Description
Return ((x - second)/(first - second)) × ((x - third)/(first - third)).
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.
Return ((x - second)/(first - second)) × ((x - third)/(first - third)).
When to use this tool
Use Quadratic Interpolation One Abscissa Difference when you need one polynomial basis weight or residual.
How to use the tool and how it works
- Enter x, first, second, third. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read firstWeight. Use the output convention below.
Require pairwise distinct first, second, and third nodes. Return ((x - second)/(first - second)) × ((x - third)/(first - third)).1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| x | X | Required. Finite number. |
| first | First value or boundary under the stated formula. | Required. Finite number. |
| second | Second value or boundary under the stated formula. | Required. Finite number. |
| third | Third | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| firstWeight | First Weight | Required. Finite number. |
{
"input": {
"x": 1,
"first": 0,
"second": 2,
"third": 4
},
"output": {
"firstWeight": 0.375
}
}Limitations and assumptions
- This is the first quadratic Lagrange weight only; interpolation values and conditioning are not evaluated.
- 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.
- Lagrange Basis Leading Coefficient
Return the reciprocal product of selected-node minus every other node.