Talk:Numerical integration

Equation reference is broken new version of page

In the section "Conservative (a priori) error estimation" there is a refernce to function (*) this function is present in older versions of the page (i.e. oldid=976581), but not in the current version. The index "n" for the simplified formula for the Riemann sum is therefore not explained to the reader.

130.95.29.24 04:01, 1 December 2006 (UTC)Anders

this is still not fixed it should 've been fixed by now(as of dec 2010).

Introduction needed.

This subject could use an introduction that is easy for the casual reader to grasp. The true mark of higher intelligence is the ability to take a complex subject and, not only understands it, but re-presents it in terms that are simple and easy to understand. Anyone can paraphrase a textbook, but to truly understand the concept means being able to re-craft it, eliminating what is esoteric and teaching the core principles. Any layman can appreciate that approach and, if they are intrigued by the simple introduction, can pursue the more formal instruction further in the article.

do it yourself, then What's up Dr. Strangelove 06:17, 7 August 2007 (UTC)

Distinction between numerical integration and numerical solution of diff. eqs.

There needs to be a clear distinction made between numerical integration and numerical solution of differential equations. Which is a subset of which? What determines when one method can be applied but not another? The current explanation does not make this clear. Jdpipe 00:43, 26 March 2007 (UTC)

Reasons for numerical integration

in this section the author almost implies that exp(-t^2) is impossible , however it is infact sqrt(pi) if it varies between -inf to +inf —The preceding unsigned comment was added by 82.35.33.103 (talk) 13:54, 3 April 2007 (UTC).

That's a very special case. If it's from 3 to 5, or from −½ to 0, then we don't have a formula. -- Jitse Niesen (talk) 14:46, 3 April 2007 (UTC)

At the moment, the page refers to an antiderivitive "which is a simple combination of elementary functions". Elementary functions are defined as being closed under 'simple combination' so the phrase 'simple combination' is redundant.

Definition of "quadrature"

This continues the discussion by edit summary, which went

"quadrature is integration, numerical quadrature is the most common phrase it is used in, but o/w the 'numerical' would be redundant."

and then

"undo: the first section says otherwise (Davis and Rabinowitz quote) and we have to distinguish from symbolic integration. if you have a conflicting source, we can discuss this on the talk page..."

The Springer Encyclopaedia of Mathematics defines quadrature as "The calculation of an area or an integral (of a function of a single variable)." [1]. The Oxford English Dictionary says "the calculation of the area bounded by, or lying under, a curve; the calculation of a definite integral, esp. by numerical methods." The book by Davis and Rabinowitz is not clear, as far as I can see, but on page 2, they write "The terms mechanical or approximate quadrature are also employed for this type of numerical process [i.e., approximate integration."

Finally, the quote in the article is wrong and should be removed. There is no Greek word quadratos; in fact, the Greeks did not have the letter Q or an equivalent. As the Oxford English Dictionary says, quadrature comes from the Latin word quadratura, which derives via quadrum (a square) from quattuor (four). I don't know what the Greek word for quadrature is, but they used tetragōnos for square (which led to the English words "tetragon" and "tetragram"), from tettares (four). -- Jitse Niesen (talk) 18:16, 16 January 2008 (UTC) Also, the English word "tetrahedron", which is a lot more familiar one. For one thing, carbon atoms in chemical compounds like to have their four chemical bonds distributed as a regular tetrahedron. Three classic examples of this are in diamonds, methane, and methyl alcohol.98.67.97.108 (talk) 00:27, 13 August 2010 (UTC)

I concur that quadrature means the finding of area, not necessarily by (numerical) integration. But I also agree that its most common use today is in reference to numerical integration (e.g., the MATLAB quad* commands, or quadrature points). As I look at the num. analysis books on my shelves I see books that have "Numerical Quadrature" in their titles (e.g. Stroud) but not just "Quadrature". The "numerical" is not redundant, in my opinion, though it's arguably somewhat pedantic; the text should read "numerical quadrature" (not merely "quadrature").

