Talk:Euler method

Convergence?
It would be nice to see a section on the convergence of the method as $$h\to0$$.

See also has links to approximation methods which show sequence series convergence. This method is presented as applying as an introduction to numerical methods for numeric solving of (ordinary) differential equations and in that context advanced calculus is a pre-requisite. Error is exactly specified in this article for this method (which is convergence). It is the awful source code which ruins that (because as it converges the source code fails, meaning it will give wildly wrong solutions to whomever gives it a problem requiring a very small h). — Preceding unsigned comment added by 2601:143:400:547B:2529:40EF:1365:4F7 (talk) 16:54, 6 November 2019 (UTC)

Error in the first illustration
I believe there is an error in the illustration (http://en.wikipedia.org/wiki/File:Euler_method.png) because in this image the solution points are not equally spaced while, apparently, the segments have the same length. I think this is wrong, because the (plain) Euler method computes points at t+n*h, so they are evenly spaced. — Preceding unsigned comment added by 78.12.252.87 (talk) 22:31, 5 June 2013 (UTC)

this looks like it has already been corrected some time ago - they look equal to me and are labeled so — Preceding unsigned comment added by 2601:143:400:547B:2529:40EF:1365:4F7 (talk) 17:02, 6 November 2019 (UTC)

Informal Geometric Description
Whoever wrote this section is a saint. Every math-related wikipedia article needs a section like this to spur understanding in readers. Thank you. — Preceding unsigned comment added by 76.29.46.158 (talk) 03:10, 27 September 2013 (UTC)

SN-order?
was is das? Some1Redirects4You (talk) 17:10, 27 April 2015 (UTC)

it doesn't seem to mean anything. it should be replace by "first order"

"not often used"
In the numerical stability section, it reads: This limitation —along with its slow convergence of error with h— means that the Euler method is not often used, except as a simple example of numerical integration. This is wrong. The Euler method is probably the most used method. Most computer games use forward Euler to simulate kinematics and mechanics. Backward Euler is also widely used for its robustness, simplicity and high speed. Higher-order methods work fine for simple problems, but complex simulations often rely on first-order methods. Italo Tasso (talk) 23:04, 7 June 2017 (UTC)

The Euler method is not used by advanced Mathematical softwares such as Mathematica (perhaps free wolfram alpha) (arbitrary precision, etc). The Euler method used often in school books. Most ODE books present that these methods are intended to be tabulated by computer and may use more advanced algorithms (giving a few examples of those, but not attempting to list all).

The point of the The Euler method is that the student understands that many ODE can be solved by approximation rather than lengthy algebraic calculus solving: to give them the skill needed to use software (reguardless but aware of the ramifications of underlying methods). — Preceding unsigned comment added by 2601:143:400:547B:2529:40EF:1365:4F7 (talk) 17:01, 6 November 2019 (UTC)

Popular Culture: "Hidden Figures"
It would improve the quality of the article if it made the statement if Euler's Method actually played the role described in the movie, or if that was pure "poetic license", or possibly something in between.66.25.171.16 (talk) 02:35, 25 June 2017 (UTC)


 * To do so, we would need a reliable source directly discussing Euler's Method in the movie vs. history. - Sum mer PhD v2.0 16:00, 25 June 2017 (UTC)

*sigh*
Anyone see what's wrong with the highlighted line below? Anyone? Care to guess? Here's a hint: it's the math version of "burying the lede" in journalism.

Example
Given the initial value problem


 * $$y'=y, \quad y(0)=1, $$

we would like to use the Euler method to approximate $$y(4)$$.

Using step size equal to 1 (h = 1)
The Euler method is


 * $$ y_{n+1} = y_n + hf(t_n,y_n). \qquad \qquad$$

Joeedh (talk) 06:07, 3 April 2019 (UTC)

it has been fixed already, h=1 is in the sub-title — Preceding unsigned comment added by 2601:143:400:547B:2529:40EF:1365:4F7 (talk) 16:48, 6 November 2019 (UTC)

AWFUL SOURCE CODE HACKS DO NOT BELONG IN AN ENCYCLOPEDIA of Math
The source code is INCORRECT HACK CODE because it does not use arbitrary precision math and contains no method to compensate for it's lack of doing so AND NO WARNING OF LIMITATIONS (there may be, likely are, other source code problems). This is, of course, A CRITICAL MISTAKE for an algorithm which is "supposed to become more accurate, not less" as terms progress. It is a hack.

Algorithms belong in anthologies (such as "The Art of Computer Programming", Donald E. Knuth).

Also: modern mathematics, ex Mathematica, have approximation methods built-in to handle the above mentioned and many other important factors of modern mathematics.

The worst burden acceptable would be a URL to "computer algo section containing code" or an external link.

Another issue is that the code doesn't make the math easier to understand or more complete - so again it is useless.

THE SOURCE CODE DOES NOT WORK USING 'R'. It reads like an advertisement "you must upgrade to recent ubuntu to run this code". (why: R on older linux didn't support all the methods used in the presented code, therefore, it would refuse to run without upgrade - and there is no free online way to evaluate it otherwise - such a process of upgrade or installation of linux to run R, desktop, and graphing can take months or years IN MANY CASES HAS). On the other hand, as an example of low level programming (C) it is far from it (lacking structure such as function definition), and of higher (Swift) it is vacant, and of high level programming: it isn't anywhere near high, so it is a limited semi-mid-tier generation language (3rd) that is inaccessible without total devotion to "ubuntu-ism").

— Preceding unsigned comment added by 2601:143:400:547B:2529:40EF:1365:4F7 (talk) 16:24, 6 November 2019 (UTC)

Institutionum calculi integralis publication date??
The introduction states that Institutionum calculi integralis was published 1768–1870 (which is over a century.) Is that correct? I tried to track down the dates but didn't find the date of publication of the last volume. The-erinaceous-one (talk) 04:22, 23 July 2020 (UTC)


 * That's a typo. Vol. 3 was from 1770. –jacobolus (t) 19:08, 1 September 2023 (UTC)

that section on rounding errors
I think it can be removed. Reason: - It claims (without source) that rounding errors had not been considered. I can call this out as being a false claim. - It gives some handwavy arguments that are insignificant because they are naive and unsharp. Thereby, it gives the wrong impression that smaller step sizes are likely to damage the accuracy. - As a method for non-stiff problems, rounding errors can be considered as perturbation of the initial data and have negligible effect on the result --especially because the method is only first-order. — Preceding unsigned comment added by 2A02:908:1657:A860:5D55:726:2640:C0F3 (talk) 08:54, 28 October 2021 (UTC)
 * I trimmed the section. It is now well supported by references.--Srleffler (talk) 12:05, 28 October 2021 (UTC)

Tautology?
"The Euler method can also be numerically unstable, especially for stiff equations" seems redundant or at least self evident, and problematically so, because a stiff equation is by definition one which has numerically unstable behavior under a mathematical method. I don't consider myself knowledgeable enough on the subject to make the edit myself or I would. ZephyrCubic (talk) 18:32, 1 September 2023 (UTC)