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Each Lagrange basis polynomial () can be rewritten as the product of three parts, a function () = common to every basis polynomial, a node-specific constant = (called the barycentric weight), and a part representing the displacement from to : [4]
In number theory, Lagrange's theorem is a statement named after Joseph-Louis Lagrange about how frequently a polynomial over the integers may evaluate to a multiple of a fixed prime p. More precisely, it states that for all integer polynomials f ∈ Z [ x ] {\displaystyle \textstyle f\in \mathbb {Z} [x]} , either:
Furthermore, there is a Lagrange remainder form of the error, for a function f which is n + 1 times continuously differentiable on a closed interval , and a polynomial () of degree at most n that interpolates f at n + 1 distinct points , …,.
Lagrange himself did not prove the theorem in its general form. He stated, in his article Réflexions sur la résolution algébrique des équations, [3] that if a polynomial in n variables has its variables permuted in all n! ways, the number of different polynomials that are obtained is always a factor of n!.
A restricted form of the theorem was proved by Michel Rolle in 1691; the result was what is now known as Rolle's theorem, and was proved only for polynomials, without the techniques of calculus. The mean value theorem in its modern form was stated and proved by Augustin Louis Cauchy in 1823. [ 2 ]
In a somewhat opposite manner, the approximation for the costate (adjoint) is performed using a basis of Lagrange polynomials that includes the final value of the costate plus the costate at the N LG points. These two approximations together lead to the ability to map the KKT multipliers of the nonlinear program (NLP) to the costates of the ...
The Gershgorin circle theorem applies the companion matrix of the polynomial on a basis related to Lagrange interpolation to define discs centered at the interpolation points, each containing a root of the polynomial; see Durand–Kerner method § Root inclusion via Gerschgorin's circles for details.
The Lagrange resolvent may refer to the linear polynomial = where is a primitive nth root of unity. It is the resolvent invariant of a Galois resolvent for the identity group. It is the resolvent invariant of a Galois resolvent for the identity group.