BTW, I don't think of symbolic integration as quadrature as it's the finding of an antiderivative, not an area. Quadrature harkens back to the days when to find the area of something meant to have a geometric means of constructing a square with the same area as the object ("squaring the circle" being the most notable example of such a problem). JJL (talk) 18:42, 16 January 2008 (UTC)

I'm also very much into the area myself -- I'm currently writing a PhD on adaptive quadrature -- and I agree that "quadrature" is not formally defined as "numerical integration", yet I don't think I have ever seen it used for anything other than numerical integration.

May I suggest wording this as

• "The term quadrature is more or less a synonym for integration, especially as applied to one-dimensional integrals and is generally used to mean numerical integration.

Unless, of course, I'm wrong and it is also used for analytical integration...

Cheers and thanks, pedro gonnet - talk - 17.01.2008 07:14

I am the one who made the original edit. I was taught that quadrature is the calculus of the area under the curve and that numerical quadrature was the same done numerically. Unfortunately, neither of the books I used (Trefethen and Bau as well as Stoer and Bulirsch) explicitly define the term. However both only talk about numerical quadrature using the entire term. I think that JJLJitse Niesen's comment is illuminating, and that he gets it right he also highlights why its rare to talk about quadrature when it is not numerical. I also strongly agree with JJLJitse Niesen that the quote should be purged (if there is no greek Q and the OED disagrees on etymology) and that it does not define numerical quadrature as the replying author did. Pdbailey (talk) 00:40, 18 January 2008 (UTC) (edited by pdbailey to replace JJL with Jitse Niesen, sorry for any confusion. Pdbailey (talk) 04:52, 19 January 2008 (UTC)

The use of the word "quadrature" for non-numerical integration has a long history in in applied technology and engineering. For example, back in the 1920s and 1930s, automatic pilots were developed and produced that were not electronic at all. The were mechanical devices, perhaps with some electic motors incorporated into them. Someone figured out how to make mechanisms with spinning wheels, etc., that could do quadrature. In fact, I think that it was called "mechanical quadrature". This was vital for autopilots because one of those needs to find a accelerations, and then integrate those to find a velocities (up, down, forwards, sideways, etc.); and other parts of it need to integrate velocities to find distances. Such autopilots were indedpensible for certain missions. For example Wiley Post wanted to fly around the world solo, and at a record speed. The only way for him to accomplish this was to put his airplane on automatic pilot on a predetermined course, and then take naps along the way. [Actually, Post "only" flew about 14,000 miles because he flew over the United States, Europe, Russia, China, and maybe Japan, and then back to the U.S., possibly via Alaska and Canada. Later, the rule for around the world flights was that they had to be at least as long as the Tropic of Cancer, which is about 20,000 miles. Amelia Earhart was trying to approximate the Equator better that that, which is why she and Fred Newnan flew via Brazil, Central Africa, India, Singapore, Darwin, Australia, and Lae, New Guinea. Then, her next destination was to be Howland Island, which is close to the Equator, but she never made it. It is known that she flew over the island of Bougainville, which is definitely south of the Equator. If she and Noonan had made it, their next two destinations were going to be Honolulu and California, which would have completed their circumnavigation.]

After mechanical autopilots had been developed, then next came electronic ones that utilized electronic analog computers. Given the right electronics, integrators are rather easy to make. However, analog computers are sensitive to changes in temperature, and a lot of other problems. Once digital electronics were invented, there was a big push to start making digital computer autopilots. Primitive ones of those came into use during the 1960s, but now they can do nearly everything.

Still, at the hearts of those are numerical routines that integrate accelerations to give velocities and others that integrate velocities to give positions (in three dimensions, remember). 98.67.97.108 (talk) 00:17, 13 August 2010 (UTC)

diff eq comparison

It's my opinion that the comparison with differential equations could go at the bottom because it stands on its own, is useful primarily if you know math from more advanced classes than the rest of the article requires. Does anyone object? Pdbailey (talk) 15:02, 19 January 2008 (UTC)

oddness in the figure

I have always felt there is something "off" about this figure presently on the main page and I just put my finger on it. What does the grey that outlines the trapezoids between the lines represent? If it is the integral, shouldn't it go the the edges (in some way, like a line with zero slope, or the edges should be sampled to allow for this). If it isn't the integral, how is it interesting, and why is it there? Pdbailey (talk) 20:02, 6 April 2008 (UTC)

Uppon closer inspection, I see the the integral boundaries are sampled first, by now I wonder, why is there only one additional point in the first three innovations, but then four in the last one? What exactly is going on in this figure? Is it really the best figure to explain this idea? Pdbailey (talk) 20:05, 6 April 2008 (UTC)

Is there a name for this iterative method of numerical integration?

