Questions tagged [na.numerical-analysis]
Numerical algorithms for problems in analysis and algebra, scientific computation
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numerical scheme for SDE and empirical estimation of rate of convergence
Consider $\{X_t , t \geq 0 \}$ real valued diffusion satisfying
$$
d X_t = b(X_t) d t + \sigma (X_t) d W_t, \quad X_0 = x \in \mathbb{R}
$$
where $b, \sigma$ are well-behaved functions and $W$ is a ...
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Asymptotics for the number of digits of the ratio of binomial coefficients
Let $a$ and $b$ be distinct positive real numbers. Let $(a_n)$ and $(b_n)$ be sequences of natural numbers such that $a_n\sim an$ and $b_n\sim bn$. All the limit relations here are for $n\to\infty$. ...
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Numerical algorithm for extracting the coefficients of transseries
Assume a function $f(x)$ is given numerically for $x>0$, i.e. for any $x>0$ there is a numerical procedure to obtain $f(x)$ to any desired precision.
Also assume that the function $f(x)$ has a ...
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Distance of distributions of random variables, without PDF
Consider an interval $I$ with a smooth probability measure $d\mu (x) = c(x) dx$ and two known real measurable functions $f_1(x)$,$f_2(x)$. Both functions define a distribution on $X = {\rm Im} \, [f_1]...
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Eigenvalues of monomial matrices
Let $M = PD$, where $P$ is a permutation matrix and $D$ diagonal.
If $P$ is also symmetric, then does $M$ have all real eigenvalues?
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Time discretization of the variational formulation of the Navier-Stokes equation
Let
$T>0$
$I:=(0,T]$
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be nonempty and open, $$\mathcal V:=\left\{\phi\in C_c^\infty(\Lambda,\mathbb R^d):\nabla\cdot\phi=0\right\}$$ and $$V:=\overline{...
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Time discretization of the (stochastic) Navier-Stokes equation
Let
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be nonnempty and open
$\langle\;\cdot\;,\;\cdot\;\rangle:=\langle\;\cdot\;,\;\cdot\;\rangle_{L^2(\Lambda,\:\mathbb R^d)}$
I've found a thesis where ...
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Benchmark Systems for ODE Solvers - Reference Request
I would like to have a basic models as a ground truth for the numerical solvers. I am looking for systems which have available analytic solution. As an example I know that the closed form solution of ...
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Multi-dimensional moment problem
Let $\mu$ be a measure on $\def\r{\mathbb{R}}\r^n$, $1\le n \le \infty$. Given a (finite) multi-index $\bar{i} = (i_1, i_2, \ldots)$, one can define the moment
$$ m_{\bar i} = \int x_i^{i_1} x_2^{i_2}...
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How to solve $y''+y'/x+f(x)y=0$ using B.C.s $y(0)=0$ and $y'(0)=1$ [closed]
The term $f(x)$ is available numerically. It was curve fitted to some function of $x$. I've used dsolve in Matlab. It reported that solution can't be found.
I tried solving the above equation using
...
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1
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Lower bounds for finite difference formulas
I'm interested in approximating higher derivatives of a function via values of the function only. I guess the following question has been studied, but I haven't been able to find a reference. I know ...
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Approximating a function with sums of powers
One can approximate an analytic $f: \mathbb R\to\mathbb R$ with Chebyshev polynomials $T_n$ or with Taylor polynomials. In applications one usually prefers Chebyshev ones because they would converge ...
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Exponential of large matrices
I want to make a diffusion kernel, which involves $e^{\beta A}$, where A is a large matrix (25k by 25k). It is an adjacency matrix, so it's symmetric and very sparse.
Does anyone have a ...
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Ask for reference about finite difference method on HJB equation
I am a fresh PhD student in numerical analysis. Recently I am considering finite difference methods and their error analysis for solving HJB equation of the following form:
$$
v_t=g(a(x)v_x),\quad x\...
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Discontinuity of solutions to approximation schemes in the Barles-Souganidis framework
I have attached pg. 275 and pg. 276 of [BS91]. My concern is with the claim (2.7) on pg. 276. To prove this claim, I require the following additional assumption, which is not made by the authors:
...
