All Questions
1,732 questions
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votes
3
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Properties of rational functions
Hi everyone. It is well known that a polynomial of degree $n$ is completely determined by $n+1$ points. Now, is there any similar result for rational functions?
- 105
5
votes
2
answers
2k
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Evaluating elliptic integrals
I am interested in evaluating some elliptic integrals, and I have not been able to secure a reference to do exactly what I need. Most of the references I've found seem to focus on reducing more ...
- 26.9k
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votes
3
answers
2k
views
Square root algorithm
I would like an efficient algorithm for square root of a positive integer. Is there a reference that compares various square root algorithms?
5
votes
5
answers
9k
views
Characterizing convex polynomials
Let $p=\sum_{i=0}^{n}a_ix^i$. Under what conditions on the coefficients $a_i$ is $p$ convex? Strictly convex?
- 51
5
votes
5
answers
3k
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
- 153
5
votes
2
answers
352
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Locating the maximum point $x_n$ of $f_n(x):=e^{-1/x}\Bigl(1+\frac{1}{n^2 x^n} \Bigr)$ in $(0,1)$
I am trying to observe the behavior of $x_n \in (0,1)$ defined such that the function
\begin{equation}
f_n(x):=e^{-1/x}\Bigl(1+\frac{1}{n^2 x^n} \Bigr)
\end{equation}
attains its maximum inside the ...
- 3,477
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votes
3
answers
2k
views
Automatic vs numerical differentiation of a function known from samples
Suppose I have $n$ samples $(x_i, f(x_i))_{i=1}^n$ from an unknown function $f$. I need to approximate (estimate) the derivative $f'(x^*)$ at some new test point $x^*$, that is not necessarily one of ...
- 710
5
votes
2
answers
691
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What is state of the art for the Shooting Method?
I am interested in examples where the Shooting Method has been used to find solutions to systems of ordinary differential equations that are either
reasonably large systems, or
the search algorithm ...
- 2,123
5
votes
2
answers
301
views
Euler–Maclaurin formula in $\mathbb{Z}^d$
I was wondering whether there is a Euler–Maclaurin formula of sorts for expressions such as
$$
\sum_{x \in [a,b]^d\cap \mathbb{Z}^d} f(x) - \int_{[a,b]^d}f(x)
$$
where $d\ge 2$ is an integer, $a,b \...
- 446
5
votes
3
answers
1k
views
Chudnovsky formula vs. Machin type formulae for calculating $\pi$
In honor of recent $\pi$ Day, I found myself drawn into thinking about how modern calculation of $\pi$ is done. Both the approach via Machin-like formulae and the approach via the Chudnovsky formula ...
- 5,827
5
votes
1
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569
views
Existence of solutions to a nonlinear algebraic equation
How can we prove that equation (1) has solutions for every $p$. I mean, is there an analytic method that can be used to show that there exist solutions for every $p$ for this nonlinear equation:
...
- 313
5
votes
2
answers
919
views
Analytic Solution to SDEs
Are there any example of SDEs with constant diffusion terms, other than the Ornstein Uhlenbeck process, which have exact solutions? I'm thinking of something of the form:
\begin{equation}
dX_t = f(...
- 379
5
votes
2
answers
4k
views
sparsity of QR decomposition
Hi, everyone!
I have a sparse $n \times n$ matrix $A$ with $nnz(A)$ denoting the number of non-zero entries in $A$. Now I use QR factorization to decompose $A$ into an orthogonal matrix $Q$ and ...
- 51
5
votes
3
answers
550
views
closest equidistant point to N points in M dimensions
Is there a formula/algorithm/etc. to find the closest equidistant point (assuming it exists) to a set of points, allowing that the number of dimensions of the space is independent of the number of ...
- 425
5
votes
2
answers
2k
views
Solving for Moore Penrose pseudo inverse
I have a system to solve, set up as :
$$Ax = b$$
with a square rank deficient matrix $A$. The paper suggests to use a Moore Penrose pseudo inverse, which in my case can be computed using the ...
- 481
5
votes
1
answer
481
views
Stability of root-finding near the unit circle
It is stated in several sources on numerical analysis that the general problem of polynomial root-finding is ill-conditioned, but that it is well-conditioned if the roots are near the unit circle. (e....
- 173
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1
answer
2k
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Are piecewise linear functions dense in $W^{1,\infty}$?
Are piecewise linear functions dense in $W^{1,\infty}$ ?
- 393
5
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2
answers
2k
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Multiplicative gradient descent?
