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I want to understand why, according to several sources, Gauss's discovery of Gaussian quadrature in his 1814 article was "the most significant event of the 19th century in the field of numerical integration and perhaps in all of numerical analysis" (this quotation is taken from the article "Construction and applications of Gaussian quadratures with nonclassical and exotic weight functions").The level of my knowledge at numerical analysis is very low, and the most advanced method that i know is Newton-Cotes method for approximate integration by using an $n-1$ degree interpolating polynomial (where $n$ is the number of sample points on the function).

According to several sources, Gauss proved that for an unknown function $f(x)$, it's better (in the sense of the size of remainder) to integrate the function in the interval $(-1,+1)$ by using a weighted sum of it's values at abscisas which are roots of Legendre-polynomials, rather than uniformly spaced abscisaas. Gauss originally proved this by using his method of continued fractions derived from hypergeometric series, and later (1826) Jacobi gave an elegant interpretation using orthogonal polynomials.

Therefore my questions are:

  • Did Gauss prove that for arbitrary function, the best way to numerically integrate it (i.e the difference between the area under the function and the weighted sum will be minimal) is to use Gauss-Legendre quadrature? obviously for some functions this will be a better method while for others this will be less-suitable, so if this statement is true, than it must be true "in average". This seems to me like a very general functional statement, and a very difficult theorem to prove... so do i understand Gauss's statement correctly?
  • If the answer to the previous question is yes, than obviously i understand the historic significance of Gauss's article on numerical integration. If the answer is no, than i'd also like to know what was so significant in Gauss's article?
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Here is the assessment of Trefethen, Is Gauss Quadrature Better than Clenshaw–Curtis?, 2008:

Gauss formulas are defined by choosing the nodes optimally in the sense of maximizing the degree of polynomials that [the Gaussian quadrature formula] integrates exactly. Since there are n + 1 nodes, it is not surprising that the attainable degree is n + 1 orders higher than in the generic case, i.e., degree 2n + 1. Gauss quadrature formulas were discovered by Gauss in his mid-thirties [25], and it is a measure of the extraordinary productivity of Gauss’s career that this great discovery merits hardly a mention in many accounts of his life. For these formulas are truly powerful. Their impact was mainly theoretical before the advent of computers, because of the difficulty of determining the nodes and weights and also of evaluating integrands at irrational arguments, but in the computer era, especially since the appearance of a paper by Golub and Welsch in 1969 (following earlier work of Goertzel in 1954, Wilf in 1962, and Gordon in 1968), it has been practical as well.

Another way of looking at the $n$-point Gauss-Legendre rule is that it integrates the first 2n terms of the Taylor expansion of an analytic function exactly and approximates the rest. Gauss's objective was the highest degree of approximation (as in terms up to $x^{2n-1}$), but it turns out to approximate the remainder the Taylor series of a function analytic in a neighborhood of $[-1,1]$ well and one with an infinite radius of convergence extremely well.

The theoretical importance of Gaussian quadrature received a boost by Jacobi 1826 and others, who connected Gaussian quadrature to orthogonal polynomials, and by the subsequent development of spectral methods.


"...perhaps in all of numerical analysis."

Since the full significance of Gaussian integration arrives in the 1960s by Trefethen's account, I cannot resist mentioning another major development in the 1960s that has origins in the work of Gauss: The FFT (Cooley and Tukey, An algorithm for the machine calculation of complex Fourier series, 1965).

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So, I'm not a historian of math, but I am a recovering numerical linear algebraist. What Gauss proved, or at least that is true, is that the n-point rule is EXACT for a function that happens to be a polynomial of degree up to $2 n - 1$, IIRC. Thus, unless your function has some incredibly high frequency noise, or is outright not differentiable or even continuous, you can usually do just fine IRL with pretty small n. Now you ask, how do you get a PERFECT fit for $2 n - 1$ polynomials with just n points? Sounds impossible given there are $2 n - 1$ constraints but n degrees of freedom (optimizing location of points). Well there are n other DOF, namely the sizes of the weights. So, there are $2 n$ DOF, the same as the number of linearly independent polynomials of degree $\leq 2 n - 1$.

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