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Solving a boundary value problem numerically, with high precision
In the paper ON A PAINLEVÉ-TYPE BOUNDARY-VALUE PROBLEM, the authors consider the BVP given by the ODE
$$y''=y^2-x \tag{1} $$
with the boundary conditions
$$\begin{align} y(0)&=0, \tag{2a} \\ y(x)&...
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Approximating a function with sums of gaussians
For an application I have to approximate a continuous (and hopefully smooth) positive even function that decays at infinity with a sum of sums of gaussians, preferably orthogonal ones.
That is, a ...
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Polynomial interpolants in quadrature points and L2 convergence spectral rate
We recall that the Lagrange Interpolation Polynomial $p_n(x)$ of a function $f\in C^n(\Omega )$ for some $\Omega \subseteq \mathbb{R}$ and $n\in \mathbb{N}$, has a pointwise error term of the form $$|...
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Existence and uniqueness of solution for nonlinear system
Under what conditions will a solution $y \in \Re^n$ exist for this system of nonlinear equations? If it exists, will it be unique?
$$ \mu_k = \int_0^\infty g_k(x) \, f(x, y) \, dx \qquad \forall \, k ...
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Numerical Computation of Orthogonal Polynomials Recurrence Relations
Background and notations: Given an interval $I\subseteq \mathbb{R}$ and a continuous finite measure $d\mu = w(x)dx$, and denote $p_n(x)$ the orthogonal polynomials with respect to $d\mu$. We have the ...
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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:
...
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Finding a solution to a simple geometric set of equalities
Let $p_1,\dots,p_n$ be a collection of points in the plane $\mathbb{R}^2$ and let $a$ be a positive number such that $a<1$. Is there a good numerical algorithm to find points $x_1,\dots,x_n$ in ...
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3
answers
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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 ...
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$ 4 + \sqrt{17} \approx \frac{2}{9} e^{(5/18) \pi \sqrt{17}}$ and other formulas
I found this formula attributed to Kronecker relating solutions of Pell equation to exponential sum:
$$ 4 + \sqrt{17} \approx \frac{2}{9} e^{(5/18) \pi \sqrt{17}} \text{ and } \frac{1}{\sqrt{5}}e^{(1/...
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complexity of computing the singular vector corresponding to the smallest singular value
It is known that the singular value decomposition of an $m \times n$ matrix $A$ is in general of complexity of the order $m n^2$, assuming that $m \ge n$. But what if we only want to compute say the ...
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Dual basis of Lagrange nodal variables in $R^d$
I am studying some theories around FEM method in 2D, and I am trying to solve this problem from Ciarlet's book (the proof was not provided): Consider a simplex $T$ in $R^d$ with $N_1(T) = \left\{N_i\...
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"Designing" Nodal sets of Laplacians in 2 or 3 dimensional domains
The properties of nodal sets (i.e. zero level sets of eigenfunctions) for the first non-trivial eigenfunction for Laplacians have been studied extensively.
My rough understanding is that one could ...
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Are there any explicit probability conserving solvers for Pauli equation?
I know that there exist probability conserving explicit solvers for time-dependent Schrödinger's equation, for example, Visscher's one.
But when I tried to take into account spin and magnetic field (...
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Question about preconditioning
I posted the following question on stackexchange but didn't get any replies; I'm hoping perhaps someone can help me here.
I understand that for many iterative methods, convergence rates can be shown ...
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Numerical scheme for $ \int_{S^2} f(x) \, dS \approx \sum_P f(P) $
The midpoint rule or trapezoid rule is only good up to an error, for some $c \in [a,b]$ we have that:
$$ \left| \int_a^b f(x) \, dx - \frac{1}{2}\big[\,f(b)+f(a)\,\big](b-a)\right| < \frac{1}{12}\,...
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Sign correction for SVD in Matlab [closed]
I have a Matlab function that runs a SVD. Unfortunately, the function [U,S,V] = svd(A) has a sign ambiguity which could give misleading results in my application. ...
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Generalized eigenvalue problem with nonnegative eigenvector constraint
Consider the following problem that is known to be non-convex but can be solved as a generalized eigenvalue problem (i.e. has a global optimum solution):
$\underset{w}{\text{maximize}}\quad w^{\top}...
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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 ...
