Chebyshev Polynomial Approximation

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

Description

Chebyshev polynomials come in first- and second-kind families.

Chebyshev polynomials come in first- and second-kind families. This tool uses first-kind Tn, with T1=x.^[1]

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

When to use Chebyshev Polynomial Approximation

Use this tool to evaluate chebyshev family polynomial by three-term recurrence for degree 0..128 and |x|≤100; first-kind T_n convention.

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.

How to use Chebyshev 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 first-kind recurrence and return the finite polynomial value.

T0=1, T1=x; T_(k+1)=2xT_k−T_(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": -1
  }
}

Limitations and assumptions

This is not second-kind Un evaluation. Values outside [−1,1] can grow rapidly and no 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 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 Hermite Polynomial Approximation: Evaluate hermite family polynomial by three-term recurrence for degree 0..128 and |x|≤100; physicists H_n convention.

References

Chebyshev 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.
Hermite Polynomial Approximation
Evaluate hermite family polynomial by three-term recurrence for degree 0..128 and |x|≤100; physicists H_n convention.