Questions tagged [na.numerical-analysis]
Numerical algorithms for problems in analysis and algebra, scientific computation
1,268 questions
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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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votes
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Equivalence of continuous and discrete $L^{1/2}$ "norms"
Recall the iterated Trapezoidal rule of quadrature:
$$
\int_0^1 f(x) \, dx \approx I_n f := {1 \over 2n} \left(f(0) + f(1) + \sum_{k=1}^{n-1} 2f(k/n) \right).
$$
Recall also the $L^{1/2}$ "norm&...
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How to share algorithms for testing a conjecture?
I am preparing a paper where some results involve computational verification of a conjecture. Of course, I am not proving the conjecture in full, but I verify it for some large values of the involved ...
- 1,359
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Probability that five random unit disks fit in a disk of radius 3
In Ten Digit Problems (in An Invitation to Mathematics: From Competitions to Research, Springer, 2011, pages 119–136), Lloyd Trefethen considers putting disks of unit radius randomly inside a disk of ...
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Solving equations on a high dimensional torus
I want to establish some useful criteria for uniqueness of solutions to the following:
$$Mx=b,\\ \text{subject to}\ ||x(2k-1:2k)||=1, k=1,2,\cdots,5,$$ where $M\in\mathbb{R}^{10\times10},\ x\in\mathbb{...
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Lebesgue vs Riemann numerically
Let $f : [0,T] \to \mathbb R$ be a continuous function. We are interested in computing the integral
$$
I_{\mathrm{Riemann}} := \int_0^T f(t)\,dt,
$$
which is the standard Riemann integral. ...
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3
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Solution space of linear and quadratic equations
Denote $x=[x_{1},x_{2},\cdots,x_{n}]\in\mathbb{R}^{nd}$, where $x_{i}\in\mathbb{R}^{d}$ for $i=1,2\cdots,n$. Suppose a matrix $A\in\mathbb{R}^{k\times n}$,$B=A\otimes I_{d}\in\mathbb{R}^{kd\times nd}$ ...
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How to check if numbers are close to be zeros of a polynomial?
I've got a lot of polynomials presented in the basis $(1, X, X^2, \cdots)$ and their corresponding zeros in a Python file. I would like to check that there is no mistake in these data, i.e., I would ...
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Reducing boundary artifacts in discrete‐Fourier (integer or fractional) derivatives
I am interested in calculating integer and fractional derivatives of a experimental data using discrete Fourier transform. There is a paper Calculating numerical derivatives using Fourier transform: ...
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Practical applications where one needs $L^p$ with $p\not\in\{1,2,\infty\}$
It appears to me that in practical applications one only ever needs the $L^1$, $L^2$ and $L^\infty$ norms, which are rather special cases among the $L^p$ norms. However, I am virtually sure that this
...
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Taking limits of Gaussian Processes - call for references
I'm investigating a particular topic and I'd like to get some references on it.
The idea is as follows: pick some natural $d$ and let $\mathcal{F}_d$ be a Gaussian Process on $\mathbb{R}^d$ with mean ...
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Handling defective eigenvalues in shifted block Lanczos algorithm
I’m implementing a shifted version of the block Lanczos algorithm, following the approach described in the paper by Lewis, Simon, and Grimes , to solve generalized eigenvalue problems. My ...
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Bounded discrete derivative for heat equation
Consider a heat equation on a discrete box B=$[0,1]\times[0,1]\cap\epsilon\mathbb{Z}\times\epsilon\mathbb{Z}$:$$\partial_tV_t=\Delta^{\epsilon}V_t.$$ With initial condition $V^{\epsilon}_0=g$ and ...
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Synthetic Differential Geometry and Machine Precision: Is there a conceptual link?
This is a naive question, acknowledging a speculative analogy between two formally distinct domains. I aim to explore whether this perspective could be heuristically fruitful.
Synthetic Differential ...
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Convergence rate of Hermitian QR iteration
Suppose $A$ is an $n \times n$ dimensional Hermitian matrix with $\|A\| \le 1$. I now consider the QR algorithm. I set $A_0 = A$ and at the $k$th step compute the QR decomposition $A_k = Q_k R_k$ and ...
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