Nonlinear eigenproblem: Difference between revisions

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{{Infobox Website
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| name            = Project Euler
| commercial      = No
| logo            =
| screenshot      = [[Image:Leonhard Euler.jpg|Euler|125px]]
| caption        =
| url            = [http://projecteuler.net/ projecteuler.net]
| alexa = {{IncreaseNegative}} 44,162 ({{as of|2014|2|2|alt=February 2014}})<ref name="alexa">{{cite web|url= http://www.alexa.com/siteinfo/projecteuler.net |title= Projecteuler.net Site Info | publisher= [[Alexa Internet]] |accessdate= 2014-02-02 }}</ref><!--Updated monthly by OKBot.-->
| type            = Problem Solving Website
| registration    = Free
| owner          =
| author          = Colin Hughes (aka euler)
| launch date    = October 5, 2001
| current version =
| revenue        =
| employees      =
}}
'''Project Euler''' (named after [[Leonhard Euler]]) is a website dedicated to a series of computational problems intended to be solved with computer programs. The project attracts adults and students interested in [[mathematics]] and [[computer programming]]. Since its creation in 2001 by Colin Hughes, Project Euler has gained notability and popularity worldwide.<ref>{{cite news
| url=http://www.theatlantic.com/technology/print/2011/06/how-i-failed-failed-and-finally-succeeded-at-learning-how-to-code/239855/
| title=How I Failed, Failed, and Finally Succeeded at Learning How to Code - Technology
| work=The Atlantic
| author=James Somers
| date=June 2011
| accessdate=2013-12-14 }}
</ref> It includes over 450<ref>{{cite web|url=http://projecteuler.net/problems |title=Project Euler (list of problems) |accessdate = 2014-01-25}}</ref> problems, with a new one added every weekend. Problems are of varying difficulty but each is solvable in less than a minute using an efficient algorithm on a modestly powered computer. A forum specific to each question may be viewed after the user has correctly answered the given question.<ref>{{cite web|url=http://projecteuler.net/about |title=Project Euler - About |accessdate=2008-04-04}}</ref> {{as of|January 2014}} Project Euler has over 360000 users from all over the world (who solved at least one problem).<ref>{{cite web|url=http://projecteuler.net/statistics |title=Project Euler (Statistics) - not accessible for anonymous users |accessdate = 2012-11-16}}</ref>
 
Participants can track their progress through seventeen achievement levels based on number of problems solved. A special Eulerians level exists to track achievement based on the fastest fifty solvers of recent problems so that newer members can compete without solving older problems.<ref>{{cite web|url=http://projecteuler.net/news=archives |title=Project Euler (News Archives) |accessdate = 2010-09-13}}</ref>
 
A subset of the Project Euler problems was used in an [[APL (programming language)|APL]] programming contest.<ref>{{cite web|url=http://www.dyalog.com/contest2009/qanda.html|title=APL programming contest|accessdate=2010-11-02}}</ref>
 
There are 68 sequences<ref>{{cite web|url=http://oeis.org/search?q=%22Project+Euler%22&sort=&language=english&go=Search|title=OEIS sequences referencing Project Euler problems|accessdate=2011-04-15}}</ref> in the [[On-Line Encyclopedia of Integer Sequences|On-Line Encyclopedia of Integer Sequences (OEIS)]] referencing Project Euler problems.
 
== Example problem and solutions ==
The first Project Euler problem is
<blockquote>If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.
 
Find the sum of all the multiples of 3 or 5 below 1000.<ref group="note">This is the [[Logical disjunction|inclusive OR]], not the [[exclusive OR]]</ref>
</blockquote>
 
Though this problem is much simpler than the typical problem, it serves to illustrate the potential difference that an efficient algorithm makes. The [[brute-force search|brute-force]] algorithm examines every natural number less than 1000 and keeps a running sum of those meeting the criteria. This method is simple to implement, as shown by the following [[pseudocode]]:
 
<pre>
Set TOTAL to 0;
for every number NUM from 1 to 999 do
  if NUM mod 3 = 0 or if NUM mod 5 = 0 then
    add NUM to TOTAL;
output TOTAL
</pre>
 
For harder problems, it becomes increasingly important to find an efficient algorithm. For this problem, we can reduce 1000 operations to a handful by using the [[inclusion-exclusion principle]] and a [[Closed-form expression|closed form]] [[summation]] formula.
 
: <math>\begin{align}
\mathrm{sum}_{\text {3 or 5}}(n) & = \mathrm{sum}_3(n) + \mathrm{sum}_5(n) - \mathrm{sum}_{15}(n) \\
 
\mathrm{sum}_k(n) & = \sum_{i=1}^{\left \lfloor \frac{n-1}{k} \right \rfloor} ki \\
 
\sum_{i=1}^n ki & = k\frac{(n)(n+1)}{2}
\end{align}</math>
Here, <math>\mathrm{sum}_k(n)</math> denotes the sum of multiples of <math>k</math> below <math>n</math>.
In [[Big O notation]], the brute-force algorithm is O(n) and the efficient algorithm is O(1) (assuming constant time arithmetic operations).
 
==Notes==
{{reflist|group="note"}}
 
==See also==
* [[List of topics named after Leonhard Euler]]
 
== References ==
{{primary sources|date=July 2013}}
{{reflist}}
 
== External links ==
*[http://projecteuler.net/ Home page]
*[http://euler.jakumo.org/ Jakumo : Project Euler In Russian]
*[http://projecteuler.javafling.org/ Project Euler in Romanian]
 
[[Category:Mathematics education]]
[[Category:Computer science competitions]]
[[Category:Educational websites]]
[[Category:Problem solving]]
[[Category:Puzzles]]

Latest revision as of 02:43, 8 December 2014

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