Newton's method and Cheerleading: Difference between pages

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[[Image:YouthCheerleadingPompons.jpg|thumb|250px|right|Youth Cheerleaders during a football [[halftime show]]. Youth Cheer—[[high school]] ages and younger—make up the vast majority of cheerleaders and cheer teams.]]
In [[numerical analysis]], '''Newton's method''' (also known as the '''Newton–Raphson method''', named after [[Isaac Newton]] and [[Joseph Raphson]]) is perhaps the best known method for finding successively better approximations to the zeros (or [[root (mathematics)|roots]]) of a [[Real number|real]]-valued [[function (mathematics)|function]]. Newton's method can often converge remarkably quickly, especially if the iteration begins "sufficiently near" the desired root. Just how near "sufficiently near" needs to be, and just how quickly "remarkably quickly" can be, depends on the problem. This is discussed in detail below. Unfortunately, far from the desired root, Newton's method can easily lead an unwary user astray with little warning. Thus, good implementations of the method embed it in a routine that also detects and perhaps overcomes possible convergence failures.


'''Cheerleading''' is a [[sport]]<ref name="nw20007">{{cite journal|last=Campo-Flores| first=Arian |title=A World of Cheer! |journal=[[Newsweek]] |date=[[2007-05-14]] |url=http://www.msnbc.msn.com/id/18497473/site/newsweek/ |accessdate=2007-05-17}}</ref><ref name="Forbes06">{{cite journal | last=Schoenberger | first=Chana R. | title=The Most Dangerous Sports | journal=[[Forbes]] | date=[[2006-11-16]] | url=http://www.forbes.com/forbeslife/sports/2006/11/15/sports-injuries-fitness-forbeslife_cz_cs_1114dangersports.html?partner=rss | accessdate=2007-06-29}}</ref><ref name="CBS1-07">{{cite news | last=[[CBS]]/[[Associated Press|AP]] | title=Cheerleading Injuries Increasing | work=[[The Early Show]] | publisher=[[CBS|CBS Broadcasting Inc.]] | date=[[2006-01-03]] | url=http://www.cbsnews.com/stories/2006/01/03/earlyshow/health/health_news/main1173837_page2.shtml | accessdate=2007-06-29}}</ref><ref name="IASF homepage">{{cite web |url=http://iasfonline.org/dotnetnuke/Home/tabid/37/Default.aspx |title=IASF home page |accessdate=2007-06-29 }}</ref> that uses organized routines made from elements of [[tumbling]], [[dance]], jumps and [[List of cheerleading stunts|stunting]] to direct spectators of events to cheer on sports teams at games and matches and/or compete at cheerleading competitions. The athlete involved is called a cheerleader. With an estimated 1.5 million participants in allstar cheerleading (not including the millions more in high school, college or little league participants) in the [[United States]] alone, cheerleading is, according to ''[[Newsweek|Newsweek's]]'' Arian Campo-Flores, "the most quintessential of [[United States|American]] sports."<ref name="nw20007"/> The growing presentation of the sport to a global audience has been led by the 1997 start of broadcasts of cheerleading competition by [[ESPN]] International and the worldwide release of the 2000 film ''[[Bring it On]]''. Due in part to this recent exposure, there are now an estimated 100,000 participants scattered around the rest of the world in countries including [[Australia]], [[China]], [[Colombia]], [[France]], [[Germany]], [[Japan]],<ref name="japancheer">{{cite news | last = Japan Echo, Inc | title=Three Cheers for the Champions! | work= [http://www.web-japan.org Web Japan]| date =[[2005-05-30]] | url=http://web-japan.org/trends/sports/spo050530.html | accessdate=2007-09-22}}</ref> the [[Netherlands]], [[New Zealand]], [[Sweden]] and the [[United Kingdom]]. <ref name="nw20007"/>
Newton's method can also be used to find a minimum or maximum of such a function, by finding a zero in the function's first derivative, see [[Newton's method in optimization|Newton's method as an optimization algorithm]].

The algorithm is first in the class of [[Householder's method]]s, succeeded by [[Halley's method]].

==Description of the method==

The idea of the method is as follows: one starts with an initial guess which is reasonably close to the true root, then the function is approximated by its [[tangent line]] (which can be computed using the tools of [[calculus]]), and one computes the ''x''-intercept of this tangent line (which is easily done with elementary algebra). This ''x''-intercept will typically be a better approximation to the function's root than the original guess, and the method can be [[iterative method|iterated]].

[[Image:newton_iteration.svg|alt Illustration of Newton's method|thumb|right|300px|An illustration of one iteration of Newton's method (the function ''f'' is shown in blue and the tangent line is in red). We see that <math>x_{n+1}</math> is a better approximation than <math>x_n</math> for the root <math>x</math> of the function <math>f</math>.]]

Suppose ''f'' : [''a'', ''b''] → '''R''' is a [[derivative|differentiable]] function defined on the [[interval (mathematics)|interval]] [''a'', ''b''] with values in the [[real number]]s '''R'''. The formula for converging on the root can be easily derived. Suppose we have some current approximation ''x''<sub>n</sub>. Then we can derive the formula for a better approximation, ''x''<sub>n+1</sub> by referring to the diagram on the right.
We know from the definition of the derivative at a given point that it is the slope of a tangent at that point.

That is

:<math>f'(x_{n}) = \frac{ \mathrm{rise} }{ \mathrm{run} } = \frac{ \mathrm{\Delta y} }{ \mathrm{\Delta x} } = \frac{ f( x_{n} ) - 0 }{ x_{n} - x_{n+1} }.\,\!</math>

Here, ''f''&nbsp;' denotes the [[derivative]] of the function ''f''. Then by simple algebra we can derive
:<math>x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\,\!</math>.
We start the process off with some arbitrary initial value ''x''<sub>0</sub>. (The closer to the zero, the better. But, in the absence of any intuition about where the zero might lie, a "guess and check" method might narrow the possibilities to a reasonably small interval by appealing to the [[intermediate value theorem]].) The method will usually converge, provided this initial guess is close enough to the unknown zero, and that <math>f'(x_0) \neq 0</math>. Furthermore, for a zero of [[multiplicity]] 1, the convergence is at least quadratic (see [[rate of convergence]]) in a [[neighbourhood (mathematics)|neighbourhood]] of the zero, which intuitively means that the number of correct digits roughly at least doubles in every step. More details can be found in the [[#Analysis|analysis section]] below.

==Examples==
===Example 1===
Consider the problem of finding the square root of a number. There are many [[Methods of computing square roots]], and Newton's method is one.

For example, if one wishes to find the square root of 612, this is equivalent to finding the solution to
:<math>x^2 = 612\,</math>
The function to use in Newton's Method is then,
:<math>f(x) = x^2 - 612 \,</math>
with derivative,
:<math>f'(x) = 2x \,</math>

With an initial guess of 10, the sequence given by newtons method is
:<math>\begin{matrix}
x_1 & = & x_0 - \frac{f(x_0)}{f'(x_0)} & = & 10 - \frac{10^2 - 612}{2 \cdot 10} & = & 35.6 \\
x_2 & = & x_1 - \frac{f(x_1)}{f'(x_1)} & = & 35.6 - \frac{35.6^2 - 612}{2 \cdot 35.6} & = & \underline{2}6.3955056 \\
x_3 & = & \vdots & = & \vdots & = & \underline{24.7}906355 \\
x_4 & = & \vdots & = & \vdots & = & \underline{24.7386}883 \\
x_5 & = & \vdots & = & \vdots & = & \underline{24.7386338}
\end{matrix}
</math>
Where the correct digits are underlined. With only a few iterations one can obtain a solution accurate to a few decimal places.

===Example 2===
Consider the problem of finding the positive number ''x'' with cos(''x'') = ''x''<sup>3</sup>. We can rephrase that as finding the zero of ''f''(''x'') = cos(''x'') − ''x''<sup>3</sup>. We have ''f''&nbsp;'(''x'') = −sin(''x'') − 3''x''<sup>2</sup>. Since cos(''x'') ≤ 1 for all ''x'' and ''x''<sup>3</sup> > 1 for ''x''>1, we know that our zero lies between 0 and 1. We try a starting value of ''x''<sub>0</sub> = 0.5.

