Should this begin by saying "In numerical analysis, ...", or is some other specialty appropriate? (At any rate, more context-setting and examples are needed than what is here now.) Michael Hardy 20:34 25 May 2003 (UTC)
I have this unsettling feeling that numerical stability is one of those terms in which different mathematicians and scientists use in different ways. The article as described didn't seem to have any connection with the term that people I know use, but I think this might be because different people are using it in different ways.
- We all use it in the same way. This article suffers from attitude without knowledge. — The preceding interjection was added by 126.96.36.199 19:43, 16 June 2007 (UTC)
You are right roadrunner. This article is one of the wildcards that Wikipedia has to be concerned with. It seems to be knowledgeable but isn't. The author has not bothered to define the symbols and terms being used.
Numerical stability is probably the most important mathematical concept in PDE simulations and hence in simulations of climate, hydrodynamics, etc. etc. Roadrunner
Stability is not generating significant roundoff errors?
In numerical analysis the basic phenomenon is instability rather than stability because, as you noticed, the results of unstable computation are garbage. While rigor requires a stability analysis, many computations are, in practice, done without doing the stability analysis until something goes wrong. —Preceding unsigned comment added by 188.8.131.52 (talk) 19:43, 16 June 2007
Roadrunner do you agree we shouldn't define a stable method as a method where errors get damped out but rather define a stable calculation as a calculation which doesn't generate significant cutoff/roundoff errors? It would also defend your last line in the paragraph saying that an unstable method generates garbage. I have not much experience using numerical analysis to solve differential equations but for solving sets of linear equations, interpolation, inegration, differentiation, ... we define a stable method as one which doesn't generate too much errors. We actually define 3 types of statbility
- Forward or Strong stability
- Weak stability
- Backward stability
Are you familiar with these types of stabilty and should I elaborate them?Jurgen
Who are the "we" you are referring to? Stability refers to the damping of errors. You are talking about accuracy not stability. It is clear that you do not know the subject.
-That's right; a stable numerical method does not magnify rounding or truncation errors.184.108.40.206 04:30, 29 August 2007 (UTC)
Inaccuracies in the article
This article seems to contain a lot of good ideas that should be present in a discussion of numerical stability, but they are put together poorly and there are some inaccuracies.
For example, by the definition given, every algorithm for the numerical solution of ODEs or PDEs is unstable, yet the article says: Unstable methods quickly generate garbage and are useless for numerical processing. This is not true! Rather, it's important to understand the accumulation of error and only trust results in a regime where the total error is small. This should probably be mentioned in the article, along with a brief comment about weather forecasting as a typical example.
The comments in the Notes section are a bit disjointed and should be integrated into appropriate parts of the main article. For instance, it's not clear how truncation and round-off error are relevant to the issue of numerical stability. A comment like "All numerical computation includes a certain amount of error or uncertainty, and thus it is essential to understand the numerical stability of numerical algorithms, as this determines whether the uncertainty will grow to a level that makes the results of a computation useless" would go a long way here. It's also a travesty that this article doesn't link to the article on floating point representation of real numbers.
At some point in the future I will try to implement these suggestions and also address my criticisms by revising the article. --David Dumas 05:54, 11 July 2005 (UTC)
Stability versus accuracy
I removed the following comment by User:220.127.116.11 from the article:
- "[Wrong! Stability is not accuracy. If you need to understand this subject go to a book. This site is full of mistakes.]"
Firstly, this kind of comment really does not belong in Wikipedia articles; they go on talk pages. Even better of course it to fix the article.
Secondly, stability is a nebulous concept in numerical analysis. In particular, it is not enough to say that one can look it up in a book. I actually consulted a number of Numerical Analysis books and my conclusion is that they all talk about stability, but few actually explain what it means. I found Nick Higham's book (see the references in the article) to be the clearest, so I based the article on his book. He does not explicitly distinguish between stability and accuracy, and it appears from the definitions in the book that they are largely the same.
Nick's speciality is in numerical linear algebra. The concept of "stability" being used in numerical differential equations, especially in those that are solved by time-stepping methods, is a bit different and has to do with how much errors are magnified from one step to the next. I did not find a very good discussion of this concept of stability; if anybody knows one please tell me. -- Jitse Niesen (talk) 07:31, 13 March 2006 (UTC)
However, the statement that "the algorithm is (numerically) stable if it produces a good approximation to the true solution" is indeed misleading for some definition of stability, so I tried to reformulate it. Of course, it'd be appreciated if somebody could find a better approximation. -- Jitse Niesen (talk) 07:07, 16 March 2006 (UTC)
delta x over x?
shouldn't (|x + dx| - |x|)/|x| simply be |dx/x|?
something is wrong?
it says "f(x + Δx) − y* is small", but according to the definition, f(x + Δx) = y*. Should this say "f(x + Δx) − y is small", i.e. "Δy is small"? MisterSheik 20:42, 30 June 2006 (UTC)
- This is a different Δx. I tried to reformulate the sentence to clarify. Does this help? -- Jitse Niesen (talk) 04:48, 1 July 2006 (UTC)
- Yeah, it makes perfect sense now. Those images are fantastic by the way... MisterSheik 20:00, 6 July 2006 (UTC)
Which page numbers in the book correspond to which parts of the article?-MsHyde 23:29, 6 February 2007 (UTC)
Additional content and error growth
The section on Wikipedia covering numerical stability is very thin and does not include many important areas. I think it would be beneficial to add the definition of stable and unstable in mathematical terms. There are also not any examples on the website. I would like to add one that is relevant.
Under the already existing ‘Numerical Stability’ topic, I would like to squeeze in as number 3), a section with the title ‘Error growth’. In that section will be the definition and some comments and additional explanation to linear and exponential error growth. Further below this will be an example as well as figures and graphs to go along with an example. I will attempt to box off and highlight the most important aspects to make them easier to take in. —Preceding unsigned comment added by Gl270604 (talk • contribs) 20:54, 1 October 2008 (UTC)
- I agree, there is a lot that can be added to the article. Please go ahead. I'm interested to see what you come up with. -- Jitse Niesen (talk) 11:29, 2 October 2008 (UTC)
If the computer has only two digits of precision, how come we can operate with numbers like 0.01 in the first place? And why is it that adding 0.01 to zero has an effect on the sum while adding 0.01 to 1.0 does not? 18.104.22.168 (talk) 19:40, 17 December 2008 (UTC)
- That is because the computer uses floating point numbers. See that article for further details. -- Jitse Niesen (talk) 20:23, 17 December 2008 (UTC)
Linear vs. Nonlinear PDEs
I edited the discussion of the stability of numerical methods for PDEs. As stated, the article asserted that Lax Equivalence and von Nuemann stability analysis worked for all evolution equations. These results are both limited to linear PDEs, and it's my opinion that the fact that they don't apply to nonlinear PDEs should be emphasized.
This article is exceptionally poor, it's content ranges from mis-characterizations to blatantly false statements. Isn't there some type of banner someone can put at the top of the article to emphasize that if you're trying to learn about numerical stability this is probably the last place you should go?