Questions tagged [na.numerical-analysis]
Numerical algorithms for problems in analysis and algebra, scientific computation
1
vote
4answers
296 views
how to solve $\sum_{i=0}^n (x-\mu_i)e^{-(x-\mu_i)^2} = 0$
I’d like to solve following equation for x.
If it is not possible, why I can’t?
$$\sum_{i=0}^n (x-\mu_i)e^{-(x-\mu_i)^2} = 0$$
3
votes
0answers
33 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
$$
a_g=s_0+\frac{s_1}g+\frac{s_2}{...
1
vote
0answers
80 views
On functions obtained from Gaussian Quadrature integration
Fix an integer $n \ge 2$. Let $x_1,...,x_n$ s and $w_1,...,w_n$ s be the Gauss Quadrature nodes and weights respectively in the interval $[0,1]$ (https://en.wikipedia.org/wiki/Gaussian_quadrature) . ...
4
votes
1answer
58 views
Numerical instability of the axis-angle representation of rotations in 3D
Suppose that I have $1000$ pair of points where each pair consists of a point in $\mathbb{R}^3$ and its image after a rotation in $\mathrm{SO}(3)$ with some noise. I have used RANSAC to find the ...
28
votes
5answers
3k views
Should computer code be included within publications that present numerical results?
Many research papers include numerical results obtained through computation. Most of the time such computations are performed using software that is used by many mathematicians, i.e., Maple, ...
2
votes
1answer
77 views
Variation of steepest descent/Laplace methods for non-exponential integrands
I was wondering if versions of the Laplace/steepest descent methods exists for integrals of the type
$$\int_C f(z) M(\lambda g(z)) dz$$
for $\lambda >>0$ functions $f(z), g(z): \mathbb C \...
2
votes
1answer
123 views
On the continuity and injective-ness of Gauss quadrature scheme for numerical integration, with weight function identically $1$
Fix an integer $n\ge 2$. Let $[a,b]$ be an interval and $f: [a,b]\to \mathbb R$ be a continuous function and for $x_1,...,x_n$ being the Gaussian Quadrature nodes in $[a,b]$, and Gaussian Quadrature ...
2
votes
2answers
98 views
Approximation of half-integers modified Bessel function of the second kind
I am trying to optimise the calculation of the probability distribution poisson-inverse-gaussian
its calculation involves a half-integers modified Bessel function of the second kind. Here's a formula ...
6
votes
1answer
257 views
Celestial mechanics and Runge Kutta methods
I am working on an example here to simulate the orbit of Earth for one year.
As you can see in the notebook, RK45 doesn't conserve energy, and after one simulated year it has spiraled in substantially....
0
votes
0answers
42 views
The limit of infinite ODE solver iteration with zero time step
Suppose I am trying to find a solution of an ordinary differential equation:
\begin{equation}
\begin{aligned}
y'(x) &= f(y(x))\\
y(0) &= y_0
\end{aligned}
\end{equation}
on ...
0
votes
0answers
17 views
Numerical solution of two coupled nonlinear eigenvalue problems
I would like to numerically solve the following system of coupled nonlinear differential equations:
$$
-\frac{\hbar^2}{2m_a} \frac{\partial^2}{\partial x^2}\psi_a + V_{ext}\psi_a +
\left( g_a |...
1
vote
1answer
117 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
78 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
139 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
81 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
160 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
95 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
165 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
30 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
477 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
117 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
138 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
63 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,$$ ...
7
votes
1answer
135 views
Is there any sort of higher-order SVD (quadratic and above) for dimensionality reduction?
(Posted this on math.stackexchange and cross.correlated over more than a week ago, but didn't get an answer, and this is a question in my research so this seems like it might have been the better ...
1
vote
0answers
53 views
Calculation of $\chi^2$ for very small covariance matrices
I produced simulations of data for my experiments, changing each time my initial parameter $\theta$ , $n$ times: $\theta_1\, \theta_2, \cdots \theta_N$. For each realisation of my data I calculated ...
0
votes
0answers
32 views
Unstable convergence of a Poisson equation
What could be the reason that the solution of a Poisson equation is smooth when obtained by an iterative solver, only if the maximum residual is set to a high value (e.g. 0.1)? When the maximum ...
1
vote
0answers
50 views
Reference request on numerical integration
Let $\rho:\mathbb R^d\to\mathbb R_+$ be a density function with finite first moment, i.e.
$$\int_{\mathbb R^d}~ \rho(x)dx~=~1 \quad \mbox{ and }\quad \int_{\mathbb R^d}~ |x|\rho(x)dx<+\infty.$$
...
1
vote
0answers
55 views
Distance between quadric surface and point or Intersection of sphere and quadric surface
I asked a similar question on math.stackexchange, but the answer wasn't quite ideal for my application. Apparently analytic solutions are surprisingly rare for general quadric distances.
