Talk:Integral
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I have removed this:
Integration, however, is particularly hard for computerized algebra systems. Although newer systems have improved, even the best systems are not nearly as effective as an experienced human.
The first time I read it I was a bit surprised, but then decided the person who wrote it must know something I don't. However, when I accidentally read it again today, I decided it either needs to be supported by some hard data, or be removed. In my experience, Maple is very good at finding antiderivatives. The Risch-Norman algorithm is very general and efficient. See On the Risch-Norman integration method and its implementation in MAPLE by Geddes and Stefanus.
I have replaced the above statement with a generic note about how it's hard to find antiderivatives.
If you're talking about certain special definite integrals which can be solved by residue calculus (say), I've also had good experience integrating those in Maple.
Please clarify if you want to restore this text. Loisel 19:28, 21 May 2004 (UTC)
- Maybe we can make this statement more precise. I believe it's uncontroversial to say that integration is hard because there is no "one size fits all" algorithm. The current revision says as much. Can we make this more precise by enumerating mentioning some general symbolic integration algorithms, and mentioning what they don't cover? -- FWIW, I've tried to solve integrals, arising in Bayesian statistical inference, using Mathematica, and as often as not Mathematica can't find a symbolic solution. So I'm inclined to think that symbolic integration is still hard. Regards & happy editing, Wile E. Heresiarch 02:00, 23 May 2004 (UTC)
- It would be fantastic if you could give the specific example, and even better if you could give a hint as to why the standard algorithms don't work. If the integral was indefinite, we'd need to have the antiderivative to show that we're better than the computers. Loisel 11:09, 1 Jun 2004 (UTC)
- Well, here are some examples. I used the Mathematica integrator web interface (http://integrals.wolfram.com), which can find indefinite integrals but not definite integrals. Some of these problems are more usefully stated as definite integrals but I didn't try that. Mathematica just returned the integral for all of these four.
- (1) Exp [-(y-x)^2] BesselK [0, Abs[x]]
- (2) Exp [-(y-x)^2] Exp [-Log [x]^2]/x
- (3) x Exp[-x^2/2] (1/2) (1+Erf[x/Sqrt[2]])^2
- (4) Exp[-x^2]/(Exp[-x^2]+Exp[-(x-3)^2]) Exp[-(v-x)^2] Exp[-(x-5)^2]
- Nos. 1 and 2 originated from finding the sum of two variables with different distributions, so those are convolutions. No. 3 is the expected value of the largest of three Gaussian variables. No. 4 came up in trying to find a marginal distribution in a case where some conditional distributions were mixtures of conditional Gaussians. -- I don't know the solutions for any of these, which is why I was trying Mathematica. I can't claim to be better than Mathematica; however, what is of interest here is that Mathematica is no better than me, for these four problems. Well, I suppose I might have entered the commands wrong or something; if so someone will soon straighten it out. Comments? Wile E. Heresiarch 04:42, 2 Jun 2004 (UTC)
- Ah, I was hoping for an example where a closed-form primitive was available. I'm not completely certain, but I suspect the functions you offer do not have closed-form primitives. As you most probably know, the function exp(x^2) does not have a closed-form primitive either (and one can prove so.) Depending on your definition of "closed form", the Risch-Norman algorithm should always be able to integrate formulae which do have closed-form primitives. I'm not exactly sure what your background is, so in case you're not a mathematician, I'm saying that I think one could prove that it is impossible to write a formula whose derivative exp(x^2) (or one of the functions you give above.) While your functions definetly have antiderivatives, these antiderivatives will never be writable in a nice way. Hence, claiming that the computer is less good than people at antidifferentiating based on such evidence would be misleading. Loisel 07:22, 2 Jun 2004 (UTC)
- I see what you're getting at, but I think it's beside the point. "Integration is hard in principle" is entirely consistent with "integration is hard for computer algebra systems". If there's no possible solution, then say so; Mma doesn't distinguish between "too hard for Mma" and "impossible", and that's definitely a shortcoming. Even in this case, there are avenues of attack, such as finding an expression containing simpler integrands, or attempting to construct some sequence of approximations -- those are things a human could try, and there's no particular reason algebra systems couldn't do it too. Getting back to the article, it would help to outline the scope of the integration problems that are known to have solutions and point out how easy it is to go beyond those boundaries. For what it's worth, Wile E. Heresiarch 15:36, 2 Jun 2004 (UTC)
- On rereading the comment history, I see that you (Loisel) and I seem to be addressing different points. "Even the best systems are not nearly as effective as an experienced human" is an interesting statement, and might even be true, but I'm not concerned with defending it. To improve the article, I think we can steer away from that contentious assertion and just describe why integration is hard, what's possible, how much of the possible is now handled by computer systems, etc. Happy editing, Wile E. Heresiarch 16:14, 2 Jun 2004 (UTC)
