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Notation query: f ~ g can be written with the keyboard character ~. HTML isn't needed. For my browser this is fine.
Charles Matthews 11:15, 20 Oct 2004 (UTC)
On my current browser it looks fine, but whatever I was using before showed it up too high. I guess it is pot-luck according to the font one is using. Now I see there is a html entity: ∼ aka ∼ -- Zero 11:43, 20 Oct 2004 (UTC)
To Michael's question: "Do some people prefer \ldots to \cdots between plus signs? Why?", my answers are (1) they have no artistic taste, (2) they are so old that they learned their typography on typewriters where temporarily shifting the baseline by a fraction of a line was too much trouble, (3) none of the above cos I just made them all up. ;-) -- Zero 03:59, 6 Nov 2004 (UTC)
Could somebody please include some layman's language in this article at the top; just a sentence saying that asymptotic analysis covers how long something takes to be processed, or whatever the definition is. Thanks. -- Ian Howlett 19:30, 4 Jul 2005 (UTC)
I would like to second the idea of a layman's explanation of Asymptotic "equality." As an outsider looking in on the math community, a plain-English explanation of the concept would go a long ways for understanding of what you're trying to describe. I recommend that you ask someone outside the math community if they understand what you're writing before you post it; that way you've taken care of your "lowest common denominator." 76.111.8.135 ( talk)
ISO 31-11 uses the Unicode character ≃ to denote "is asymptotically equal to". Is this character actually in usage? Wouldn't it then be a good addition to the this article? -- Abdull 21:16, 28 May 2006 (UTC)
& sim; ∼ tilde operator = varies with = similar to = proportional = asympt.equiv. & cong; ≅ approximately equal to / (geometrically) congruent & asymp; ≈ almost equal to / asymptot.equal (\approx in latex) & equiv; ≡ identical to (used by mathematicians for (arithmetic) congruence)
I agree there are different "fashions", but it would be good to clarify that. For example what is the LaTeX symbol \asymp good for? -- Christian.Mercat ( talk) 15:22, 15 February 2010 (UTC)
Someone did not like my more general definition of asymptotic equivalence, which I took to be f=g(1+o(1)). He argued, that "my" definition (which is common in asymptotic analysis) was not more general, because asymptotic equivalence only deals with limits... Please, explain to me, by using the present definition of the article (lim f/g = 1), whether or not 0 is asymptotically equivalent to 0, and then explain to me, why my definition is not more general than the present definition. According to my definition, 0 ist asymptotically equivalent to 0. ASlateff 128.131.37.74 05:27, 11 February 2007 (UTC)
The current text "This defines an equivalence relation (on the set of functions being nonzero for all n large enough (most mathematicians prefer the definition f-g=o(g) in terms of Landau notation, which avoids this restriction))." is incorrect. There is no function which is o(g) unless g is nonzero for large enough argument, and then f would need to have that property too. Actually the version f = (1+o(1))g is the only one mentioned here that allows f and g to be 0 infinitely often (provided all but finitely many of these places are in common). I'm changing it. Zero talk 10:46, 16 September 2009 (UTC)
I come long after the discussion, and understand your arguments in favour of clarity, but let me stress out that when I came and had a look at this article, I saw many completely wrong statements. One may wish to have a simple definition like . But then the next paragraph was a complete confusion between and (in your definition, this means you do not divide by the same thing). The worse help one can do for non-mathematicians who will read the article is to write wrong contradictory things in the hope it will require less effort from readers. I hence changed the section "Asymptotic expansion", and I would also strongly appreciate that the consensual mathematical definition of equivalent (ie ) would be added to the article. Sleurent ( talk) 14:27, 22 August 2015 (UTC)
I understand that asymptotic analysis is an important tool in the analysis of computational complexity, but there are other fields besides CS which also regularly use this. (Physics is an obvious example, but you could also include all physical sciences, theoretical biology and ecology, etc.) I suggest removing the emphasis on CS and leaving Applied Mathematics as the principal area of application. Tim 136.186.1.187 ( talk) 02:50, 4 May 2010 (UTC)
Does anyone else find the example in the "Method of dominant balance" section to be hard to follow? Assumptions seem to be pulled out of a hat; also the constant A that appears along the way is mysteriously discarded. If y(x) is a solution, then so is Ay(x), so this is an actual error. Zero talk 09:11, 17 October 2010 (UTC)
I agree that this section has big issues. To start with, what was written there was completely wrong, containing mis-uses of the <\math>\sim\/math> notation. I edited it to at least remove some obviously wrong statements, and I'm in favor of removing this section, or rewritting it completely, as I do not regard it as satisfactory.
