Rational approximation

Rational approximation represents a function by a ratio of polynomials. It can capture poles and steep changes that polynomial approximations may need many terms to describe.

Padé approximant: A rational function whose power-series expansion matches a target function through a specified order. It builds a rational approximation by matching the target function’s series coefficients.
Polynomial approximation: The representation of a function by a polynomial chosen to match or approximate its values. Rational approximation extends this approach by allowing a nonconstant denominator.
Analytic continuation: The extension of an analytic function beyond the domain where its original representation is defined. Rational forms can reveal behavior beyond a function’s local power-series region.
Minimax approximation: Approximation chosen to minimize the largest error over a specified domain. Rational minimax approximation optimizes the worst-case error within a rational family.
Barycentric rational interpolation: A representation of an interpolating rational function as a quotient of weighted sums. Its stable formula makes rational interpolants practical to compute from sampled values.
Interpolation: The construction of a function that passes through a specified set of data points. Many rational approximants are defined by matching given function values exactly.
Numerical quadrature: The approximation of definite integrals by weighted sums of function values. Rational approximation can supply accurate integrands or accelerate quadrature calculations.
Polynomial interpolation: The unique polynomial passing through a prescribed set of distinct data points. Unlike rational interpolation, it cannot represent poles through its denominator.
Pole (complex analysis): An isolated singularity where a complex function diverges like a finite negative power of the distance. Approximating a function’s poles can determine where rational approximants converge.
Taylor series: A power series built from a function’s derivatives at one point. Series coefficients provide the local data matched by Padé approximants.

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