example, some computer algebra systems allow you to define your own functions and operations, which could potentially include error term handling. It may also be possible to use numerical methods or programming languages to handle error terms in a more customized manner.

In conclusion, while there may not be a specific computer algebra system that can handle error terms, there are likely alternative approaches that can achieve similar results. It may require some creativity and customization, but with the right tools and techniques, you can likely accomplish your desired manipulations with big O error terms.