:<math>\begin{matrix}
x_1 & = & x_0 - \frac{f(x_0)}{f'(x_0)} & = & 0.5 - \frac{\cos(0.5) - 0.5^3}{-\sin(0.5) - 3 \times 0.5^2} & = & 1.112141637097 \\
x_2 & = & x_1 - \frac{f(x_1)}{f'(x_1)} & = & \vdots & = & \underline{0.}909672693736 \\
x_3 & = & \vdots & = & \vdots & = & \underline{0.86}7263818209 \\
x_4 & = & \vdots & = & \vdots & = & \underline{0.86547}7135298 \\
x_5 & = & \vdots & = & \vdots & = & \underline{0.8654740331}11 \\
x_6 & = & \vdots &= & \vdots & = & \underline{0.865474033102}
\end{matrix}
</math>

The correct digits are underlined in the above example. In particular, ''x''<sub>6</sub> is correct to the number of decimal places given. We see that the number of correct digits after the decimal point increases from 2 (for ''x''<sub>3</sub>) to 5 and 10, illustrating the quadratic convergence.
<!--
Consider the following example in pseudocode.

'''function''' newtonIterationFunction(x) {
'''return''' x - (cos(x) - x^3) / (-sin(x) - 3*x^2)
}
'''var''' x := 0.5
'''for''' i '''from''' 0 '''to''' 99 {
print "Iteration: " + i
print "Guess: " + x
xold := x
x := newtonIterationFunction(x)
'''if''' x = xold {
print "Solution found!"
'''break'''
}
}

Here is the same using a calculator.

<math>\begin{matrix}
\mbox{What to type} & & & \\
0.5\;\mbox{(enter)} & = & x_0 & = & 0.5 \\
Ans - \frac{\cos (Ans) - Ans^3}{-\sin (Ans) - 3 * Ans^2}\;\mbox{(enter)} & = & x_1 & = & 1.1121416371 \\
\mbox{(enter)} & = & x_2 & = & 0.909672693736 \\
\mbox{(enter)} & = & x_3 & = & 0.867263818209 \\
\vdots & & \vdots & & \vdots
\end{matrix}
</math>
-->


==History==
==History==
[[Image:JCampbell.jpg|thumb|right|150px|[[Minnesota Golden Gophers Spirit Squads|Minnesota Gopher]] cheerleader Johnny Campbell]]
Cheerleading first appeared in the United States in the late 1880s with the crowd chanting as a way to encourage school spirit at athletic events. The first organized, recorded cheer was yelled "Ray, Ray, Ray! TIGER, TIGER, SIS, SIS, SIS! BOOM, BOOM, BOOM! Aaaaah! PRINCETON, PRINCETON, PRINCETON!" at [[Princeton University]] in 1884.<ref name="ICFHandbook">{{Citation | last=Neil | first=Randy L. | last2=Hart | first2 =Elaine | title=The Official Cheerleader's Handbook | publisher =[[Simon & Schuster]] | year =1986 | edition =Revised Fireside Edition 1986 | isbn =0-671-61210-7}}</ref> A few years later, Princeton graduate [[Thomas Peebles]] introduced the idea of organized crowds cheering at football games to the [[Minnesota Golden Gophers Spirit Squads|University of Minnesota]]. However, it was not until 1898 that [[Minnesota Golden Gophers Spirit Squads|University of Minnesota]] student Johnny Campbell directed a crowd in cheering "Rah, Rah, Rah! Sku-u-mar, Hoo-Rah! Hoo-Rah! Varsity! Varsity! Varsity, Minn-e-So-Tah!”, making Campbell the very first cheerleader and [[November 2]], [[1898]] the official birth date of organized cheerleading. Soon after, the [[Minnesota Golden Gophers Spirit Squads|University of Minnesota]] organized a "yell leader" squad of 6 male students, who still use Campbell's original cheer today<ref name="ICFHandbook"/> In 1903 the first cheerleading [[Fraternities and sororities|fraternity]], Gamma Sigma was founded.<ref name="ACFeb2003">{{Citation | last=Walker | first=Marisa | title=Cheer Milestones | journal=American Cheerleader | volume= 11 | issue=1 | pages=41–43 | date=February 2005 | year=2005 | id = ISSN 1079-9885 }}</ref> Cheerleading started out as an all-male activity, but females began participating in 1923, due to limited availability of female collegiate sports. At this time, gymnastics, tumbling, and megaphones were incorporated into popular cheers, and are still used today.<ref name="ACFeb2003"/> Today it is estimated that 97% of cheerleading participants overall are female, but males still make up 50% of cheering squads at the collegiate level. <ref name="popwarner">{{cite web | last =Balthaser | first =Joel D. | title=Cheerleading – Oh How far it has come! | publisher =[[Pop Warner Little Scholars|Pop Warner]] | date = [[2005-01-06]] | url=http://www.popwarner.com/articles/phenomenon.asp | accessdate=2007-01-11}}</ref>
[[Image:PostcardCornellUniversityCheerleader1906.jpg|thumb|left|150px|[[Cornell University]] cheerleader on a 1906 postcard]]


In 1948, Lawrence "Herkie" Herkimer, of Dallas, TX and a former cheerleader at [[Southern Methodist University]] formed the ''National Cheerleaders Association (NCA)'' as a way to hold cheerleading clinics. In 1949, The NCA held its first clinic in Huntsville, TX with 52 girls in attendance.<ref name="popwarner"/> "Herkie" contributed many "firsts" to the sport including the founding of ''Cheerleader & Danz Team'' uniform supply company, inventing the [[herkie]], (where one leg is bent towards the ground and the other is out to the side as high as it will stretch in the toe touch position)<ref name="cheerleading.about">{{cite web | title=Cheerleading Jump Herkie | url=http://cheerleading.about.com/cs/jumps/g/herkie.htm | accessdate=2007-08-06}}</ref> and creating the "Spirit Stick".<ref name="ACFeb2003"/> By the 1960s, college cheerleaders began hosting workshops across the nation, teaching fundamental cheer skills to eager high school age girls. In 1965, Fred Gastoff invented the vinyl [[pom-pon]] and it was introduced into competitions by the International Cheerleading Foundation (now the World Cheerleading Association or WCA). Organized cheerleading competitions began to pop up with the first ranking of the "Top Ten College Cheerleading Squads" and "Cheerleader All America" awards given out by the International Cheerleading Foundation in 1967. In 1978, America was introduced to competitive cheerleading by the first broadcast of [[Collegiate Cheerleading Championships]] on [[CBS]]<ref name="ICFHandbook"/><ref name="ACFeb2003"/>
Newton's method was described by [[Isaac Newton]] in ''De analysi per aequationes numero terminorum infinitas'' (written in 1669, published in 1711 by [[William Jones (mathematician)|William Jones]]) and in ''De metodis fluxionum et serierum infinitarum'' (written in 1671, translated and published as ''[[Method of Fluxions]]'' in 1736 by [[John Colson]]). However, his description differs substantially from the modern description given above: Newton applies the method only to polynomials. He does not compute the successive approximations <math>x_n</math>, but computes a sequence of polynomials and only at the end, he arrives at an approximation for the root ''x''. Finally, Newton views the method as purely algebraic and fails to notice the connection with calculus. Isaac Newton probably derived his method from a similar but less precise method by [[François Viète]]. The essence of Viète's method can be found in the work of the [[Mathematics in medieval Islam|Persian mathematician]], [[Sharaf al-Din al-Tusi]], while his successor [[Jamshīd al-Kāshī]] used a form of Newton's method to solve <math>x^P - N = 0</math> to find roots of ''N'' (Ypma 1995). A special case of Newton's method for calculating square roots was known much earlier and is often called the [[Methods of computing square roots#Babylonian method|Babylonian method]].
In the 1960s [[National Football League]] (NFL) teams began to organize professional cheerleading teams. The [[Indianapolis Colts|Baltimore Colts]] (now the [[Indianapolis Colts]]) was the first NFL team to have an organized cheerleading squad.<ref name="colts_football.com">{{cite web | title=Indianapolis Colts - www.football.com | url=http://www.football.com/nfl/indianapoliscolts/ | accessdate=2007-11-04}}</ref> It was the [[Dallas Cowboys Cheerleaders]] who gained the spotlight with their revealing outfits and sophisticated dance moves, which debuted in the 1972-1973 season, but were first seen widely in [[Super Bowl X]] (1976). This caused the image of cheerleaders to permanently change, with many other NFL teams emulating them. Most of the professional teams' cheerleading squads would more accurately be described as dance teams by today's standards; as they rarely, if ever, actively encourage crowd noise or perform modern cheerleading moves.