Given a ...
3
votes
0answers
43 views
For noisy or fine-structured data, are there better quadratures than the midpoint rule?
Only the first two sections of this long question are essential. The others are just for illustration.
Background
Advanced quadratures such as higher-degree composite Newton–Cotes, Gauß–Legendre, ...
1
vote
0answers
36 views
Lower semicontinuity of proximal gradient descent sequence
I'm using an alternate optimization scheme to minimize a function $F(a,b,c) = G(a,b,c) + H(c)$ where $G$ is continuous and differentiable, $H$ is not differentiable but lower semi-continuous and ...
0
votes
0answers
55 views
Numerical error on the spectrum of a matrix
Let $Q=(q_{j,k})_{1\le j,k\le N}$ be a (Hermitian) $N\times N$ matrix with complex-valued entries. The matrix $Q$ is given numerically and the absolute error on each entry is bounded above by a (small)...
2
votes
1answer
196 views
Minimizing the expectation of a functional of probability distribution subject to an entropy constraint
Consider a PDF $\pi(x)$ for $x\in[0,1]$, and the following functional
$$
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
153 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
101 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)$ ...
3
votes
1answer
76 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
104 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 ...
4
votes
0answers
59 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
62 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 ...
3
votes
1answer
100 views
Propagation error for ODEs
I am looking for a generic estimate to the following problem coming from biology:
I am solving the ODE
$$y'(t)=Ay(t)+zf(t), y(0)=0.$$
where $f$ is an external force determined by us and $z$ a ...
1
vote
1answer
203 views
Are there any numerical packages solving Volterra integral equations?
I am looking for numerical packages (ideally python) to solve second kind Volterra integral equations, such as
$$u(t)=g(t)+\int_0^tK(t,s)u(s) ds$$
or Volterra-Fredholm integral equations
$$u(x,t)=g(...
1
vote
1answer
61 views
Polynomial Eigenvalue Problem with few non-zero coefficients
Let us define a diagonal matrix $\mathbf{D}(\lambda) = diag(\lambda^{m_1}, \dots, \lambda^{m_n})$ with $\lambda\in\mathbb{C}$ and positive integers $m_1, \dots, m_n$. The generalized characteristic ...
0
votes
0answers
31 views
Why are iso-parametric methods popular?
When dealing with complex geometries, one approach is to use iso-geometric mappings for both subdomains and solutions. For example, in the finite element, people use iso-parametric elements (e.g., ...
2
votes
0answers
34 views
Rate of convergence of generalized polynomial chaos
Let $\eta=g(\xi_1,\ldots,\xi_M)$ be a random variable expressed as a function of the random vector $\xi=(\xi_1,\ldots,\xi_M)$. Assume that $\xi_1,\ldots,\xi_M$ are absolutely continuous and ...
4
votes
1answer
183 views
Inverse of matrix with blocks of ones
It seems that there is a nice inverse for matrices that can be written as a diagonal matrix plus a symmetric matrix consisting of scaled blocks of ones.
Consider a real matrix of the form:
$$\begin{...
0
votes
0answers
22 views
Role of polyhedral domain in convergence of finite element method
I am reading a paper by Diening and Kreuzer where they consider the convergence of finite element approximations for $p$-Laplace equation when using a certain algorithm.
In the paper, they assume ...
7
votes
1answer
269 views
Can we estimate the error $\left| \frac{1}{N^2} \sum f ( \{ \sqrt{2} m + \sqrt{3} n \} ) - \int_0^1 f(x) \, dx \right|$?
As a computer experiment I did a few Riemannian sums to see if I could quantify the density statement $\overline{\mathbb{Q}(\sqrt{2}, \sqrt{3})} = \mathbb{R}$ :
$$ \Big| \frac{1}{N^2} \sum_{0 \leq m,...
0
votes
0answers
88 views
Unique root by numerical solution
Let
$$
f(x)=\ln(b-x)-\ln x +(1-c)\ln\Bigl(\frac{\ln(b-x)}{\ln x}\Bigr)-(-\ln x)^c+(-\ln(b-x))^c,
$$
with $c\in (0,1)$ and $x\in(0,b/2)$. Numerical evidence suggests that, for each fixed value of $b$ ...
3
votes
1answer
69 views
Numerical iterative methods for Poisson equation
Given a domain $\Omega \subset \Bbb R^n$ and $\Delta\varphi=f$ where $\varphi:\Bbb R^n \to \Bbb R$ is unknown and $f:\Omega\to \Bbb R$ is a blackbox function (for each $\bf x$ it provides $f({\bf x})$,...
8
votes
3answers
199 views
Regularized linear vs. RKHS-regression
I'm studying the difference between regularization in RKHS regression and linear regression, but I have a hard time grasping the crucial difference between the two.
Given input-output pairs $(x_i,y_i)...
