Continuity correction: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
en>Sean a wallis
→‎See also: Add Wilson cc ref
en>Brwilso2
m Paragraph 2 included an equation for the variance of the binomial distribution. The variance was improperly described a s n(1-p) I amended it to read np(1-p) as it should be.
 
Line 1: Line 1:
{{
Winter season, a assortment of boots shuttle on the streets. And a pair of lovely round circle of the mind as a member snow boots can often grab the consideration of every person to do. Straightforward design is not only fancy, but also heat and at ease, common both. As a result, the comfortable winter season manner snow boots come to be the initial preference of men and women out and turn into the most well known solitary product when the stars are putting on, a short while ago found that lots of woman and boys have been all in the snow boots, UGG wearing with several kinds of clothing and trousers are extremely specific, you can put on UGG garments with range leisure, wonderful sort, star emotion. In simple fact, no make any difference how to don UGG snow boots both equally have a distinctive attraction, so we can be crazy about emotion to wear outfits UGG snow boots. Predominantly because of to use seriously at ease to put on UGG stroll, it can extend and agreement your leg muscle tissues, but would not drop above. Result in it was complicated to have on significant heels, but not for UGG snow boots. Also, bending your toes have to have to grip pressure, which can training the front thigh muscular tissues and calf. UGG snow boots are now almost to the level where by nine out of 10 is putting on. Hollywood well-liked stars in a number of a long time ago had distribute wearing UGG , and as the enhancement of people's life typical, a surge of funs to stick to the star to wear UGG. The fashion is have an enduring level of popularity. <br><br>UGG boots monopoly penetration in the star is rather superior, abroad Hilary Duff, Ashley Simpson, Jessica Simpson, Nicole Richie, Meghan Fox, Brittany SnowParis Hilton, Lauren Conrad, China Luo, Vivian Hsu, and so, they applied the sheepskin boots with T-shirt, denims, shorts or small skirts. Kirsten Dunst's totally free and effortless type snow boots generous, with shorts, youth movement, there is an alternate design and style of Eskimo boots style. Kate Moss snug and pure fashion that will bring the emotion of independence for the pursuit of self-launched girl. Mischa Barton's UGG, a tiny identify race the wind, with the flowers. <br><br><br>Chilly will come, popular stars like wearing UGG. Putting on tight denims, you can filling tall, slender display tendencies. Primarily to everyday enjoy, appeared in substantial spirits, the influx of winter season gown. If the morning up late and can blend with pajamas. Star is almost nothing but this, though there are a good deal of huge apparel end, several instances we do not see that a single product may perhaps not be very eye-catching, but it is a fantastic wild stock. UGG footwear said to be the youthful surfer from Australia, Brian produced, and introduced California to boost, what we commonly UGG AUSTRALIA, most of them are created in china, and high high quality workmanship is beautiful. In  can be ordered. <br><br>In point, in accordance with the UGG origin, that was Australia's most pure, like jumbo UGG and shearers UGG Australia are produced by, jumbo UGG model does not however have entered the domestic current market, only purchasing, but this is definitely a hazard, a assertion claimed "Browsing at possibility, the buy require to be cautious", just after all, expend so a lot dollars to obtain footwear is not simple. The shearers UGG in  have sold, and children styles, type is very massive, low-priced, also licensed, but also provides authentic protection or realistic to acquire a lot more. There are quite a few fake UGG ranges: the worst variety, chamois leather outdoors, suede, pretend fur within. Not undesirable one particular, exterior chamois leather, suede wool insied, which is correct, but the wool was pasted. Marginally greater, chamois leather and wool are integrative, but not tender shoes, due to the fact the essence of wool is terrible. The finest of Australia's indigenous Sheepskin , quite soft and heat. At last, say, distinguished for their very own ft, assistance the comfortable and warm UGG!<br><br>If you loved this write-up and you would like to obtain far more details pertaining to [http://tinyurl.com/k7shbtq uggs on sale] kindly stop by our website.
About|the quadrature rule for approximating integrals|the implicit trapezoidal rule for solving initial value problems|Trapezoidal rule (differential equations)|the explicit trapezoidal rule for solving initial value problems|Heun's method}}
[[Image:trapezoidal rule illustration.png|right|thumb|The function ''f''(''x'') (in blue) is approximated by a linear function (in red).]]
In [[numerical analysis]], the '''trapezoidal rule''' (also known as the '''trapezoid rule''' or '''trapezium rule''') is a technique for approximating the [[integral|definite integral]]
 