The 1980s saw the onset of modern cheerleading with more difficult stunt sequences and gymnastics being incorporated into routines. ESPN first broadcasted the National High School Cheerleading Competition nationwide in 1983. Cheerleading organizations such as the American Association of Cheerleading Coaches and Advisors (AACCA) started applying universal safety standards to decrease the number of injuries and prevent dangerous stunts, pyramids and tumbling passes from being included in routines. <ref name="AACCA">{{cite web | title=About the AACCA | url=http://www.aacca.org/about.asp | accessdate=2007-01-11}}</ref> In 2003, the National Council for Spirit Safety and Education (NCSSE) was formed to offer safety training for youth, school, all star and college coaches. The NCAA requires college cheer coaches to successfully complete a nationally recognized safety-training program. The NCSSE or AACCA certification programs are both recognized by the NCAA.
Newton's method was first published in 1685 in ''A Treatise of Algebra both Historical and Practical'' by [[John Wallis]]. In 1690, [[Joseph Raphson]] published a simplified description in ''[[Analysis aequationum universalis]]''. Raphson again viewed Newton's method purely as an algebraic method and restricted its use to polynomials, but he describes the method in terms of the successive approximations ''x''<sub>''n''</sub> instead of the more complicated sequence of polynomials used by Newton. Finally, in 1740, [[Thomas Simpson]] described Newton's method as an iterative method for solving general nonlinear equations using fluxional calculus, essentially giving the description above. In the same publication, Simpson also gives the generalization to systems of two equations and notes that Newton's method can be used for solving optimization problems by setting the gradient to zero.


Today, cheerleading is most closely associated with [[American football]] and [[basketball]]. Sports such as [[football (soccer)|soccer]], [[ice hockey]], [[volleyball]], [[baseball]], and [[wrestling]] sometimes sponsor cheerleading squads. The ICC Twenty20 Cricket World Cup in South Africa in 2007 was the first international cricket event to have cheerleaders. The [[Florida Marlins]] were the first [[Major League Baseball]] team to have cheerleaders. Debuting in 2003, the "Marlin Mermaids" gained national exposure and have influenced other MLB teams to develop their own cheer/dance squads.
[[Arthur Cayley]] in 1879 in ''The Newton-Fourier imaginary problem'' was the first who noticed the difficulties in generalizing the Newton's method to complex roots of [[polynomials]] with degree greater than 2 and complex initial values. This opened the way to the study of the theory of iterations of rational functions.


== Types of teams ==
==Practical considerations==
=== School-sponsored ===
[[Image:Georgia Tech cheerleaders 2006.jpg|thumb|right|400px|[[Georgia Institute of Technology|Georgia Tech]] [[Georgia Tech Yellow Jackets|Yellow Jackets]] cheerleaders at a [[college basketball]] game.]]
Most American high schools and colleges have organized cheerleading squads made up solely of students. Several colleges that compete at cheerleading competitions offer cheerleading scholarships. Some [[military academies]] use their [[drill team]] or [[color guard]] team instead of a cheersquad at athletic events, but some [[military academies]] have traditional cheerleading squads just like other everyday [[universities]]. Normally, the main reason for school-sponsored cheerleading is to promote school spirit and motivate the players and fans. They may compete outside of sporting events (local, regional, and national competitions), but their main task is to cheer for sporting events and get the crowd involved. Cheerleading is quickly becoming a year round sport starting with tryouts during the preceding school year, practices, attendance at various sporting events and ending with Competition season from winter through spring.


===Youth league===
In general the [[rate of convergence|convergence]] is '''quadratic''': the error is essentially squared at each step (that is, the number of accurate digits doubles in each step). There are some caveats, however. First, Newton's method requires that the derivative be calculated directly. (If the derivative is approximated by the [[slope]] of a line through two points on the function, the [[secant method]] results; this can be more efficient depending on how one measures computational effort.) Second, if the initial value is too far from the true zero, Newton's method can fail to converge. Because of this, most practical implementations of Newton's method put an upper limit on the number of iterations and perhaps on the size of the iterates. Third, if the [[root (mathematics)|root]] being sought has [[multiplicity]] greater than one, the convergence rate is merely linear (errors reduced by a constant factor at each step) unless special steps are taken.
Many organizations that sponsor youth league [[American football|football]] or [[basketball]] sponsor cheerleading squads as well. [[Pop Warner Little Scholars|Pop Warner]] organizations are an example of this. The YMCA (Young Men's Christian Association) is also a popular sponsor for youth cheerleading leagues, as well as numerous other sports leagues.


===All-Star cheerleading===
Since the most serious of the problems above is the possibility of a failure of convergence, Press et al. (1992) present a version of Newton's method that starts at the midpoint of an interval in which the root is known to lie and stops the iteration if an iterate is generated that lies outside the interval.
In the early 1980s, cheerleading squads not associated with a schools or sports leagues, whose main objective was competition, began to emerge. The first organization to call themselves all stars and go to competitions were the Q94 Rockers from [[Richmond, Virginia]], founded in 1982 by Hilda McDaniel.<ref name="ACFeb2007">{{Citation | last = Smith | first = Jennifer Renèe | title = The All-Star Chronicles | journal = American Cheerleader | volume = 13 | issue = 1 | pages = 40–42 | date= February 2007 | year = 2007 | id = ISSN 1079-9885 }}</ref> All-star teams competing prior to 1987 were place into the same divisions as teams that represented schools and sports leagues. In 1986 National Cheerleaders Association (NCA) decided to address this situation by creating a separate division for these teams lacking a sponsoring school or athletic association, calling it the 'All-Star Division' and debuting it at their 1987 competitions. As the popularity of these types of teams grew, more and more of them were formed, attending competitions sponsored by many different types of organizations and companies, all using their own set of rules, regulations and divisions. This situation became one of the chief concerns of gym owners. These inconsistencies caused coaches to keep their routines in a constant state of flux, detracting from time that should be utilized to develop skills and provide personal attention to their athletes. More importantly, because the various companies were constantly vying for the competitive edge, safety standards had becoming more and more lax. In some cases, unqualified coaches and inexperienced squads are attempting dangerous stunts as a result of these “expanded” sets of rules.<ref name="varsity2004">{{cite web | title = The Cheerleading Worlds Administered by the USASF | publisher = Varsity Brands, Inc. | url = http://www.varsity.com/index.asp?article=1514 | accessdate = 2007-09-20 }}</ref>


The USASF was formed in 2003 by these various competition companies to act as the national governing body for all star cheerleading and to create a standard set of rules and judging standards to be followed by all competitions sanctioned by the Federation and ultimately leading to the Cheerleading Worlds. The USASF hosted the first Cheerleading Worlds on Saturday, April 24, 2004.<ref name="varsity2004"/> At the same time, cheerleading coaches from all over the country organize themselves for the same rule making purpose, calling themselves the National All Star Cheerleading Coaches Congress (NACCC). In 2005, the NACCC was absorbed by the USASF to become their rule making body.<ref name="ACFeb2007"/> By late 2006, the [[United States All Star Federation|USASF]] was ready to expand its reach even further, by facilitating the creation of the International All-Star Federation (IASF), the first international governing body for the sport of cheerleading. <ref name="USASFDec06Newsletter">{{cite web | title = USASF Insider | url = https://secure.usasf.net/Documents/Newsletters/December%202006.pdf | format = pdf | accessdate = 2007-09-15}}</ref>
Developers of large scale computer systems involving root finding tend to prefer the [[secant method]] over Newton's method because the use of a difference quotient in place of the derivative in Newton's method implies that the additional code to compute the derivative need not be maintained. In practice, the advantages of maintaining a smaller code base usually outweigh the superior convergence characteristics of Newton's method.


Currently all-star cheerleading as sanctioned by the USASF involves a squad of 6-36 females and/or males. The squad prepares year-round for many different competition appearances, but they only actually perform for up to 2½ minutes during their routines. The numbers of competitions a team participates in varies from team to team, but generally, most teams tend to participate in eight-twelve competitions a year. These competitions include locals, which are normally taken place in school gymnasiums, nationals, hosted in big venues all around the U.S. with national champions, and the Cheerleading Worlds, taken place at Disney World in Orlando, Florida. During a competition routine, a squad performs carefully choreographed stunting, tumbling, jumping and dancing to their own custom music. Teams create their routines to an eight-count system and apply that to the music so the team members execute the elements with precise timing and synchronization.
==Counter examples==
[[Image:NewtonsMethodConvergenceFailure.svg|thumb|300px|The tangent lines of ''x''^3 - 2''x'' + 2 at 0 and 1 intersect the x-axis at 1 and 0, respectively, illustrating why Newton's method oscillates between these values for some starting points.]]
===Starting point too far away===
If the starting point is not close to a root then convergence may fail to occur. Let
:<math>f(x) = x^3 - 2x + 2 \!</math>
and take 0 as the starting point. The first iteration produces 1 and the second iteration returns to 0 so the sequence will oscillate between the two without converging to a root. In general, the behavior of the sequence can be very complex. (See [[Newton fractal]].)


Judges at the competition watch for illegal moves from the group or any individual member. Here, an illegal move is something that is not allowed in that division due to difficulty and safety restrictions. More generally, judges look at the difficulty and execution of jumps, stunts and tumbling, synchronization, creativity, the sharpness of the motions, showmanship, and overall routine execution.
''In the following examples, the desired root is at zero for simplicity—it could have been placed elsewhere.''


