Weighted Manhattan Axis Subgradient Tool

Description

For weight*abs(difference), return lower = upper = -weight when difference < 0, +weight when positive, and the interval [-weight,+weight] at zero.

Weighted coordinate distances depend on whether weights multiply absolute differences or squared differences. Zero weights can make different vectors equivalent, and derivatives at a zero distance or absolute-value corner need explicit treatment.

For weight*abs(difference), return lower = upper = -weight when difference < 0, +weight when positive, and the interval [-weight,+weight] at zero.

When to use this tool

Use Weighted Manhattan Axis Subgradient when checking one weighted-axis contribution, support rule or derivative.

How to use the tool and how it works

Enter difference, weight. Use the units and conventions below.
Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
Read lower, upper. Use the output convention below.

Require nonnegative weight. For weight*abs(difference), return lower = upper = -weight when difference < 0, +weight when positive, and the interval [-weight,+weight] at zero.^[1]

Input field	Meaning and units	Accepted input
difference	Difference	Required. Finite number.
weight	Weight	Required. Finite number; minimum 0.

Output field	Interpretation	Returned shape
lower	Lower endpoint of the absolute-value axis subgradient.	Required. Finite number.
upper	Upper endpoint of that subgradient; endpoints differ at zero.	Required. Finite number.

json

{
  "input": {
    "difference": 0,
    "weight": 3
  },
  "output": {
    "lower": -3,
    "upper": 3
  }
}

Limitations and assumptions

This is one axis's subgradient with respect to its difference coordinate, not a full vector distance or optimizer. At zero the derivative is set-valued.
Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Numeric signed-zero results are exported as zero.

Alternative or Complementary methods

Use a complete distance implementation with an explicit weighting and missing-coordinate convention for vector comparison and optimization.

References

Calculation and input contract: distance/weighted-manhattan-axis-subgradient — tool.io

Similar or alternative tools

Weighted Euclidean One Axis Gradient
Return one axis gradient of a supplied positively supported weighted Euclidean distance.
Weighted Interval Gap Tool
Calculate weighted interval gap from explicit caller-supplied inputs.