# Tagged Questions

**1**

vote

**2**answers

108 views

### Runge-Kutta with all nodes at n+1 or zero weights otherwise

So, lets say for the family of the explicit Runge-Kutta methods:
$$y_{n+1} = y_n + \sum_{i=1}^s b_i k_i$$
where,
$$k_1 = hf(t_n, y_n)$$
$$k_2 = hf(t_n+c_2h, y_n+a_{21}k_1)$$
$$\vdots$$
$$k_s = ...

**2**

votes

**0**answers

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=
...

**4**

votes

**1**answer

105 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

**1**answer

160 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

**3**answers

257 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

**1**answer

122 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

**2**answers

267 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

**1**answer

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

**1**answer

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

**1**answer

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

**3**answers

842 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

**7**answers

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

**1**answer

380 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

**1**answer

370 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

**2**answers

224 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

**0**answers

79 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

**0**answers

318 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

**2**answers

352 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

**0**answers

80 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

**1**answer

396 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

**2**answers

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

**5**answers

940 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

**2**answers

511 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

**6**answers

589 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

**17**

votes

**6**answers

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

**0**answers

545 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

**4**answers

335 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

**4**answers

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

**1**answer

596 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 ...

**11**

votes

**0**answers

665 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

**1**answer

257 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

**2**answers

397 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 ...