Simpson's Rule
Simpson's Rule is a numerical method used to approximate the integral, or the area under a curve. It utilizes a specific formula that combines the values of the function at selected points to estimate this area. The rule can be expressed mathematically as \( \frac{1}{3} h (Y_D + 4Y_E + Y_F) \), where \( h \) is the width of the intervals and \( Y_D, Y_E, Y_F \) are the function values at specific points. This method is based on the principle that a quadratic curve can be fitted through three points, allowing for a more accurate approximation compared to simpler methods like rectangles or trapezoids.
Thomas Simpson, an English mathematician who lived in the 18th century, is credited with popularizing this rule, although its foundational concepts trace back to earlier mathematicians such as Johannes Kepler. Simpson's Rule is particularly noted for its precision when applied to functions that are smooth and continuous, though it requires that the number of intervals used be even for optimal results. Overall, Simpson's Rule is a valuable tool in numerical analysis, providing a structured approach to integrating functions when analytical solutions are difficult to obtain.
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Simpson's Rule
Simpson’s rule is a numerical method of integration achieved by calculating the area under the curve. The rule states that 1/3h(sum of end ordinates + 4 × sum of odd ordinates + 2 × sum of remaining even ordinates).
Overview
Thomas Simpson was born August 20, 1710, and died May 14, 1761. Simpson’s rule developed from work done by himself, in collaboration with Roger Cotes and Sir Isaac Newton. It is believed, however, that this rule was established about 100 years earlier by Johannes Kepler.
It is always possible to fit a quadratic curve to three points. See Figure 1. Here points D, E and F define a curve with area delimited by +/– h.
Use y = ax2 + bx + c (the formula for a quadratic) and use simultaneous equations to find expressions for a, b and c. Put each point’s coordinates into this quadratic expression in turn.
Replace c by YE(from (2)) into equation (1)
Replace c by YE in (3)
Add (4) and (5): YD + YF = 2ah2 + 2YE and so a = (YD - 2YE + YF) / 2h2
The shaded area under the curve is given by the integral of ax2 + bx + c between h and –h.
Substituting the expressions for a and c (see above) the shaded area is
This expression for the area simplifies to 1/3h(YD + 4YE + YF).
The shaded shape could be translated any distance along the x-direction without the area changing. The formula applies for any pair of strips, each of width h, with the given y-ordinates. By convention, the y-ordinates are identified by numbers, so the formula becomes 1/3h(y0 + 4y1 + y2).
In Figure 2 a given area (shaded in grey) has been divided into three pairs of strips. The formula derived above is applied separately to each pair (so a separate quadratic is fitted to each of the three sections.
It is apparent that the total area here is
,
which simplifies to
There are alternative methods to finding the area, such as using trapeziums or rectangles. Simpson’s rule, however, is found to be extrememly accurate. Note that an even number of strips of equal width are to be used.
Bibliography
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