Questions tagged [na.numerical-analysis]
Numerical algorithms for problems in analysis and algebra, scientific computation
1,272 questions
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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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1
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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
$$ H_d(m) := \left( \frac{B_{i+j+1}(m)}{i+j+1} \right)...
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How to project a function represented by a neural network into finite element spaces
Suppose we trained a neural network to fit a solution of a PDE, but we want to do something in a Finite Element Space, so we need transform our neural network to the latter. What is the way to do this ...
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How can referees verify computationally intensive results when HPC resources are required?
This question is somewhat related to my previous question and is also inspired from this other question concerning the credibility of extensive computations (although from a different perspective).
In ...
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3
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1
answer
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Equivalence of continuous and discrete $L^{1/2}$ "norms"
Recall the iterated Trapezoidal rule of quadrature:
$$
\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).
$$
Recall also the $L^{1/2}$ "norm&...
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1
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How to share algorithms for testing a conjecture?
I am preparing a paper where some results involve computational verification of a conjecture. Of course, I am not proving the conjecture in full, but I verify it for some large values of the involved ...
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489
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Probability that five random unit disks fit in a disk of radius 3
In Ten Digit Problems (in An Invitation to Mathematics: From Competitions to Research, Springer, 2011, pages 119–136), Lloyd Trefethen considers putting disks of unit radius randomly inside a disk of ...
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Solving equations on a high dimensional torus
I want to establish some useful criteria for uniqueness of solutions to the following:
$$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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5
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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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2
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2
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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.$$ With initial condition $V^{\epsilon}_0=g$ and ...
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Synthetic Differential Geometry and Machine Precision: Is there a conceptual link?
This is a naive question, acknowledging a speculative analogy between two formally distinct domains. I aim to explore whether this perspective could be heuristically fruitful.
Synthetic Differential ...
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Convergence rate of Hermitian QR iteration
Suppose $A$ is an $n \times n$ dimensional Hermitian matrix with $\|A\| \le 1$. I now consider the QR algorithm. I set $A_0 = A$ and at the $k$th step compute the QR decomposition $A_k = Q_k R_k$ and ...
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1
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Characterization of the probability density induced by an antisymmetric wave function
Suppose real-valued wave function $\psi(r_1,\dots,r_N)\in L^2$ (or $H^1$) is unnormalized and antisymmetric, that is:
$$\psi(r_{\sigma(1)},\dots,r_{\sigma(N)}) = \text{sgn}(\sigma)\psi(r_1,\dots,r_N),\...
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1
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Confusion in creating bounds over step size in numerical integration for error calculation?
This question is linked to:
Computational complexity of integration in two dimensions
Here in a suggested answer by John Gunnar Carlsson it is mentioned that
$|E| \leq K_1 h^2 |\Omega| M_2$
for some ...
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3
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1
answer
304
views
Can we have finite-dimensional approximations for weak convergence?
Heuristic version of my question:
Can the nearness of two probability measures on a compact hypercube [or other nice compact shape] be approximated by the Euclidean distance of suitable linear ...
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3
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0
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207
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Simple way to simulate SDE solutions?
To simulate an Ito diffusion,
$$dX_t = f(t,X_t)dt + g(t,X_t)dW_t,$$
you can discretize this in time by using an equidistant partition $(t_0^N,t_1^N,\dots,t_N^N)$ of the time interval $[0,T]$. The ...
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Lower bound of the condition number of scaled positive definite matrix
whether we can find postive definite $\{A_k\}$ such that $\frac{\kappa(\lambda_{\max}((I+\alpha D_k)^{-1}(I-\alpha(A_k-D_k))))}{\kappa(A_k)}=\Theta(n)$. Here $A_k\in\mathbb{R}^{n\times n}$ is positive ...
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Efficiently finding spanning independent subsets from overlapping vector sets
We are given a set of $n$ vectors $X := \{x_{1}, \dots, x_{n}\}$, where each $x_{i} \in \mathbb{R}^r$ and $r = O(\log n)$. We also have $n$ subsets $S_{1}, \dots, S_{n} \subset X$ such that $\sum_{i} |...
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Bound minimum singular value of a triangular matrix
For an upper triangular matrix $T$, one can bound from above the minimum singular value with
$$
\sigma_{\min}(T) \leq \min_i |T_{ii}|,
$$
and it is well known that this bound can be very loose; for ...
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1
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146
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Condition number can be arbitrarily worse after diagonally preconditioning?
This question is a generalization of a previous question of mine.
Can we find a symmetric positive definite matrices sequence $\{A_k\}$ with fixed dimension, such that
$$ \frac{\kappa \left( D_k^{-\...