Link to description of algorithm: iterative_method.htm Jeffareid (talk) 06:46, 20 July 2009 (UTC)

That's not about computing integrals but computing the solution of a differential equation; see Numerical ordinary differential equations. The predictor is forward Euler and the corrector is the trapezoidal rule, so I'd call it an Euler-trapezoidal method, iterated till convergence. It's the first one in a series of predictor-corrector methods called Adams-Bashforth-Moulton or AB/AM because they use an Adams-Bashforth method as predictor and an Adams-Moulton method as corrector (see linear multistep method). -- Jitse Niesen (talk) 10:46, 20 July 2009 (UTC)

Thanks, I'll make a copy of this.Jeffareid (talk) 19:41, 20 July 2009 (UTC)

You omitted to say anything at all about the Runge-Kutta method of numerical integration.98.67.97.108 (talk) 23:50, 12 August 2010 (UTC)

Another meaning of numerical integration

There is another significant meaning of numberical integration - one used in digital signal processing. Given any time series of samples of a function, by using seveal different methods (depending on the accuracy needed), I can use a compluter (or a hand calculor if I am patient enough) to calculate (approximately) the indefinite integral of that function.
Methods of this range from very simple to quite complicated:
A. "Boxcar integration", which is an approximation to Riemann integration by using slender rectangles. ("rectangular increments")
B. Somewhat more accurate is approximating the "increments" by giving them tops that are right trangles. (I wish that I could draw you a picture.)
C. Increasingly more complicated integration methods, such as the 4th-order Runge-Kutta integration method.

Such integration methods are quite useful for use in doing computer simulations and in implementing such things as automatic pilots, digital flight control systems, computerized navigation systems, and so forth. Some people visualize these applications as merely solving families of differential equations, but this is not so. There are such practical systems that have so many nonlinearities in them that it is impractical to describe them in terms of differential equations. In fact, sometimes the reason for resorting to computer simulations is because of all of the nonlinearities. Furthermore, many of the above systems in modern applications are "discrete time" systems in which differential equations do not apply, and in which the nonlinearities are particularly strong. Still, some things remain true:
A. integrate acceleration and get velocity
B. integrate velocity and get position
C. integrate power and get energy D. integrate instantaneous frequency and get instantaneous phase E. and so on, and so on... 98.67.97.108 (talk) 02:33, 13 August 2010 (UTC) 98.67.97.108 (talk) 02:11, 13 August 2010 (UTC)

What makes you think this is a "different" meaning of numerical integration? Integration in time is still integration. Of course, there are many algorithms for numerical integration, but the existence of different algorithms does not mean a different meaning of the term "numerical integration". (And in any case the algorithms you mention are already discussed, e.g. what you call "boxcar" integration is the composite rectangle rule, and the second method you mentioned sounds like the composite trapezoidal rule.)

Things like Runge-Kutta are for "integrating" ODEs, which are a generalization of the concept of "numerical integration", in that the solution of an ODE isn't in general just an integral of known functions unless the ODE is very special. If this is what you mean, then that definition is already mentioned in the lede. (If you are just doing ordinary integration and want high-order accuracy, there are actually far better methods than Runge-Kutta.) — Steven G. Johnson (talk) 00:08, 14 May 2011 (UTC)

There a very short bit in this article pointing at Monte Carlo methods which is what you are referring to. This has become a hot topic nowadays with high dimensional spaces, look for Metropolis methods and quasi Monte Carlo if you are interested in that sort of thing. Financial statistics have for instance really got into this sort of thing with Bayesian statistics as even with supercomputers a bit of help is needed. Dmcq (talk) 09:33, 14 May 2011 (UTC)