Explanation why it was wrong: the statements and were claimed to simultaneously hold. That woud imply , which is false if
Another issue with this section is that it seems to take the derivative of an equivalent. Which is not legitimate thing at all! Sleurent ( talk) 14:17, 22 August 2015 (UTC)
The comment(s) below were originally left at Talk:Asymptotic analysis/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
The article gives the impression that asymptotic analysis is mostly about the big-O notation but there's much more to it. -- Jitse Niesen ( talk) 03:24, 2 June 2007 (UTC) |
Last edited at 03:24, 2 June 2007 (UTC). Substituted at 01:47, 5 May 2016 (UTC)
I just added a section on "Construction", because I've thought several times in the past about how to construct such a function and finally identified a solution.
Some of the primary contributors to this page may know a better solution to this problem than what I supplied and / or may know some literature that could be cited here. If so, I hope they will improve what I wrote.
In particular, can a hyperbola be constructed asymptotic to two different but arbitrary straight lines, one as as , and a different one as as ?
Thanks, DavidMCEddy ( talk) 21:18, 2 February 2020 (UTC)
Hello, The current introduction only partially defines asymptotic analysis. I propose this edit below. The definition closely corresponds to the definition provided in the references who are recognized authorities in this field. Please discuss if you disagree. Thanks
In mathematical analysis, asymptotic analysis focuses on the development and application of methods to approximate analytical solutions to a wide range of mathematical problems. Asymptotic analysis has been applied to computer science, analysis of algorithms, differential equations, integrals, functions, series, partial sums, and difference equations. [1] [2] [3]
TMM53 ( talk) 19:04, 8 April 2024 (UTC)
I have a proposed revision. The revision contains almost everything in the current document but adds significant content. If you have concerns, please send me a message or comment below. I will revise the document in the future.
Revisions
Thanks TMM53 ( talk) 22:50, 9 May 2024 (UTC)
![]() | This ![]() It is of interest to the following WikiProjects: | ||||||||||
|
Notation query: f ~ g can be written with the keyboard character ~. HTML isn't needed. For my browser this is fine.
Charles Matthews 11:15, 20 Oct 2004 (UTC)
On my current browser it looks fine, but whatever I was using before showed it up too high. I guess it is pot-luck according to the font one is using. Now I see there is a html entity: ∼ aka ∼ -- Zero 11:43, 20 Oct 2004 (UTC)
To Michael's question: "Do some people prefer \ldots to \cdots between plus signs? Why?", my answers are (1) they have no artistic taste, (2) they are so old that they learned their typography on typewriters where temporarily shifting the baseline by a fraction of a line was too much trouble, (3) none of the above cos I just made them all up. ;-) -- Zero 03:59, 6 Nov 2004 (UTC)
Could somebody please include some layman's language in this article at the top; just a sentence saying that asymptotic analysis covers how long something takes to be processed, or whatever the definition is. Thanks. -- Ian Howlett 19:30, 4 Jul 2005 (UTC)
I would like to second the idea of a layman's explanation of Asymptotic "equality." As an outsider looking in on the math community, a plain-English explanation of the concept would go a long ways for understanding of what you're trying to describe. I recommend that you ask someone outside the math community if they understand what you're writing before you post it; that way you've taken care of your "lowest common denominator." 76.111.8.135 ( talk)
ISO 31-11 uses the Unicode character ≃ to denote "is asymptotically equal to". Is this character actually in usage? Wouldn't it then be a good addition to the this article? -- Abdull 21:16, 28 May 2006 (UTC)
& sim; ∼ tilde operator = varies with = similar to = proportional = asympt.equiv. & cong; ≅ approximately equal to / (geometrically) congruent & asymp; ≈ almost equal to / asymptot.equal (\approx in latex) & equiv; ≡ identical to (used by mathematicians for (arithmetic) congruence)
I agree there are different "fashions", but it would be good to clarify that. For example what is the LaTeX symbol \asymp good for? -- Christian.Mercat ( talk) 15:22, 15 February 2010 (UTC)
Someone did not like my more general definition of asymptotic equivalence, which I took to be f=g(1+o(1)). He argued, that "my" definition (which is common in asymptotic analysis) was not more general, because asymptotic equivalence only deals with limits... Please, explain to me, by using the present definition of the article (lim f/g = 1), whether or not 0 is asymptotically equivalent to 0, and then explain to me, why my definition is not more general than the present definition. According to my definition, 0 ist asymptotically equivalent to 0. ASlateff 128.131.37.74 05:27, 11 February 2007 (UTC)