:<math> \int_{a}^{b} f(x)\,dx.</math>
 
The trapezoidal rule works by approximating the region under the graph of the function
<math>f(x)</math> as a [[trapezoid]] and calculating its area. It follows that
 
:<math> \int_{a}^{b} f(x)\, dx \approx (b-a) \left[\frac{f(a) + f(b)}{2} \right].</math>
 
== Applicability and alternatives ==
The trapezoidal rule is one of a family of formulas for [[numerical integration]] called [[Newton–Cotes formulas]], of which the [[midpoint rule]] is similar to the trapezoid rule. [[Simpson's rule]] is another member of the same family, and in general has faster convergence than the trapezoidal rule for functions which are twice continuously differentiable, though not in all specific cases. However for various classes of rougher functions (ones with weaker smoothness conditions), the trapezoidal rule has faster convergence in general than Simpson's rule.<ref name="cun02">{{Harv|Cruz-Uribe|Neugebauer|2002}}</ref>
 
Moreover, the trapezoidal rule tends to become extremely accurate when [[periodic function]]s are integrated over their periods, which can be [[#Periodic functions|analyzed in various ways]].<ref name="rs90">{{Harv|Rahman|Schmeisser|1990}}</ref><ref name="w02">{{Harv|Weideman|2002}}</ref>
 
For non-periodic functions, however, methods with unequally spaced points such as [[Gaussian quadrature]] and [[Clenshaw–Curtis quadrature]] are generally far more accurate; Clenshaw–Curtis quadrature can be viewed as a change of variables to express arbitrary integrals in terms of periodic integrals, at which point the trapezoidal rule can be applied accurately.
 
== Numerical implementation ==
 
=== Uniform grid ===
 
For a domain discretized into ''N'' equally spaced panels, or ''N''+1 grid points ''a'' = ''x''<sub>1</sub> < ''x''<sub>2</sub> < ... < ''x''<sub>N+1</sub> = ''b'', where the grid spacing is ''h''=(''b''-''a'')/''N'', the approximation to the integral becomes
 
:<math> \int_{a}^{b} f(x)\, dx \approx \frac{h}{2} \sum_{k=1}^{N} \left( f(x_{k+1}) + f(x_{k}) \right)</math> <math>{}= \frac{b-a}{2N}(f(x_1) + 2f(x_2) + 2f(x_3) + \ldots + 2f(x_N) + f(x_{N+1})).</math>
 
=== Non-uniform grid ===
When the grid spacing is non-uniform, one can use the formula
 
:<math> \int_{a}^{b} f(x)\, dx \approx \frac{1}{2} \sum_{k=1}^{N} \left( x_{k+1} - x_{k} \right) \left( f(x_{k+1}) + f(x_{k}) \right).</math>
 
==Error analysis==
 
The error of the composite trapezoidal rule is the difference between the value of the integral and the numerical result:
 
:<math> \text{error} = \int_a^b f(x)\,dx - \frac{b-a}{N} \left[ {f(a) + f(b) \over 2} + \sum_{k=1}^{N-1} f \left( a+k \frac{b-a}{N} \right) \right]</math>
 
There exists a number ''ξ'' between ''a'' and ''b'', such that<ref>{{harvtxt|Atkinson|1989|loc=equation (5.1.7)}}</ref>
 
:<math> \text{error} = -\frac{(b-a)^3}{12N^2} f''(\xi)</math>
 
It follows that if the integrand is [[concave up]] (and thus has a positive second derivative), then the error is negative and the trapezoidal rule overestimates the true value. This can also be seen from the geometric picture: the trapezoids include all of the area under the curve and extend over it. Similarly, a [[concave-down]] function yields an underestimate because area is unaccounted for under the curve, but none is counted above. If the interval of the integral being approximated includes an inflection point, the error is harder to identify.
 