Reading the question properly I think the closest article to what is wanted is PID controller and associated things in control theory. I'd been looking at Witsenhausen's counterexample a little while ago and it definitely isn't a simple area if you try doing anything complicated! Dmcq (talk) 09:43, 14 May 2011 (UTC)

Methods with guaranteed precision

Another user just removed the section Methods with guaranteed precision I would like to second this. In order to "guarantee" a precision, one would have to specify that the function be Lipshitz continuous and the user would have to provide the Lipschitz constant. If you don't know this, there can be no guarantee. 0¹⁸ (talk) 15:33, 7 July 2011 (UTC)

links to open source software (OSS)

I don't really understand why an article on Numerical integration would have links to Numerical Integration packages that are OSS. If I wanted that, I would google it, or reviews of the same and trust the results a lot more. Also, why exclude non-OSS? If this article was about software for numerical integration, the links might make more sense, but it isn't. I propose deleting the section. 0¹⁸ (talk) 21:58, 7 July 2011 (UTC)

Many numerical-analysis topics on Wikipedia have links to code implementing the algorithms. This is a natural and relevant thing to link in such a context, since it is likely that a many readers want to know about the algorithms because they want to use them. As to linking open-source specifically, was there a particular non-open-source integration package you wanted to link? Pedagogically, linking packages where source code is at least available (whether or not it is FLOSS) seems more helpful to readers who want to find out how the algorithms work. — Steven G. Johnson (talk) 00:06, 9 July 2011 (UTC)

(PS. Note that WP:NOT is not relevant here. It says that Wikipedia articles should not normally be merely lists of links and cautions against link lists that "dwarf articles", which is clearly not the case here, and specifically states that context-relevant links are appropriate.)

(Your other argument would seem to argue against any content on Wikipedia at all: "If I wanted to know about X, I would google it, or reviews of the same, and trust the results a lot more." — Steven G. Johnson (talk) 00:13, 9 July 2011 (UTC))

The fact that other pages do something is a terrible reason to do something. However, your other arguments make sense to me. At the same time, the link list doesn't seem like a good format. Perhaps notes on what languages they are written in and what they focus on. And, I have no non OSS package in mind but wonder, why not change the title to "codes with source code" or allow links to non-OSS that also reveals the source.

How about if every listing said the language the main quadrature routine was written in (some but not all do this now) and removing the license detail (this doesn't really seem apperopos to your argument). 0¹⁸ (talk) 23:28, 9 July 2011 (UTC)

That seems reasonable. — Steven G. Johnson (talk) 04:15, 10 July 2011 (UTC)

integration by taylor series expansion

How does integration by taylor series expansion compare to the other mentioned numerical methods? I know already that not every function can be approximated by a taylor series -- but I've found only one example -- exp(-1/x^2). In my university they always mentioned real(z) as a function which is not analytical. Now I've to say that for the first function the derivative does not exist for x=0 -- but I guess one can get one via limit calculation. And the second function is plainly not a function in Z.

Kind Regards ExcessPhase — Preceding unsigned comment added by ExcessPhase (talk • contribs) 00:20, 24 July 2011 (UTC)

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numerical double integration using simpson method in fortran .

to evaluate the double integration ∬f(x,y)dxdy using simpson method in fortran we should builde a programe as this programe forexample to evaluate the double integration to this function . ∬x^2+y dxdy where x changed from 0 to (y+3) and y changed from 0 to 1 .

external f

a=0.0;b=1;eps=1.e-2

call simpison(f,a,b,eps,val)

write(*,*)' the value of integtation = ',val

end

function f(y)

external g

common yy;yy=y

a=0.0;b=y+3;eps=1.e-2

call simpison(g,a,b,eps,val)

f=val

return;end

function g(x)

common y

g=x**2+y

return

end

recursive subroutine simpison(f,a,b,eps,val)

n=6.0;so=f(a)+f(b)

do k=1,100

h=(b-a)/n

s1=0.0;do i=1,n-1,2;s1=s1+f(a+i*h);enddo

s2=0.0;do j=2,n-2,2;s2=s2+f(a+j*h);enddo

val=h*(so+4*s1+2*s2)/3.0

if(k>1)then

e=abs(val-oldval)

if(e<eps)return

endif

write(*,*)'E =',e

oldval=val

n=n*2

enddo

write(*,*)' max no conv:last error =',e

return

end A.S — Preceding unsigned comment added by Ahmed said math (talk • contribs) 10:45, 26 December 2012 (UTC)

