# Which sequence acceleration method should I use?

• japplepie
In summary, the sequence being worked with is mostly monotonic, with the exception of rare instances like 0.0001001. The Richardson extrapolation method is being questioned as the best approach for this type of sequence. Additionally, there is concern about the accuracy of using R(10) on a 30 term sequence, with potential for large discrepancies. It is suggested to drop the outlier terms and work with the remaining well-behaved terms for better results. High order schemes may also not be effective on this type of sequence due to irregularities.
japplepie
The sequence that I'm working with is sort of monotonic.

It's monotonic most of the time, sometimes there's one number that ruins the trend like 0.0001001 in
0.1, 0.01, 0.001, 0.0001001, 0.0001, ...
but those are very rare. Btw, the sequence above is just an example.

Is Richardson extrapolation the best method to use in this case?

I also have a question about the Richardson extrapolation .
I'm going to call the nth Richardson extrapolation, R(n)

Why is it that when I test R(10) on a 30 term sequence, it gives me ridiculous approximations like -5.24 when the actual answer is 1.68?

I think you'll be better off just dropping those parasite terms and working with the other terms alone, assuming they are well behaved, e.g. as in your example ##|x_{n+1}-x_n|\sim a^n## - typical acceleration schemes work only if the convergence is regular, what they do is leverage that regularity to predict the limit. They will produce worse results than no acceleration if applied to a generic random sequence.

If you applied a high order scheme to your original sequence, the result is not surprising - that sequence has wild variations for instance in the ratio ##\frac{x_{n+1}-x_n}{x_n-x_{n-1}}## and extrapolating based on an assumption of regular behaviour is bound to give poor results.

## 1. What is the difference between sequential and parallel acceleration methods?

Sequential acceleration methods involve completing one task at a time, while parallel acceleration methods involve completing multiple tasks simultaneously. Sequential methods are generally simpler and easier to implement, but parallel methods can be more efficient for certain types of tasks.

## 2. How do I determine which acceleration method is best for my specific project?

The best acceleration method for a project depends on factors such as the type of task, available resources, and desired speed-up. It is important to carefully evaluate these factors and consider the trade-offs between different methods before making a decision.

## 3. Can I combine sequential and parallel acceleration methods?

Yes, it is possible to combine sequential and parallel acceleration methods for optimal performance. This can be done by dividing a task into smaller sequential steps and then using parallel processing to complete each step simultaneously.

## 4. Are there any limitations to using acceleration methods?

Yes, there are some limitations to using acceleration methods. Some tasks may not be suitable for acceleration, and certain methods may not work well with certain types of hardware or software. It is important to carefully consider these limitations before implementing an acceleration method.

## 5. How do I measure the effectiveness of an acceleration method?

The effectiveness of an acceleration method can be measured by comparing the time it takes to complete a task with and without the acceleration method. Other factors such as resource usage and scalability should also be considered when evaluating the effectiveness of an acceleration method.

Replies
65
Views
3K
Replies
7
Views
2K
Replies
1
Views
1K
Replies
1
Views
961
Replies
20
Views
1K
Replies
24
Views
3K
Replies
2
Views
1K
Replies
1
Views
2K
Replies
2
Views
2K
Replies
1
Views
1K