In general, three techniques are used in the analysis of error:<ref name="w0223">{{Harv|Weideman|2002|loc=p. 23, section 2}}</ref>
# [[Fourier series]]
#[[Residue calculus]]
#[[Euler–Maclaurin summation formula]]:<ref>{{harvtxt|Atkinson|1989|loc=equation (5.1.9)}}</ref><ref>{{harvtxt|Atkinson|1989|loc=p. 285}}</ref>
An asymptotic error estimate for ''N'' → ∞ is given by
:<math> \text{error} = -\frac{(b-a)^2}{12N^2} \big[ f'(b)-f'(a) \big] + O(N^{-3}). </math>
Further terms in this error estimate are given by the Euler–Maclaurin summation formula.
 
It is argued that the speed of convergence of the trapezoidal rule reflects and can be used as a definition of classes of smoothness of the functions.<ref name="rs90" />
 
=== Periodic functions ===
The trapezoidal rule often converges very quickly for periodic functions.<ref name="w02" /> This can be explained intuitively as:
:When the function is periodic and one integrates over one full period, there are about as many sections of the graph that are concave up as concave down, so the errors cancel.<ref name="w0223" />
More detailed analysis can be found in.<ref name="rs90" /><ref name="w02" />
 
=== "Rough" functions ===
{{Expand section|date=January 2010}}
For various classes of functions that are not twice-differentiable, the trapezoidal rule has sharper bounds than [[Simpson's rule]].<ref name="cun02" />
 
==See also==
* [[Gaussian quadrature]]
* [[Newton–Cotes formulas]]
* [[Rectangle method]]
* [[Romberg's method]]
* [[Simpson's rule]]
 
==Notes==
<references/>
 
==References==
{{wikibooks|A-level Mathematics|C2/Integration#Trapezium Rule|Trapezium Rule}}
{{refbegin}}
* {{Citation | last1=Atkinson | first1=Kendall E. | title=An Introduction to Numerical Analysis | publisher=[[John Wiley & Sons]] | location=New York | edition=2nd | isbn=978-0-471-50023-0 | year=1989}}.
* {{ Citation | first1 = Qazi I. | last1 = Rahman | first2 = Gerhard | last2 = Schmeisser | title = Characterization of the speed of convergence of the trapezoidal rule | journal = Numerische Mathematik | issn = 0945-3245 | volume = 57 | issue = 1 |date=December 1990 | doi = 10.1007/BF01386402 | pages = 123–138 }}
* {{citation | first=Richard L. | last=Burden | coauthors=J. Douglas Faires | year=2000 | title=Numerical Analysis | edition=7th | publisher=Brooks/Cole | isbn=0-534-38216-9}}.
* {{ Citation | title = Numerical Integration of Periodic Functions: A Few Examples | first = J. A. C. | last = Weideman | journal = [[The American Mathematical Monthly]] | volume = 109 | issue = 1 |date=January 2002 | pages = 21–36 | doi = 10.2307/2695765 | jstor = 2695765 }}
* {{ Citation | journal = [http://jipam.vu.edu.au/ Journal of Inequalities in Pure and Applied Mathematics] | volume = 3 | issue = 4 | year = 2002 | title = Sharp Error Bounds for the Trapezoidal Rule and Simpson's Rule | first1 = D. | last1 = Cruz-Uribe | first2 = C.J. | last2 = Neugebauer | url = http://www.emis.de/journals/JIPAM/images/031_02_JIPAM/031_02.pdf }}
{{refend}}
 
