Questions tagged [na.numerical-analysis]
Numerical algorithms for problems in analysis and algebra, scientific computation
-3
votes
0answers
35 views
Finding a Pattern Between n and c [on hold]
I have a sequence in which I need to relate n (column 1) and c (column 2) with one formula. I found the pattern between the even and odd n's for c, but I need a formula connecting n only to its c.
...
0
votes
0answers
45 views
Runge-Kutta 4th order for predator-prey model [on hold]
I'm trying to compute the numerical solution for a Predator-prey model with 3 equations. This is the model:
$$\frac{dx}{dt} =x(1-\frac{x}{k_1})-\frac{pxz}{1+ax+chy}\\
\frac{dy}{dt} =y(1-\frac{y}{k_2})...
0
votes
0answers
68 views
Why does this numerical scheme work on this nonlinear PDE?
i am currently solving a nonlinear PDE of mixed parabolic/hyperbolic type of the Form
\begin{align*}
\frac{\partial}{\partial x} \left(A \frac{\partial p}{\partial x} \right)
+\frac{\partial}{\...
2
votes
0answers
246 views
How to show that the sinc function series $\sum_{n=-\infty}^\infty\text{sinc}(x+n)$ is equal to pi for all $x$? [closed]
We know the very well-known identity:
$$\sum_{n=-\infty}^\infty\text{sinc}(n)=\pi.$$
But how to show that
$$\sum_{n=-\infty}^\infty\text{sinc}(x+n)=\pi, \qquad \forall x?$$
6
votes
0answers
121 views
Can we approximate any eigenvalue of an infinite matrix via eigenvalues of some sequence of submatrices which approximates the matrix?
Let $T:\ell^2\to\ell^2$ be a compact linear operator. Let $[T]=(a_{i,j})_{i,j=1}^{\infty}$ be the representing infinite matrix of $T$ with respect to the canonical base. Let $T_n$ be the finite rank ...
1
vote
0answers
55 views
explanation of reverse automatic differentiation with language of manifold?
I heard that forward mode automatic differentiation correspond to tangent vectors,and reverse mode ad correspond to cotangent vectors,is this correct? And what is the detailed construction?
2
votes
0answers
47 views
discrete parabolic Harnack inequality
I am currently looking for a discrete version of the parabolic Harnack inequality in the following "$L^1$ to $L^\infty$" form:
If $u(t,x)\geq 0$ is a (say, smooth) subsolution of
\begin{equation}
...
7
votes
2answers
321 views
Is it possible to prove unboundedness of 3rd order ODE?
Consider the 3rd order ODE
$$\dddot{x}+A\ddot{x}-\dot{x}^{2}+x=0$$ where $\dot{x}\equiv \frac{dx}{dt},\ddot{x}\equiv \frac{d^{2}x}{dt^{2}}, etc$. $A$ is a constant.
If we multiply this equation by $\...
2
votes
1answer
32 views
why there is no relaxation method for Jacobi linear system iterative methods?
I found that the relaxation methods for solving linear system as an iterative sequence are derived from the Gauss-Seidel method and not from the Jacobi method. I understand that the Gauss-Seidel ...
0
votes
0answers
25 views
Reference request: Numerical methods for Hamilton-Jacobi-Bellman equations with state constraints
I have two questions on numerical methods for solving Hamilton-Jacobi-Bellman (HJB) equations with state constraints.
Consider an optimal control problem given by
$$
v(x) = \max_{\{u(t)\}_t} \int_o^\...
3
votes
1answer
204 views
A certain generalisation of the golden ratio
Consider a real number $a \ge 1$, and let $g(a)$ be the unique positive solution $x>0$ of $x^a - x^{a-1} - 1 = 0.$
We have $g(1) = 2$, $g(2) = {1+\sqrt{5}\over2}$ (the golden ratio), and $g$ is ...
1
vote
0answers
76 views
When are quadratic integer programs “easy to solve”?
