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Newton method for polynomials with random starting points

I know that this question exists, but unfortunately it doesn't cover my issue sufficiently. Assume that we have a polynomial $p(x)$ of degree $n$ with real coefficients, we can assume that all its ...
Oleksandr  Kulkov's user avatar
3 votes
1 answer
132 views

How to maximize the variance of a subset of integers?

$\DeclareMathOperator{\Var}{Var}$Given the set of numbers $\Omega := \{1, \ldots, n\}, n \in \mathbb{Z}^+$, how can I choose a subset, $A$ of $\Omega$ , such that $\min(\Var(A), \Var(\Omega \setminus ...
Hasan Zaeem's user avatar
2 votes
0 answers
46 views

Convergence of finite-difference method for Cauchy-Riemann equations

Let $I\subseteq \mathbb{R}$ an open interval. Let $f:I\rightarrow \mathbb{C}$ real analytic. Suppose we want to numerically compute an analytic extension of $f$. We will assume the following: we are ...
Plemath's user avatar
  • 312
1 vote
0 answers
29 views

Integral hull of a polyhedron Q is polyhedron

Let $Q \subseteq R^n$ be a rational polyhedron and let $Q_I=Convexhull(Q \cap Z^n)$. By finite basis theorem, we have $Q=P+C$ for some rational polytope $P$ and finitely generated cone $C$ where $C=R_+...
Sowbarnika R's user avatar
-1 votes
0 answers
41 views

Is it possible to backtrack an optimization solver? [closed]

I have an optimization problem and was using a linear programming optimizer to find solutions. However, I find that past a certain size, the problem becomes "infeasible" and has no solutions....
Bamboozle's user avatar
0 votes
0 answers
57 views

Complexity of evaluation of analytic functions

Given an analytic function $f(x)$ (say as combination of elementary functions and operators), is it possible to compute $n$ first bits of the value of the function on the whole interval $[a, b]$ ...
roignoirewg's user avatar
0 votes
0 answers
21 views

Easy instance of set cover

I am trying to prove that a natural greedy algorithm solves the following instance of the set cover problem: for a set of elements $e\in U$ with a set of weights $w_e$, we define the cost of a subset ...
Tom Solberg's user avatar
  • 4,049
7 votes
2 answers
242 views

Prove that $ n \leq d+1 $ under ordering constraints in $\mathbb{R}^d$

Let $x_1, \dotsc, x_n \in \mathbb{R}^d$ and $\theta_1, \dotsc, \theta_n \in \mathbb{R}^d$ be vectors such that for every $k \in [n]$, the following inequality holds: $$ \langle x_k, \theta_k \rangle &...
Alireza Bakhtiari's user avatar
1 vote
0 answers
58 views

Method of characteristics, the characteristic lines follow gradient, is this significant?

The PDE I am working on comes from geology, which I do not have much background on. Said equation aims to describe describes the erosion by describing it as an advection phenomena: the advection ...
betelgeuse's user avatar
1 vote
1 answer
141 views

Understanding quadrature rule of a function multiplied by another $C^{\infty}$ function

Define a function $f \in C^m[-1,1],m \in \mathbb{N}$ and $g\in C^{\infty}[-1,1]$. Also define a quadrature rule $Q$ for approximating the integral $\int_{-1}^1 h(x)dx$ for some function integrable ...
Sam's user avatar
  • 69
0 votes
2 answers
97 views

Optimization algorithms for Kronecker approximation of high-dimensional covariance matrices

I'm working with a high-dimensional covariance matrix and exploring Kronecker product approximations to make it computationally manageable. Here's the setup: I have a graph $G$ represented by a $D\...
JJbox's user avatar
  • 1
5 votes
0 answers
75 views

What is the maximal advantage of randomized over deterministic algorithms for approximation in the worst-case?