==External links==
*[http://math.fullerton.edu/mathews/n2003/TrapezoidalRuleMod.html Trapezoidal Rule for Numerical Integration]
*[http://dedekind.mit.edu/~stevenj/trapezoidal.pdf Notes on the convergence of trapezoidal-rule quadrature]
 
{{DEFAULTSORT:Trapezoidal Rule}}
[[Category:Numerical integration (quadrature)]]

Latest revision as of 18:10, 22 April 2014

Winter season, a assortment of boots shuttle on the streets. And a pair of lovely round circle of the mind as a member snow boots can often grab the consideration of every person to do. Straightforward design is not only fancy, but also heat and at ease, common both. As a result, the comfortable winter season manner snow boots come to be the initial preference of men and women out and turn into the most well known solitary product when the stars are putting on, a short while ago found that lots of woman and boys have been all in the snow boots, UGG wearing with several kinds of clothing and trousers are extremely specific, you can put on UGG garments with range leisure, wonderful sort, star emotion. In simple fact, no make any difference how to don UGG snow boots both equally have a distinctive attraction, so we can be crazy about emotion to wear outfits UGG snow boots. Predominantly because of to use seriously at ease to put on UGG stroll, it can extend and agreement your leg muscle tissues, but would not drop above. Result in it was complicated to have on significant heels, but not for UGG snow boots. Also, bending your toes have to have to grip pressure, which can training the front thigh muscular tissues and calf. UGG snow boots are now almost to the level where by nine out of 10 is putting on. Hollywood well-liked stars in a number of a long time ago had distribute wearing UGG , and as the enhancement of people's life typical, a surge of funs to stick to the star to wear UGG. The fashion is have an enduring level of popularity.

UGG boots monopoly penetration in the star is rather superior, abroad Hilary Duff, Ashley Simpson, Jessica Simpson, Nicole Richie, Meghan Fox, Brittany SnowParis Hilton, Lauren Conrad, China Luo, Vivian Hsu, and so, they applied the sheepskin boots with T-shirt, denims, shorts or small skirts. Kirsten Dunst's totally free and effortless type snow boots generous, with shorts, youth movement, there is an alternate design and style of Eskimo boots style. Kate Moss snug and pure fashion that will bring the emotion of independence for the pursuit of self-launched girl. Mischa Barton's UGG, a tiny identify race the wind, with the flowers.


Chilly will come, popular stars like wearing UGG. Putting on tight denims, you can filling tall, slender display tendencies. Primarily to everyday enjoy, appeared in substantial spirits, the influx of winter season gown. If the morning up late and can blend with pajamas. Star is almost nothing but this, though there are a good deal of huge apparel end, several instances we do not see that a single product may perhaps not be very eye-catching, but it is a fantastic wild stock. UGG footwear said to be the youthful surfer from Australia, Brian produced, and introduced California to boost, what we commonly UGG AUSTRALIA, most of them are created in china, and high high quality workmanship is beautiful. In can be ordered.

In point, in accordance with the UGG origin, that was Australia's most pure, like jumbo UGG and shearers UGG Australia are produced by, jumbo UGG model does not however have entered the domestic current market, only purchasing, but this is definitely a hazard, a assertion claimed "Browsing at possibility, the buy require to be cautious", just after all, expend so a lot dollars to obtain footwear is not simple. The shearers UGG in have sold, and children styles, type is very massive, low-priced, also licensed, but also provides authentic protection or realistic to acquire a lot more. There are quite a few fake UGG ranges: the worst variety, chamois leather outdoors, suede, pretend fur within. Not undesirable one particular, exterior chamois leather, suede wool insied, which is correct, but the wool was pasted. Marginally greater, chamois leather and wool are integrative, but not tender shoes, due to the fact the essence of wool is terrible. The finest of Australia's indigenous Sheepskin , quite soft and heat. At last, say, distinguished for their very own ft, assistance the comfortable and warm UGG!

If you loved this write-up and you would like to obtain far more details pertaining to uggs on sale kindly stop by our website.

Retrieved from "https://en.formulasearchengine.com/index.php?title=Continuity_correction&oldid=232214"