Let $N_i=\{0,1,\dots,\bar{n}_i\}$ and define $N=N_1\times \dots \times N_I$. I want to maximize $f$ on $N$. $f$ has the following form
$$
f(n) = \sum_i A_i n_i -\sum_i \sum_{j\neq i} B_{ij} (n_i-n_j)^...
1
vote
1answer
145 views
Time discretization in the Feynman-Kac formula with boundary conditions
I am applying the Feynman-Kac theory for solving a PDE with boundary conditions.
For the SDE simulation I use the Euler-approximation, which introduces a time-step $h$ for the Brownian Motion, and ...
3
votes
1answer
93 views
Boundary condition for elliptic problems and domain decomposition
This question is motivated by one that has been previously asked on this website: Elliptic problem on a domain split in two subdomains
Consider an open domain $U$ split in two non-overlapping ...
1
vote
4answers
452 views
how to solve $\sum_{i=0}^n (x-\mu_i)e^{-(x-\mu_i)^2} = 0$ [closed]
I’d like to solve following equation for $x.$
If it is not possible, why I can’t?
$$\sum_{i=0}^n (x-\mu_i)e^{-(x-\mu_i)^2} = 0$$
4
votes
0answers
67 views
Convergence acceleration of successions with logarithms
I have a numerical question regarding acceleration of a succession.
A preliminary: suppose that I have a succession $a_g$ that, for high $g$, asymptotically goes as
$$
a_g=s_0+\frac{s_1}g+\frac{s_2}{...
1
vote
0answers
82 views
On functions obtained from Gaussian Quadrature integration
Fix an integer $n \ge 2$. Let $x_1,...,x_n$ s and $w_1,...,w_n$ s be the Gauss Quadrature nodes and weights respectively in the interval $[0,1]$ (https://en.wikipedia.org/wiki/Gaussian_quadrature) . ...
4
votes
1answer
69 views
Numerical instability of the axis-angle representation of rotations in 3D
Suppose that I have $1000$ pair of points where each pair consists of a point in $\mathbb{R}^3$ and its image after a rotation in $\mathrm{SO}(3)$ with some noise. I have used RANSAC to find the ...
30
votes
5answers
3k views
Should computer code be included within publications that present numerical results?
Many research papers include numerical results obtained through computation. Most of the time such computations are performed using software that is used by many mathematicians, i.e., Maple, ...
3
votes
1answer
107 views
Variation of steepest descent/Laplace methods for non-exponential integrands
I was wondering if versions of the Laplace/steepest descent methods exists for integrals of the type
$$\int_C f(z) M(\lambda g(z)) dz$$
for $\lambda >>0$ functions $f(z), g(z): \mathbb C \...
2
votes
1answer
133 views
On the continuity and injective-ness of Gauss quadrature scheme for numerical integration, with weight function identically $1$
Fix an integer $n\ge 2$. Let $[a,b]$ be an interval and $f: [a,b]\to \mathbb R$ be a continuous function and for $x_1,...,x_n$ being the Gaussian Quadrature nodes in $[a,b]$, and Gaussian Quadrature ...
2
votes
2answers
102 views
Approximation of half-integers modified Bessel function of the second kind
I am trying to optimise the calculation of the probability distribution poisson-inverse-gaussian
its calculation involves a half-integers modified Bessel function of the second kind. Here's a formula ...
6
votes
1answer
285 views
Celestial mechanics and Runge Kutta methods
I am working on an example here to simulate the orbit of Earth for one year.
As you can see in the notebook, RK45 doesn't conserve energy, and after one simulated year it has spiraled in substantially....
1
vote
1answer
125 views
Expected value of sin(X) for Gamma r.v. X in closed form (approximation is fine)
I have a random variable $X \sim \operatorname{Gamma}(\alpha, \beta)$.
How can I compute or approximate $\mathbb{E} \sin(X)$ very quickly? Iterative quadrature would be too slow, I need some closed ...
