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In numerical analysis, why do we use the best square approximation to fit a known function?

This may be a stupid question for professionals (I know it is reasonable in books). In numerical analysis, the discrete data fitting method is extended to continuous function, that is, the so-called best square approximation of continuous function. I'm confused. I know the exact expression of this function. Why should I fit it and get an approximate polynomial with errors? o？ )

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Shelo

For the convenience of examples, the expressions of specific functions are given in the book for readers to understand. But in general, we don't know the expression of the function, only the value of the function at some points. In order to infer the value of the function at other points, fitting is a better method. In order to understand the fitting effect, we need some methods to estimate the error. Because we only have the information of the function at some points, in order to estimate the error of the function in other places, we need to assume that the function meets some conditions, such as continuity and differentiability, and thus derive some error estimation formulas.

Edited in June 2022-65438+May. The copyright belongs to the author.

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Jackie Lee

Computational mathematics and mathematics. mechanics

If this is the case in the topic and the function is known, I think the main advantage is that polynomials are easy to store, evaluate, derive and integrate.

Another common situation is that we don't know this function, but it is the solution of a partial differential equation/variational problem (such as heat conduction equation). After selecting a set of polynomial bases, we can get the best square approximation of the unknown function by Ritz/Galerkin method.

Edited in June 2022-10. The copyright belongs to the author.

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Know what is wrong and purify your mind.

Sin(x), cos(x), exp(x) and other functions, although the function value of a given x can be calculated by pressing the calculator. But if you think about it carefully, you will find that if you want me to calculate it by hand, this thing can't be worked out at all, such as sin( 1.6) and exp(2.3). In fact, all we can do is add, subtract, multiply and divide these four operations. You may say "Taylor expansion", but Taylor expansion may converge slowly and its accuracy may not meet the requirements.

If at this time, we can find a polynomial approximation of a function, so that my initial problem can be turned into a polynomial calculation problem, and polynomials are something we can calculate, which is much more convenient. And as long as the polynomial is chosen well, it is easy to achieve the accuracy we want. You don't have to worry about the problem that the convergence effect is still not good after the brainless Taylor expands and calculates many terms.

Edited in June 2022-10. The copyright belongs to the author.

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Shaer fish restaurant

Fresh and lively dumb fish specialty shop

It is difficult for computers to directly calculate non-polynomial functions. So what you know is knowing in the sense of "existence", not knowing in the sense of "numerical calculation". Considering the limited storage space and calculation time, it is impossible to calculate the exact value of the function value such as Exp(2). Numerical analysis is mainly a branch of mathematics that provides theoretical support for this kind of numerical calculation.

Edited in June 2022-65438+May. The copyright belongs to the author.

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