Talk:Numerical integration

message: 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)
thread_title:  Equation reference is broken new version of page 

message: 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)
thread_title:  Introduction needed. 

message: 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)
thread_title:  Distinction between numerical integration and numerical solution of diff. eqs. 

message: 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)
replies: text: 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)
thread_title:  Reasons for numerical integration 

message: 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)
thread_title:  diff eq comparison 

message: 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)
replies: text: 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)
thread_title:  oddness in the figure 

message: Link to description of algorithm:
iterative_method.htm
Jeffareid (talk) 06:46, 20 July 2009 (UTC)
replies: text: 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)
replies: text: Thanks, I'll make a copy of this.Jeffareid (talk) 19:41, 20 July 2009 (UTC)

text: 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)
thread_title:  Is there a name for this iterative method of numerical integration? 

message: 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)
thread_title:  Methods with guaranteed precision 

message: 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)
thread_title:  integration by taylor series expansion 

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thread_title: File:Filename.jpeg Nominated for speedy Deletion

message: 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)
thread_title:  numerical double integration using simpson method in fortran . 

message: 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)
replies: text: 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)
thread_title:  Algorithm in the Adaptive section 

message: Substituted at 20:09, 1 May 2016 (UTC)
thread_title: Assessment comment

message: 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)
replies: text: 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)
thread_title:  Explanation of nesting is not clear 

message: 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)
replies: text: 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)
thread_title:  What does the term dx mean? 

message: 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}}}}} \\=&2\sum\limits_{m=1}^M{\sum\limits_{n=0}^\infty{\frac{{{{\left({2M}\right)}^{2n+1}}\left({2n+1}\right)!}}{{\left.{{f^{(2n)}}(x)}\right|}_{x=\frac{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)
thread_title:  Is the generalized midpoint rule a proper name?