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.

Description

Legendre polynomials are a specific orthogonal-polynomial family.

Legendre polynomials are a specific orthogonal-polynomial family. Their three-term recurrence fixes normalization without requiring caller coefficient expansion.^{[1, 2]}

Evaluate legendre 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 Legendre Polynomial Approximation

Use this tool to evaluate legendre 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 Legendre 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 real x with |x|≤100. Advance the three-term recurrence and return the floating-point value; degree zero returns 1.

P0=1, P1=x; P_(k+1)=[(2k+1)xP_k−kP_(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

json

{
  "input": {
    "degree": 3,
    "x": 0.5
  },
  "output": {
    "polynomialValue": -0.4375
  }
}

Limitations and assumptions

The domain extends beyond the usual orthogonality interval, where values can grow rapidly. No arbitrary-precision error certificate is supplied.

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 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. 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

Legendre Polynomial Approximation — tool.io
Classical orthogonal polynomials: recurrence relations — NIST Digital Library of Mathematical Functions

Similar or alternative tools

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.
Chebyshev Polynomial Approximation
Evaluate chebyshev family polynomial by three-term recurrence for degree 0..128 and |x|≤100; first-kind T_n convention.