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Lagrange and other interpolation at equally spaced points, as in the example above, yield a polynomial oscillating above and below the true function. This behaviour tends to grow with the number of points, leading to a divergence known as Runge's phenomenon ; the problem may be eliminated by choosing interpolation points at Chebyshev nodes .
For example, 4 equally spaced data points ,,, of a quadratic () obey = + +, and solving for gives the same interpolation equation obtained above using the Lagrange method. Interpolation error: Lagrange remainder formula
One can use Lagrange polynomial interpolation to find an expression for ... The formula above is obtained by combining the composite Simpson's 1/3 rule with the one ...
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]
Bicubic interpolation can be accomplished using either Lagrange polynomials, cubic splines, or cubic convolution algorithm. In image processing , bicubic interpolation is often chosen over bilinear or nearest-neighbor interpolation in image resampling , when speed is not an issue.
An example of a nonlinear delay differential equation; applications in number theory, ... d'Alembert's equation or Lagrange's equation 1 = + ...
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables). [1] It is named after the mathematician Joseph-Louis ...
As explained above in Applications, the polynomial interpolation problem for () = + + + + satisfying () =, …, = is equivalent to the matrix equation =, which has the unique solution =. There are other known formulas which solve the interpolation problem, which must be equivalent to the unique a = V − 1 y {\displaystyle a=V^{-1}y} , so they ...