Let $X\subset Y$ be Banach spaces and $B_X:=\{x\in X: \|x\|_X\le1\}$ be the unit ball of $X$. The goal is to find an approximation of every element from $B_X$ with error measured in $Y$ by using at ...
Mario Ullrich's user avatar
2 votes
4 answers
212 views

Efficient algorithm for graph problem

Let $D=(V,E)$ be a directed graph, $S,T\subset V$ and $f:V\rightarrow \{1,\ldots, k\}$ a positive, bounded weight-function and $l\in \mathbb{N}$, find a path $v_1,\ldots, v_l\in V$ with $v_1\in S$ and ...
Martin Clever's user avatar
1 vote
1 answer
152 views

Linear interpolation vs L2-projection

I'm reading the book "The Finite Element Method: Theory, Implementation, and Applications" by Larson and Bengzon. In the first chapters there are presented two methods for approximating ...
Cymek3's user avatar
  • 19
1 vote
1 answer
106 views

Iterated optimal transport

Suppose we are interested in two consecutive transport plans (in the Kantorovich formulation). That is, we are given finite sets $X$, $Y$ and $Z$, endowed with probability measures $\mu_X$, $\mu_Y$ ...
tex.support's user avatar
4 votes
1 answer
253 views

Reference to formal approach to homotopy analysis method

I'm currently reading a book about the Homotopy Analysis Method (HAM), but it isn't very rigorous (it explains most things with a single example), which is bothering me. I'm searching for papers where ...
BobTheThird's user avatar
3 votes
1 answer
138 views

Handling absolute value and other discontinuities in numerical optimization methods that use gradients

Suppose we have difficult peak fitting problems where the the users wish to fit asymmetric peaks to their experimental data by the least squares method. One such function is illustrated below: Here $...
ACR's user avatar
  • 879
4 votes
0 answers
198 views

Pricing zero coupon bonds through PDE

I'm currently studying Paul Wilmott on quantitative finance and saw an interesting idea for an interest rate model that went unexplored in the book. The idea is to model the market price of risk as a ...
David Hunt's user avatar
0 votes
2 answers
178 views

Inversion formula for discrete sine and cosine transforms

$\newcommand{\wh}[1]{{\widehat{#1}}} \newcommand{\R}{{\mathbb{R}}} $I am looking for a proof of the inversion formulas for the discrete sine and cosine transforms, i.e. a proof of the fact that these ...
Bettina's user avatar
  • 113
4 votes
1 answer
88 views

Simulation of SDEs using Karhunen Loeve expansion

A very common and easy way to simulate the solution of a SDE is to use the Euler-Maruyama method. At each time step the only random part comes from the realization of the increment of the Brownian. It ...
happy and healthy's user avatar
3 votes
1 answer
79 views

How to deal with singularities in thin plate splines?

Follow up from this question Thin-Plate-Spline understanding and solution. In the general case of $\mathbb{R}^N$ the following problem (interpolant which minimizes the Thin Plate Energy, specifically ...
user8469759's user avatar
1 vote
0 answers
69 views

Newton-Kantorovich: theorem geometric

This post is cross-posted from Math StackExchange where I did not receive any response after 5 days. I guess this question might be targeted more towards research level mathematics, so I decided that ...
Victor Liu's user avatar
0 votes
0 answers
25 views

Numerical computation of spectral abscissa of operator

I would like to numerically compute the spectral abscissa of an unbounded linear operator $A$ on a Hilbert space. To give you an idea my operator has the form: $$Af(x,y) = a(y) \partial_x f(x,y) - b(x)...
toaster's user avatar
  • 143
0 votes
0 answers
48 views

A question on a quantitative form of Farkas' lemma

Suppose A is an $m \times n$ matrix whose entries are non-negative integers and $\mathbf{b}$ is a vector with rational entries. A version of Farkas lemma implies that if the equation $$A\mathbf{x}=\...
Keivan Karai's user avatar
  • 6,214
1 vote
1 answer
133 views

Chebyshev approximation via iterated weighted least squares fits

I have the task of finding a Chebyshev approximation for a time-series; I want to check different types of functions, e.g. polynomials, rational functions, harmonics, etc. I know that the Remez ...
Manfred Weis's user avatar
  • 13.2k
0 votes
0 answers
54 views

How to deal with minimizing a flat objective function

Problematic (Debiased Sinkhorn barycenter, proposed by H.Janti et al.): Let $\alpha_1, \ldots, \alpha_K \in \Delta_n$ and $\mathbf{K}=e^{-\frac{\mathrm{C}}{\varepsilon}}$. Let $\pi$ denote a sequence ...
Tung Nguyen's user avatar
1 vote
0 answers
59 views