3
votes
0answers
82 views
Partial Liouville equation
In my master's thesis, I worked on mathematical multi-scale models for muscle tissue. Now after finishing it, I would like to find out if one direction could be a research topic for my PhD.
At one ...
6
votes
1answer
141 views
Adding constraints as penalty with $\| \cdot \|_0$ norm
In the paper Image Denoising Via Sparse and Redundant Representations Over Learned Dictionaries (page 2), the authors rewrite the minimization problem
\begin{align}
\min_{\alpha \in \mathbb R^k} \| \...
1
vote
1answer
90 views
Accurate estimate/lower bound for the l2 approximation error of trigonometric polynomial approximation
This is a restated version of my original very broad question.
Let $P$ be probability a measure on an interval $[a,b]$ ($-\infty<a<b<\infty$) that's dominated by Lebesgue measure. Let $\...
1
vote
0answers
28 views
Normed Scalar Error/Objective Function vs Vectorial Error/Objective Function in Newton Raphson
I am working on an harmonic balance solver which involves finding the zeros of the objective/error function using the Newton Raphson method. Presently I am using a vectorial error function but I have ...
1
vote
2answers
169 views
A min-max approximation
Let $n\ge 1$ be an integer, $\mathcal P_n$ be the vector space of all polynomial functions over $[a,b]$, of degree at most $n$.
My question is : Is it true that
$$\inf_{x_0,x_1,...,x_n\in[a,b], x_0&...
1
vote
1answer
111 views
Any inequalities / estimates for a lower bound of the $L^2$ inner product of a quantity and its derivative?
For some numerical analysis of a fluid, I am wondering if there is any inequality that provides a lower bound (in the $L^2$ norm), for the $L^2$ inner product of a quantity with its derivative. In ...
3
votes
2answers
171 views
Find parameter values for which a 3x3 matrix has a triple eigenvalue
An Exceptional point generally occurs in eigenvalue problems in which the matrix is dependent on some parameter(s). The particular point in which the eigenvalues become degenerate for the parameter(s) ...
1
vote
0answers
31 views
Point-wise invertible non-linearity to reduce matrix rank [closed]
Suppose $A$ is a matrix, and its rank-$r$ SVD approximation looks like $A \approx U \Sigma V^\top$.
I want to apply some invertible point-wise non-linear function $f$ and apply it to $A$ to make a new ...
15
votes
1answer
484 views
How can I distinguish a genuine solution of polynomial equations from a numerical near miss?
Cross-posted from MSE, where this question was asked over a year ago with no answers.
Suppose I have a large system of polynomial equations in a large number of real-valued variables.
\begin{align}
...
6
votes
1answer
121 views
Any convergence rule for ${\mathbf X}_k={\mathbf A}{\mathbf X}_{k-1}{\mathbf B}$?
We know iteration ${\mathbf X}_k=\mathbf{A}{\mathbf X}_{k-1}$ converges if the spectral radius of $\mathbf A$ is smaller than 1 (see here). Is there any known rule for iteration ${\mathbf X}_k={\...
3
votes
1answer
149 views
Exponential map/ Lie derivative in variation for constant formula for ODE
In short: The question is how to go from the first equation on page 8, of this paper to the second equation.
Some background
I'm working in optimization and I am currently reading a paper
see page ...
0
votes
1answer
71 views
Computing spectrum of convex combination of SPD matrices given individual spectral decompositions
Given the spectral decompositions of a non-commuting collection of symmetric positive definite $N\times N$ matrices $$\left\{ K_{i}\right\} _{i=1}^{M}, U_{i}D_{i}U_{i}^{T}=K_{i},\quad i=1,\dots,M,$$ ...
7
votes
1answer
147 views
Is there any sort of higher-order SVD (quadratic and above) for dimensionality reduction?
(Posted this on math.stackexchange and cross.correlated over more than a week ago, but didn't get an answer, and this is a question in my research so this seems like it might have been the better ...