The normal gradient descent is additive: $w_{t+1}=w_t-\lambda_t\nabla f(w_t)$, but is there a multiplicative gradient descent that looks something like $w_{t+1}=w_t[-\lambda_t\nabla f(w_t)]$?
I know ...
- 211
5
votes
2
answers
3k
views
Continuous Linear Programming: Estimating a Solution
I have a "continuous" linear programming problem that involves maximizing a linear function over a curved convex space. In typical LP problems, the convex space is a polytope, but in this case the ...
- 373
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votes
3
answers
5k
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measure of quality of curve fit
I am interested in a measure for the quality of fit to a curve which would distinguish the two cases shown in the following image (without addressing the fact that incidentally the right one has more ...
- 307
5
votes
2
answers
731
views
Do any two hermitian matrices A and B commute with the support of their commutator?
Let $A$ and $B$ be Hermitian matrices. Let $[A,B] = AB - BA$ be their commutator and let $[A,B]^+$ be the Moore-Penrose pseudoinverse of $[A,B]$.
Is it then true that $A$ and $B$ both commute with the ...
- 1,947
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votes
2
answers
2k
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Real world example of use of Monte Carlo method for high dimensional integrals
The Monte Carlo method for numerical integration is usually presented as a method invented to efficiently compute high dimensional integrals numerically. However, I haven't found any source which has ...
- 141
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2
answers
1k
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Roots of the Chebyshev polynomials of the second kind
It is known that the roots of Chebyshev polynomials of the second kind, denote it by $U_n(x)$, are in the interval $(-1,1)$. I have noticed that, by looking at the low values of $n$, the roots of $(...
- 373
5
votes
2
answers
192
views
Accuracy of the formulas for angles between almost colinear vectors
Assume $x$ and $y$ are two vectors in $\mathbb{R}^3$ and we want to compute the acute angle $\alpha\in(0,\pi/2]$ between these two (noncolinear) vectors. There are (at least) two possibilities:
In ...
- 215
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1
answer
1k
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 ...
- 173
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votes
3
answers
1k
views
Algorithm for the intersection of a vector subspace with a cone of non-negative vectors
Hi,
I would like to know whether there is some more effective way of how to compute an intersection of a vector subspace of $\mathbb{R}^{n}$ with a cone of vectors with non-negative entries than the ...
5
votes
3
answers
8k
views
Linear programming - uniqueness of optimal solution
Is it possible to build such an objective function for a given set of constraints, so that there will be only one optimal solution?
My general problem is to get any vertex of a polytope formed by a ...
- 85
5
votes
2
answers
716
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 ...
- 580
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1
answer
5k
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Is there a general method for determing the domain of dependence of (higher-order) PDEs?
Is there a general method for determining the domain of dependence of (higher-order) PDEs? I would be plenty happy with a reference to a paper or textbook I could look at; despite much reading around,...
- 846
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1
answer
2k
views
Inverting a covariance matrix numerically stable
Given an $n\times n$ covariance matrix $C$ where $n$ around $250$, I need to calculate $x\cdot C^{-1}\cdot x^t$ for many vectors $x \in \mathbb{R}^n$ (the problem comes from approximating noise by an $...
- 51
5
votes
3
answers
1k
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Interpolation by rational functions reference
I have been hearing a lot about a theory of interpolation using rational function, parallel to that of polynomial interpolation.
I'm looking for a book chapter, or even short lecture notes, that will ...
- 3,574
5
votes
1
answer
407
views
Is this inverted integral transform valid?
I have the following transform:
$$F(y) = \int_{0}^{\infty} y\exp{\left[-\frac{1}{2}(y^2 + x^2)\right]} I_0\left(xy\right)f(x)\;\mathrm{d}x$$
with the following conditions:
$f(x)$ and $F(y)$ must be ...
- 153
5
votes
2
answers
1k
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What is the definition of an antilimit?
I've seen some references to antilimits in the numerical analysis literature, but no definition of the term. The impression I get is that in specific contexts where every sequence $x_0,x_1,x_2,\dots$ ...
- 19.7k
5
votes
2
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4k
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Are there any algorithms for solving nonlinear matrix equations over $\mathbb{C}$?
Are there any algorithms for solving nonlinear matrix equations over $\mathbb{C}$?
I am especially interested in solving polynomial nonlinear matrix equations.
For instance, let $X$ be some matrix ...
- 1,503
5
votes
5
answers
1k
views
solving series of linear systems with diagonal perturbations
I would like to solve a series of linear systems Ax=b as quickly as quickly as possible. However, the systems are related. Specifically, each matrix A is given by:
cI + E
where E is a fixed sparse, ...
