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This page was last edited on 29 December 2024, at 22:10 (UTC).; Text is available under the Creative Commons Attribution-ShareAlike 4.0 License; additional terms may apply.
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. What is meant by best and simpler will depend on the application.
When using approximation equations or algorithms, especially when using finitely many digits to represent real numbers (which in theory have infinitely many digits), one of the goals of numerical analysis is to estimate computation errors. [5] Computation errors, also called numerical errors, include both truncation errors and roundoff errors.
Let [,] be the space of all functions that are differentiable on (,) that are of bounded variation on [,], and let be a linear functional on [,].Assume that that annihilates all polynomials of degree , i.e. =, [].
Linear approximations in this case are further improved when the second derivative of a, ″ (), is sufficiently small (close to zero) (i.e., at or near an inflection point). If f {\displaystyle f} is concave down in the interval between x {\displaystyle x} and a {\displaystyle a} , the approximation will be an overestimate (since the ...
In numerical analysis, catastrophic cancellation [1] [2] is the phenomenon that subtracting good approximations to two nearby numbers may yield a very bad approximation to the difference of the original numbers.
Suppose we have a continuous differential equation ′ = (,), =, and we wish to compute an approximation of the true solution () at discrete time steps ,, …,.For simplicity, assume the time steps are equally spaced:
In numerical analysis and applied mathematics, sinc numerical methods are numerical techniques [1] for finding approximate solutions of partial differential equations and integral equations based on the translates of sinc function and Cardinal function C(f,h) which is an expansion of f defined by