60
votes
Recent observation of gravitational waves
Short version: LIGO matches their data onto waveforms calculated in numerical relativity. The mathematical study of black hole solutions plays a significant role in this; we couldn't trust our ...
- 5,879
40
votes
Accepted
How does Mathematica do symbolic integration?
An overview by one of the developers of Mathematica, focusing on definite integrals, is at Symbolic definite integration: methods and open issues.
Mathematica knows all the entries in Gradshteyn-...
- 178k
33
votes
Should computer code be included within publications that present numerical results?
My answer is:
Don't put code in your paper. Do: put pseudocode in your paper, version control your code on Github, and add a link to your Github repository to your paper.
The purpose of a paper ...
27
votes
Accepted
show that $ \frac{\Gamma(\frac{1}{24})\Gamma(\frac{11}{24})}{\Gamma(\frac{5}{24})\Gamma(\frac{7}{24})} = \sqrt{3}\cdot \sqrt{2 + \sqrt{3}} $
This formula can actually be proved using only properties of the Gamma
function already known to Gauss, with no need to invoke special values of
Dirichlet series. The relevant identities are
$$
\...
- 77.5k
25
votes
How does Mathematica do symbolic integration?
Maple uses the Risch algorithm; see Keith Geddes and George Lebahn, Symbolic and numeric integration in Maple
- 25.6k
23
votes
Recent observation of gravitational waves
The numbers 36,29,62 were obtained as the best match between the received signals and the output of computer simulations. The 90% confidence intervals on these numbers are about $\pm 4$. The details (...
- 37.2k
23
votes
Accepted
Did human computers use floating-point arithmetics?
In the field of hydrodynamics the first calculation by a human computer was carried out around 1920 for a project to transform an open sea into a closed lake, with the aim to protect Holland from ...
- 178k
21
votes
Accepted
Numerical integration using interval arithmetic, nowadays
I develop Arb, an arbitrary-precision interval library with special functions support. There is some code included for integration of complex analytic functions using Taylor series (documentation), ...
- 2,205
17
votes
Approximation of sum of the first binomial coefficients for fixed N
One of the more convenient and popular approximations of the sum is
$$\frac{2^{nH(\frac{k}{n})}}{\sqrt{8k(1-\frac{k}{n})}} \leq \sum_{i=0}^k\binom{n}{i} \leq 2^{nH(\frac{k}{n})}$$
for $0< k < \...
- 3,662
17
votes
How does Mathematica do symbolic integration?
People usually mention the Risch algorithm first, as other answers have.
Another approach, which is surprisingly successful, is to do what you or I would when solving integrals: look for patterns for ...
- 891
16
votes
Should computer code be included within publications that present numerical results?
At least in my field (numerical linear algebra), the current standard is that including the full source code is not mandatory for a publication. That said, there are many reasons why sharing your code ...
16
votes
Methods of solving linear system of equations, how to select the appropriate method
Disclaimer 1: Treating these topics properly would require a quick course in numerical analysis.
Disclaimer 2: If you are using any sane computer system, it's already going to have a library function ...
- 19.4k
14
votes
The unreasonable effectiveness of Padé approximation
The function used in the example has an asymptotic value of $1/2$ as $x\to \infty$
A Maclaurin expansion will only either go to infinity or negative infinity as x goes to infinity. In order to match ...
- 241
14
votes
Accepted
How to compute $\sum_{x \in \mathbb{Z}^n} e^{-x^TMx}$ efficiently
You are trying to compute a multi-dimensional theta function, and this question is studied in depth in this 2003 Math. Comp. article by Deconinck, Heil, Bobenko, van Hoeij,and Schmies.
- 95.6k
14
votes
Accepted
Is it possible to prove unboundedness of 3rd order ODE?
Actually, no matter what $A$ is, there will be nonzero solutions that will converge to zero, so you can't prove unboundedness, even though it may be true that the 'generic' solution is unbounded. ...
- 106k
13
votes
show that $ \frac{\Gamma(\frac{1}{24})\Gamma(\frac{11}{24})}{\Gamma(\frac{5}{24})\Gamma(\frac{7}{24})} = \sqrt{3}\cdot \sqrt{2 + \sqrt{3}} $
After reading the original paper of Chowla-Selberg, I think it is a problem which can be solved with Kummer's Fourier Expansion of Gamma Function
$$\log\Gamma(x)=(\frac{1}{2}-x)(\gamma+\log2)+(1-x)\...
