3
votes
0answers
75 views

What do we know about the generalized eigenvalue problem involving a projector?

Consider a matrix $A\in\mathbb{R}^{n\times n}$ and a projector $P\in\mathbb{R}^{n\times n}$. Are there results regarding the generalized eigenpairs $(v,\lambda)$ of the generalized eigenproblem ...
4
votes
1answer
155 views

Local positivity of solutions to linear differential inequalities (Chaplygin's theorem)

According to the entry "Differential inequality" of the Encyclopedia of Mathematics http://www.encyclopediaofmath.org/index.php/Differential_inequality the following result is due to Chaplygin ...
6
votes
1answer
893 views

Mathematical study of Mpemba effect?

It has been known since the days of Aristotle and Descartes that under certain circumstances warm water freezes faster than cold water. This effect is now commonly known as the Mpemba effect, named ...
1
vote
1answer
145 views

Literature on root finding of convex Functions

I am interested in using a result about Newton's method, which basically states that if f is convex on $[a,b]$ and it holds $f(a)<0$ and $f(b)>0$, then the Newton iteration converges to ...
10
votes
1answer
313 views

Who is Petrov of the Petrov-Galerkin method?

I was not able to find the origin of the name Petrov in the Petrov-Galerkin method for the numerical approximation of PDEs. Wikipedia refers to a certain Alexander G. Petrov, but it is still not ...
1
vote
0answers
205 views

Reference request for parallel transport

I am learning about parallel transport on a Riemannian manifold equipped with an affine connexion. It seems (if I understand it well) that, in general, we might not be able to compute the parallel ...
2
votes
0answers
25 views

In what paper was the shrinkage parameter introduced to the nelder-mead simplex direct search algorithm?

I have read lots of papers referencing a 4th shrinkage parameter when talking about the Nelder Mead Simplex method. However, I cannot see any shrinkage parameter in the flow chart of the original ...
2
votes
1answer
217 views

Approximation theory under $L_1$-error

Is there a reference for results in approximation theory of bounded functions of one (and multiple) variables under $L_1$-error? Formal definitions for functions of one variable are below. Let $C$ ...
5
votes
1answer
406 views

Numerical Methods for ODEs - History

Wikipedia presents a timeline of important developments in Numerical Methods for ODEs, namely: ...
2
votes
1answer
219 views

Numerical integration for functions of symmetric matrices

This is mostly a reference request. I have integrals of the type \begin{equation} \int_C f(A) (dA) \end{equation} where $f$ is a real-valued function of a positive-(semi)definite matrix ...
28
votes
11answers
5k views

“Must read” papers in numerical analysis

In 1993, Prof. L.N. Trefethen published a NA-net posting with a list of thirteen paper he used for teaching the seminar Classic Papers in Numerical Analysis. In Trefethen's words, ... this course ...
5
votes
0answers
125 views

reference for perturbation of projection result

Let $A$ and $B$ have the same rank and dimensions. If $P_A$ denotes the projection onto the range space of $A$, then $$ \|P_A - P_B\|_2 \leq \|A - B\| \cdot \min (\|A^\dagger\|_2, \|B^\dagger\|_2). $$ ...
1
vote
1answer
258 views

Stieltjes convolution with white noise

I'm looking for a reference that would discuss a Stieltjes convolution between a wiener process and a function of bounded variation. Additionally, I had a question about this sort of convolution. Is ...
4
votes
2answers
282 views

Convolutive noise removal

I have the time domain signal $$ u_o(t) = u(t)e^{-t/\tau}\eta(t) + \sigma(t) $$ where $\tau$ is known, $\eta$ is non-Gaussian noise, and $\sigma$ is Gaussian noise. The distribution of $\eta(t)$ is ...
0
votes
2answers
326 views

Discrete Wavelet Transform and L2 Basis

Using the mother wavlet $phi$ one obtains an orthonormal basis $\phi_{j,k}(x):=2^{j/2}\,\phi(2^j\,x-k)$of L^2 (on the unit interval say). Given a function $f$ on can calculate the coefficients using ...
8
votes
0answers
388 views

How to evaluate binomial coefficients efficiently and as correctly as possible?

This question is more precisely about evaluation with a computer, of a binomial coefficient of the form $ \binom{x}{m}$ where $x$ is a real number and $m$ a rational integer. The reason why I ask is ...
4
votes
3answers
611 views

Quanitative de Moivre–Laplace theorem (reference request)

The classical de Moivre-Laplace theorem states that we can approximate the normal distribution by discrete binomial distribution: $${n \choose k} p^k q^{n-k} \simeq \frac{1}{\sqrt{2 \pi ...
4
votes
1answer
572 views

Is $O(10^{-6})$ an acceptable notation in numerical analysis? [closed]

The following question has been on math.SE for several days. Without having a satisfying answer, I'd like to ask the experts here. In mathematics, the big $O$ notation is used to describe the ...
14
votes
1answer
808 views

Who introduced the notion of “stability” in numerical analysis?

I am preparing a lecture course on the applications of operator theory where I intended to make some numerical analysis application. I was wondering about this question while browsing the literature I ...
2
votes
2answers
304 views

Integral Fredholm equation of the second type

There is an equation $$ w(x) = g(x)+\int\limits_0^M w(y)f(x-y)\,dy $$ where $f\geq 0$, $f\in C^\infty(\mathbb R\setminus\{c\})$ for some point $c$ and $\int\limits_{-\infty}^\infty f(t)\,dt\leq 1$. ...
0
votes
1answer
165 views

Partitioned Runge-Kutta (Lobatto IIIAB)

I am wondering, if anybody knows some paper, that study convergence and stability of Partitioned Rung-Kutta Methods (especially Lobatto IIIAB) applied on separable Hamiltonian system.
1
vote
1answer
489 views

numerical methods for matrices (method of full reduction)

Hello, Could you tell me where can I read about method of full reduction (it is method for system of linear equations with triangular matrix).
3
votes
5answers
770 views

n-widths and Kolmogorov's entropy

Most of the authors of research papers in compressed sensing use n-widths and Kolmogorov's entropy extensively, which are kind of hard for me to understand. Any suggestion on books or expository ...
7
votes
1answer
1k views

What are “variational crimes” and who coined the term?

I just caught sight on arXiv a paper by Holst and Stern titled Geometric Variational Crimes. Apparently a Variational Crime is an approach to solve problems using a finite element method (e.g. ...
7
votes
4answers
817 views

Reference request for conceptual numerical analysis

I am interested in clean algorithms for approximating solutions and so I am interested in numerical analysis, but most of the books I have seen get bogged down in error analysis or they spend a lot of ...
1
vote
3answers
330 views

Numerical algorithms on mixed-precision computational models.

I want to learn more about numerical algorithms that use mixed-precision computational models (where instead of everything being 32/64 bit floating points, we can do lower precision calculations at ...