Which vectorial norm should I use?

  • Thread starter RicardoMP
  • Start date
  • Tags
    Norm
In summary, the conversation discusses studying the convergence of an iterative method for nonlinear systems of equations. The speaker mentions using a formula to evaluate the rate of convergence and asks which vectorial norm to use. They also inquire about measuring the order of convergence and mention that using the infinity norm is common. The other norms should give the same results and the order of convergence is often displayed as the slope on a log scale. Ultimately, all the mentioned norms are equivalent in a finite dimensional vector space.
  • #1
RicardoMP
49
2
I am to study how fast an iterative method for nonlinear system of equations converges to a certain root and found out that I can evaluate my rate of convergence by using the following formula: ##r^{(k)}=\frac{||x^{(k+1)}-x^{(k)}||_V}{||x^{(k)}-x^{(k-1)}||_V}##. My question is which vectorial norm should I use? ##||.||_{\infty}##? ##||.||_1##? Etc... Is there any criteria in choosing them? And after getting my rate of convergence, how do I measure my order of convergence?
 
Physics news on Phys.org
  • #2
I think it is common to use the infinity norm. All the norms should be related by a constant multiple anyway, so when you talk about order, it should not matter too much.
The order of convergence is often displayed as the slope of your convergence plot on a log scale.
 
  • #3
If you work in a normed vector space with finite dimension, all these norms are equivalent. They give rise to the same topology.
 

FAQ: Which vectorial norm should I use?

1. What is a vectorial norm?

A vectorial norm is a mathematical tool used to measure the magnitude or size of a vector. It is a function that assigns a non-negative value to each vector, representing its length or distance from the origin.

2. Why do I need to use a vectorial norm?

Vectorial norms are useful in many applications, such as optimization problems, data analysis, and machine learning. They help to quantify the size or magnitude of a vector, which can provide important information about the data or system being studied.

3. How do I choose the right vectorial norm?

The choice of vectorial norm depends on the specific problem or application. Some common norms include the Euclidean norm, the Manhattan norm, and the maximum norm. It is important to consider the properties and characteristics of each norm to determine which one is most suitable for your problem.

4. Can I use more than one vectorial norm?

Yes, it is possible to use multiple vectorial norms in a problem. In fact, it is often beneficial to use a combination of norms to gain a more comprehensive understanding of the data or system being studied. However, using too many norms can also lead to information overload, so it is important to choose the most relevant ones.

5. Are there any drawbacks to using a vectorial norm?

While vectorial norms are useful tools, they also have limitations. For example, some norms may be more sensitive to outliers in the data, while others may not accurately represent the true distance between vectors. It is important to be aware of these limitations and choose the appropriate norm for each specific problem.

Similar threads

I Looking for Guarantees that the method of fixed-point iteration will work
Replies
22
Views
2K
I Error bounds for Newton's Method
Replies
13
Views
2K
Does each norm on vector space become discontinuous when restricted to S^1?
Replies
1
Views
848
I Understanding the nth-term test for Divergence
Replies
17
Views
3K
V Space With Norm $||*||$ - Fourier Series
Replies
26
Views
2K
MHB Which test do we have to apply at each case?
Replies
6
Views
2K
I Lipschitz continuity of vector-valued function
Replies
3
Views
2K
I Gradient With Respect to a Set of Coordinates
Replies
3
Views
2K
I How does the ratio test fail and the root test succeed here?
Replies
6
Views
2K
MHB Power Series (Which test can i use to determine divergence at the end points)
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top