All-star cheerleaders are placed into divisions, which are grouped based upon age, size of the team, gender of participants, and ability level. The age levels vary from under 4 year of age to 18 years and over. The divisions used by the [[United States All Star Federation|USASF/IASF]] are currently Tiny, Mini, Youth, Junior, Junior International, Junior Coed, Senior, Senior coed, Open International and Open.<ref name="USASF_Ages_07-08">{{cite web | title = USASF All-Star Cheer Divisions for 2007-2008 | url = https://secure.usasf.net/Documents/Rules/07-08/Age%20Grid%2007-08%20--%20Rev%203-23-07.pdf | format = pdf | accessdate = 2007-06-29}}</ref>
=== No feasible starting points ===
Some functions can be constructed on which Newton's method will always diverge, unless the solution is guessed on the first try; a simple one is the cubic root, which is continuous and infinitely differentiable with continuity (except for ''x''=0, where its derivatives are infinite, but that is the one point that Newton's method will never use as it is the solution):
:<math>f(x) = \sqrt[3]{x}</math>
For any iteration point ''x<sub>n</sub>'', the next iteration point will be:
:<math>x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} = x_n - \frac{x_n^{1/3}}{1/3\,x_n^{1/3-1}} = x_n - 3\,x_n = -2\,x_n</math>
The algorithm overshoots the solution and lands on the other side of the ''y'' axis, farther away than it initially was; applying Newton's method actually doubles the distances from the solution at each iteration.


If a team places high enough at selected [[United States All Star Federation|USASF/IASF]] sanctioned national competitions, they could be included in the [[United States All Star Federation|Cheerleading Worlds]] and compete against teams from all over the world, as well as receive money for placing.<ref name="nw20007"/>
This consideration holds true for every <math>f(x) = \sqrt[n]{|x|}</math>, where ''n''>2. In the limiting case of ''n''=2 (square root), the iterations will oscillate indefinitely between points ''x<sub>0</sub>'' and ''-x<sub>0</sub>''.


==Cheerleaders==
===Discontinuous derivative===
{{See also|list of cheerleaders}}
If the derivative is not continuous at the root, then convergence may fail to occur in any neighborhood of the root. Consider the function
===Famous Former Cheerleaders===
{{Cleanup-reason|August 2008|This section is becoming too lengthy compared to the rest of the article. Most of these names should be added to the [[List of cheerleaders]] page and removed from here.}}
Many prominent people have been cheerleaders, including:
*[[Madonna (entertainer)|Madonna]]
*[[Paula Abdul]]
*[[Robin Williams]]
*[[Tom Hanks]]
*[[Lindsay Lohan]]
*[[Ronald Reagan]]
*[[Franklin D. Roosevelt]]
*[[Dwight D. Eisenhower]]
*[[George W. Bush]]<ref name="popwarner"/><ref>[http://www.cnn.com/SPECIALS/2000/democracy/bush/stories/bush/ CNN report with picture of future President Bush as a cheerleader]</ref>
*[[Meryl Streep]]
*[[Steve Martin]]
*[[Eddie Van Halen]]
*[[Halle Berry]]
*[[Angus Young]]
*[[Teri Hatcher]]
*[[Sandra Bullock]]
*[[Cameron Diaz]]
*[[Reese Witherspoon]]
*[[Hilarie Burton]]
*[[Sally Field]]
*[[Rachael Ray]]
*[[Alicia Silverstone]]
*[[Laci Peterson]]
*[[Stacy Ferguson]]
*[[Jamie Lynn Spears]]
*[[Bon Scott]]
*[[Dana Plato]]
*[[Felicity Pollard]]
*[[Tina Turner]]
*[[Miley Cyrus]]
*[[Rihanna]]
*[[Paula Deanda]]
*[[Anjelah Johnson]]
*[[Kellie Pickler]]
*[[Mandy Moore]]
*[[Ciara]]
*[[Tiffany Pollard]]
*[[Allie DiMeco]]


==Cheerleading in popular culture==
:<math>f(x) = \begin{cases}
===Movies and television===
0 & \text{if } x = 0,\\
''Also see [[List of cheerleaders#Fictional|List of cheerleaders in fiction]]''
x + x^2\sin\left(\frac{2}{x}\right) & \text{if } x \neq 0.
[[Image:Claire the cheerleader.jpg|thumb|right|[[Claire Bennet]],[[Heroes (TV series)|''Heroes'']]]]
\end{cases}</math>


Cheerleading's increasing popularity in recent decades has made it a prominent feature in high-school themed movies and television shows. The 2000 film ''[[Bring It On (film)|Bring It On]]'', about a [[San Diego]] high school cheerleading squad called "The Toros", starring real-life former cheerleader [[Kirsten Dunst]]. ''Bring It On'' was a surprise hit and earned nearly $70 million domestically. It spawned two direct-to-video [[sequels]] (''[[Bring It On Again]]'' in 2003 and ''[[Bring It On: All or Nothing]]'' in 2006). The fourth film in the franchise, ''[[Bring It On: In It to Win It]]'', was released on December 18, 2007. ''Bring It On'' was followed in 2001 by another teen cheerleading comedy, ''[[Sugar & Spice]]''. In 1993, ''[[The Positively True Adventures of the Alleged Texas Cheerleader-Murdering Mom]]'' was an acclaimed [[TV movie]] which told the true story of [[Wanda Holloway]], the Texas mother whose obsession with her daughter's cheerleading career made headline news.
Its derivative is:
:<math>f'(x) = \begin{cases}
\nexists & \text{if } x = 0,\\
1 + 2\,x\,\sin\left(\frac{2}{x}\right) + 2\,\cos\left(\frac{2}{x}\right) & \text{if } x \neq 0.
\end{cases}</math>


In 2006, [[Hayden Panettiere]], star of ''Bring It On: All or Nothing'', took another cheerleading role as [[Claire Bennet]], the cheerleader with an accelerated [[healing factor]] on [[NBC|NBC's]] hit [[sci-fi]] [[TV series]] [[Heroes (TV series)|''Heroes'']], launching cheerleading back into the limelight of pop culture. Claire was the main focus of the show's first story arc, featuring the popular catchphrase, "Save the cheerleader, save the world." Her prominent, protagonist role in ''Heroes'' was supported by a strong fan-base and provided a positive image for high school cheerleading.
The derivative does not exist at ''x''=0 since both ''f''<nowiki>'</nowiki>(0<sup>-</sup>) and ''f'''(0<sup>+</sup>) are undefined.


In 2006, [[Cheerleader Nation]], was a reality show featured on the channel, Lifetime. Cheerleader Nation is a 60 minute television series based on the Paul Laurence Dunbar High School cheerleading team's ups and downs on the way to nationals, of which they are the three time champions. This show also explains how cheerleading is a tough sport. This show takes place in Lexington, Kentucky. The team is on a quest to win a third national championship.
Within any neighborhood of the root, this derivative keeps changing sign as ''x'' approaches 0 from the right (or from the left) while ''f(x)''≥''x''-''x²''>0 for 0<''x''<1.


===Video games===
So ''f(x)/f'(x)'' is unbounded near the root, and Newton's method will not converge, even though:
Nintendo has released a pair of video games in Japan for the [[Nintendo DS]], ''[[Osu! Tatakae! Ouendan]]'' and its sequel ''[[Moero! Nekketsu Rhythm Damashii Osu! Tatakae! Ouendan 2|Moero! Nekketsu Rhythm Damashii]]'' that star teams of male cheer squads, or [[Ouendan]] that practice a form of cheerleading. Each of the games' most difficult modes replaces the male characters with female cheer squads that dress in western cheerleading uniforms. The games task the cheer squads with assisting people in desperate need of help by cheering them on and giving them the motivation to succeed.
*the function is everywhere differentiable with continuity, except at the root;
*''f'' is infinitely differentiable except at the root; and
* the derivative is bounded in a neighborhood of the root (unlike ''f(x)/f'(x)'').