1
vote
0answers
54 views
Calculation of $\chi^2$ for very small covariance matrices
I produced simulations of data for my experiments, changing each time my initial parameter $\theta$ , $n$ times: $\theta_1\, \theta_2, \cdots \theta_N$. For each realisation of my data I calculated ...
1
vote
0answers
55 views
Reference request on numerical integration
Let $\rho:\mathbb R^d\to\mathbb R_+$ be a density function with finite first moment, i.e.
$$\int_{\mathbb R^d}~ \rho(x)dx~=~1 \quad \mbox{ and }\quad \int_{\mathbb R^d}~ |x|\rho(x)dx<+\infty.$$
...
1
vote
0answers
63 views
Distance between quadric surface and point or Intersection of sphere and quadric surface
I asked a similar question on math.stackexchange, but the answer wasn't quite ideal for my application. Apparently analytic solutions are surprisingly rare for general quadric distances.
Given a ...
3
votes
0answers
54 views
For noisy or fine-structured data, are there better quadratures than the midpoint rule?
Only the first two sections of this long question are essential. The others are just for illustration.
Background
Advanced quadratures such as higher-degree composite Newton–Cotes, Gauß–Legendre, ...
1
vote
0answers
38 views
Lower semicontinuity of proximal gradient descent sequence
I'm using an alternate optimization scheme to minimize a function $F(a,b,c) = G(a,b,c) + H(c)$ where $G$ is continuous and differentiable, $H$ is not differentiable but lower semi-continuous and ...
0
votes
0answers
56 views
Numerical error on the spectrum of a matrix
Let $Q=(q_{j,k})_{1\le j,k\le N}$ be a (Hermitian) $N\times N$ matrix with complex-valued entries. The matrix $Q$ is given numerically and the absolute error on each entry is bounded above by a (small)...
2
votes
1answer
210 views
Minimizing the expectation of a functional of probability distribution subject to an entropy constraint
Consider a PDF $\pi(x)$ for $x\in[0,1]$, and the following functional
$$
F(\pi) = \mathbb{E}_\pi |x-y| $$
It is minimized by any point mass, so to avoid such degeneracy I'd like to lower-bound the ...
2
votes
1answer
160 views
Integration of hypergeometric product for legendre polynomials
I'm looking for a general solution to the integral:
$\int_{0}^1 {_2}F_1(m-n,m+n+1,m+1;z){_2}F_1(m-k+1,m+k+2,m+2;z) dz$
where $m,n,k\in \mathbb{N}$ and $m\leqslant n$ and $m+1 \leqslant k$.
To give ...
5
votes
1answer
104 views
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 $P_m(z)e^z-Q_n(z)$ ...
2
votes
1answer
126 views
Lagrangian interpolation at Chebyshev points - estimate on coefficients in monomic basis
First, let us fix some Notation:
Let $n\in\mathbb{N}$ and $x_i=\cos(\tfrac{(i+1/2)\pi}{(n+1)})$, $i=0,\dots,n$, be the Chebyshev points. Let
\begin{align}L_i(x)={\displaystyle\prod_{\substack{0\leq j\...
1
vote
1answer
120 views
Compute the eigenvectors corresponding to the $k$ smallest eigenvalues w.r.t high dimensional symmetric sparse matrix?
So far as I know:
The power iteration method can only get the eigenvector corresponding to the largest eigenvalue;
The inverse power iteration method requires that the matrix is invertible;
The QR ...
5
votes
0answers
63 views
Numerical and computational approaches to limit cycle theory
I am searching for a big list of papers or researches devoted to limit cycles of planar polynomial vector fields based on a computational and numerical approach.
I would like to ask ...
0
votes
1answer
64 views
A System of generalized Abel's integral equation
Is there a method for solving the following system of generalized Abel's integral equation:?
$(x^2 -1)\int_0^x \frac{u(t)}{(x-t)^{\frac{1}{2}}}\; dt + x\int_0^x \frac{v(t)}{(x-t)^{\frac{1}{3}}}\; dt ...
