Search results
Results from the WOW.Com Content Network
For example, given a function defined on the interval [,] and a degree bound , a minimax polynomial approximation algorithm will find a polynomial of degree at most to minimize max a ≤ x ≤ b | f ( x ) − p ( x ) | . {\displaystyle \max _{a\leq x\leq b}|f(x)-p(x)|.} [ 3 ]
The polynomial of best approximation within a given subspace is defined to be the one that minimizes the maximum absolute difference between the polynomial and the function. In this case, the form of the solution is precised by the equioscillation theorem .
One can obtain polynomials very close to the optimal one by expanding the given function in terms of Chebyshev polynomials and then cutting off the expansion at the desired degree. This is similar to the Fourier analysis of the function, using the Chebyshev polynomials instead of the usual trigonometric functions.
In mathematics, least squares function approximation applies the principle of least squares to function approximation, by means of a weighted sum of other functions.The best approximation can be defined as that which minimizes the difference between the original function and the approximation; for a least-squares approach the quality of the approximation is measured in terms of the squared ...
In its most common form, the given function satisfies the condition to the Brouwer fixed-point theorem: that is, is continuous and maps the unit d-cube to itself. The Brouwer fixed-point theorem guarantees that has a fixed point, but the proof is not constructive. Various algorithms have been devised for computing an approximate fixed point.
This is often the case for algorithms that work by solving a convex relaxation of the optimization problem on the given input. For example, there is a different approximation algorithm for minimum vertex cover that solves a linear programming relaxation to find a vertex cover that is at most twice the value of the relaxation. Since the value of ...
Henri Padé. In mathematics, a Padé approximant is the "best" approximation of a function near a specific point by a rational function of given order. Under this technique, the approximant's power series agrees with the power series of the function it is approximating.
It was first proved by Hassler Whitney in 1957, [1] and is an important tool in the field of approximation theory for obtaining upper estimates on the errors of best approximation. Statement of the theorem