Approximating a symmetric function in S(R) by Gaussians

The comments on this question point to a paper which demonstrates that the Gaussians $$E_b(x)=e^{-bx^2}$$ are dense in the even functions in $\mathcal{S}(\mathbb{R})$. I'm wondering if there's a nice ...
user3137493's user avatar
0 votes
0 answers
83 views

Interpolating points and orthogonal polynomials on varying intervals

In general, a Lebesgue-Stieltjes integral, $\int (\cdot) \, d \alpha(x)$, that defines an inner product on the space of polynomials establishes a notion of orthogonality. Suppose we have a sequence of ...
NewUser's user avatar
  • 13
1 vote
2 answers
156 views

Numerical evaluation of monomial divided differences

Suppose $f(x)=x^{n+1}$ for some $n\in\mathbb{N}$, and define the divided difference $$f[a,b]=\frac{a^{n+1}-b^{n+1}}{a-b}.$$ I am wondering about the best way to numerically evaluate $f[a,b]$ to high ...
Stephen Berg's user avatar
1 vote
0 answers
99 views

Minimum of the maximum element frequency given the family size and the universe size

[Crossposted at math.stackexchange]. Consider families of sets $\mathcal{F}$ with size $n = |\mathcal{F}|$ and universe $U(\mathcal{F})$ with size $q = |U(\mathcal{F})|$. I have written and solved ...
Fabius Wiesner's user avatar
1 vote
1 answer
60 views

Does Monotone (linear) convergence of iterates imply monotone (linear) convergence of function values?

I am considering a proof that would require a certain connection between convergence of iterates and corresponding function values: Consider an algorithm with iterates $\left\{{\mathbf{x}}^k\right\}_{...
AY Wer's user avatar
  • 13
0 votes
0 answers
32 views

Finding measure representation for rank 2 moment matrices

Assuming the following equation has a solution, I'm interested in finding any concrete values of $x_{1},\dots x_{n},y_{1},\dots y_{n},c_{1},c_{2},R$ that fulfills it. $$ \begin{bmatrix} 1 & 1 \\ ...
patchouli's user avatar
  • 275
3 votes
1 answer
122 views

Upper bound of $-r’’_\nu(x)/r’_\nu(x)$ where $r_\nu(x)=I_{\nu+1}(x)/I_\nu(x)$

Please see the bottom of the post for an updated version of the question. Original Question. I’m looking for an analytical proof of an upper bound for $-r’’_\nu(x)/r’_\nu(x)$ for $\nu \ge 0$ and $x \...
fancidev's user avatar
0 votes
0 answers
101 views

Simulation of Markov processes with exponential timestepping

Let $(Y_t)_{t\ge0}$ be a time-homogeneous Markov process with transition semigroup $(\kappa_t)_{t\ge0}$. Numerical simulation of $(Y_t)_{t\ge0}$ can be done in the following way: Choose an initial ...
0xbadf00d's user avatar
  • 167
0 votes
0 answers
115 views

Software for computing polytopes

As can be inferred from the title, I want to do some computation on the facets representation of the polytopes given the vertices. My advisor recommended me Polymake, which is indeed useful even with ...
AlexiosF's user avatar
0 votes
0 answers
72 views

Probability of being inside a convex n-dimensional polytop

I am currently conducting some post-grad research about wireless transmissions with uncertain transmission delays. As part of the research, each individual transmission is modelled using a probability ...
Florian Bauer's user avatar
6 votes
1 answer
234 views

Stopping criteria for damped Newton iterations with backtracking line search

Are there better criteria than the Armijo criterion for damped Newton iteration with backtracking line search, when the objective is standard self-concordant? (See Boyd and Vandenberghe.) Let $F(x)$ ...
Sébastien Loisel's user avatar
5 votes
1 answer
176 views

Efficient counting of integer solutions to linear system

In my research, I have a particular 18x18 matrix $\mathbf{A}$ which defines the linear system $\mathbf{A}\cdot \mathbf{x} \leq \mathbf{-1}$ over the nonnegative integers. And I'm interested in ...
user326210's user avatar
0 votes
0 answers
30 views