- 3,317
13
votes
Approximation of sum of the first binomial coefficients for fixed N
A well-known upper bound, for $k\le N/2$, is
$$ \sum_{i=0}^k {N\choose i} \le 2^{N H(k/N)},$$
where $H$ is the binary entropy function
$$ H(x) = -x\log_2(x)-(1-x)\log_2(1-x).$$
This bound was ...
- 6,061
13
votes
Should computer code be included within publications that present numerical results?
I think that Federico Poloni's answer gives good advice as of 2018, but as a mathematical community I think we should be thinking harder about this question. Simply making source code available, even ...
13
votes
Accepted
Is there a systematic theory for Gibbs measures (better if on Hilbert spaces)?
Any probability measure $\mu_1$ absolutely continuous with respect to $\mu_1$ can be written as a Gibbs measure if you allow $G$ to take values $\pm \infty$. If the density is bounded above and below,...
- 53.6k
13
votes
Accelerating convergence for some double sums
With Mathematica I can first sum the series over $\ell$ to get a closed-form expression in terms of polygamma functions,
$$Z(2,2)= \sum_{k,\ell \geq 0} \frac{2k+3}{\binom{k+2}{2}^2} \frac{2\ell+3}{(\...
- 178k
13
votes
Mathematics of sustainable development and energy sobriety in the classroom
My former colleague David Mond at Warwick has developed some materials on this issue. He has some talks (at school and u/g level) on Climate change and game theory. He also lists some links there, to ...
13
votes
Accepted
On a fast high precision numerical analysis C library
Since you speak about mathematical proofs, probably you don't want an arbitrary-precision library, but a verified computation library based on interval arithmetic.
Maybe Arb? Or boost-interval?
And ...
- 19.4k
12
votes
Finding all roots of a polynomial
Edit - Big News: Below I give an algorithm and show that it works, in a specified sense. I acknowledge that it does not work perfectly, in a sense. We have fix for that - a bit of pre-calculation and ...
- 223
12
votes
Accepted
Evaluating elliptic integrals
This question has been out for a while, and I think it deserves a thorough answer, even tho I came quite late to this party.
As both the classical Legendre-Jacobi theory and the Carlson theory have ...
12
votes
Accepted
Resultant of linear combinations of Chebyshev polynomials of the second kind
Since $U_n + t U_{n-1}$ is of degree $n$ and $\sum_{k=0}^{n-1} U_k$ is of degree $n-1$ with leading coefficient $2^{n-1}$, the resultant factors as
$$ 2^{n(n-1)} (-1)^{n(n-1)} \prod_{j=1}^{n-1} (U_n(...
- 109k
12
votes
Accepted
Guaranteed correct digits of elementary expressions
There is a certain confusion in the answers, so let me try to dispel this confusion.
There are two different issues here. One is “computing an approximation with arbitrary precision” and one is “...
- 30.2k
11
votes
"Wild" solutions of the heat equation: how to graph them?
I know this is a late answer, but I think that the other answers posted so far completely fail to illustrate the wildness of the function in question, since they plot it on a domain where not much ...
- 511
11
votes
Accepted
A centralised website for computational attempts in graph theory and metric geometry?
There are a few websites with lists and/or databases of graphs, maps and polytopes.
House of graphs has a searchable database of interesting graphs and aims to serve as a repository for lists of ...
- 126
11
votes
Accepted
Is there a standard name for (non-square) matrices with orthonormal columns?
Orthonormal $\boldsymbol n$-frames : https://en.wikipedia.org/wiki/Stiefel_manifold.
Added: This terminology of Hirzebruch (1966), Steenrod (1951) translates the $\boldsymbol n$-Systeme of Stiefel (...
- 29.5k
11
votes
Accepted
“Taylor series” is to “Volterra series” as “Padé approximant” is to _________?
"I would be at least happy to know the answer for the discrete Volterra series, which (I think) would be equivalent to something like a multivariate Padé approximant."
Multi-variate ...
- 436
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