Gegenbauer Polynomial Approximation
Evaluate gegenbauer family polynomial by three-term recurrence for degree 0..128 and |x|≤100; family parameters/domain explicit, not arbitrary caller polynomial evaluation.
Description
Gegenbauer polynomials form an alpha-parameterized family related to Legendre and Chebyshev polynomials.
Gegenbauer polynomials form an alpha-parameterized family related to Legendre and Chebyshev polynomials. The positive-alpha convention determines the normalization used here.1
Evaluate gegenbauer family polynomial by three-term recurrence for degree 0..128 and |x|≤100; family parameters/domain explicit, not arbitrary caller polynomial evaluation.
When to use Gegenbauer Polynomial Approximation
Use this tool to evaluate gegenbauer family polynomial by three-term recurrence for degree 0..128 and |x|≤100; family parameters/domain explicit, not arbitrary caller polynomial evaluation.
Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
How to use Gegenbauer Polynomial Approximation and understand its results
- Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
- Enter degree, x, alpha using the field table and calculation rules, then run the tool.
- Read polynomialValue. The explanation below defines the units, signs, ordering and any flags or placeholder values.
- If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.
| Input field | Meaning | Value type; see rules below | Requirement |
|---|---|---|---|
| degree | Degree | number | Required |
| x | X | number | Required |
| alpha | Alpha | number | Required |
Use integer degree 0–128, |x|≤100 and 0<alpha≤100. Advance the three-term recurrence and return the finite value.
C0=1, C1=2alpha x; C_(k+1)=[2(k+alpha)xC_k−(k+2alpha−1)C_(k−1)]/(k+1) Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| polynomialValue | Polynomial value | number |
{
"input": {
"degree": 2,
"x": 0.5,
"alpha": 1
},
"output": {
"polynomialValue": 0
}
}Limitations and assumptions
No arbitrary-precision accuracy guarantee is supplied. Negative or zero alpha is outside this implementation.
Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.
Alternative or Complementary methods
Use Legendre Polynomial Approximation: Evaluate legendre family polynomial by three-term recurrence for degree 0..128 and |x|≤100; family parameters/domain explicit, not arbitrary caller polynomial evaluation. For a complementary task, use Jacobi Polynomial Approximation: Evaluate jacobi family polynomial by three-term recurrence for degree 0..128 and |x|≤100; family parameters/domain explicit, not arbitrary caller polynomial evaluation.
References
-
Gegenbauer Polynomial Approximation — tool.io
Similar or alternative tools
- Legendre Polynomial Approximation
Evaluate legendre family polynomial by three-term recurrence for degree 0..128 and |x|≤100; family parameters/domain explicit, not arbitrary caller polynomial evaluation.
- Jacobi Polynomial Approximation
Evaluate jacobi family polynomial by three-term recurrence for degree 0..128 and |x|≤100; family parameters/domain explicit, not arbitrary caller polynomial evaluation.