This case covers approximation, interpolation and numerical integration, moving from Lagrange polynomials and Lebesgue functions to modern rational and adaptive methods as well as extrapolation techniques.
Polynomial interpolation
Discussion of Lagrange polynomials and the Lebesgue function as a measure of interpolation stability. Node placement strategies and conditioning are explored.
Rational interpolation
Floater–Hormann rational interpolants give better numerical behavior for high-degree approximations; practical implementation notes are provided.
Adaptive rational approximation (AAA)
Introduce the Antoulas–Anderson adaptive algorithm for constructing rational approximants with good accuracy and stability.
Regression splines & LSPIA
Least-squares progressive iterative approximation (LSPIA) for efficient spline regression and fitting large datasets.
Extrapolation & Euler–Maclaurin
Euler–Maclaurin expansion and its role in extrapolation methods for accelerating quadrature convergence and numerical series summation.