Simpson,s Rule
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
...
8-12-2021
1935
Simpson's Rule
Simpson's rule is a Newton-Cotes formula for approximating the integral of a function 
using quadratic polynomials (i.e., parabolic arcs instead of the straight line segments used in the trapezoidal rule). Simpson's rule can be derived by integrating a third-order Lagrange interpolating polynomial fit to the function at three equally spaced points. In particular, let the function 
be tabulated at points 
, 
, and 
equally spaced by distance 
, and denote 
. Then Simpson's rule states that
Since it uses quadratic polynomials to approximate functions, Simpson's rule actually gives exact results when approximating integrals of polynomials up to cubic degree.
For example, consider 
(black curve) on the interval ![[0,pi/2]](https://mathworld.wolfram.com/images/equations/SimpsonsRule/Inline15.gif)
, so that 
, 
, and 
. Then Simpson's rule (which corresponds to the area under the blue curve obtained from the third-order interpolating polynomial) gives
whereas the trapezoidal rule (area under the red curve) gives 
and the actual answer is 1.
In exact form,
where the remainder term can be written as
 |
(8)
|
with 
being some value of 
in the interval ![[x_0,x_2]](https://mathworld.wolfram.com/images/equations/SimpsonsRule/Inline37.gif)
.
An extended version of the rule can be written for 
tabulated at 
, 
, ..., 
as
![int_(x_0)^(x_(2n))f(x)dx=1/3h[f_0+4(f_1+f_3+...+f_(2n-1))
+2(f_2+f_4+...+f_(2n-2))+f_(2n)]-R_n,](https://mathworld.wolfram.com/images/equations/SimpsonsRule/NumberedEquation2.gif) |
(9)
|
where the remainder term is
 |
(10)
|
for some ![x^* in [x_0,x_(2n)]](https://mathworld.wolfram.com/images/equations/SimpsonsRule/Inline42.gif)
.
REFERENCES:
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 886, 1972.
Horwitz, A. "A Version of Simpson's Rule for Multiple Integrals." J. Comput. Appl. Math. 134, 1-11, 2001.
Jeffreys, H. and Jeffreys, B. S. Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 286, 1988.
Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, p. 105, 2004. http://www.mathematicaguidebooks.org/.
Whittaker, E. T. and Robinson, G. "The Trapezoidal and Parabolic Rules." The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 156-158, 1967.
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