Questions tagged [na.numerical-analysis]
Numerical algorithms for problems in analysis and algebra, scientific computation
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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 ...
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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 ...
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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 ...
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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 ...
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Controlling the constant in central difference schemes
Let $d,k$ be positive integers. For each $\delta>0$ and every $i\in \{1,\dots,k\}$ define the finite (central) difference operator $\Delta_i:C(\mathbb{R}^d)\to C(\mathbb{R}^d)$ by
$$
\Delta_i^{\...
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Does Monotone (linear) convergence of iterates imply monotone (linear) convergence of function values?
I am considering a proof that would require a certain connection between convergence of iterates and corresponding function values: Consider an algorithm with iterates $\left\{{\mathbf{x}}^k\right\}_{...
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Finding measure representation for rank 2 moment matrices
Assuming the following equation has a solution, I'm interested in finding any concrete values of $x_{1},\dots x_{n},y_{1},\dots y_{n},c_{1},c_{2},R$ that fulfills it.
$$
\begin{bmatrix}
1 & 1 \\
...
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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 \...
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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 ...
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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 ...
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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 $F(x)$ ...
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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 ...
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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 ...
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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 ...
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Can numerical differentiation be applied to tensor derivatives?
I know that for a 1D function, I can calculate the numerical derivative at every point, (x1,y1), with dy/dx where dy = y2 - y0 ...
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