Description
Inspect a supplied leading-to-trailing recurrence coefficient list without solving the recurrence.
A sequence rule specifies an index origin, state order, and relation among terms. Advancing one step, mapping an index, and checking a local residual answer different questions. Supplied coefficients or periods define a model; they do not establish that observations follow it.
Inspect a supplied leading-to-trailing recurrence coefficient list without solving the recurrence.
When to use this tool
Use Recurrence Coefficient Leading Gap when you need this particular index map, recurrence step, count, or local algebraic term under supplied sequence conventions.
How to use the tool and how it works
- Enter coefficients, referenceLeading. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read leadingIndex, leadingCoefficient, gap, allZero. Inspect allZero before interpreting the associated values.
Find the first nonzero coefficient, return its zero-based leadingIndex, leadingCoefficient, and difference from referenceLeading, plus allZero. Use index minus one and coefficient zero when no nonzero entry exists.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| coefficients | Coefficients | Required. Array of finite number; maximum items 256. |
| referenceLeading | Reference Leading | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| leadingIndex | First nonzero position or minus one for all-zero/empty list. | Required. Safe integer. |
| leadingCoefficient | Leading Coefficient | Required. Finite number. |
| gap | Gap | Required. Finite number. |
| allZero | All Zero | Required. Boolean. |
{
"input": {
"coefficients": [
0,
0,
3,
2
],
"referenceLeading": 2
},
"output": {
"leadingIndex": 2,
"leadingCoefficient": 3,
"gap": 1,
"allZero": false
}
}Limitations and assumptions
- The first array position is the caller's leading-order convention. An empty list is allZero, and no polynomial degree, recurrence validity, or significance threshold is inferred.
- The field tables state structural bounds; the mechanics also specify cross-field constraints. Malformed inputs and unsupported domains cause errors. A successful false flag, empty branch value, or zero placeholder has the meaning stated above and is distinct from an input error.
- 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. Signed zero is exported as zero.
Alternative or Complementary methods
Choose exact integer arithmetic when the index or coefficient range requires it. A complete sequence analysis needs actual terms and a consistent index and state-order convention beyond one local step.
References
Similar or alternative tools
- Linear Recurrence Coefficient Impulse Term
Return contribution=coefficients[index]*impulse for a valid zero-based index.
- Sequence: Linear Recurrence First Difference Step
Advance one short real linear recurrence with oldest-to-newest state and matching coefficient order, returning next state and its first difference; no symbolic recurrence solving.