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Rounding errors in images of Julia sets
One typically computes Julia sets by iterating a complex function, such as a polynomial or rational function.
How do rounding errors affect the results?
I'm looking for references on this issue, ...
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1
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Methods to compute the Green's function for the 1D wave equation with nonsmooth coefficient?
I am seeking advice on the best available numerical methods to compute the Green's function for a 1D wave equation with rough coefficient.
Suppose that the coefficient $c(x)$ in the 1D wave equation ...
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2
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How to apply Neuman boundary condition to Finite-Element-Method problems?
I have a 2D rectangular domain. The governing equation on this domain is Laplace equation:
$\nabla^2 f = 0$
In the left edge there is Neumann boundary conditon :
$\frac{\partial f}{\partial n} = -a$...
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Analytical formula for numerical derivative of the matrix pseudo-inverse?
Is there a simple numerical procedure for obtaining the derivative (with respect to $x$) of the pseudo-inverse of a matrix $A(x)$, without approximations (except for the usual floating-point ...
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algorithm for finding the minimizer of a almost convex function
Let $f(t)$ be a function from $(0,1)$ to $\mathbb R$. If $f$ is strictly convex, then finding the minimizer is an easy task. For example, newton's method would be able to do the job.
However, if my ...
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Which functions satisfy the following condition on discrete Fourier coefficients?
Background
I am studying a numerical method for solving 2D heat transfer problem (a splitting scheme in time combined with certain space approximation using FEM with rectangular meshes).
On the ...
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Compatible Finite Elements [closed]
I have a basic question as a matter of definition. I am wondering what is meant by compatible finite elements? Does it has to do with the spaces over which the trial functions are defined?
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Least squares problem with constrained solution [closed]
If $a_{m\times 1}$ and $Q_{m\times n}$ ($m<n $) are known, and we know every element of $b$ is between $[-1\ \ 1]$, how to determine $b$ to minimize $\|a+Qb\|_2$?
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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 ...
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How to solve Ax=b incrementally ?
Hi, everyone.
What I am struggling is the following problem. I have a linear matrix equation $Ax=b$, where $A$ is a known $n \times n$ large sparse real matrix, $x$ and $b$ are known $n \times 1$ ...
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Approximating a $C^1$ function in $Lip$ norm with piecewise linear
For a continuous function $f:[a,b]\to R$ there is a natural and obvious procedure to approximate it with a sequence of continuous, piecewise linear functions: take $N$ equally spaced points in $[a,b]$ ...
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Unknown bias in a distribution related to prime numbers
If $n$ is composite then $\phi(n) < n-1$, hence there is at least one divisor $d$ of $n-1$ which does not divide $\phi(n)$. We call $d$ as the totient divisor of $n$. Trvially, if $n$ is prime then ...
- 4,638
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Lattice Boltzman derivation for vorticity eqn $\omega_{t}+ v\cdot \nabla \omega=\mu \Delta \omega$
So as showed by Frisch et al. (a), the 2D Euler equation $$v_{t}+ v\cdot \nabla v=\mu \Delta v$$ can be derived by the Hexagonal-placed automaton (for low velocity).
I am curious about the existence ...
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Euler Schemes in Stochastic Differential Equations
So i am trying to understand what happens in Implicit (backward) and Explicit (forward) Euler in Stochastic Differential Equations
I'll start with explicit. Say I have the following SDE known as ...
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votes
2
answers
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Numerical integration using interval arithmetic, nowadays
This is an update to my question Rigorous numerical integration from three years ago.
Is there now a package for rigorous numerical integration that uses interval arithmetic and has access to a well-...
CommunityBot
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1
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PDF and CDF using Gauss-Legendre quadrature
Consider the unit interval $I$ with a continuous probability measure $\mu$, and consider a smooth random variable $f:I\to \mathbb{R}$. We can define its cumulative distribution function and ...
CommunityBot
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votes
1
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$C^\infty$ Periodic Pole-free Rational Interpolation
let $\quad-1=x_0 < x_1 <\ ...\ < x_n<1\quad$ be a set of abscissas
and $\quad(y_0, y_1,\ ...\,y_n)\quad$ a sequence of the corresponding ordinates.
Question:
what can be said ...
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