# Questions tagged [na.numerical-analysis]

Numerical algorithms for problems in analysis and algebra, scientific computation

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### Galerkin scheme in $H^s_0(G)\subset L^2(G)\subset H^{-s}(G)$ ($s>0$)

What basis functions are usually choosen if one attempts to conduct a Galerkin finite element method given an evolution triplet $H^s_0(G)\subset L^2(G)\subset H^{-s}(G)$. Where $G$ is a sufficiently ...
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1 vote
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### Discretization of oscillating integral

Suppose I am interested in computing $$I \equiv \int_0^B dx \, g(x) f(x)$$ where $B$ is a known upper bound for the integral, $g(x)$ is a known oscillating function and $f(x)$ is a smooth function ...
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### Antiderivatives via Taylor series and the FT of Calculus

If $f$ is a real function on an interval $[a,b]$ such that $f$ is computationally tractable on $[a,b]$: you can calculate $f(x)$ to $n$ bits of precision using an algorithm which is polynomial in $n$ ...
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### Concentration of bilinear forms

This is a bit vague so I'll begin by indicating the motivation. I am looking for ways to [do something interesting or useful] with the self-attention in transformer models. Ultimately the self-...
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• 111
1 vote
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### Average distance between points of lower dimensional simplices in $\mathbb R^n$

Notation: By a simplex, we mean the convex hull of a finite set of distinct points in $\mathbb R^n$, which are called the vertices of the simplex. $\mathcal H^n$ will denote the $n$-dimensional ...
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### Is polynomial interpolation with RKHS in some way more advantageous than simple Lagrange interpolation?

[Question originally posted here but maybe it is more suitable for this site.] The reproducing kernel Hilbert space associated with the polynomial kernel $K(x,z)=(1+xz)^{d-1}$ (or other similar ...
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### Inflection point calculation for cubic Bézier curve encounters division by zero

I've been working on finding the inflection points of a cubic Bezier curve using the method described in a paper Hain, Venkat, Racherla, and Langan - Fast, Precise Flattening of Cubic Bézier Segment ...
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• 49
1 vote
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### Practical calculation of Canterbury approximants

I'm looking for references on how to compute Canterbury approximants numerically from a practical point of view. The references on Canterbury approximants that I am aware of all appear rather abstract ...
• 2,758
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### Reshaping data vector into a matrix for deconvolution using a circulant matrix

Suppose we have a circulant matrix S made from pseudorandom binary sequence of length $N$ consisting of $0$'s or/and $1$'s. $1$ means that we can inject something for chemical analysis and $0$ means ...
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1 vote
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### Resolving singularities in numerical integration

I am now trying to compute numerically the following integral.  \begin{split} L_1^s(\hat{\phi}_s)(r,\zeta,\theta_\zeta) &=\frac{1}{\sqrt{2}\pi} \int\limits_{0}^{2\pi} d\varphi \int\limits_0^{\pi}...
• 11
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### Newton type method for finite fields?

I have a polynomial $p(x)$ in $\mathbb{Z}/q\mathbb{Z}$ that is easy to compute for any $x$ but has an absurdly large degree $d > 2^{256}$. I know for a fact that it has a zero and I would like to ...
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### Quadrature error estimates for $n$-rectangular finite elements in the context of elliptic second order problems

In Ciarlet's book "Finite Element Methods for Elliptic Problems" from 1978, in Chapter 4.1 "The Effect of Numerical Integration", the following Theorem is stated and proved: #######...
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### Is quadrature still considered part of numerical analysis?

This question may admittedly sound strange, but having received several desk-rejects (all of them being based on being "out of scope" for the journal in question) from numerical analysis ...
• 2,758
It is well known that using fast Fourier transform it's possible to multiply a vector by a Toeplitz matrix $A \cdot v = w$ in $n\cdot\log(n)$ operations. I read somewhere that also the product of a ...