Difference between Quadrature Rules and Riemann Sum

Question

I am reading and learning about Quadrature rules. Would it be correct to say that a Riemann Sum method enters in the category of quadrature rules? I am asking because the Riemann sum evaluates the "area under the curve by a series of small rectangles" and in the with quadrature rules you seem to be able to do do the same thing? I find strange the wikipedia page on numerical integration doesn't say more about the Riemann sum.

Thank you.

Answer 1

You're right that if you just 'truncate' (use a partial sum rather than the infinite sum) then the Riemann Sum is a 'numerical Quadrature'. However, this is also how integration is defined many times so it isn't reiterated at that point. Those rules from the Wikipedia page are just a more sophisticated way to do the numerical integration. There are many many ways. See for example the Euler Maclaurin_formula and it's use for numerical integration.

A method favoured by Nick Trefethen is to approximate the function with chebyshev polynomials (by doing a discrete cosine transform), and then doing standard Gaussian Quadrature The chebyshev polynomials have many nice properties other than being orthogonal. For example they don't suffer from Runge's phenomenon. Have a look Trefethen's expositions on these topics - it is all very readable.

tl;dr

If your function is just a sequence of boxes this is a great idea, otherwise there are better methods.

Answer 2

By Riemann sum methods, I suspect you mean estimating the area by a finite number of rectangles. This is a sort of quadrature, although it is perhaps the 'trivial' sort of quadrature. Good quadratures, such as Gaussian Quadrature have great error estimates because they involve clever weighting schemes. An $n$-term Gaussian quadrature has zero error on degree $2n-1$ polynomials, for example. But $n$-term Riemann sums have zero error on linear functions, and have error on everything else in general.

If you want to know more about using Riemann sums to approximate integrals, you should look into the rectangle method. Really, though, this is equivalent to the 'midpoint approximation' method that many people learn in their first semester of calculus. This is the first of the Newton-Cotes formulas (first meaning the worst). The second is Simpson's Rule - something else that many students learn in first semester calculus.

The wiki article on numerical integration mentions both the rectangle method and Newton-Cotes, and those pages are perfectly adequate. But it doesn't give them more credence because they are comparatively worse and harder to calculate.