Rate of convergence for functions

In summary, the rate of convergence is a measure of how quickly a function G approximates another function F, with the convergence rate being determined by the speed at which the difference between the two functions approaches zero. This is typically measured using a norm, such as the maximum value over all x.
  • #1
user11
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I am not very familiar with terms from numerical analysis, thus I do understand the definition for convergence rate from http://en.wikipedia.org/wiki/Rate_of_convergence" [Broken]. Still, here the definition appears only for sequences.

Which is the definition for rate of convergence for functions? For instance: for I closed and bounded set, and for O discrete set, a function F:I->O, x-> F(x) is approximated by a set of functions G: I X R+ ->O , (x,p(k)) ->G(x,p(k)), where p: R+->R+, k ->p(k) is a monotonic decreasing function, and R+ denotes the positive real numbers. The set of functions G converge towards F, i.e. lim_{k->0} G(x,p(k))=F(x). Which is the convergence rate for G?
Any idea on how rate of convergence would be defined in this way? What does it mean if the rate of convergence is infinity in this case?

Thank you very much for your help.
 
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  • #2
The rate of convergence would be how fast |G(x,p(k))-F(x)| -> 0 as a function of k. Since we are dealing with functions, you need to define a norm, for example max over all x.
 

1. What is the rate of convergence for a function?

The rate of convergence for a function is a measure of how quickly a sequence of numbers generated by the function approaches a limiting value. It tells us how fast the function converges towards its limit as the input values get closer to a particular value.

2. How is the rate of convergence calculated?

The rate of convergence is typically calculated by taking the ratio of the difference between two consecutive terms in the sequence to the difference between the previous two terms. This ratio is then raised to a power, which represents the speed at which the terms in the sequence are approaching the limit.

3. What is a "fast" rate of convergence?

A "fast" rate of convergence refers to a function that has a high rate of convergence, meaning that the terms in the sequence are approaching the limit at a relatively quick pace. In other words, the function is converging rapidly towards its limiting value.

4. How does the rate of convergence affect the accuracy of a function?

The rate of convergence is directly related to the accuracy of a function. A higher rate of convergence means that the function is approaching its limit at a faster pace, resulting in a more accurate estimation of the limit. Conversely, a lower rate of convergence may result in a less accurate estimation of the limit.

5. Can the rate of convergence be improved?

Yes, the rate of convergence can be improved by using more advanced techniques and algorithms in the function. Additionally, changing the input values or using different methods of approximation can also help improve the rate of convergence.

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