Polynomial evaluation

Polynomial evaluation substitutes specified values for a polynomial’s variables and computes the resulting value. It maps each input to an output using the polynomial’s coefficients and operations.

Horner's method: An algorithm that evaluates a polynomial by repeatedly multiplying by the input and adding a coefficient. It reorganizes evaluation into nested operations, often using fewer multiplications.
Substitution (mathematics): Replacing a variable or expression with another expression in a mathematical expression. Polynomial evaluation begins by replacing each variable with its assigned input.
Polynomial interpolation: The construction of a polynomial that passes through a specified set of data points. Interpolation repeatedly evaluates candidate polynomials to match given observations.
Exact arithmetic: Arithmetic that represents numbers and operations without rounding or approximation. It gives exact polynomial values where finite-precision computation may round.
Synthetic division: A shortcut for dividing a polynomial by a linear factor using its root and coefficient arithmetic. Its remainder directly gives the polynomial’s value at the divisor’s root.
Coefficient: A numerical or symbolic factor multiplying a variable or term in an algebraic expression. The coefficients determine the polynomial’s contributions at each input.
Lagrange polynomial: A polynomial expressed as a weighted sum of basis polynomials that equal one at one data node and zero at others. Its basis terms make evaluation at interpolation nodes especially transparent.
Symbolic computation: Computer manipulation of mathematical expressions in exact or algebraic form rather than solely as numerical values. It can preserve a polynomial expression instead of returning a value at a chosen input.
Estrin's scheme: A polynomial evaluation algorithm that groups terms into powers to enable parallel computation. It offers a parallel alternative to Horner’s serial sequence of operations.
Monomial: An algebraic expression consisting of a coefficient multiplied by variables raised to nonnegative integer powers. A polynomial is a sum of monomials, each evaluated separately or through an algorithm.

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