6.2: Composite Rules

• Jeffrey R. Chasnov
• Hong Kong University of Science and Technology

We now use our elementary formulas obtained for (6.2) to perform the integral given by \((6.1)\)

6.2.1. Trapezoidal rule

6.2.2. Simpson’s rule

We here consider the composite Simpson’s rule for evenly space points. We apply Simpson’s rule over intervals of \(2 h\) , starting from \(a\) and ending at \(b\) : \[\begin \int_^ f(x) d x=\frac\left(f_+4 f_+f_\right)+\frac\left(f_+4 f_+f_\right)+& \ldots \\ &+\frac\left(f_+4 f_+f_\right) \end \nonumber \] Note that \(n\) must be even for this scheme to work. Combining terms, we have \[\int_^ f(x) d x=\frac\left(f_+4 f_+2 f_+4 f_+2 f_+\cdots+4 f_+f_\right) \nonumber \] The first and last terms have a multiple of one; the even indexed terms have a multiple of \(2 ;\) the odd indexed terms have a multiple of \(4 ;\) and the entire sum is multiplied by \(h / 3\) .

This page titled 6.2: Composite Rules is shared under a CC BY 3.0 license and was authored, remixed, and/or curated by Jeffrey R. Chasnov via source content that was edited to the style and standards of the LibreTexts platform.

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2. • 6.1: Elementary Formulas
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   1. source@https://www.math.hkust.edu.hk/~machas/numerical-methods.pdf
   2. Trapezoidal Rule