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Results tagged with na.numerical-analysis
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user 11260
Numerical algorithms for problems in analysis and algebra, scientific computation
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-Ryzhi …
- 188k
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 floo …
- 188k
18
votes
Accepted
Who introduced the notion of "stability" in numerical analysis?
John von Neumann is credited as having pioneered the stability analysis of finite difference schemes. Crank and Nicholson [1] acknowledge Von Neumann when they demonstrate the stability of their schem …
- 188k
17
votes
Accepted
Who is Petrov of the Petrov-Galerkin method?
The Petrov you are looking for is: Georgii Ivanovich Petrov (1912-1987), biographies are here and here and here.
I quote from the third biography:
G.I. Petrov was a prominent Russian scientist in …
- 188k
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,
$$Z(2,2)= \sum_{k,\ell \geq 0} \frac{2k+3}{\binom{k+2}{2}^2} \frac{2\ell+3}{(\ …
- 188k
13
votes
"Wild" solutions of the heat equation: how to graph them?
Eq. 1.1 of this 1994 paper gives an explicit example in the form of a series expansion that seems tractable for numerical approximation. At least, I had no difficulty plotting a few terms of the serie …
- 188k
11
votes
Accepted
Is quadrature still considered part of numerical analysis?
Math journals that have recently published papers on "quadrature" include
Applied Numerical Mathematics
IMA Journal of Numerical
Analysis
Journal of Approximation Theory
Journal of Computational and …
- 188k
8
votes
Accepted
Algorithm to produce random number with a gamma distribution
The difficulty mentioned in Wikipedia refers to gamma distributions with small shape parameter; this has been addressed in arXiv:1302.1884:
The gamma distribution with small shape parameter can be …
- 188k
8
votes
Accepted
Stability of root-finding near the unit circle
The issue is explained nicely in Six Myths of Polynomial Interpolation and Quadrature. It is not a stability problem of polynomial root finding, but a problem of finding the proper representation of t …
- 188k
8
votes
On mathematical studies of the Mpemba effect
Since this question is still open, I take the liberty of pointing to a recent survey of the status of the Mpemba effect, Pathological Water Science -- Four Examples and What They Have in Common, which …
- 188k
8
votes
Accepted
Locating the maximum point $x_n$ of $f_n(x):=e^{-1/x}\Bigl(1+\frac{1}{n^2 x^n} \Bigr)$ in $(...
The maximum $x_n$ of
$$f_n(x):=e^{-1/x}\Bigl(1+\frac{1}{n^2 x^n} \Bigr)$$
is the smallest solution in $(0,1)$ of the equation
$$x=n x^n+\frac{1}{n}.$$
For $n\gg 1$ this gives $x_n\rightarrow 1/n$.
The …
- 188k
7
votes
Accepted
What is an extragradient method?
This is the key reference: G.M. Korpelevich, "The extragradient method for finding saddle points and other problems." Ekonomika i Matematicheskie Metody 12 (1976): 747-756.
I have not found this arti …
- 188k
7
votes
A numerical calculation for an integral
Nemo's representation of $F(\eta)$ in terms of a hypergeometric function can be evaluated without difficulty for large $\eta$:
$$F(\eta)=\frac{\sqrt{3} {\eta}^2 \Gamma \left(\frac{2}{3}\right) \; _1F_ …
- 188k
7
votes
Accepted
Reporting inconclusive experimental searches
An easy and reliable way to share code is via Zenodo --- works much like arXiv, you get a DOI, can update your files, and it's free. We use it regularly to document computer simulations in physics, I …
- 188k
6
votes
Chudnovsky algorithm and Pi precision
[I'm following up on my comment, in response to Wojowu's query:]
The number of digits $d$ of $1/\pi=\sum_{k=0}^\infty c_k$ produced per iteration by the Chudnovsky algorithm, which has a linear conve …
- 188k