Approximation theory

Approximation theory is a field of mathematics which is concerned with approximating the values of a function. Usually, the approximation is only needed for a range of values or domain. A good approximation will only have a small error in the domain of interest.

There are different ways in which this approximation is done: common ones are using a combination of simpler functions or using polynomials.

Approximation Theory Media

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  Error between optimal polynomial and log(x) (red), and Chebyshev approximation and log(x) (blue) over the interval [2, 4]. Vertical divisions are 10−5. Maximum error for the optimal polynomial is 6.07 × 10−5.

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  Error between optimal polynomial and exp(x) (red), and Chebyshev approximation and exp(x) (blue) over the interval [−1, 1]. Vertical divisions are 10−4. Maximum error for the optimal polynomial is 5.47 × 10−4.

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  Error P(x) − f(x) for level polynomial (red), and for purported better polynomial (blue)

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  Error of the polynomial produced by the first step of Remez's algorithm, approximating ex over the interval [−1, 1]. Vertical divisions are 10−4.

Related pages

• Curve fitting
• Rounding
• Taylor series

Further reading

• Trefethen, L. N. (2019). Approximation theory and approximation practice. Society for Industrial and Applied Mathematics.