algebraic-range

Generate ranges of algebraic numbers.

Python port of the Wolfram Language resource function AlgebraicRange 2.0, by the same author. Requires SymPy ≥ 1.12.

pip install algebraic-range

Overview

algebraic_range creates ranges made of algebraic numbers: it extends the basic concept of range() to include, besides rational numbers, also roots — always restricted to the real domain.

The first two arguments are the bounds of the range, while the optional third and fourth arguments (by default 1 and 0) set the upper and lower bounds of the steps, the differences between successive elements:

algebraic_range(x)             # Sqrt[Range[1, x²]],   for x ≥ 1
algebraic_range(x, y)          # Sqrt[Range[x², y²]],  for 0 ≤ x ≤ y
algebraic_range(x, y, s)       # steps bounded above by s,  0 < s ≤ y − x
algebraic_range(x, y, s, d)    # steps bounded below by d,  0 ≤ d ≤ |s|

If no step is given, the elementary range is the square-root grid anchored at the bounds — Sqrt[Range[x², y²]] — since among irrationals no constant step exists. Negative bounds are handled by reflection (real roots are defined for positive arguments), and a negative s produces a descending range:

>>> from algebraic_range import algebraic_range
>>> algebraic_range(3)
[1, sqrt(2), sqrt(3), 2, sqrt(5), sqrt(6), sqrt(7), 2*sqrt(2), 3]

>>> algebraic_range(-3, -1)
[-3, -2*sqrt(2), -sqrt(7), -sqrt(6), -sqrt(5), -2, -sqrt(3), -sqrt(2), -1]

With a step upper bound the range is filled by rational multiples of the elementary roots — conceptually the outer product of algebraic_range(min(x, x/s), y) with the step-s rational grid, restricted to [x, y]:

>>> from sympy import Rational
>>> algebraic_range(0, 2, Rational(1, 2))
[0, 1/2, sqrt(2)/2, sqrt(3)/2, 1, sqrt(2), 3/2, sqrt(3), 2]

Taken literally, that outer product scales quadratically; like the Wolfram 2.0 implementation, this port instead selects for each root only the admissible multipliers, by binary search over the sorted factor grid, and effectively scales linearly (see benchmark/BENCHMARK.md).

The fourth argument imposes a minimum absolute difference between successive elements, taming the accumulation of algebraics towards certain points and producing nearly uniform distributions:

>>> algebraic_range(2, 5, Rational(1, 3), Rational(1, 4))
[2, 4*sqrt(3)/3, 2*sqrt(15)/3, 2*sqrt(19)/3, sqrt(10), 2*sqrt(3),
 5*sqrt(5)/3, 4, sqrt(19), 8*sqrt(3)/3, 2*sqrt(6)]

Options

root_order

An integer r includes all roots up to order r; [r] only order r; [r1, r2, …] all listed orders.

algebraic_range(2, root_order=3)               # square and cubic roots
algebraic_range(2, root_order=[3])             # cube roots only:
# [1, 2**(1/3), 3**(1/3), 2**(2/3), 5**(1/3), 6**(1/3), 7**(1/3), 2]
algebraic_range(1, Rational(3, 2), root_order=[3, 5])

step_method

The default "Outer" uses the rational-multiplier construction above; "Root" steps directly in the power domain, (Range[x^n, y^n, s^n])^(1/n) — generally a superset:

>>> algebraic_range(0, 2, Rational(2, 3), step_method="Root")
[0, 2/3, 2*sqrt(2)/3, 2*sqrt(3)/3, 4/3, 2*sqrt(5)/3, 2*sqrt(6)/3,
 2*sqrt(7)/3, 4*sqrt(2)/3, 2]

farey_range

Generalizes the resource function FareyRange: the range combines the algebraic ranges of all steps in the Farey sequence of the given order (an integer step s means order s; a step 1/n means order n):

>>> algebraic_range(0, 3, Rational(1, 3), farey_range=True)  # F₃ steps: 1/3, 1/2, 2/3, 1

equals the union of the plain ranges with steps 1/3, 1/2, 2/3 and 1, and contains FareyRange[0, 3, 3] as its rational backbone.

formula_complexity

An alternative way to thin a range: discard elements whose symbolic form exceeds a heuristic complexity threshold.

algebraic_range(4, root_order=4, formula_complexity=8)

algebraics_only

By default, transcendental parameters are rejected (NotAlgebraicError); disable to deliberately allow them:

from sympy import sqrt, E
algebraic_range(0, 5, sqrt(E), algebraics_only=False)

working_precision

With steps below machine resolution, nearby algebraics may collide numerically and be omitted; raising the precision keeps them distinct — it can be pushed arbitrarily high:

algebraic_range(1 - Rational(1, 10**13), 1 + Rational(1, 10**13),
                Rational(1, 10**17), working_precision=30)

Behaviour notes

• Approximate inputs are exactified before use (the WL RootApproximant step): algebraic_range(0.1, 3.1) starts at 1/10.
• Results are plain lists of exact SymPy expressions, sorted ascending (descending for negative s), without duplicates.
• Errors mirror the WL failure modes: NotRealError, NotAlgebraicError, FareyStepError, LowerBoundError (negative d), StepBoundError (d > |s|).

Performance

The five reference cases of the Wolfram verification suite run in about 1–7 s each (40 000–80 000 elements) — up to ~1.7× faster than the Wolfram Language 2.0 implementation on the stepped cases, and ~40×–100× faster than algebraic-range 0.8. Details and methodology: benchmark/BENCHMARK.md.

Applications

A natural application is the search for closed forms of floating-point numbers, providing structured search ranges of algebraic candidates. The sibling package find-closed-form (port of the Wolfram resource function FindClosedForm, by the same author) accepts these ranges since version 0.5.0 through its search_range option, searching functions of exact algebraic arguments — here recovering a formula from its bare machine digits:

from find_closed_form import find_closed_form

find_closed_form(4.1132503787829275, search_range="Algebraic")
# exp(sqrt(2))

Generator options reach algebraic_range through search_range_options (e.g. {"root_order": 3} to search cube-root arguments), and an explicit algebraic_range(...) output can be passed as fixed search_arguments.

See also

• Full documentation of the original AlgebraicRange (contributed by the same author and vetted by the Wolfram Review Team), whose 2.0 verification suite this package transcribes in tests/.
• The paclet GeneralizedRange for extended functionality and syntax.
• The farey package, by the same author, for standalone Farey sequences and ranges.
• CHANGELOG.md for the 0.9 rewrite notes.