2
votes
0answers
95 views

Stationary Distribution for Markov-like system?

Let \begin{equation} A= \begin{pmatrix} 0 & a_{1,2} & a_{1,3} \\ a_{2,1} & 0 & a_{2,3} \\ a_{3,1} & a_{3,2} & 0 \end{pmatrix}, \end{equation} \begin{equation} B= ...
3
votes
1answer
86 views

Delay Differential Equations Numerical methods

I have a general question about delay differential equations. I know that even simple ones hardly have analytic solutions and mine clearly doesn't have any as it is a system of non-linear delay ...
4
votes
1answer
144 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 ...
1
vote
3answers
254 views

What are the basis functions for a product space?

Let $X=L^1\left([0,1]^3\right)$, for numerical purpose, what are the possible basis function for $X$? In finite element method, the basis functions are tooth functions, or polynomial functions. Is ...
2
votes
1answer
116 views

elliptic integral with singularities

I need to calculate elliptic integrals with singularities, up to a huge number of digits (250-1000). The problem is that Wolfram Mathematica can't do so many digits, and Pari intnum doesn't handle ...
6
votes
2answers
266 views

Rigorous numerics for maxima and minima (one variable)

Let $f:\mathbb{R}_0^+\to \mathbb{R}$ be defined by some combination of the four basic operations and square roots. (The argument of square-roots is assumed is to be non-negative, and the value of ...
1
vote
1answer
126 views

For what values of the parameter does this function have an elementary anti-derivative?

I am a grad student working on some independent research trying to derive some exact formulas for a particular class of power series. During my study I came across the following integral which would ...
13
votes
1answer
2k views

The unreasonable effectiveness of Pade approximation

I am trying to get an intuitive feel for why the Pade approximation works so well. Given a truncated Taylor/Maclaurin series it "extrapolates" it beyond the radius of convergence. But what I can't ...
3
votes
1answer
124 views

approximation methods in integral equations

Recently I was reading about integral equations and I am a beginner in it. There was a constant reference to the non-availability of methods to find the exact solutions and hence lot of approximation ...
16
votes
3answers
840 views

Easy functions ?

Let $f$ be an analytic function, and suppose that we want to compute $f(x)$. The input consists of the digits of $x$ and the output of a rational number approximating $f(x)$. A function $f$ is called ...
8
votes
7answers
2k views

Any good books on numerical methods for ordinary differential equations?

I need to find some masters-level exercises about numerical methods for solving ODEs. Are there any good references?
1
vote
1answer
377 views

Series acceleration for more complicated types of oscillating series

Question: It is well known that the greatest integer function has a Fourier series representation. Since the greatest integer function itself is not periodic, the representation is derived from the ...
2
votes
1answer
369 views

Valid use of Laplace's method?

I am trying to say something about the asymptotics of $$\int_{\mathbb{R}} e^{cx - x^{4/3}}dx$$ as $c \to +\infty$, and need a sanity check. As I understand it, Laplace's method is to write $$q(x) = ...
1
vote
2answers
213 views

Approximation by polynom 1) with respect to supremum-norm 2) I need F_{approx} > F_{exact}

Given a function F, how to find polynom which is best/good approximate with respect supreremum-norm, i.e. minimize over P_{approx} sup|F-P_{approx}| ? I am intersted in polynoms in two variables of ...
1
vote
0answers
78 views

numerical methods for discontinuous ODEs

Greetings, what are state of art methods for numerical solution of ODEs with discontinuous right side? I'm mostly interested piecewise-smooth right side functions, e.g. sign.
2
votes
0answers
315 views

Gaussian type integral with inverse square root

Hi, I have encountered an integral of the following type in an engineering application: $\int_{-\infty}^\infty dx \frac{1}{\sqrt{x^2+a^2}}\exp(-x^2/2+i x b)$, where $a$ and $b$ are real ($a$ could ...
3
votes
2answers
346 views

ODE with non-continuous right hand side

My brother asked me a question which I didn't know the answer to. Are there theorems about existence, uniqueness and stability of solutions of ODEs of the followin type $$ \frac{d^2 y}{dt^2} = ...
2
votes
0answers
79 views