Algorithm in the Adaptive section

In the adaptive algorithm section an adaptive algorithm is given. This "algorithm" consists of the word "def". I haven't seen "def" in any algorithm in any book. An algorithm is supposed to be clear. So, I am suggesting replacing "def" with "define function". Condmatstrel (talk) 14:29, 5 March 2013 (UTC)

That "algorithm" is written in pseudo-Python, following some of the conventions of that programming language. (For example, "def" is the Python keyword for a function definition.) Since this page is about a general topic in numeric analysis, not Python programming, it would be a good idea to rewrite the algorithm in something closer to ordinary English. --Colin Douglas Howell (talk) 09:52, 5 August 2017 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Numerical integration/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.

No discussion of order. Article could use some pictures and perhaps an example. Section on adaptivity is weak. Perhaps discuss more quadrature rules; definitely, compare them to each other. -- Jitse Niesen (talk) 13:28, 2 August 2007 (UTC)

Last edited at 13:28, 2 August 2007 (UTC). Substituted at 20:09, 1 May 2016 (UTC)

Explanation of nesting is not clear

The article says "Quadrature rules with equally spaced points have the very convenient property of nesting. The corresponding rule with each interval subdivided includes all the current points, so those integrand values can be re-used." The second sentence doesn't seem clear to me. Bayesiangame (talk) 15:27, 17 February 2017 (UTC)

It could be better written. Here's what I think it means. Suppose you numerically compute an integral by some rule using equally-spaced points, and that while doing so (this is important), you save all the integrand values you evaluated to compute the integral. Then you estimate your error and find that it's too high, so you decide to further subdivide each interval between the points to compute a more accurate result. While computing this new result, you can use all your saved integrand values to lessen the computing work you have to do, only performing new evaluations of the integrand at the new subpoints. --Colin Douglas Howell (talk) 10:05, 5 August 2017 (UTC)

What does the term dx mean?

The symbols dx used in the integral formula shown in the introduction does not have an associated definition. Is there a convention for what d(x) is supposed to mean? The Wikipedia manual of style recommends defining all variables used in a formula. Help me understand what the term d stands for. Stephen Charles Thompson (talk) 18:31, 2 October 2018 (UTC)

As explained the the integral article, "the symbol dx," is "called the differential of the variable x," and it "indicates that the variable of integration is x." Anyone who knows what "∫" means should also know what the "dx" means, and it would make no more sense to define dx than it would to define ∫, the integral sign. -AndrewDressel (talk) 18:49, 2 October 2018 (UTC)

Is the generalized midpoint rule a proper name?

The formula

${\begin{aligned}\int \limits _{0}^{1}{f(x)dx}=&\sum \limits _{m=1}^{M}{\sum \limits _{n=0}^{\infty }{{\frac {{{\left({-1}\right)}^{n}}+1}{{{\left({2M}\right)}^{n+1}}\left({n+1}\right)!}}{{\left.{{f^{\left(n\right)}}\left(x\right)}\right|}_{x={\frac {m-1/2}{M}}}}}}\\=&2\sum \limits _{m=1}^{M}{\sum \limits _{n=0}^{\infty }{{\frac {1}{{{\left({2M}\right)}^{2n+1}}\left({2n+1}\right)!}}{{\left.{{f^{(2n)}}(x)}\right|}_{x={\frac {m-1/2}{M}}}}}}\,\,\end{aligned}}$

is very efficient for numerical integration as it provides much flexibility in accuracy by choosing a combination of integers $M$ and $N$ in truncation. Even though this formula is a generalization of the conventional midpoint method, the survey shows that name "the generalized midpoint rule" has been reserved for completely different integration methods. What is the proper name for this integration formula? Math2know (talk) 19:28, 18 March 2019 (UTC)

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