# Questions tagged [na.numerical-analysis]

Numerical algorithms for problems in analysis and algebra, scientific computation

**1**

vote

**4**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**5**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**2**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**3**answers

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)...