Description
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.
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.
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.
When to use this tool
Use Sequence: Linear Recurrence First Difference Step 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 state, coefficients. 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 result. Expand result to inspect the returned fields.
Dot aligned nonempty state and coefficients for nextValue, subtract the last state entry from nextValue for firstDifference, and shift the oldest-first state left before appending next, inside result.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| state | Oldest-first values aligned with coefficients. | Required. Array of finite number.. |
| coefficients | Coefficients | Required. Array of finite number.. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| result | Result | Required. Object. |
{
"input": {
"state": [
2,
3
],
"coefficients": [
1,
1
]
},
"output": {
"result": {
"nextValue": 5,
"firstDifference": 2,
"nextState": [
3,
5
]
}
}
}Limitations and assumptions
- Coefficient order must match the supplied state. The tool advances one caller-defined linear rule without fitting it, checking earlier terms, or proving stability or convergence.
- Outputs are nested inside result. 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.
- Recurrence Initial State Shift Step
Emit state[0] and return the nonempty state's remaining entries followed by supplied next.