Description
Multiply two ascending-power polynomial coefficient lists by convolution and normalize high-power exact zeros; no FFT or symbolic integer arithmetic.
A coefficient list here is constant-first: index j multiplies x to power j. Thus [1, 2, 3] represents 1 + 2x + 3x². Trailing list zeros are high-power zeros; stored length can differ from mathematical degree, and the zero polynomial needs a separate degree convention.
Multiply two ascending-power polynomial coefficient lists by convolution and normalize high-power exact zeros; no FFT or symbolic integer arithmetic.
When to use this tool
Use this tool to multiply two ascending-power polynomial coefficient lists by convolution and normalize high-power exact zeros; no FFT or symbolic integer arithmetic.
How to use the tool and how it works
- Enter left, right using the field definitions below.
- Run the operation. Correct input errors before interpreting an output.
- Read result.coefficients. Use the ordering, boundary rules and absence conditions described below.
Convolve constant-first coefficient lists and remove redundant high-power zeros from the product.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| left | Left factor’s constant-first coefficient list. | Dense list of at most 128 finite numbers; integer/shape constraints above also apply. |
| right | Right factor’s constant-first coefficient list. | Dense list of at most 128 finite numbers; integer/shape constraints above also apply. |
| Output field | Interpretation |
|---|---|
| result | Object containing the named fields below. |
| result.coefficients | Result coefficients in ascending power order. |
{
"input": {
"left": [
1,
2
],
"right": [
3,
4
]
},
"output": {
"result": {
"coefficients": [
3,
10,
8
]
}
}
}Limitations and assumptions
- Both inputs must be nonempty and the unnormalized product length must not exceed 128.
- Coefficient lists use ascending powers and ordinary floating-point arithmetic. Nonempty input is required; individual operations impose tighter size limits. The tool does not attach or check physical units for x and its coefficients.
- Every declared top-level input field is required and unexpected fields are rejected. Ordinary strings are limited to 8,192 UTF-16 code units and lists to 128 entries, with tighter operation-specific bounds where stated. JSON text has bounded nesting and node counts.
- Numeric outputs must be finite and no larger in absolute magnitude than the safe-integer limit; operations returning exact large integers use decimal text. Positions are zero-based and spans end exclusively unless the operation explicitly states a different convention.
Alternative or Complementary methods
Check coefficient order before importing data from another algebra library. Use separate root-finding or symbolic algebra methods when an operation requires more than the stated coefficient transformation.
References
Similar or alternative tools
- Polynomial Truncated Coefficient Product
Return polynomial product coefficients only through an explicit maximum power, equivalent to reduction modulo x^(degree+1); no coefficient rounding.
- Polynomial Antiderivative Coefficient List
Integrate ascending-power polynomial coefficients with an explicitly supplied integration constant; bounded number arithmetic, no symbolic rational simplification.