Minimum time planar paths under a bound on magnitude of acceleration

On a plane, given initial position (x1,y1), initial velocity (u1,v1), final position (x2,y2), and final velocity (u2,v2), compute the solution to x''= cos(z), y''=sin(z) that has these endpoint ...
5
votes
1answer
395 views

Approximating high-dimensional integrals by low-dimensional ones

This question is motivated by the following naive one: suppose we have a nice subset $X$ of some Euclidean space, say a polyhedron, and a nice $\mathbb{R}$-valued function $f$ on this subset, say a ...
2
votes
2answers
1k views

Stone-Weierstrass theorem applied to Fourier series

This is a question on Fourier series convergence. The problem is, in the applications of the Stone Weierstrass approximation theorem on wikipedia, there's stated that as a consequence of the theorem ...
4
votes
5answers
928 views

An inequality on concave functions

Could somebody help me to answer the following question? Let $f:R_+ \rightarrow R_+$ be a nonindentically zero, nondecreasing, continuous, concave function with $f(0)=0$. Do we have that for any ...
5
votes
2answers
508 views

Runge-Kutta method with c<1

In trying to solve an ODE $y'=f(y,t)$ with a function f that is discontinuous at a subset (codim=1) of $\mathbb R^n$, I am looking for a Runge-Kutta ODE method whose stages do not evaluate $f(x,t)$ at ...
5
votes
6answers
573 views

best approximation to the LambertW(x) or exp(LambertW(x))

what is the best available approximation ( say up to 10 digits ) for LambertW(x) or exp(LambertW(x)) for x > 2000
16
votes
6answers
2k views

Why not evaluate integrals using ODE-solvers?

Hello! I have a question about the relationship between numerical integration and the solution of ordinary differential equations (ODE). Suppose I want to evaluate the integral $I(x) = \int_{0}^{x} ...
2
votes
0answers
530 views

Multi-variate secant method for solving $F(x)=0$

The secant method for solving an equation $F(x)=0$ in one variable is much older than Newton's one. Recall that given two iterates $x_{k-1}$ and $x_k$, it provides an update $x_k$ by taking the ...
2
votes
4answers
332 views

When do functions near F have zeros near a zero of F?

Consider a sequence of functions $F_n : \mathbb{R}^d \to \mathbb{R}^d$, a function $F: \mathbb{R}^d \to \mathbb{R}^d$, and an $\mathbf{x} \in \mathbb{R}^d$ so that $F(\mathbf{x}) = \mathbf{0}$. In ...
1
vote
4answers
1k views

The maximum of a real trigonometric polynomial

Given the coefficients $a_0,\ldots,a_N$, $b_1,\ldots,b_N$ of a real trigonometric polynomial: $ f(x) = a_0 + \sum_{n=1}^N a_n \cos(nx) + \sum_{n=1}^N b_n \sin(nx) $ is there any efficient way to ...
2
votes
1answer
587 views

Root Finding for Raytracing (Ray and Meta-Ball Intersection)

The motivation behind this is to find the points of intersection between a ray and a level set of a potential function $g$, built in terms of a basic potential function $f$ (the building is explained ...
10
votes
0answers
650 views

Constructive aspects of Caratheodory's theorem in convex analysis

Let me paraphrase Caratheodory's theorem in a probabilistic setup: Let $X$ be a real-valued random variable. For $k = 1, \ldots, m$, let $f_k: \mathbb{R} \to \mathbb{R}$ be a continuous function such ...
2
votes
1answer
256 views

Can we find an l-2 sequence if we know all l-p norms?

I'm wondering if there is a way to approximate the first $M$ terms of a non-increasing $\ell^2$ sequence $\{c_n\}$ if we know $|c|_p^p = \sum c_n^p$ for $p=2,3,4,\dots$? I've tried truncating the ...
1
vote
2answers
395 views

“Misbehaved” differential equations

I have always been fascinated by the so called taxicab geometry first considered by Hermann Minkowski. The metric that has to be used here is a L1 distance which e.g. means that the lenght of the ...