Lagrange interpolation

Lagrange interpolation constructs the unique polynomial of degree at most one less than the number of distinct input points that passes through given data values.

Lagrange basis polynomial: A polynomial equal to one at one interpolation node and zero at every other node. These polynomials isolate each data value so their weighted sum passes through every point.
Lagrange interpolation formula: A sum of data values multiplied by basis polynomials that produces the interpolating polynomial. It is the defining construction behind the method's name.
Numerical analysis: The study of algorithms for approximate computation and their errors. Interpolation is a standard tool for approximating functions from sampled values.
Newton polynomial: An interpolating polynomial expressed using products of factors based on successive nodes. Newton's form represents the same unique polynomial but is easier to update when adding points.
Interpolation node: An input value at which an interpolating function is required to match a specified value. Each node determines where one basis polynomial is one and all the others vanish.
Polynomial degree: The highest exponent with a nonzero coefficient in a polynomial. With n distinct nodes, the interpolation polynomial can be chosen with degree at most n−1.
Runge's phenomenon: Large oscillations near interval endpoints that can arise in polynomial interpolation at equally spaced nodes. It demonstrates that adding more equally spaced samples can worsen approximation.
Spline interpolation: Interpolation by piecewise polynomials joined with specified smoothness conditions. Splines avoid relying on one global polynomial, often reducing oscillation for many data points.
Polynomial interpolation: The construction of a polynomial that matches prescribed values at selected input points. Lagrange interpolation is an explicit formula within this broader family of methods.
Polynomial remainder theorem: A theorem stating that division of a polynomial by x−a leaves remainder equal to its value at a. It helps establish that a polynomial vanishing at all nodes is divisible by their product.

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