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.
Description
Jacobi polynomials depend on two shape parameters and a specified normalization.
Jacobi polynomials depend on two shape parameters and a specified normalization. Legendre polynomials are the alpha=beta=0 special case.1, 2
Evaluate jacobi 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 Jacobi Polynomial Approximation
Use this tool to evaluate jacobi 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 Jacobi 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, beta 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 |
| beta | Beta | number | Required |
Use integer degree 0–128, |x|≤100 and both parameters in (−1,100]. Evaluate the family recurrence, rejecting a numerically singular denominator. Return the finite polynomial value.
P0=1; P1=[alpha−beta+(alpha+beta+2)x]/2; higher terms use Jacobi three-term recurrence 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,
"beta": 1
},
"output": {
"polynomialValue": 0.1875
}
}Limitations and assumptions
Floating-point evaluation is not arbitrary precision; extreme parameters or degree can overflow.
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 Laguerre Polynomial Approximation: Evaluate laguerre family polynomial by three-term recurrence for degree 0..128 and |x|≤100; family parameters/domain explicit, not arbitrary caller polynomial evaluation.
Further background: Classical orthogonal polynomials: recurrence relations.
References
-
Jacobi Polynomial Approximation — tool.io
-
Classical orthogonal polynomials: recurrence relations — NIST Digital Library of Mathematical Functions
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.
- Laguerre Polynomial Approximation
Evaluate laguerre family polynomial by three-term recurrence for degree 0..128 and |x|≤100; family parameters/domain explicit, not arbitrary caller polynomial evaluation.