Two comments about the current revision (2004/02/09). (1) A few days ago I put "The integral of a function is, roughly speaking, an area, mass, ..." which was modified to "...can be used to represent an area, mass, ...". I deliberately chose "is" because the article needs to say what the integral -is-. I'll propose reversion unless someone else has a better candidate to say what an integral -is-. (2) There is some discussion about Riemann and Lebesgue integrals. This replicates material found in the articles on Riemann and Lebesgue integrals, so I'm inclined to suggest it be cut back to a summary and reference to those other articles. Happy editing, Wile E. Heresiarch 03:16, 11 Feb 2004 (UTC)
- I'm the one who wrote most of these articles (Riemann integral, Lebesgue integral and Integral.) The text in Integral does duplicate a bit of information, but the point in Integral was to be able to give a very coarse idea of how the two mainstream area-based theories of integration differ. If you want to remove this text, that's okay with me, but I still think that the nuance between the Riemann and Lebesgue integral should at least be outlined in this article. The reader should not have to be familiar with either integral in order to get a basic explanation of a few paragraphs. Loisel 19:33, 21 May 2004 (UTC)
- I agree that the integral article ought to mention Riemann and Lebesgue. Maybe instead of mechanics, this article can outline why there is not a single definition of the integral, and then leave the details to the Lebesgue integral article. I'll give it a try in the next day or two and we can see how successful that is. -- To go back to point (1) above, I've attempted to state a definition using the word "is". I may not have been completely successful with what I wrote today ("In calculus, the integral of a function is a generalization of area, mass, volume, total, and average"), but if that's still off the mark, I'd like to suggest that it be replaced by something which likewise says what the integral -is-. FWIW, & happy editing, Wile E. Heresiarch 01:48, 23 May 2004 (UTC)
The notation for the floor function is incorrect - I'll look into this to see if it can be done more effectively. -- User:David Martland
Something like <math>\lfloor x \rfloor <math>? -- The Anome 07:52 25 Jun 2003 (UTC)
Or ⌊ x ⌋ ? - The Anome
New Subject:
When someone types in "integration" or "integrate", they get Wikipedia rules for page integration for integrate and Mathematics Integral for Integration. In technology, integrate means "to integrate to application or purpose" >" to create a set of interdependencies to make a primary function possible." An integration in technology is "A number of dissimilar systems or components interrelated and interdependent in such a manner as to make a primary function possible." Examples are the computer and the automobile. If one studies a chip on a motherboard or the brakes on an automobile as a standalone and not as an interdependency of integration, the interpretive lens will be disoriented and the understanding will be distorted. It follows that if one studies a process in the natural world, outside the integrated whole in which it resides and views it as a standalone and not as an interdependency of a much larger integration, the interpretive lens will be disoriented and the understanding and thus the attempted explanation will be distorted. This may lead one to build an entire thesis off of a tangent that does not exist.
An integration may be static or dynamic. It may be lateral or dynamically layered. The lens of Logic requires a lens of an enginner or architect or Information Specialist; preferably all three.
Interdisciplinary Study, connected learning and "thinking outside the box" are buzz terms used in the academic world, but so far have not been integrated and applied in the academic environment. A biochemistry professor should take a course in Information Science to better understand the "automation run from a genetic database" he [she] is studing.
The terms "integration" and "interdependencies of integration" turn a light on. They bring into focus an understanding that is hidden from view now. Integrations are information rich. But the information is embedded into the application or purpose. So without a view of the integrated whole, most of the information is invisible to the interpretive lens and not available for extraction. Integration brings a whole world to light.
Interpreting the word "integration," as abstractly as possible, it would appear that in one way or another it refers to the way in which distinct ideas or entities form a whole. The the differing values falling within the range of an integrand each play a part in the evaluation of the integration. While derivatives involve division of subtractions, integrals can be evaluated with a summation of multiplications. The word "integral" has been used in differing contexts by mathematicians: The integral can be seen as the the evaluation of the a whole antiderivation, ergo an integral symbol presented in conjunction with an integrand and the differential of the variable upon which the integrand depends; It can also be seen as the very integral symbol itself. The meaning of the integral is not heavily crucial to engineers who use the tool mathematics to implement, whereas those who define mathematics are a bit more philosophical, exempli gratia, Isaac Newton.
Personally, I do not have as high of a praise of the nomenclature of integration as I did once ago. I more inclined to used the following in works:
- <math>
\left[\left({d \over dx}\right)^{-1} f(x)\right]_{u_0}^u <math>
instead of the traditonal and accepted
- <math>
\int_{u_0}^u f(x)\,dx <math>
Lindberg G Williams Jr 04:35, 22 May 2004 (UTC)
Ah, today I finanlly stumbled upon fractional calculus which I have shared ideas similar though less developed. It is a concept I have never seen in a text book but which I have often mingled. Througout the course of my mathematical career I surmise that pregraduate scholars are not allowed to invent new ideas in math.
