# Tag Info

25

You can derive all of the integrals $\int_0^1 x^p dx$ by chopping them in half, and rescaling each half to fit in [0,1] again. The proof is by induction on p, and by "recovering the integral back". To give you an idea of how this will go, I will just derive $\int_0^1 x^2 dx$ assuming we already have the results for p=0,1 (which are apparent geometrically ...

19

This may not be in the spirit of what you want, but... by scaling arguments it suffices to establish that $\int_0^1 x^p dx = \frac{1}{p + 1}$. Consider the following probabilistic argument (not entirely rigorous but very suggestive): the integral describes the probability that if you choose $p + 1$ points uniformly at random in the interval $[0, 1]$, then ...

7

To add to Fredrik Johansson's answer: A nice history of algorithms for computing Gauss quadrature rules can be found in this SIAM News article by Alex Townsend. Therein, it is stated that the "final chapter" was written by Ignace Bogaert in this SISC paper, which gives an algorithm that is even faster and more accurate than the algorithm of Hale & ...

7

There are asymptotic methods that essentially give you $N$ nodes and weights in $O(N)$ time if the precision is assumed to be fixed (e.g. at double precision). See Nicholas Hale and Alex Townsend, "Fast and Accurate Computation of Gauss-Legendre and Gauss-Jacobi Quadrature Nodes and Weights", SIAM J. Sci. Comput., 35(2) (a PDF is available at http://eprints....

6

It is a conjecture of Stieltjes, apparently still open, see T.J. Stieltjes, Letter No. 275 of Oct. 2, 1890, in Correspondance d'Hermite et de Stieltjes, vol 2, Gauthier-Villars, Paris, 1905. that Legendre polynomials of different degrees have no common roots, except $x=0$ when both degrees are odd. Laguerre polynomials $L_{n}^{\alpha}$, $\alpha>-1$, ...

5

I assume you mean maximum degree of accuracy for polynomials, that is, you require that the formula is exact for polynomial functions of degree up to $d$, and you look for the maximum possible $d$. This is the most usual requirement, although not necessarily the best one (see for instance http://eprints.maths.ox.ac.uk/1116/1/NA-06-07.pdf). The maximum $d$ ...

4

Here is a very simple proof for nonnegative integer $p$. By elementary combinatorial reasoning, we have $$\sum_{j=0}^{n-1} \binom{j}{p} = \binom{n}{p+1},$$ which is the same as $$\sum_{j=0}^{n-1} j(j-1)\cdots(j-p+1) = \frac{n(n-1)\cdots(n-p)}{p+1}.$$ After scaling that becomes a lower bound for $\int_0^1 x^p dx$. Similarly, $$\sum_{j=0}^{n-1} j(j+1)\... 3 This is a particular implementation of a more general method, described in John Boyd's Why Eigenvalues Are Roots: A Derivation of the One-Dimensional Companion Matrix for General Orthogonal Polynomials (restricted access). Gauss-Legendre quadrature of order n needs the roots of the Legendre polynomial P_n(x) and a numerical root solver must guarantee ... 3 This is theorem 2.1 in On the convergence rates of Legendre approximation (2012) [yes, with a proof in English] 2 Here's another approach. I'm assuming p > 0. For b > 0, let F(b) be the area under y = x^p for x from 0 to b. The transformation (x,y) \to (tx, t^p y) maps the region under the curve for x from 0 to b to the region under the curve from 0 to tb. This scales by t in the x direction and t^p in the y direction; we ... 2 As Federico Poloni said, it is a special case of Gauss-Jacobi quadrature. I am in a good mood, so here is your homework. The formula is$$\int_{-1}^{1}\sqrt{\frac{1-x}{1+x}}f(x)dx= \frac{\pi(5+\sqrt{5})}{10}f\bigg(\frac{-1-\sqrt{5}}{4}\bigg)+\frac{\pi(5-\sqrt{5})}{10}f\bigg(\frac{-1+\sqrt{5}}{4}\bigg)+R(f).

2

Numerical evidence suggests that the nonzero roots of the Legendre polynomials do not repeat. The numerical experiment I performed is simple: I gathered all of the nonzero roots of the first 100 Legendre polynomials. I sorted them in ascending order. I then graphed the first order difference of this sorted vector with logarithmic scaling in the vertical ...

2

In "Error bounds for the Clenshaw-Curtis quadrature formulas" the error bounds are obtained without relying on analytic function theory (and no Fourier transforms).

1

Here, three possible references for the formula: P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Computer Science and Applied Mathematics. Academic Press, New York, 1984 (see p.88) J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, CRC Press, New York, 2003 (see section 8.3.2 for the case of Chebyshev Polynomials) J. ...

1

Computation of the coefficients of the recurrence relations of orthogonal polynomials is studied in details in the standart reference: Gautschi, W., Orthogonal polynomials: computation and approximation. Numerical Mathematics and Scientific Computation. Oxford University Press, New York, 2004 In particular, discretization methods are described in Section ...

1

Gauss quadrature approximations of probability distributions undergird some quite new numerical methods for approximating the solution of SDEs of McKean type; see Section 2 of https://arxiv.org/abs/1608.06741 and references therein. In such SDEs, the coefficients depend on both the current state of the process and the distribution of the solution.

1

You are describing the Golub-Welsch algorithm. As described, it costs O(n^2) operations to compute n-point Gauss-Legendre quadrature rule. http://www.ams.org/journals/mcom/1969-23-106/S0025-5718-69-99647-1/ If you are interested in large Gauss-Legendre quadrature rule, then better (faster and more accurate) algorithms now exist. See your previous question: ...

1

I think it is also worth looking at the case $p=-1$, in the spirit of your context. It also provides a nice way to introduce the logarithm and the exponential function (quite closely to the historical development). Premise: Suppose we have just proved the first elementary facts about the Riemann integral, including integrability of monotonic functions and ...

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