Quadratic Barycentric Pole Residue Tool

Description

For poleIndex, compute the reciprocal product of its two node differences; return that denominator residue, its product with the node value, and the removable limit.

Barycentric interpolation expresses a polynomial as a quotient of weighted sums. Standard weights depend on node differences; common nonzero scaling cancels only when applied consistently to the full quotient.

For poleIndex, compute the reciprocal product of its two node differences; return that denominator residue, its product with the node value, and the removable limit.

When to use this tool

Use Quadratic Barycentric Pole Residue when you need one barycentric weight, sign, or influence component.

How to use the tool and how it works

Enter nodes, values, poleIndex. Keep parallel arrays aligned. Use the units and conventions below.
Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
Read denominatorResidue, numeratorResidue, removableLimit. Use the output convention below.

Require three distinct nodes and matching values. For poleIndex, compute the reciprocal product of its two node differences; return that denominator residue, its product with the node value, and the removable limit.^[1]

Input field	Meaning and units	Accepted input
nodes	Nodes	Required. Array of finite number; maximum items 256.
values	Values	Required. Array of finite number; maximum items 256.
poleIndex	Pole Index	Required. Safe integer; minimum 0; maximum 2.

Output field	Interpretation	Returned shape
denominatorResidue	Denominator Residue	Required. Finite number.
numeratorResidue	Numerator Residue	Required. Finite number.
removableLimit	Removable Limit	Required. Finite number.

json

{
  "input": {
    "nodes": [
      0,
      1,
      2
    ],
    "values": [
      2,
      3,
      6
    ],
    "poleIndex": 1
  },
  "output": {
    "denominatorResidue": -1,
    "numeratorResidue": -3,
    "removableLimit": 3
  }
}

Limitations and assumptions

The limit is the supplied node value; this does not evaluate the rational formula at a singular denominator or estimate conditioning.
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 barycentric evaluator with exact-node handling and appropriate weight scaling for actual interpolation.

References

Calculation and input contract: interpolation/quadratic-barycentric-pole-residue — tool.io

Similar or alternative tools

Barycentric Single Node Influence Ratio
At an exact node hit return one for that selected node and zero for others; otherwise normalize selected weight/(x-node) by the sum of all such terms.
Lagrange Basis Leading Coefficient
Return the reciprocal product of selected-node minus every other node.