Peak search for NLS equation solutions

By solving the initial value problem of the NLS equation, I can get a matrix of numerical solutions. I hope to perform a peak search on the matrix of this numerical solution, that is, there are ...
miao zhengwu's user avatar
4 votes
1 answer
341 views

rank of an integer valued matrix

I make some numerical experiments, involving rank of integer valued matrices of the size about $14\times 24$. As the matrix is integer valued, theoretically there should be no room for errors. However ...
Dmitri Scheglov's user avatar
3 votes
1 answer
118 views

Restate equality between random variables in a numerical stable way

Let $X$ and $Y$ be two random variables drawn from two distinct normal distributions, $\mathcal{N} (\mu_X, \sigma_X^2)$ and $\mathcal{N} (\mu_Y, \sigma_Y^2)$, that correspond to the measurements of an ...
ElectricPhysiscist's user avatar
1 vote
0 answers
124 views

Can numerical differentiation be applied to tensor derivatives?

I know that for a 1D function, I can calculate the numerical derivative at every point, $\DeclareMathOperator{\d}{d\!} (x_1,y_1)$, with $\d y/\d x$ where $\d y = y_2 - y_0$ and $\d x = x_2 - x_0$. If ...
Jesse Feng's user avatar
0 votes
0 answers
72 views

Initial guess in shifted QR algorithm

I'm comparing timings of two implementations of algorithms for the computation of Gauss-Legendre nodes. 1 - The first is a Newton algorithm to find the roots of the Gauss-Legendre polynomials. Quite ...
G. Fougeron's user avatar
2 votes
0 answers
58 views

Is there any way to approximate a function based on a series of logarithm [closed]

Is there a way to expand a function in terms of logarithmic series? Like when we expand a function in terms of Taylor or Fourier series? I tried to answer this question this way: Let's assume it can ...
epol's user avatar
  • 21
0 votes
0 answers
49 views

Galerkin scheme in $H^s_0(G)\subset L^2(G)\subset H^{-s}(G)$ ($s>0$)

What basis functions are usually choosen if one attempts to conduct a Galerkin finite element method given an evolution triplet $H^s_0(G)\subset L^2(G)\subset H^{-s}(G)$. Where $G$ is a sufficiently ...
Perelman's user avatar
  • 163
1 vote
0 answers
65 views

Discretization of oscillating integral

Suppose I am interested in computing $$ I \equiv \int_0^B dx \, g(x) f(x) $$ where $B$ is a known upper bound for the integral, $g(x)$ is a known oscillating function and $f(x)$ is a smooth function ...
knuth's user avatar
  • 33
0 votes
0 answers
136 views

Antiderivatives via Taylor series and the FT of Calculus

If $f$ is a real function on an interval $[a,b]$ such that $f$ is computationally tractable on $[a,b]$: you can calculate $f(x)$ to $n$ bits of precision using an algorithm which is polynomial in $n$ ...
Joe Shipman's user avatar
1 vote
0 answers
95 views

Vandermonde-type factorization of moment matrix?

Consider $n,d \in \mathbb{N}_{>0}$, there are many functions $y:\mathbb{N}^{n} \to \mathbb{R}$. Now for simplicity, we denote $y(\alpha)$ to be $y_{\alpha}$. Let $|\alpha| = \sum_{i=1}^{n}\alpha_{i}...
patchouli's user avatar
  • 275
1 vote
0 answers
38 views

Formulation of multipoint constraints using Lagrange multipliers for a time dependent problem (with the Finite Element Method)

Intro Suppose we have the following static linear equations (e.g. of an elastostatic problem): $$\mathbf{K}\boldsymbol{u}=\boldsymbol{f}$$ We want a multipoint constraint of the type $$\boldsymbol{\...
Breno's user avatar
  • 111
3 votes
1 answer
204 views

Converting an algebraic equation into a ODE

I'm working on a method to solve algebraic equations by converting them into ordinary differential equations (ODEs) and then integrating these ODEs over time. Given an algebraic equation $f(x(t), t) = ...
Joe's user avatar
  • 31

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