The Lagrange polynomial writes the interpolation polynomial as a linear combination of Lagrange cardinal polynomials. Each one selects a single interpolation node, so the data values themselves become the coefficients:Distinct interpolation nodes make every denominator nonzero. At , all terms vanish except . This gives an explicit construction without solving for monomial coefficients. The elegance of the Lagrange polynomial is that it builds all constraints into the basis, leaving no equations to solve for the data values.
For and , substitution givesAt , the Lagrange cardinal polynomial values are , so . A negative Lagrange cardinal polynomial value is allowed: these weights sum to one but need not be probabilities.
The product form is especially useful for deriving differentiation weights from nodal evaluations and quadrature rules. For repeated numerical evaluation, the barycentric interpolation formula reorganizes the same Lagrange polynomial into precomputed weights and sums; it avoids evaluating every full product at every argument.
The Lagrange polynomial is a form of polynomial interpolation used to find a polynomial that passes through a given set of points.
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