Questions tagged [na.numerical-analysis]
Numerical algorithms for problems in analysis and algebra, scientific computation
1,280 questions
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A problem about the vector-valued error estimate for multivariate Hermite interpolation
I am reading an Hermite interpolation method on manifold which is in Section4 in HERE, the core idea is as follows.
Set $dim(\mathcal{M})=m$. We construct an interpolation $\hat{f}_{\tan }: \mathbb{R}^...
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Efficient construction of an orthogonal matrix with prescribed diagonal quadratic forms
Let $\boldsymbol{\lambda}=(\lambda_1,\dots,\lambda_n)$ and $\mathbf{d}=(d_1,\dots,d_n)$ be two vectors of real numbers sorted in non‑increasing order, satisfying the majorization conditions
$$
\sum_{i=...
- 4,190
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Approximation of multivariant Lipschitz functions by piecewise linear functions
Let $f\colon \mathbb R^n \to \mathbb R$ be a Lipschitz functions.
Is it possible to approximate $f$ by piece-wise linear functions $f_k$ with almost the same Lipschitz constant as $f$?
More precisely, ...
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Sharper concentration bounds for Gram matrices of Müntz–Szász systems?
Let $\mathcal{D}_K = \{x^\alpha\} \cup \{p_k\}_{k=0}^K$ be a dictionary on $[0,1]$, where $p_k$ are the orthonormal Jacobi polynomials with respect to the measure $d\mu = x^\beta dx$ (with $\beta > ...
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How many iterations to solve this 'ultimate challenge' equation?
I've been benchmarking root-finding methods on this equation :
$$f(x) = e^{\sin(10x)} \cdot \arctan(100x) + \ln(x+1) \cdot \cos\left(\frac{1}{x+0.1}\right) - 5$$
Properties :
Contains exponentials, ...
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Reducing multivariate root-finding to univariate root-finding
This question has to do with the numerical solution of nonlinear systems of equations (not differential equations).
Let me give you some background. In computational fluid dynamics, the state of the ...
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Asymptotic behavior of an alternating multidimensional harmonic sum series
Motivation: As a personal side project I have been working with an inclusion-exclusion formulation that is counting weighted powers $x^a$ between consecutive squares $[n^2, (n+1)^2]$. The function $f(...
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Interpolation Stability
I have a mapping $u:\Omega_t\to H^1_0(\Omega_x)$ where $\Omega_t\subset \mathbb{R}^N$ is bounded, convex with Lipschitz boundary and $\Omega_x\subset \mathbb{R}^d$ is bounded, convex with Lipschitz ...
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Which features of integer approximation error in nonlinear activations matter for training stability?
When approximating nonlinear activations in integer arithmetic, which error features actually matter for training stability: peak magnitude, symmetry, or localization?
This is in the context of end-to-...
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How to efficiently calculate higher logarithmic derivatives of the Riemann zeta function?
I know how to calculate higher derivatives of the Riemann zeta function $$\left(\frac{d}{ds}\right)^i\zeta(s)=\left(\frac{d}{ds}\right)^i\sum_{n=1}^{\infty}\frac1{n^s}=\sum_{n=1}^{\infty}\frac{(-\log ...
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Simple GMRES and standard GMRES coefficient matrices
I am studying Homer F. Walker and Lu Zhou's "A simpler GMRES", Numer. Linear Algebra Appl. 1, No. 6, 571-581 (1994), MR1310986, Zbl 0838.65030, and if I understand this correctly, what one ...
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Decomposition $PLUP^{−1}$
This is a cross-post as I didn't get any answer. Let $A\in\mathrm{M}_n(\mathbb{C})$. If the leading principal minors (namely the determinants of the top left submatrices) of $A$ are non-zero, then ...
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Define a domain of ball in finite element anlaysis using fenics
It seems strange to me that, in Fenics, there is no available API to define a domain of a ball. What is the optimal way to define this ? I checked the web but didn't find a clear solution.
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Is this Hankel matrix involving Bernoulli polynomials positive definite?
Let $(B_n(X))_{n \ge 0}$ denote the sequence of Bernoulli polynomials. For any couple of integers $(d,m) \in \mathbb{N}$, define the Hankel matrix
$$ H_d(m) := \left( \frac{B_{i+j+1}(m)}{i+j+1} \right)...
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How can referees verify computationally intensive results when HPC resources are required?
This question is somewhat related to my previous question and is also inspired from this other question concerning the credibility of extensive computations (although from a different perspective).
In ...
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