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[parent] proof of Simpson's rule (Proof)

We want to derive Simpson's rule for

\begin{displaymath} \int_a^b f(x) \,dx. \end{displaymath}

We will use Newton and Cotes formulas for $n=2$. In this case, $x_0=a$, $x_2=b$ and $x_1 = (a+b)/2$. We use Lagrange's interpolation formula to get a polynomial $p(x)$ such that $p(x_j)=f(x_j)$ for $j=0,1,2$.

The corresponding interpolating polynomial is

\begin{displaymath} p(x)=f(x_1)\frac{(x-x_2)(x-x_3)}{(x_1-x_2)(x_1-x_3)}+ f(x_2)... ...1)(x_2-x_3)} +f(x_3)\frac{(x-x_1)(x-x_2)}{(x_3-x_1)(x_3-x_2)}. \end{displaymath}

and thus
\begin{displaymath} \int_a^b f(x) \,dx\approx \int_a^b f(x_1)\frac{(x-x_2)(x-x_3... ..._2-x_3)} +f(x_3)\frac{(x-x_1)(x-x_2)}{(x_3-x_1)(x_3-x_2)}\,dx. \end{displaymath}

Since integration is linear, we are concerned only with integrating each term in the sum. Now, taking $x_j = a + hj$ where $j=0,1,2$ and $h=\vert b-a\vert/2$, we can rewrite the quotients on the last integral as

\begin{displaymath} \int_a^b p(x)\, dx = hf(x_0)\int_0^2\frac{(t-1)(t-2)}{(0-1)(... ...(1-2)}\,dt + hf(x_2)\int_0^2\frac{(t-0)(t-1)}{(2-0)(2-1)}\,dt. \end{displaymath}

and if we calculate the integrals on the last expression we get
\begin{displaymath} \int_a^b p(x)\,dx=hf(x_0)\frac{1}{3} + hf(x_1)\frac{4}{3}+hf(x_2)\frac{1}{3}, \end{displaymath}

which is Simpson's rule:
\begin{displaymath} \int_a^b f(x)\,dx \approx \frac{h}{3}(f(x_0) + 4f(x_1) + f(x_2)). \end{displaymath}



"proof of Simpson's rule" is owned by drini. [ owner history (1) ]
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Cross-references: expression, calculate, integral, quotients, sum, term, polynomial, Lagrange's Interpolation formula, Newton and Cotes formulas, Simpson's rule
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This is version 1 of proof of Simpson's rule, born on 2004-11-22.
Object id is 6509, canonical name is ProofOfSimpsonsRule.
Accessed 8035 times total.

Classification:
AMS MSC: 65D32 (Numerical analysis :: Numerical approximation and computational geometry :: Quadrature and cubature formulas)
 41A55 (Approximations and expansions :: Approximate quadratures)
 26A06 (Real functions :: Functions of one variable :: One-variable calculus)
 28-00 (Measure and integration :: General reference works )
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