Section 15.1 Motivation: search for better evaluation points
What is the most accurate approximation to \(\int_{-1}^1 f(x)\,dx\) using two points of evaluations? So far we saw the trapezoidal rule, which is \(\int_{-1}^1 f(x)\,dx\approx f(-1)+f(1)\text{.}\) For example, if \(f(x)=e^x\) we have \(\int_{-1}^1 f(x)\,dx = 2.3504\) while \(f(-1)+f(1) = 3.0862\text{,}\) so the rule makes a significant error, overestimating the integral by \(0.7358\text{.}\) The concavity of the function is responsible for this error.
It turns out there is a much more accurate two-point integration rule:
\begin{equation*}
\int_{-1}^1 f(x)\,dx\approx f(-1/\sqrt{3}) + f(1/\sqrt{3})
\end{equation*}
For the exponential function it gives \(2.3427\text{,}\) the error of only 0.0077. This is about 100 times more accurate than the trapezoidal rule, with the same number of function evaluations.
Where does \(1/\sqrt{3}\) come from? It will take some time to develop a general theory leading to this choice, but a quick way to justify it is to consider small powers of \(x\text{:}\)
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If \(f(x)=x^0\text{,}\) then \(\int_{-1}^1 x^0\,dx = 2 = f(x_1)+f(x_2)\) holds for any choice of points \(x_1, x_2\)
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If \(f(x)=x^1\text{,}\) then \(\int_{-1}^1 x^1\,dx = 0 = f(x_1)+f(x_2)\) holds whenever the points are symmetric about 0, meaning that \(x_2=-x_1\text{.}\)
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If \(f(x)=x^2\text{,}\) then \(\int_{-1}^1 x^2\,dx = 2/3\) and \(f(x_1)+f(x_2) = 2x_1^2\text{.}\) The two things are equal when \(x_1=1/\sqrt{3}\text{.}\)
As a bonus, the rule is also accurate for \(f(x)=x^3\text{,}\) by virtue of symmetry, so its order of accuracy is 4.
There is a general method of choosing evaluation points \(x_1,\dots,x_n\) in an optimal way: they will be the roots of some special \(n\)-th degree polynomial. This method is called Gaussian integration and it is built into TI-83 calculators and many software packages, including Matlab (
quadgk command). But before we get to actual integration, we have to study the orthogonal polynomials themselves. The process of using them for integration will be covered in next chapter.
