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The '''Gauss–Legendre algorithm''' is an [[algorithm]] to compute the digits of [[Pi|π]]. It is notable for being rapidly convergent, with only 25 iterations producing 45 million correct digits of π. However, the drawback is that it is memory intensive and it is therefore sometimes not used over [[Machin-like formulas]].
It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>Should you cherished this article in addition to you desire to get more info relating to [http://www.youtube.com/watch?v=90z1mmiwNS8 Best Dentists in DC] kindly pay a visit to our page.
 
The method is based on the individual work of [[Carl Friedrich Gauss]] (1777–1855) and [[Adrien-Marie Legendre]] (1752–1833) combined with modern algorithms for multiplication and [[square root]]s. It repeatedly replaces two numbers by their [[arithmetic mean|arithmetic]] and [[geometric mean]], in order to approximate their [[arithmetic-geometric mean]].
 
The version presented below is also known as the '''Gauss–Euler, Brent–Salamin (or Salamin–Brent) algorithm''';<ref>[[Richard Brent (scientist)|Brent, Richard]] ''Old and New Algorithms for pi'', Letters to the Editor, Notices of the AMS 60(1), p. 7</ref> it was independently discovered in 1975 by [[Richard Brent (scientist)|Richard Brent]] and [[Eugene Salamin (mathematician)|Eugene Salamin]]. It was used to compute the first 206,158,430,000 decimal digits of π on September 18 to 20, 1999, and the results were checked with [[Borwein's algorithm]].
 
== Algorithm ==
 
1. Initial value setting:
 
:<math>a_0 = 1\qquad b_0 = \frac{1}{\sqrt{2}}\qquad t_0 = \frac{1}{4}\qquad p_0 = 1.\!</math>
 
2. Repeat the following instructions until the difference of <math>a_n\!</math> and <math>b_n\!</math> is within the desired accuracy:
 
<math> \begin{align} a_{n+1} & = \frac{a_n + b_n}{2}, \\
                      b_{n+1} & = \sqrt{a_n b_n}, \\
                      t_{n+1} & = t_n - p_n((a_{n+1})^2 - (b_{n+1})^2), \\
                      p_{n+1} & = 2p_n.
        \end{align}
</math>
 
3. π is then approximated as:
 
<math>\pi \approx \frac{(a_{n+1}+b_{n+1})^2}{4t_{n+1}}.\!</math>
 
The first three iterations give (approximations given up to and including the first incorrect digit):
 
:<math>3.140\dots\!</math>
:<math>3.14159264\dots\!</math>
:<math>3.1415926535897932382\dots\!</math>
 
The algorithm has second-order convergent nature, which essentially means that the number of correct digits doubles with each step of the algorithm.
 
== Mathematical background ==
 
=== Limits of the arithmetic–geometric mean ===
 
The [[arithmetic–geometric mean]] of two numbers, a<sub>0</sub> and b<sub>0</sub>, is found by calculating the limit of the sequences
 
:<math>\begin{align} a_{n+1} & = \frac{a_n+b_n}{2}, \\
                    b_{n+1} & = \sqrt{a_n b_n},
      \end{align}
</math>
 
which both converge to the same limit.
If <math>a_0=1\!</math> and <math>b_0=\cos\varphi\!</math> then the limit is <math>{\pi \over 2K(\sin\varphi)}\!</math> where <math>K(k)\!</math> is the [[Elliptic integral#Complete elliptic integral of the first kind|complete elliptic integral of the first kind]]
 
:<math>K(k) = \int_0^{\pi/2} \frac{d\theta}{\sqrt{1-k^2 \sin^2\theta}}.\!</math>
 
If <math>c_0 = \sin\varphi\!</math>, <math>c_{i+1} = a_i - a_{i+1}\!</math>. then
 
:<math>\sum_{i=0}^\infty 2^{i-1} c_i^2 = 1 - {E(\sin\varphi)\over K(\sin\varphi)}\!</math>
 
where <math>E(k)\!</math> is the [[Elliptic integral#Complete elliptic integral of the second kind|complete elliptic integral of the second kind]]:
 
:<math>E(k) = \int_0^{\pi/2}\sqrt {1-k^2 \sin^2\theta}\, d\theta.\!</math>
 
Gauss knew of both of these results.<ref name="brent">{{Citation
| last=Brent
| first=Richard
| author-link=Richard Brent (scientist)
| publication-date=
| date=
| year=1975
| title=Multiple-precision zero-finding methods and the complexity of elementary function evaluation
| periodical=Analytic Computational Complexity
| series=
| publication-place=New York
| place=
| publisher=Academic Press
| editor-last=Traub
| editor-first=J F
| volume=
| issue=
| pages=151–176
| url=http://wwwmaths.anu.edu.au/~brent/pub/pub028.html
| issn=
| doi=
| oclc=
| accessdate=8 September 2007
}}</ref>
<ref name="salamin1">[[Eugene Salamin (mathematician)|Salamin, Eugene]]. ''Computation of pi'', Charles Stark Draper Laboratory ISS memo 74–19, 30 January 1974, Cambridge, Massachusetts</ref>
<ref name="salamin2">{{Citation
| last=Salamin
| first=Eugene
| author-link=Eugene Salamin (mathematician)
| publication-date=
| year=1976
| title=Computation of pi Using Arithmetic–Geometric Mean
| periodical=Mathematics of Computation
| series=
| publication-place=
| place=
| publisher=
| editor-last=
| editor-first=
| volume=30
| issue=135
| pages=565–570
| url=
| issn=0025--5718
| doi=
| oclc=
| accessdate=
}}</ref>
 
=== Legendre’s identity ===
For <math>\varphi\!</math> and <math>\theta\!</math> such that <math>\varphi+\theta={1 \over 2}\pi\!</math> Legendre proved the identity:
:<math>K(\sin \varphi) E(\sin \theta ) + K(\sin \theta ) E(\sin \varphi) - K(\sin \varphi) K(\sin \theta) = {1 \over 2}\pi.\!</math><ref name="brent" />
 
=== Gauss–Euler method ===
 
The values <math>\varphi=\theta={\pi\over 4}\!</math> can be substituted into Legendre’s identity and the approximations to K, E can be found by terms in the sequences for the arithmetic geometric mean with <math>a_0=1\!</math> and <math>b_0=\sin{\pi \over 4}=\frac{1}{\sqrt{2}}\!</math>.<ref>Adlaj, Semjon ''An eloquent formula for the perimeter of an ellipse'', Notices of the AMS 59(8), p. 1096</ref>
 
== See also ==
* [[Numerical approximations of π]]
 
== References ==
{{reflist}}
 
{{DEFAULTSORT:Gauss-Legendre algorithm}}
[[Category:Pi algorithms]]
 
[[pt:Algoritmo de Gauss-Legendre]]

Revision as of 14:44, 28 February 2014

It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.

Here are some common dental emergencies:
Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.

At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.

Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.

Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.

Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.

Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.

Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.

In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.

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