- 135
5
votes
1
answer
260
views
Approximate Sobolev embedding
It is well-known in $H^2(\mathbb R^3)$ embeds into $L^{\infty}(\mathbb{R}^3).$ Now consider a function $u \in \ell^{\infty}(h\mathbb Z^3)$ and a grid of points $x \in h\mathbb{Z}^3.$
We then define ...
- 124
5
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1
answer
384
views
Examples of Polyhedra with Large Shadows
Let $P \subseteq \mathbb{R}^n$ be a polyhedron described by $\mathcal{O}(n^{c_1})$ inequalities, where $c_1$ is a constant. Moreover, let $M\colon P \to \mathbb{R}^2$ be a linear mapping. I'm looking ...
- 321
5
votes
2
answers
428
views
Evaluating a limit similar to the Euler constant
In the course of studying a certain complex-valued functional equation, I have had a need to evaluate the following limit:
$$\gamma_\mathcal{T}=\lim_{n\to\infty}\left(-\frac{i}{2}\sum_{k=1}^n \frac1{...
5
votes
1
answer
503
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 ...
- 23.5k
5
votes
2
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456
views
Inverting products of matrices
I need to compute a large number of inverses of the following form:
$(A \Lambda_k A^\top)^{-1}$
where $A \in \mathbb{R}^{m \times n}$, $n > m$ and $\Lambda_k = \text{diag}(\lambda_1, ..., \...
- 153
5
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1
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177
views
Can you use Oseledet's theorem to numerically approximate the Lyapunov spectra?
Let's say you have a set of first order differential equations with known Jacobian $J$. Let $x_0, x_1, ..., x_n$ be sampled points on the trajectory near the attractor.
Let $T_n = J(x_{n-1})J(x_{n-2})....
- 188
5
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1
answer
471
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Padé multipoint approximants of the exponential function
One says that a pair of polynomials $(P_m,Q_n)$ over $\mathbb C[z]$, with
$\text{deg }P_m=m$, $\text{deg }Q_n=n$, is a "multipoint Padé approximant of the exponential function" if $P_m(z)e^z-Q_n(z)$ ...
- 3,998
5
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1
answer
183
views
Resource Constrained Routing with Refueling
What are good algorithms (resp. models) for calculating optimal or near optimal routes while taking into account fuel consumption, options for refueling and, limited tank capacity?
Especially modeling ...
- 13.2k
5
votes
2
answers
2k
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Bounding the minimal maximum norm of a solution of a linear system.
I would be grateful for pointing me out a reference to some general bound on the $\ell_{\infty}$ norm of a solution of a linear system. To be specific, suppose that we have an underdetermined linear ...
- 191
5
votes
2
answers
1k
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Applications of minmax theorem(s)
Intro We suppose $X$ and $Y$ are nonempty sets and f: $X\times Y \rightarrow \mathbb{R}$. A minimax theorem is a theorem that asserts that, under certain conditions,
$$ \inf_Y \sup_X f = \sup_X \...
5
votes
1
answer
644
views
A conjecture about the submatrix of orthogonal matrix
Let $U$ be an $n\times n$ orthogonal matrix, i.e. $U\in\mathbb{R}^{n \times n}$. For any non-empty ordered sets $S_1,S_2\subset\{1,2,...,n\}$, define $U_{S_1S_2}$ to be an $|S_1|\times|S_2|$ submatrix ...
- 1,495
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2
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1k
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Pade approximation of gaussian distribution to given precision
Apologies if the question is too elementary here.
For a certain computational application I need to approximate Gaussian distribution $e^{-x^2}$ with specific absolute precision (within $10^{-7}$ ...
- 2,205
5
votes
2
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584
views
Continuous Transportation Problem
Hi all, I'm trying to formulate an infinite linear program to prove optimality (via duality) for the Continuous Transportation Problem, e.g. the Kantorovich-Wasserstein distance. This is the ...
- 133
5
votes
1
answer
284
views
Unbounded solution but bounded Euler discretization
Is there an ordinary differential equation in $\mathbb{R}^d$ induced by a gradient vector field with unbounded solutions, for which the difference equations obtained by using the forward Euler method ...
- 311
5
votes
1
answer
394
views
Explicit Formula of Delsarte's Linear Programming Upper Bound for $A_q(n,3)$
The problem of giving an explicit formula for $A_q(n,d)$ is sometimes referred to as "the main problem in coding theory." The value of $A_q(n,d)$ is given by the maximum number of codewords in a q-ary ...
- 297