Search results
Results from the WOW.Com Content Network
The Lagrange form of the interpolation polynomial shows the linear character of polynomial interpolation and the uniqueness of the interpolation polynomial. Therefore, it is preferred in proofs and theoretical arguments.
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 , …,.
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 ...
When =, the constraint equations generated by the Lagrange multipliers reduce () to the minimum polynomial that passes through all points. At the opposite end, lim N → ∞ P N ( x ) {\displaystyle \lim _{N\to \infty }P_{N}(x)} will approach a form very similar to a piecewise polynomials approximation.
In matrix theory, Sylvester's formula or Sylvester's matrix theorem (named after J. J. Sylvester) or Lagrange−Sylvester interpolation expresses an analytic function f(A) of a matrix A as a polynomial in A, in terms of the eigenvalues and eigenvectors of A. [1] [2] It states that [3]
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 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:
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!.