Polynomial interpolation

Polynomial interpolation constructs a polynomial whose values match a given function or data set at specified points. With distinct points, there is a unique interpolating polynomial of degree less than the number of points.

Lagrange interpolation: A formula expressing an interpolating polynomial as a sum of basis polynomials that each equal one at a single node and zero at the others. It constructs the polynomial directly from the specified points without solving for coefficients.
Polynomial: An expression formed from constants and variables using addition, multiplication, and nonnegative integer powers. The interpolant belongs to this family of functions.
Runge's phenomenon: The divergence near interval endpoints that can occur when high-degree polynomial interpolation uses equally spaced nodes. It shows that adding more equally spaced data can worsen an interpolant between nodes.
Least squares: A method that chooses parameters to minimize the sum of squared residuals across data points. Unlike interpolation, it can tolerate inconsistent data without passing through every point.
Newton interpolation: A polynomial representation built from nested factors and divided differences at the interpolation nodes. Its nested form makes adding a new data point simpler than rebuilding a Lagrange formula.
Lagrange basis polynomial: A polynomial equal to one at one interpolation node and zero at every other node. These basis functions isolate each data value in the Lagrange construction.
Chebyshev nodes: Interpolation points given by the extrema or roots of Chebyshev polynomials, clustered more densely near interval endpoints. Their endpoint clustering can reduce large interpolation errors seen with equally spaced nodes.
Taylor series: A power series whose coefficients are determined by a function's derivatives at one point. It matches local derivative information rather than prescribed values at multiple nodes.
Divided differences: Recursive quantities that encode successive changes in function values across interpolation nodes. They supply the coefficients in Newton's interpolation formula.
Interpolation node: An input value at which an interpolating function is required to match specified data. The placement and number of nodes determine the interpolation constraints.

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