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The objective is to make the approximation as close as possible to the actual function, typically with an accuracy close to that of the underlying computer's floating point arithmetic. This is accomplished by using a polynomial of high degree , and/or narrowing the domain over which the polynomial has to approximate the function.
Since a Padé approximant is a rational function, an artificial singular point may occur as an approximation, but this can be avoided by Borel–Padé analysis. The reason the Padé approximant tends to be a better approximation than a truncating Taylor series is clear from the viewpoint of the multi-point summation method.
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 .
The primary difference between a computer algebra system and a traditional calculator is the ability to deal with equations symbolically rather than numerically. The precise uses and capabilities of these systems differ greatly from one system to another, yet their purpose remains the same: manipulation of symbolic equations.
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 ...
FX-9750GII Graphing Calculator. One of our favorite graphing calculators, the Casio FX-9750GII offers a lot of machine for the money. It can handle the needs of most students all the way through ...
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
The Weierstrass approximation theorem states that every continuous function defined on a closed interval [a,b] ...