algebraic-range 0.8.0

Generate ranges of algebraic numbers — a Python port of the Wolfram Language AlgebraicRange resource function.

algebraic-range

Generate ranges of algebraic numbers.

Python port of the Wolfram resource function "AlgebraicRange" resource function.

Requires SymPy ≥ 1.12.

Overview

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

The first two arguments represent the bounds of the range (minimum and maximum values), while the optional third and fourth arguments (by default equal to 1 and 0) regulate the upper and lower bounds of the steps (differences between successive elements).

Usage

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)          # step upper bound s,  0 < s ≤ y
algebraic_range(x, y, s, d)       # step lower bound d,  0 ≤ d ≤ s

Parameters

Parameter	Default	Description
r1	(required)	Start of range (or single argument)
r2	None	End of range
s	None	Step upper bound (negative → descending)
d	0	Step lower bound
root_order	2	int r → orders 2..r; [r] → only order r; [r1,r2,…] → listed orders
step_method	"Outer"	"Outer" or "Root"
farey_range	False	Use Farey-sequence–based rational multipliers
formula_complexity_threshold	inf	Discard elements above this complexity
algebraics_only	True	Reject transcendental inputs

Options

root_order

# Include square and cubic roots
algebraic_range(2, root_order=3)

# Only cubic roots
algebraic_range(2, root_order=[3])

# Cubic and fifth roots
algebraic_range(1, Rational(3, 2), root_order=[3, 5])

step_method

# "Root" method: Sqrt[Range[x², y², s²]]
algebraic_range(0, 3, Rational(1, 3), step_method="Root")

The default "Outer" method uses the outer product construction. The "Root" method is generally a superset.

farey_range

algebraic_range(0, 3, Rational(1, 3), farey_range=True)

Generalises FareyRange by combining algebraic ranges over all Farey-sequence steps.

formula_complexity_threshold

# Only keep simple expressions
algebraic_range(4, root_order=4, formula_complexity_threshold=8)

algebraics_only

from sympy import sqrt, E

# This raises NotAlgebraicError:
# algebraic_range(0, 5, sqrt(E))

# Allow transcendental step:
algebraic_range(0, 5, sqrt(E), algebraics_only=False)

Properties

• Extends range(): set(range(x, y+1)) ⊆ set(algebraic_range(x, y))
• Negative reflection: algebraic_range(-y, -x) = list(reversed([-v for v in algebraic_range(x, y)]))
• All outputs are algebraic (when algebraics_only=True) and real
• Sorted in ascending order (or descending for negative step)
• No duplicates

See also

More details and examples can be found in the documentation for the original Wolfram Language resource function AlgebraicRange, contributed by the same author and vetted by the Wolfram Review Team.