Description
Build bounded dense ascending coefficients by summing explicit sparse power occurrences into a supplied maximum degree; duplicates are summed, missing powers are zeros.
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.
Build bounded dense ascending coefficients by summing explicit sparse power occurrences into a supplied maximum degree; duplicates are summed, missing powers are zeros.
When to use this tool
Use this tool to build bounded dense ascending coefficients by summing explicit sparse power occurrences into a supplied maximum degree; duplicates are summed, missing powers are zeros.
How to use the tool and how it works
- Enter termsJson, degree using the field definitions below. For JSON text, serialize the explicit records in the shown shape.
- Run the operation. Correct input errors before interpreting an output.
- Read result.coefficients. Use the ordering, boundary rules and absence conditions described below.
Create degree + 1 zero coefficients, then add each supplied term’s coefficient at its bounded integer power.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| termsJson | JSON array of {power, coefficient} records; repeated powers are summed. | Text, at most 8,192 UTF-16 code units; the operation’s grammar also applies. |
| degree | Nonnegative degree used for the requested output extent or truncation. | A finite safe integer within the operation’s stated bounds. |
These are the decoded structures used in the example’s JSON text fields. Supply the corresponding serialized JSON strings as input.
{
"termsJson": [
{
"power": 0,
"coefficient": 1
},
{
"power": 2,
"coefficient": 3
}
]
}| Output field | Interpretation |
|---|---|
| result | Object containing the named fields below. |
| result.coefficients | Result coefficients in ascending power order. |
{
"input": {
"termsJson": "[{\"power\":0,\"coefficient\":1},{\"power\":2,\"coefficient\":3}]",
"degree": 2
},
"output": {
"result": {
"coefficients": [
1,
0,
3
]
}
}
}Limitations and assumptions
- Repeated powers are summed; terms above the requested degree are rejected rather than truncated.
- 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 Sparse Power Pair Extractor
Extract nonzero ascending polynomial power/coefficient records; explicit zero powers are intentionally omitted and exact equality is used.
- Polynomial Low Power Remainder
Split ascending polynomial coefficients into remainder modulo x^k and quotient by x^k; not general polynomial long division.