Description
For a_n = firstCoefficient*a_(n-1)+secondCoefficient*a_(n-2), return characteristic-root sum and product.
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.
For a_n = firstCoefficient*a_(n-1)+secondCoefficient*a_(n-2), return characteristic-root sum and product.
When to use this tool
Use Second Order Recurrence Characteristic Product 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 firstCoefficient, secondCoefficient. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read rootSum, rootProduct. Use the output convention below.
Return rootSum=firstCoefficient and rootProduct=-secondCoefficient.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| firstCoefficient | First Coefficient | Required. Finite number. |
| secondCoefficient | Second Coefficient | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| rootSum | Root Sum | Required. Finite number. |
| rootProduct | Negative secondCoefficient under r²-a*r-b convention. | Required. Finite number. |
{
"input": {
"firstCoefficient": 3,
"secondCoefficient": 2
},
"output": {
"rootSum": 3,
"rootProduct": -2
}
}Limitations and assumptions
- The characteristic convention is r²-firstCoefficient*r-secondCoefficient, corresponding to next=firstCoefficient*current+secondCoefficient*previous. Roots are not solved and recurrence data, real-root existence, or stability are not checked.
- 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
- Second Order Recurrence State Determinant
Return determinant=previous*next-current² for three supplied terms.
- Linear Recurrence Coefficient Impulse Term
Return contribution=coefficients[index]*impulse for a valid zero-based index.