==Sport debate and Leadership Issues==
===No second derivative===
Cheerleading among others has had debate on whether or not it truly is a ''sport''. Supporters consider cheerleading as a whole as a sport citing the heavy use of athletic talents <ref>{{cite journal|title= The road to recognition |journal=[[BCA resources]]|date=[[2008-08-02]]|url=http://www.cheerleading.org.uk/news/recognition.htm |accessdate=2008-08-02}}</ref> while critics do not see it as deserving of that status since ''sport'' implies a competition among squads and not all squads compete along with subjectivity of competitions.<ref name="stanford">{{cite journal|title=Sport, not a sport: consider Dan the expert |journal=[[The Stanford Daily]]|date=[[2004-09-29]]|url=http://daily.stanford.edu/article/2004/9/29/sportNotASportConsiderDanTheExpert |accessdate=2007-12-15}}</ref><ref name="SFU">{{cite journal|title= No, Cheerleading is not a Sport |journal=[[SFU Cheer Resources]]|date=[[2004-09-29]]|url=http://www.sfucheerleading.ca/index.php?page=10&sub=9&method=view&id=10 |accessdate=2007-12-15}}</ref>
If there is no second derivative at the root, then convergence may fail to be quadratic. Indeed, let
:<math>f(x) = x + x^{4/3} \!</math>
Then
:<math>f'(x) = 1 + (4/3)x^{1/3} \!</math>
And
:<math>f''(x) = (4/9)x^{-2/3} \!</math>
except when <math>x = 0 \!</math> where it is undefined. Given <math>x_n \!</math>,


In the UK, there is less of a debate, as the sports councils recognise cheerleading as a sport, they have however yet to assign a national governing body, there are currently two groups applying for the position: British Cheerleading Association, and British Gymnastics. <ref>{{cite journal|title= The road to recognition |journal=[[BCA resources]]|date=[[2008-08-02]]|url=http://www.cheerleading.org.uk/news/recognition.htm |accessdate=2008-08-02}}</ref>
:<math>x_{n+1} = x_n - \frac{f(x_n)}{f '(x_n)} = \frac{(1/3)x_n^{4/3}}{(1 + (4/3)x_n^{1/3})} \!</math>


==Dangers of cheerleading==
which has approximately 4/3 times as many bits of precision as <math>x_n \!</math> has. This is less than the 2 times as many which would be required for quadratic convergence. So the convergence of Newton's method (in this case) is not quadratic, even though: the function is continuously differentiable everywhere; the derivative is not zero at the root; and <math>f \!</math> is infinitely differentiable except at the desired root.
{{Expand-section|date=March 2008}}
The risk of cheerleading was highlighted when Kristi Yamaoka, a cheerleader for Southern Illinois University, suffered from a fractured vertebra after she hit her head after falling from a [[human pyramid]].<ref>{{citation|url=http://www.athleticmanagement.com/2007/02/cheer_bans_continue.html|title=Cheerbans continue|publisher=Athletic Management|issue=Aug/Sept 2006}}</ref> She also suffered from a concussion, and a bruised lung.<ref name=CWT/> The fall occurred when Yamaoka lost her balance during a basketball game between [[Southern Illinois University]] and [[Bradley University]] at the [[Savvis Center]] in [[St. Louis, Missouri|St. Louis]] on [[March 5]] [[2006]].<ref name=CWT>{{cite web
| title= Cheerleader worried for team, not herself
| url=http://www.msnbc.msn.com/id/11699607/
| accessdate=2007-09-29 }}</ref> The fall gained "national attention"<ref name=CWT/> , because Yamoka continued to perform from a stretcher as she was moved away from the game.<ref name=CWT/> The cheerleader has since made a full recovery.


The accident caused the Missouri Valley Conference to ban its member schools from allowing cheerleaders to be "launched or tossed and from taking part in formations higher than two levels" for one week during a women's basketball conference tournament, and also resulted in a recommendation by the [[National Collegiate Athletic Association|NCAA]] that conferences and tournaments do not allow pyramids two and one half levels high or higher, and a stunt known as basket tosses, during the rest of the men's and women's basketball season. <ref>{{cite web
===Zero derivative===
| title=CHEERLEADING PROGRAMS GOING ALL-OUT FOR SAFETY
If the first derivative is zero at the root, then convergence will not be quadratic. Indeed, let
| url=http://www.aacca.org/CollegePressRelease.doc
:<math>f(x) = x^2 \!</math>
| accessdate=2006-12-19 }}</ref> On [[July 11]] [[2006]], the bans were made permanent by the AACCA rules committee:
Then <math>f'(x) = 2x \!</math> and consequently <math>x - f(x)/f'(x) = x/2 \!</math>. So convergence is not quadratic, even though the function is infinitely differentiable everywhere.
<blockquote>
The committee unanimously voted for sweeping revisions to cheerleading safety rules, the most major of which restricts specific upper-level skills during basketball games. Basket tosses, 2½ high pyramids, one-arm stunts, stunts that involve twisting or flipping, and twisting tumbling skills may only be performed during halftime and post-game on a matted surface and are prohibited during game play or time-outs.<ref>{{cite web
| title=CHEERLEADING PROGRAMS GOING ALL-OUT FOR SAFETY
| url=http://www.aacca.org/CollegePressRelease.doc
| accessdate=2006-12-19 }}</ref>
</blockquote>


However, there have been far worse catastrophes in the world of cheerleading. Last October, at Los Angeles' John Marshall High School, 17 year-old Patty Phommanyvong became a comatose quadriplegic as the result of a fall to the ground. Eighteen year-old Jessica Smith suffered a fractured back and neck after hitting the ground during practice at Sacramento City College in California. Most tragically, 20 year-old Lauren Chang died this April from being kicked in the chest at a cheerleading competition in Worcester, Massachusetts.
==Analysis==


Out of the nation's 2.9 million female high school athletes, only 3% are cheerleaders, yet cheerleading accounts for 65% of all catastrophic injuries in girls' high school athletics. Since the NCAA has yet to recognize cheerleading as an official college sport, there are no solid numbers on college cheerleading, yet when it comes to injuries, 67% of female athlete injuries at the college level are due to cheerleading mishaps.
Suppose that the function ''f'' has a zero at α, i.e., ''f''(α) = 0.


<ref>{{cite web
If ''f''&nbsp; is continuously differentiable and its derivative does not vanish at α, then there exists a [[topological neighborhood|neighborhood]] of α such that for all starting values ''x''<sub>0</sub> in that neighborhood, the [[sequence]] {''x''<sub>''n''</sub>} will [[limit of a sequence|converge]] to α.
| title=Cheerleading is by far the riskiest sport for girls

| url=http://today.msnbc.msn.com/id/26293199/?GT1=43001
If the function is continuously differentiable and its derivative does not vanish at α and it has a second derivative at α then the convergence is quadratic or faster. If the second derivative does not vanish at α then the convergence is merely quadratic.
| accessdate=2008-09-11 }}</ref>

If the derivative does vanish at α, then the convergence is usually only linear. Specifically, if ''f'' is twice continuously differentiable, <math>f'(\alpha) = 0 \!</math> and <math>f''(\alpha) \ne 0 \!</math>, then there exists a neighborhood of α such that for all starting values ''x''<sub>0</sub> in that neighborhood, the sequence of iterates converges linearly, with [[rate of convergence|rate]] log<sub>10</sub>&nbsp;2 (Süli & Mayers, Exercise 1.6). Alternatively if <math>f'(\alpha) = 0 \!</math> and <math>f'(x) \ne 0 \!</math> elsewhere, in a [[topological neighborhood|neighborhood]] ''U'' of α, α being a zero of [[multiplicity]] ''r'' and if <math>f \in C^r(U)</math> then there exists a neighborhood of α such that for all starting values ''x''<sub>0</sub> in that neighborhood, the sequence of iterates converges linearly.

However, even linear convergence is not guaranteed in pathological situations.

In practice these results are local and the neighborhood of convergence are not known ''a priori'', but there are also some results on global convergence, for instance, given a right neighborhood ''U<sub>+</sub>'' of α, if ''f'' is twice differentiable in ''U<sub>+</sub>'' and if <math>f' \ne 0 \!</math>, <math>f \cdot f'' > 0 \!</math> in ''U<sub>+</sub>'', then, for each ''x<sub>0</sub>'' in ''U<sub>+</sub>'' the sequence ''x<sub>k</sub>'' is monotonically decreasing to α.

==Generalizations==
===Nonlinear systems of equations===

[[Image:ComponentNewton.jpg|right|thumb|Solving system of nonlinear equations with Newton method]]
One may use Newton's method also to solve systems of ''k'' (non-linear) equations, which amounts to finding the zeros of continuously differentiable functions ''F'' : '''R'''<sup>''k''</sup> <tt>&rarr;</tt> '''R'''<sup>''k''</sup>. In the formulation given above, one then has to left multiply with the inverse of the ''k''-by-''k'' [[Jacobian]] [[matrix (mathematics)|matrix]] ''J''<sub>''F''</sub>(''x''<sub>''n''</sub>) instead of dividing by ''f''&nbsp;'(''x''<sub>''n''</sub>). Rather than actually computing the inverse of this matrix, one can save time by solving the [[system of linear equations]]

:<math>J_F(x_n) (x_{n+1} - x_n) = -F(x_n)\,\!</math>

for the unknown ''x''<sub>''n''+1</sub> − ''x''<sub>''n''</sub>.
Again, this method only works if the initial value ''x''<sub>''0''</sub> is close enough to the true zero. Typically, a region which is [[well-behaved]] is located first with some other method and Newton's method is then used to "polish" a root which is already known approximately.

