Description
Sum their quadratic Lagrange contributions at x and return observed minus that interpolant.
Lagrange basis functions equal one at their own node and zero at the other distinct nodes. Their weighted sum interpolates node values, but closely spaced nodes can create numerical sensitivity.
Sum their quadratic Lagrange contributions at x and return observed minus that interpolant.
When to use this tool
Use Quadratic Interpolant One Node Residual when you need one polynomial basis weight or residual.
How to use the tool and how it works
- Enter x, nodes, values, observed. 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 residual. Use the output convention below.
Require three distinct nodes and three values. Sum their quadratic Lagrange contributions at x and return observed minus that interpolant.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| x | X | Required. Finite number. |
| nodes | Nodes | Required. Array of finite number; maximum items 256. |
| values | Values | Required. Array of finite number; maximum items 256. |
| observed | Observed | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| residual | Signed residual using the convention in the calculation paragraph. | Required. Finite number. |
{
"input": {
"x": 3,
"nodes": [
0,
1,
2
],
"values": [
0,
1,
4
],
"observed": 10
},
"output": {
"residual": 1
}
}Limitations and assumptions
- Extrapolation is permitted; this is a residual against a supplied observation, not a certified interpolation-error bound.
- 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 stable full interpolation evaluator and inspect node conditioning before relying on high-order or extrapolated values.
References
Similar or alternative tools
- Interpolation: Quadratic Lagrange Basis At Point
Multiply (x - otherNode) / (selectedNode - otherNode) across the two other nodes for basisValue.
- Quadratic Barycentric Pole Residue
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.