Hurwitz Approximation Calculator

Finite Hurwitz-quality witness search among denominator1…Q, earliest denominator ties; report actual absolute error and strict1/(sqrt5 q²) bound independently for selected observed witness. Does not certify representedinputirrationality or theorem infinitely many rational approximants; bounded float observations only.

Description

Hurwitz-quality rational approximation can be examined for a finite observed real number and denominator budget.

Hurwitz-quality rational approximation can be examined for a finite observed real number and denominator budget.^[1]

This tool finds the closest supplied-budget witness and compares its error with a strict bound without certifying irrationality or infinitely many approximants.

When to use Hurwitz Approximation Calculator

This tool finds the closest supplied-budget witness and compares its error with a strict bound without certifying irrationality or infinitely many approximants.

How to use Hurwitz Approximation Calculator and how it works

Prepare finite real, maximum denominator, using the input definitions and matching units below.
Enter the required fields and run Hurwitz Approximation Calculator.
Read result.numerator, result.denominator, result.absoluteError, result.hurwitzStrictErrorBound, result.observedWitnessWithinStrictBound using the units, signs and ordering explained below.
Correct any rejected inputs and rerun; interpret the result within the stated model and its assumptions.

Input field	Meaning	Value type	Requirement
actualFiniteReal	Finite observed x with absolute value no greater than 100000000.	number	Required; finite; other constraints below
actualMaximumDenominator	Positive integer denominator-search budget, at most 10000.	number	Required; finite; other constraints below

Require finite real x with absolute value<=1e8 and maximum denominator integer 1–10000. For each q from one through budget, choose p=round(x*q), calculate absolute error and keep a witness only when its error is strictly smaller, so earliest denominator wins equal-error ties. Return numerator, denominator, error, 1/(sqrt(5)*q^2) and the strict comparison flag. Ratios are not explicitly reduced and no irrationality or infinite witness theorem is certified.

witness=argmin_over_q<=Q(abs(x-round(x*q)/q)); bound=1/(sqrt(5)*q^2); within=(error<bound)

Apply the input ordering, units and branch conditions stated above.

Output field	Meaning	Value type
result	Object containing the calculated or selected fields below.	object
result.numerator	Integer numerator chosen by the bounded rational-witness search.	number
result.denominator	Positive selected denominator; equal-error ties retain the earliest denominator.	number
result.absoluteError	Absolute observed floating-point difference between input and selected rational witness.	number
result.hurwitzStrictErrorBound	The value 1/(sqrt(5)*denominator squared) for the selected witness.	number
result.observedWitnessWithinStrictBound	True when observed absolute error is strictly smaller than the bound.	boolean

json

{
  "input": {
    "actualFiniteReal": 0.3,
    "actualMaximumDenominator": 10
  },
  "output": {
    "result": {
      "numerator": 3,
      "denominator": 10,
      "absoluteError": 0,
      "hurwitzStrictErrorBound": 0.00447213595499958,
      "observedWitnessWithinStrictBound": true
    }
  }
}

Limitations and assumptions

The calculation uses the supplied observations and model inputs. It does not collect measurements, select a reference population or coefficient, or establish that the model applies to the actual case. The stated branch conditions and output units remain part of the result.

All fields require explicit values. Unexpected fields, wrong types, nonfinite numbers and sparse arrays are rejected. Alignment, bounds and supported-token rules are explained above; extremely large inputs or nonfinite results can also fail. Input errors differ from successful false comparisons, unavailable-domain flags and empty match lists.

Alternative or Complementary methods

Use Khinchin Approximation Calculator for a complementary operation: A Khinchin approximation statement needs its specific definition and measure-theoretic assumptions.

Use Khinchin Constant Approximation for a complementary operation: Khinchin's geometric continued-fraction constant concerns a measure-theoretic limiting behavior.

References

Hurwitz Approximation Calculator — tool.io

Similar or alternative tools

Khinchin Approximation Calculator
A Khinchin approximation statement needs its specific definition and measure-theoretic assumptions.
Khinchin Constant Approximation
Khinchin's geometric continued-fraction constant concerns a measure-theoretic limiting behavior.