Hermite Polynomial Approximation
Evaluate hermite family polynomial by three-term recurrence for degree 0..128 and |x|≤100; physicists H_n convention.
Description
Hermite polynomials have physicists' and probabilists' normalizations.
When to use Hermite Polynomial Approximation
Use this tool to evaluate hermite family polynomial by three-term recurrence for degree 0..128 and |x|≤100; physicists H_n convention.
Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
How to use Hermite 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 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 |
Enter integer degree 0–128 and finite |x|≤100. Advance the physicists' recurrence and return the polynomial value.
H0=1, H1=2x; H_(k+1)=2xH_k−2kH_(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": 3,
"x": 1
},
"output": {
"polynomialValue": -4
}
}Limitations and assumptions
Do not substitute probabilists' He coefficients. Large values can overflow and are rejected.
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 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. For a complementary task, use Chebyshev Polynomial Approximation: Evaluate chebyshev family polynomial by three-term recurrence for degree 0..128 and |x|≤100; first-kind T_n convention.
Further background: Classical orthogonal polynomials: recurrence relations.
References
-
Hermite Polynomial Approximation — tool.io
-
Classical orthogonal polynomials: recurrence relations — NIST Digital Library of Mathematical Functions
Similar or alternative tools
- 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.
- Chebyshev Polynomial Approximation
Evaluate chebyshev family polynomial by three-term recurrence for degree 0..128 and |x|≤100; first-kind T_n convention.