===Nonlinear equations in a Banach space===
Another generalization is the Newton's method to find a zero of a function ''F'' defined in a [[Banach space]]. In this case the formulation is
:<math>X_{n+1}=X_n-(F'_{X_n})^{-1}[F(X_n)]</math>,
where <math>F'_{X_n}</math> is the [[Fréchet derivative]] applied at the point <math>X_n</math>. One needs the Fréchet derivative to be invertible at each <math>X_n</math> in order for the method to be applicable.

===Complex functions===
[[image:newtroot_1_0_0_0_0_m1.png|thumb|Basins of attraction for x<sup>5</sup> &minus; 1 = 0; darker means more iterations to converge.]]
{{main|Newton fractal}}
When dealing with [[complex analysis|complex functions]], however, Newton's method can be directly applied to find their zeros. For many complex functions, the boundary of the set (also known as the [[basin of attraction]]) of all starting values that cause the method to converge to a particular zero is a [[fractal]].


==See also==
==See also==
* [[Artificial noise]]
{{commons | category:Newton Method}}
* [[Root-finding algorithm]]
* [[List of cheerleaders]]
* [[Secant Method]]
* [[List of cheerleading jumps]]
* [[Euler method]]
* [[List of cheerleading stunts]]
* [[National Football League cheerleading]]
* [[Integer square root]]
* [[Ōendan]], a style of Japanese cheerleading
* [[Methods of computing square roots]]
* [[Pom-pon]]
* [[UAAP Cheerdance Competition]]
* [[Cheerleader Nation]]
* [[Majorettes]]


==References==
==References==
{{reflist|2}}

* Tjalling J. Ypma, Historical development of the Newton-Raphson method, ''SIAM Review'' '''37''' (4), 531–551, 1995. {{doi|10.1137/1037125}}.
* P. Deuflhard, ''Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms.'' Springer Series in Computational Mathematics, Vol. 35. Springer, Berlin, 2004. ISBN 3-540-21099-7.
* C. T. Kelley, ''Solving Nonlinear Equations with Newton's Method'', no 1 in Fundamentals of Algorithms, SIAM, 2003. ISBN 0-89871-546-6.
* J. M. Ortega, W. C. Rheinboldt, ''Iterative Solution of Nonlinear Equations in Several Variables.'' Classics in Applied Mathematics, SIAM, 2000. ISBN 0-89871-461-3.
* W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, ''[[Numerical Recipes|Numerical Recipes in C: The Art of Scientific Computing]]'', Cambridge University Press, 1992. ISBN 0-521-43108-5 (available free online, with code samples: [http://www.nrbook.com/a/bookcpdf.html]), sections 9.4 [http://www.nrbook.com/a/bookcpdf/c9-4.pdf] and 9.6 [http://www.nrbook.com/a/bookcpdf/c9-6.pdf].
* W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, ''[[Numerical Recipes|Numerical Recipes: The Art of Scientific Computing]]'', Cambridge University Press, 2007. ISBN 0-521-88068-8 (available for a fee online, with code samples [http://www.nr.com]).
* W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, ''[[Numerical Recipes|Numerical Recipes in Fortran]]'', Cambridge University Press, 1992. ISBN 0-521-43064-X (online, with code samples: [http://library.lanl.gov/numerical/bookfpdf/f9-4.pdf])
* Endre Süli and David Mayers, ''An Introduction to Numerical Analysis'', Cambridge University Press, 2003. ISBN 0-521-00794-1.


==External links==
==External links==
{{Commons2|Cheerleaders}}
{{Wiktionary}}
*[http://www.cheerunion.org International Cheer Union - Building the 1st Cheerleading Olympic Structure (ICU) ]
*[http://www.spiritsafety.com National Council for Spirit Safety and Education (NCSSE)]
*[http://www.aacca.org American Association of Cheerleading Coaches and Advisors (AACCA)]
*[http://www.usasf.net United States All-Star Federation (USASF)]
*[http://www.iasfonline.org International All Star Federation (IASF)]
*[http://www.cheerleading.org.uk British Cheerleading Association (BCA)]
*[http://www.cheerleadinguk.com UK Cheerleading Association (UKCA)]
*[http://www.cheerleading.co.nr Cheer Ireland (CFI) ]
*[http://aescougarcheer.com/Cheer101/basics.htm Cheerleading 101: the Basics ]
*[http://www.nationalcheersafety.com National Cheer Safety Foundation ]
*[http://www.cheerleading.com.br Comissão Paulista de Cheerleading - São Paulo, Brazil ]


[[Category:Cheerleading|*]]
* [http://mathworld.wolfram.com/NewtonsMethod.html Newton's method on Wolfram.com]
[[Category:Sports culture]]
* [http://math.fullerton.edu/mathews/a2001/Animations/RootFinding/NewtonMethod/NewtonMethod.html Animations for Newton's method] by Prof. John H. Mathews
[[Category:Team sports]]
* [http://animation.yihui.name/compstat:newton_s_method Animations for Newton's method] by Yihui Xie using the [[R (programming language)|R]] package [http://cran.r-project.org/package=animation animation]
* [http://numericalmethods.eng.usf.edu/topics/newton_raphson.html Newton-Raphson Method Notes, PPT, Mathcad, Maple, Matlab, Mathematica] at [http://numericalmethods.eng.usf.edu Holistic Numerical Methods Institute]
*[http://math.fullerton.edu/mathews/n2003/Newton'sMethodMod.html Module for Newton’s Method by John H. Mathews]
* [http://plus.maths.org/issue9/puzzle/solution.html worked example]
* [http://www.hvks.com/Numerical/websolver.htm A web based Newton solver that handles polynomials with both real and complex coefficients]

[[Category:Optimization algorithms]]
[[Category:Root-finding algorithms]]


[[cs:Cheerleading]]
[[ar:طريقة نيوتن]]
[[de:Cheerleading]]
[[bg:Метод на Нютон]]
[[ca:Mètode de Newton]]
[[es:Animación (deporte)]]
[[cs:Metoda tečen]]
[[fa:هلهله‌چی]]
[[da:Newtons metode]]
[[fr:Pom-pom girl]]
[[de:Newton-Verfahren]]
[[id:Pemandu sorak]]
[[he:עידוד (ספורט)]]
[[es:Método de Newton]]
[[hu:Cheerleading]]
[[fr:Méthode de Newton]]
[[it:Metodo delle tangenti]]
[[it:Cheerleading]]
[[nl:Cheerleading]]
[[he:שיטת ניוטון-רפסון]]
[[ja:チアリーダー]]
[[nl:Newton-Raphson]]
[[no:Cheerleading]]
[[ja:ニュートン法]]
[[pl:Metoda Newtona]]
[[pl:Cheerleading]]
[[pt:Método de Newton]]
[[pt:Animadora de torcida]]
[[ru:Метод Ньютона]]
[[ru:Черлидинг]]
[[simple:Cheerleader]]
[[fi:Newtonin menetelmä]]
[[sv:Newtons metod]]
[[fi:Cheerleading]]
[[zh:牛顿法]]
[[sv:Cheerleading]]
[[th:เชียร์ลีดเดอร์]]
[[zh:競技啦啦隊]]

Revision as of 17:18, 11 October 2008

Youth Cheerleaders during a football halftime show. Youth Cheer—high school ages and younger—make up the vast majority of cheerleaders and cheer teams.