Lindberg G Williams Jr 17:09, 22 May 2004 (UTC)
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No mention of integration pertaining to simulation?
In computer graphics, when learning about physical simulations (e.g. particle systems, physically based animation), I see mention over and over again of integration as a key step in the process. Euler, Runge-Kutta and Verlet are all types of integration methods found in this context, but I'm having a hard time seeing the relationship between finding the area under a curve and calulating the coordinates for a moving particle at the next time step. Is it because both are dealing with discrete samples of a continuous value in some way?
I'd really appreciate it if someone could add a paragraph or two clarifying the relevance of integration to physics simulations a la computer animation and games.
Thanks!
- Numerical integration is the name of that technique (which doesn't appear to be linked in this article, but should be). Essentially, to compute the motion of a particle, you are solving a differential equation, and in general this is done by performing an integration. Techniques like the Runge-Kutta method are ways of approximating this integration. -- DrBob 21:53, 16 Jun 2004 (UTC)
- "Integration" is also used, by abuse of language, to discuss numerical ordinary differential equation solvers (they are called numerical integrators.) Euler is uk+1=uk+u'(xk)dx. Runge-Kutta is a family of integrators that includes the Euler scheme, but also offers higher order (more precise) methods. Verlet (or Stormer-Verlet) is a second order scheme that preserves certain important quantities such as a Hamiltonian (similar to energy.) It is even symplectic (meaning that it preserves length, surface, volume, n-dimensional volume, etc...) Such schemes are said to be geometric, because they can be viewed as a discrete iteration that preserves certain geometric and physical properties of the continuous system. High order geometric integrators are very difficult to obtain, but are available. Strangely, over extremely long integration periods, geometric integrators with a fixed, non-adaptive step size are much more precise than non-geometric integrators (or geometric integrators with variable step size.) Precisely, when the truncation error becomes quadratic in the number of steps or worse for a traditional scheme, schemes such as Stormer-Verlet will still have mainly linear error in the number of steps. Independently of this, because geometric integrators preserve many useful quantities, when the goal is to generate a credible (as opposed to precise) simulation, geometric integrators allow one to take larger step sizes. Indeed, if the scheme is stable and geometric regardless of step size, the simulation will be credible even at large step sizes, and will not explode. One popular use of the Stormer-Verlet integrator is to simulate water surfaces in computer graphics using the graphics hardware.
- By the way, the relationship between area under curve and location of particle at next step is essentially the relationship between area under curve and derivative, which is the fundamental theorem of calculus. Loisel 02:30, 17 Jun 2004 (UTC)
Thank you very much. The link to the Fundamental Theorem of Calculus was exactly what I was looking for (I had read the article on numerical integration, but that didn't help me make any connections). It's embarassing to be able to write code that implements these concepts to some degree, without understanding the concepts very well. I appreciate your responses as they help make more practical what was previously too theoretical for me to grasp.
The reader should know
Diberri removed the following sentence from the very beginning of the text:
- It is recommended that the reader be familiar with algebra, derivatives, functions, and limits
I think this deserves more discussion. When I first saw this text I was also like "other pages of wikipedia don't have this kind of introduction!" but after some consideration I changed my opinion to be more like "perhaps its the other pages that should be changed". What do other people think about this?
- IMO, it sounds odd for an encyclopedia to address the reader, which is why I moved these prerequisites to the related topics section. Wikipedia:WikiProject Mathematics is apparently in favor of these messages, though, which is why I didn't continue removing them from other math-related topics. --Diberri | Talk 23:28, Aug 25, 2004 (UTC)
- I agree that It is recommended that the reader be familiar ... sounded strange, and I agree with its removal. For what it's worth, I'm not convinced that we need to take Wikipedia:WikiProject Mathematics all that seriously; my experience with proposal pages of various kinds in Wikipedia is that they attract the attention of too small a fraction of the editors to have a serious claim to authority. Wile E. Heresiarch 04:28, 19 Oct 2004 (UTC)
I am still hesitant on the question of this page, but on pages with more advanced mathematics such a warning is IMHO mandatory. I put something like that on most of my math pages. Gadykozma 11:34, 19 Oct 2004 (UTC)
The Integral of McShane
I was curious about this integral and searched in vain for an explanation. Maybe someone could write about it? I heard it is very general.
Mass?
The introduction mentions mass. How is it relevant? Brianjd | Why restrict HTML? | 10:25, 2005 May 8 (UTC)
Integral of exp(-x2)
This is not a "nice function", according to the article. I think it is correct to say that it is not an elementary function - if somebody knows this, can they fix the article? Brianjd | Why restrict HTML? | 10:30, 2005 May 8 (UTC)
- See Error function. Charles Matthews 16:29, 8 May 2005 (UTC)