A related problem to interpolation is when we need to calculate the integral of a function
$$I = \int_{x_1}^{x_N} f(x) dx$$
given values of the function at a discrete set of points $x_1\dots x_N$, which we'll write as $f_i\equiv f(x_i)$. For simplicity here we'll assume that the spacing between points on the grid is constant $x_{i+1}-x_i = \Delta x$, but it's straighforward to generalize to non-uniform sampling if you need to.
To estimate the value of the integral, we need to model how the function behaves in each interval $x_i<x<x_{i+1}$.Aswithinterpolation,wecantake$f(x)$ineachintervaltobeapolynomial,includingadditionaltermsinthepolynomialtoincreasetheaccuracyofourapproximation.
**Rectangle rule**
The first approximation is to assume that $f(x)$ is a constant $f(x)=f_i$ in the interval $x_i\leq x\leq x_{i+1}$. The total integral is then
$$\int_{x_i}^{x_{i+1}} f(x) dx = \int_0^{\Delta x} f(x_i+y) dy \approx f_i \Delta x + (f_{i+1}-f_i){\Delta x\over 2} = {f_i+f_{i+1}\over 2}\Delta x.$$
This is known as the *trapezoidal rule* because it corresponds to the area of the trapezoid formed by connecting the points $(x_i,f_i)$ and $(x_{i+1}, f_{i+1})$ by a straight line.
When we sum over all intervals, each point gets counted twice except for the left and right boundaries, so
The reason for considering the double interval from $x_{i-1}$ to $x_{i+1}$ is that the linear term is odd in this interval (antisymmetric about $x_i$) and so does not contribute to the integral, i.e.
What kind of functions are integrated exactly (to machine precision) for each method? (Hint: polynomials of a certain order, which order?)
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## Gaussian quadrature
In the previous examples, we were given the function $f(x)$ evaluated at a pre-determined set of points $\{ x_i\}$. But if instead we are able to choose the values of $x$ where we can evaluate the function, we can do better using the method of *Gaussian quadratures*.
The simplest example is the integral
$$\int^{1}_{-1} f(x) dx$$
(which is quite general because for different integration limits you can make a change of variables to bring the limits to $-1$ and $+1$). We write the integral as
For a suitable choice of the weights $w_i$ and the locations $x_i$ it can be shown that this expression is exact when $f(x)$ is a polynomial of order $2N-1$ or less.
```{admonition} Question
This means that with 2 evaluations of $f(x)$ we can exactly evaluate the integral of a cubic polynomial! How does this compare with Simpson's method?
The proof that this works and the calculation of the values of $w_i$ and $x_i$ is quite involved. [It uses the theory of orthogonal polynomials, so for example for this particular form of the integral, the values of $x_i$ are the roots of the Legendre Polynomial $P_N(x)$.] However, we can look up $w_i$ and $x_i$ using [`numpy.polynomial.legendre.leggauss`](https://numpy.org/doc/stable/reference/generated/numpy.polynomial.legendre.leggauss.html)
to get the locations and weights for different choices of $N$, implement Gaussian quadratures.
Check that the answer is indeed exact for polynomials of degree $2N-1$ or less. What happens for higher order polynomials?
Try a function that is not a simple polynomial, e.g. $e^{-x^2}$. What is the error in the approximation and how does it scale with $N$?
Hints:
- to get the correct value of the integral to compare with to see how accurate your approximation is, you could use the general purpose integrator [`scipy.integrate.quad`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.quad.html)
- to generate a polynomial with degree `N` and random coefficients between -10 and +10 (for example) you can use
In this case, the weights and locations are given by [`numpy.polynomial.hermite.hermgauss`](https://numpy.org/doc/stable/reference/generated/numpy.polynomial.hermite.hermgauss.html). [The locations $x_i$ are the roots of the $N$th Hermite polynomial, which are the polynomials that are orthogonal under an inner product defined with weight function $W(x)$.]
Modify your code to use the Gauss-Hermite coefficients and check that you can get an exact answer for the integral of $e^{-x^2}$.
Hint: If you want to use `scipy.integrate.quad` again to get the value of the integral as a comparison, note that you can give it limits of $-\infty$ to $+\infty$ using `-np.inf` and `np.inf`.
- $W(x)=e^{-x}$ with integration limits $0$ to $\infty$. In this case, we need Gauss-Laguerre integration -- see [`numpy.polynomial.laguerre.laggauss`](https://numpy.org/doc/stable/reference/generated/numpy.polynomial.laguerre.laggauss.html)
- $W(x)=1/\sqrt{1-x^2}$ from $x=-1$ to $1$ is Gauss-Chebyshev quadrature -- see
Use Simpson's rule, Gaussian quadrature, and the general purpose integrator [`scipy.integrate.quad`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.quad.html) to evaluate the average velocity $\langle\left|v\right|\rangle$ for the 3D [Maxwell-Boltzmann distribution](https://en.wikipedia.org/wiki/Maxwell–Boltzmann_distribution).
For Simpson's rule you can use your own implementation from above or you could try [`scipy.integrate.simpson`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.simpson.html).