The current text "This defines an equivalence relation (on the set of functions being nonzero for all n large enough (most mathematicians prefer the definition f-g=o(g) in terms of Landau notation, which avoids this restriction))." is incorrect. There is no function which is o(g) unless g is nonzero for large enough argument, and then f would need to have that property too. Actually the version f = (1+o(1))g is the only one mentioned here that allows f and g to be 0 infinitely often (provided all but finitely many of these places are in common). I'm changing it. Zero talk 10:46, 16 September 2009 (UTC)
I come long after the discussion, and understand your arguments in favour of clarity, but let me stress out that when I came and had a look at this article, I saw many completely wrong statements. One may wish to have a simple definition like . But then the next paragraph was a complete confusion between and (in your definition, this means you do not divide by the same thing). The worse help one can do for non-mathematicians who will read the article is to write wrong contradictory things in the hope it will require less effort from readers. I hence changed the section "Asymptotic expansion", and I would also strongly appreciate that the consensual mathematical definition of equivalent (ie ) would be added to the article. Sleurent ( talk) 14:27, 22 August 2015 (UTC)
I understand that asymptotic analysis is an important tool in the analysis of computational complexity, but there are other fields besides CS which also regularly use this. (Physics is an obvious example, but you could also include all physical sciences, theoretical biology and ecology, etc.) I suggest removing the emphasis on CS and leaving Applied Mathematics as the principal area of application. Tim 136.186.1.187 ( talk) 02:50, 4 May 2010 (UTC)
Does anyone else find the example in the "Method of dominant balance" section to be hard to follow? Assumptions seem to be pulled out of a hat; also the constant A that appears along the way is mysteriously discarded. If y(x) is a solution, then so is Ay(x), so this is an actual error. Zero talk 09:11, 17 October 2010 (UTC)
I agree that this section has big issues. To start with, what was written there was completely wrong, containing mis-uses of the <\math>\sim\/math> notation. I edited it to at least remove some obviously wrong statements, and I'm in favor of removing this section, or rewritting it completely, as I do not regard it as satisfactory.
Explanation why it was wrong: the statements and were claimed to simultaneously hold. That woud imply , which is false if
Another issue with this section is that it seems to take the derivative of an equivalent. Which is not legitimate thing at all! Sleurent ( talk) 14:17, 22 August 2015 (UTC)
The comment(s) below were originally left at Talk:Asymptotic analysis/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
The article gives the impression that asymptotic analysis is mostly about the big-O notation but there's much more to it. -- Jitse Niesen ( talk) 03:24, 2 June 2007 (UTC) |
Last edited at 03:24, 2 June 2007 (UTC). Substituted at 01:47, 5 May 2016 (UTC)
I just added a section on "Construction", because I've thought several times in the past about how to construct such a function and finally identified a solution.
Some of the primary contributors to this page may know a better solution to this problem than what I supplied and / or may know some literature that could be cited here. If so, I hope they will improve what I wrote.
In particular, can a hyperbola be constructed asymptotic to two different but arbitrary straight lines, one as as , and a different one as as ?
Thanks, DavidMCEddy ( talk) 21:18, 2 February 2020 (UTC)
Hello, The current introduction only partially defines asymptotic analysis. I propose this edit below. The definition closely corresponds to the definition provided in the references who are recognized authorities in this field. Please discuss if you disagree. Thanks
In mathematical analysis, asymptotic analysis focuses on the development and application of methods to approximate analytical solutions to a wide range of mathematical problems. Asymptotic analysis has been applied to computer science, analysis of algorithms, differential equations, integrals, functions, series, partial sums, and difference equations. [1] [2] [3]
TMM53 ( talk) 19:04, 8 April 2024 (UTC)
I have a proposed revision. The revision contains almost everything in the current document but adds significant content. If you have concerns, please send me a message or comment below. I will revise the document in the future.
Revisions
Thanks TMM53 ( talk) 22:50, 9 May 2024 (UTC)