dependent_bterms

An extension to SageMath's module for computations with asymptotic expansions. Provides a special AsymptoticRing that allows to specify a secondary, dependent (monomially bounded) variable.

Quickstart and Summary

The package can be made available to your SageMath installation by running

$ sage -pip install dependent_bterms

from your terminal, or just

!pip install dependent_bterms

from within a SageMath Jupyter notebook. The package can then be used as follows:

sage: import dependent_bterms as dbt
sage: AR, n, k = dbt.AsymptoticRingWithDependentVariable(  # create a special AsymptoticRing
....:     'n^QQ',  # in one asymptotic (monomial) variable n with rational powers
....:     'k', 0, 1/2,  # with a symbolic variable k assumed to be in the range n^0 <= k <= n^(1/2)
....:     bterm_round_to=2  # and explicit error terms should be rounded to two decimal places
....: )
sage: k*n^2 + O(n^(3/2)) + k^3*n  # summands are ordered w.r.t. their highest potential growth
k^3*n  + k*n^2 + O(n^(3/2))
sage: asy = 1/n + AR.B(k/n^2, valid_from=10)
sage: asy_exp = dbt.taylor_with_explicit_error(lambda t: exp(t), asy, order=3, valid_from=10)
sage: asy_exp
1 + n^(-1) + B((abs(28/25*k + 73/100))*n^(-2), n >= 10)
sage: dbt.simplify_expansion(asy_exp, simplify_bterm_growth=True)
1 + n^(-1) + B(34/25*n^(-3/2), n >= 10)

One-line descriptions of the top-level members exported with this module are given below. A description of their respective input arguments and several examples are provided in the respective docstrings.

• AsymptoticRingWithDependentVariables -- A special (univariate) AsymptoticRing that is aware of a monomially bounded symbolic variable.

• evaluate -- Evaluate a symbolic expression without necessarily returning a result in the symbolic ring.

• simplify_expansion -- Simplify an asymptotic expansion by allowing error terms to try and absorb parts of exact terms.

• round_bterm_coefficients -- Round the coefficients of all B-terms in the given expansion to the next integer (or rational with respect to the provided precision).

• set_bterm_valid_from -- Changes the point from which a B-term bound is valid such that the term remains valid.

• expansion_upper_bound -- Returns an upper bound for the given asymptotic expansion by turning all B-term instances into exact terms

• taylor_with_explicit_error -- Determines the series expansion with explicit error bounds of a given function f at a specified asymptotic term.

Demo

A worksheet containing a comprehensive introduction to the capabilities of this package can be found here: toolbox_demo.ipynb.