Cheerleading is a sport[1][2][3][4] that uses organized routines made from elements of tumbling, dance, jumps and stunting to direct spectators of events to cheer on sports teams at games and matches and/or compete at cheerleading competitions. The athlete involved is called a cheerleader. With an estimated 1.5 million participants in allstar cheerleading (not including the millions more in high school, college or little league participants) in the United States alone, cheerleading is, according to Newsweek's Arian Campo-Flores, "the most quintessential of American sports."[1] The growing presentation of the sport to a global audience has been led by the 1997 start of broadcasts of cheerleading competition by ESPN International and the worldwide release of the 2000 film Bring it On. Due in part to this recent exposure, there are now an estimated 100,000 participants scattered around the rest of the world in countries including Australia, China, Colombia, France, Germany, Japan,[5] the Netherlands, New Zealand, Sweden and the United Kingdom. [1]

History

Minnesota Gopher cheerleader Johnny Campbell

Cheerleading first appeared in the United States in the late 1880s with the crowd chanting as a way to encourage school spirit at athletic events. The first organized, recorded cheer was yelled "Ray, Ray, Ray! TIGER, TIGER, SIS, SIS, SIS! BOOM, BOOM, BOOM! Aaaaah! PRINCETON, PRINCETON, PRINCETON!" at Princeton University in 1884.[6] A few years later, Princeton graduate Thomas Peebles introduced the idea of organized crowds cheering at football games to the University of Minnesota. However, it was not until 1898 that University of Minnesota student Johnny Campbell directed a crowd in cheering "Rah, Rah, Rah! Sku-u-mar, Hoo-Rah! Hoo-Rah! Varsity! Varsity! Varsity, Minn-e-So-Tah!”, making Campbell the very first cheerleader and November 2, 1898 the official birth date of organized cheerleading. Soon after, the University of Minnesota organized a "yell leader" squad of 6 male students, who still use Campbell's original cheer today[6] In 1903 the first cheerleading fraternity, Gamma Sigma was founded.[7] Cheerleading started out as an all-male activity, but females began participating in 1923, due to limited availability of female collegiate sports. At this time, gymnastics, tumbling, and megaphones were incorporated into popular cheers, and are still used today.[7] Today it is estimated that 97% of cheerleading participants overall are female, but males still make up 50% of cheering squads at the collegiate level. [8]

Cornell University cheerleader on a 1906 postcard

In 1948, Lawrence "Herkie" Herkimer, of Dallas, TX and a former cheerleader at Southern Methodist University formed the National Cheerleaders Association (NCA) as a way to hold cheerleading clinics. In 1949, The NCA held its first clinic in Huntsville, TX with 52 girls in attendance.[8] "Herkie" contributed many "firsts" to the sport including the founding of Cheerleader & Danz Team uniform supply company, inventing the herkie, (where one leg is bent towards the ground and the other is out to the side as high as it will stretch in the toe touch position)[9] and creating the "Spirit Stick".[7] By the 1960s, college cheerleaders began hosting workshops across the nation, teaching fundamental cheer skills to eager high school age girls. In 1965, Fred Gastoff invented the vinyl pom-pon and it was introduced into competitions by the International Cheerleading Foundation (now the World Cheerleading Association or WCA). Organized cheerleading competitions began to pop up with the first ranking of the "Top Ten College Cheerleading Squads" and "Cheerleader All America" awards given out by the International Cheerleading Foundation in 1967. In 1978, America was introduced to competitive cheerleading by the first broadcast of Collegiate Cheerleading Championships on CBS[6][7]

In the 1960s National Football League (NFL) teams began to organize professional cheerleading teams. The Baltimore Colts (now the Indianapolis Colts) was the first NFL team to have an organized cheerleading squad.[10] It was the Dallas Cowboys Cheerleaders who gained the spotlight with their revealing outfits and sophisticated dance moves, which debuted in the 1972-1973 season, but were first seen widely in Super Bowl X (1976). This caused the image of cheerleaders to permanently change, with many other NFL teams emulating them. Most of the professional teams' cheerleading squads would more accurately be described as dance teams by today's standards; as they rarely, if ever, actively encourage crowd noise or perform modern cheerleading moves.

The 1980s saw the onset of modern cheerleading with more difficult stunt sequences and gymnastics being incorporated into routines. ESPN first broadcasted the National High School Cheerleading Competition nationwide in 1983. Cheerleading organizations such as the American Association of Cheerleading Coaches and Advisors (AACCA) started applying universal safety standards to decrease the number of injuries and prevent dangerous stunts, pyramids and tumbling passes from being included in routines. [11] In 2003, the National Council for Spirit Safety and Education (NCSSE) was formed to offer safety training for youth, school, all star and college coaches. The NCAA requires college cheer coaches to successfully complete a nationally recognized safety-training program. The NCSSE or AACCA certification programs are both recognized by the NCAA.

Today, cheerleading is most closely associated with American football and basketball. Sports such as soccer, ice hockey, volleyball, baseball, and wrestling sometimes sponsor cheerleading squads. The ICC Twenty20 Cricket World Cup in South Africa in 2007 was the first international cricket event to have cheerleaders. The Florida Marlins were the first Major League Baseball team to have cheerleaders. Debuting in 2003, the "Marlin Mermaids" gained national exposure and have influenced other MLB teams to develop their own cheer/dance squads.

Types of teams

School-sponsored

Georgia Tech Yellow Jackets cheerleaders at a college basketball game.

Most American high schools and colleges have organized cheerleading squads made up solely of students. Several colleges that compete at cheerleading competitions offer cheerleading scholarships. Some military academies use their drill team or color guard team instead of a cheersquad at athletic events, but some military academies have traditional cheerleading squads just like other everyday universities. Normally, the main reason for school-sponsored cheerleading is to promote school spirit and motivate the players and fans. They may compete outside of sporting events (local, regional, and national competitions), but their main task is to cheer for sporting events and get the crowd involved. Cheerleading is quickly becoming a year round sport starting with tryouts during the preceding school year, practices, attendance at various sporting events and ending with Competition season from winter through spring.

Youth league

Many organizations that sponsor youth league football or basketball sponsor cheerleading squads as well. Pop Warner organizations are an example of this. The YMCA (Young Men's Christian Association) is also a popular sponsor for youth cheerleading leagues, as well as numerous other sports leagues.

All-Star cheerleading

In the early 1980s, cheerleading squads not associated with a schools or sports leagues, whose main objective was competition, began to emerge. The first organization to call themselves all stars and go to competitions were the Q94 Rockers from Richmond, Virginia, founded in 1982 by Hilda McDaniel.[12] All-star teams competing prior to 1987 were place into the same divisions as teams that represented schools and sports leagues. In 1986 National Cheerleaders Association (NCA) decided to address this situation by creating a separate division for these teams lacking a sponsoring school or athletic association, calling it the 'All-Star Division' and debuting it at their 1987 competitions. As the popularity of these types of teams grew, more and more of them were formed, attending competitions sponsored by many different types of organizations and companies, all using their own set of rules, regulations and divisions. This situation became one of the chief concerns of gym owners. These inconsistencies caused coaches to keep their routines in a constant state of flux, detracting from time that should be utilized to develop skills and provide personal attention to their athletes. More importantly, because the various companies were constantly vying for the competitive edge, safety standards had becoming more and more lax. In some cases, unqualified coaches and inexperienced squads are attempting dangerous stunts as a result of these “expanded” sets of rules.[13]

The USASF was formed in 2003 by these various competition companies to act as the national governing body for all star cheerleading and to create a standard set of rules and judging standards to be followed by all competitions sanctioned by the Federation and ultimately leading to the Cheerleading Worlds. The USASF hosted the first Cheerleading Worlds on Saturday, April 24, 2004.[13] At the same time, cheerleading coaches from all over the country organize themselves for the same rule making purpose, calling themselves the National All Star Cheerleading Coaches Congress (NACCC). In 2005, the NACCC was absorbed by the USASF to become their rule making body.[12] By late 2006, the USASF was ready to expand its reach even further, by facilitating the creation of the International All-Star Federation (IASF), the first international governing body for the sport of cheerleading. [14]

Currently all-star cheerleading as sanctioned by the USASF involves a squad of 6-36 females and/or males. The squad prepares year-round for many different competition appearances, but they only actually perform for up to 2½ minutes during their routines. The numbers of competitions a team participates in varies from team to team, but generally, most teams tend to participate in eight-twelve competitions a year. These competitions include locals, which are normally taken place in school gymnasiums, nationals, hosted in big venues all around the U.S. with national champions, and the Cheerleading Worlds, taken place at Disney World in Orlando, Florida. During a competition routine, a squad performs carefully choreographed stunting, tumbling, jumping and dancing to their own custom music. Teams create their routines to an eight-count system and apply that to the music so the team members execute the elements with precise timing and synchronization.

Judges at the competition watch for illegal moves from the group or any individual member. Here, an illegal move is something that is not allowed in that division due to difficulty and safety restrictions. More generally, judges look at the difficulty and execution of jumps, stunts and tumbling, synchronization, creativity, the sharpness of the motions, showmanship, and overall routine execution.

All-star cheerleaders are placed into divisions, which are grouped based upon age, size of the team, gender of participants, and ability level. The age levels vary from under 4 year of age to 18 years and over. The divisions used by the USASF/IASF are currently Tiny, Mini, Youth, Junior, Junior International, Junior Coed, Senior, Senior coed, Open International and Open.[15]

If a team places high enough at selected USASF/IASF sanctioned national competitions, they could be included in the Cheerleading Worlds and compete against teams from all over the world, as well as receive money for placing.[1]

Cheerleaders

See also: list of cheerleaders

Famous Former Cheerleaders

You must add a |reason= parameter to this Cleanup template – replace it with {{Cleanup|August 2008|reason=<Fill reason here>}}, or remove the Cleanup template.
 Many prominent people have been cheerleaders, including:

Cheerleading in popular culture

Movies and television

Also see List of cheerleaders in fiction

Claire Bennet,Heroes

Cheerleading's increasing popularity in recent decades has made it a prominent feature in high-school themed movies and television shows. The 2000 film Bring It On, about a San Diego high school cheerleading squad called "The Toros", starring real-life former cheerleader Kirsten Dunst. Bring It On was a surprise hit and earned nearly $70 million domestically. It spawned two direct-to-video sequels (Bring It On Again in 2003 and Bring It On: All or Nothing in 2006). The fourth film in the franchise, Bring It On: In It to Win It, was released on December 18, 2007. Bring It On was followed in 2001 by another teen cheerleading comedy, Sugar & Spice. In 1993, The Positively True Adventures of the Alleged Texas Cheerleader-Murdering Mom was an acclaimed TV movie which told the true story of Wanda Holloway, the Texas mother whose obsession with her daughter's cheerleading career made headline news.

In 2006, Hayden Panettiere, star of Bring It On: All or Nothing, took another cheerleading role as Claire Bennet, the cheerleader with an accelerated healing factor on NBC's hit sci-fi TV series Heroes, launching cheerleading back into the limelight of pop culture. Claire was the main focus of the show's first story arc, featuring the popular catchphrase, "Save the cheerleader, save the world." Her prominent, protagonist role in Heroes was supported by a strong fan-base and provided a positive image for high school cheerleading.

In 2006, Cheerleader Nation, was a reality show featured on the channel, Lifetime. Cheerleader Nation is a 60 minute television series based on the Paul Laurence Dunbar High School cheerleading team's ups and downs on the way to nationals, of which they are the three time champions. This show also explains how cheerleading is a tough sport. This show takes place in Lexington, Kentucky. The team is on a quest to win a third national championship.

Video games

Nintendo has released a pair of video games in Japan for the Nintendo DS, Osu! Tatakae! Ouendan and its sequel Moero! Nekketsu Rhythm Damashii that star teams of male cheer squads, or Ouendan that practice a form of cheerleading. Each of the games' most difficult modes replaces the male characters with female cheer squads that dress in western cheerleading uniforms. The games task the cheer squads with assisting people in desperate need of help by cheering them on and giving them the motivation to succeed.

Sport debate and Leadership Issues

Cheerleading among others has had debate on whether or not it truly is a sport. Supporters consider cheerleading as a whole as a sport citing the heavy use of athletic talents [17] while critics do not see it as deserving of that status since sport implies a competition among squads and not all squads compete along with subjectivity of competitions.[18][19]

In the UK, there is less of a debate, as the sports councils recognise cheerleading as a sport, they have however yet to assign a national governing body, there are currently two groups applying for the position: British Cheerleading Association, and British Gymnastics. [20]

Dangers of cheerleading

The risk of cheerleading was highlighted when Kristi Yamaoka, a cheerleader for Southern Illinois University, suffered from a fractured vertebra after she hit her head after falling from a human pyramid.[21] She also suffered from a concussion, and a bruised lung.[22] The fall occurred when Yamaoka lost her balance during a basketball game between Southern Illinois University and Bradley University at the Savvis Center in St. Louis on March 5 2006.[22] The fall gained "national attention"[22] , because Yamoka continued to perform from a stretcher as she was moved away from the game.[22] The cheerleader has since made a full recovery.

The accident caused the Missouri Valley Conference to ban its member schools from allowing cheerleaders to be "launched or tossed and from taking part in formations higher than two levels" for one week during a women's basketball conference tournament, and also resulted in a recommendation by the NCAA that conferences and tournaments do not allow pyramids two and one half levels high or higher, and a stunt known as basket tosses, during the rest of the men's and women's basketball season. [23] On July 11 2006, the bans were made permanent by the AACCA rules committee:

The committee unanimously voted for sweeping revisions to cheerleading safety rules, the most major of which restricts specific upper-level skills during basketball games. Basket tosses, 2½ high pyramids, one-arm stunts, stunts that involve twisting or flipping, and twisting tumbling skills may only be performed during halftime and post-game on a matted surface and are prohibited during game play or time-outs.[24]

However, there have been far worse catastrophes in the world of cheerleading. Last October, at Los Angeles' John Marshall High School, 17 year-old Patty Phommanyvong became a comatose quadriplegic as the result of a fall to the ground. Eighteen year-old Jessica Smith suffered a fractured back and neck after hitting the ground during practice at Sacramento City College in California. Most tragically, 20 year-old Lauren Chang died this April from being kicked in the chest at a cheerleading competition in Worcester, Massachusetts.

Out of the nation's 2.9 million female high school athletes, only 3% are cheerleaders, yet cheerleading accounts for 65% of all catastrophic injuries in girls' high school athletics. Since the NCAA has yet to recognize cheerleading as an official college sport, there are no solid numbers on college cheerleading, yet when it comes to injuries, 67% of female athlete injuries at the college level are due to cheerleading mishaps.

[25]

See also

References

  1. ^ a b c d Campo-Flores, Arian (2007-05-14). "A World of Cheer!". Newsweek. Retrieved 2007-05-17. {{cite journal}}: Check date values in: |date= (help)
  2. ^ Schoenberger, Chana R. (2006-11-16). "The Most Dangerous Sports". Forbes. Retrieved 2007-06-29. {{cite journal}}: Check date values in: |date= (help)
  3. ^ CBS/AP (2006-01-03). "Cheerleading Injuries Increasing". The Early Show. CBS Broadcasting Inc. Retrieved 2007-06-29. {{cite news}}: Check date values in: |date= (help)
  4. ^ "IASF home page". Retrieved 2007-06-29.
  5. ^ Japan Echo, Inc (2005-05-30). "Three Cheers for the Champions!". Web Japan. Retrieved 2007-09-22. {{cite news}}: Check date values in: |date= (help); External link in |work= (help)
  6. ^ a b c Neil, Randy L.; Hart, Elaine (1986), The Official Cheerleader's Handbook (Revised Fireside Edition 1986 ed.), Simon & Schuster, ISBN 0-671-61210-7
  7. ^ a b c d Walker, Marisa (February 2005), "Cheer Milestones", American Cheerleader, 11 (1): 41–43, ISSN 1079-9885{{citation}}: CS1 maint: date and year (link)
  8. ^ a b c Balthaser, Joel D. (2005-01-06). "Cheerleading – Oh How far it has come!". Pop Warner. Retrieved 2007-01-11. {{cite web}}: Check date values in: |date= (help)
  9. ^ "Cheerleading Jump Herkie". Retrieved 2007-08-06.
  10. ^ "Indianapolis Colts - www.football.com". Retrieved 2007-11-04.
  11. ^ "About the AACCA". Retrieved 2007-01-11.
  12. ^ a b Smith, Jennifer Renèe (February 2007), "The All-Star Chronicles", American Cheerleader, 13 (1): 40–42, ISSN 1079-9885{{citation}}: CS1 maint: date and year (link)
  13. ^ a b "The Cheerleading Worlds Administered by the USASF". Varsity Brands, Inc. Retrieved 2007-09-20.
  14. ^ "USASF Insider" (pdf). Retrieved 2007-09-15.
  15. ^ "USASF All-Star Cheer Divisions for 2007-2008" (pdf). Retrieved 2007-06-29.
  16. ^ CNN report with picture of future President Bush as a cheerleader
  17. ^ "The road to recognition". BCA resources. 2008-08-02. Retrieved 2008-08-02. {{cite journal}}: Check date values in: |date= (help)
  18. ^ "Sport, not a sport: consider Dan the expert". The Stanford Daily. 2004-09-29. Retrieved 2007-12-15. {{cite journal}}: Check date values in: |date= (help)
  19. ^ "No, Cheerleading is not a Sport". SFU Cheer Resources. 2004-09-29. Retrieved 2007-12-15. {{cite journal}}: Check date values in: |date= (help)
  20. ^ "The road to recognition". BCA resources. 2008-08-02. Retrieved 2008-08-02. {{cite journal}}: Check date values in: |date= (help)
  21. ^ Cheerbans continue, Athletic Management
  22. ^ a b c d "Cheerleader worried for team, not herself". Retrieved 2007-09-29.
  23. ^ "CHEERLEADING PROGRAMS GOING ALL-OUT FOR SAFETY". Retrieved 2006-12-19.
  24. ^ "CHEERLEADING PROGRAMS GOING ALL-OUT FOR SAFETY". Retrieved 2006-12-19.
  25. ^ "Cheerleading is by far the riskiest sport for girls". Retrieved 2008-09-11.

External links

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