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5
votes
1
answer
481
views
Is there an efficient method to solve this optimization problem on the surface of the sphere?
Let $A$ be a non-singular matrix in $\mathbb{R}^{n \times n}$. Let $S^{n-1}$ denote the surface of the unit $n$-sphere in $\mathbb{R}^{n}$. Suppose we know that there exists an $x \in S^{n-1}$ such ...
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Question about the number of correct digits of $e$ generated by a self-iterated quadratic map
For a fixed $n$, the sequence is defined as $S_n(k+1) = S_n(k) \times S_n(k)$, with $S_n(0) = 2^n + 1$. It's quite obvious that the first $n$ binary digits of $S_n(n)$ match those of $e$, give or take....
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An approach to numerical mathematics using only floating-point numbers from the outset?
When discussing the correctness, accuracy, condition, stability etc. of numerical algorithms, the arguments are usually performed in $\mathbb{R}$. Only when considering stability in the presence of ...
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Rough numerical approximation of the Bessel functions of the first kind
For $x > 0, \alpha > 0 \in \mathbb{R}$, as $x \to \infty$:
$$J_\alpha(x)\sim \sqrt{\frac{2}{\pi x}}\left(\cos \left(x-\frac{\alpha\pi}{2} - \frac{\pi}{4}\right) + \mathcal{O}\left(|x|^{-1}\...
1
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1
answer
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Diagonalizing a linear self-adjoint matrix in a rank-deficient non-standard inner product space
I want to solve $Ax=\lambda x$ in a rank-deficient non-standard inner product space $\mathcal{S}^r$.
By "rank-deficient", I mean that although $x$ is represented by a vector of $\mathbb{R}^n$...
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Runge-Kutta integration on spherical manifold for Neural ODEs (with Tensorflow implementation) [closed]
I am confused about the correct implementation of Runge-Kutta integration on a d-dimensional hypersphere manifold. (Additionally I am locked into using tensorflow and am trying to implement a ...
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12
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2
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1k
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Applications of category theory to numerical analysis?
Category theory has become pretty much the standard language in many "pure" fields such as algebra, topology, algebraic topology ..., and it is also making inroads into some applied fields, ...
1
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0
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71
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Why do error points in spectral differentiation converge in pairs?
I am studying spectral differentiation methods, particularly in the context of periodic differentiation matrices (circulant matrices). I ran a numerical experiment similar to Trefethen’s Spectral ...
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2
votes
0
answers
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views
multivariate integral equation to find a probability density function
I'm working on problem in probability that has led me to a multivariate integral equation:
$$f(x,y,z) = \frac23\int_0^x (1-t)^{-7/3} f\left(\frac{x-t}{1-t},\frac{y}{1-t},\frac{z}{1-t}\right)dt + \...
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Clarification on Calculations for $v_{1,z} $ in "A Method for Verifying the Generalized Riemann Hypothesis"
I am reading the paper "A Method for Verifying the Generalized Riemann Hypothesis" by Ghaith A. Hiary, Summer Ireland, and Megan Kyi. On page 18, $ v_{1,z} $ is calculated using Corollary 9 ...
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What is the most efficient algorithm for computing $E[x_1 x_2 \cdots x_n]$ in a multivariate normal distribution?
I am working with a multivariate normal distribution $\mathbf{x} = [x_1, x_2, \ldots, x_n] \sim \mathcal{N}(\mathbf{\mu}, \mathbf{\Sigma})$, and I need to compute the expectation $E[x_1 x_2 \cdots x_n]...
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1
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Newton method for polynomials with random starting points
I know that this question exists, but unfortunately it doesn't cover my issue sufficiently.
Assume that we have a polynomial $p(x)$ of degree $n$ with real coefficients, we can assume that all its ...
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Convergence of finite-difference method for Cauchy-Riemann equations
Let $I\subseteq \mathbb{R}$ an open interval. Let $f:I\rightarrow \mathbb{C}$ real analytic. Suppose we want to numerically compute an analytic extension of $f$.
We will assume the following: we are ...
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When is a first-order proportional delay differential equation equivalent to a higher-order ordinary differential equation?
The proportional delay differential equation
$$
xf'(x)+2xf'(x/2)+C+4f(x/2)-5f(x)=0
$$
with initial condition $f(0)=C$ expresses that Simpson's rule exactly integrates $f$ over any interval $[0,x]$ and ...
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Method of characteristics, the characteristic lines follow gradient, is this significant?
The PDE I am working on comes from geology, which I do not have much background on.
Said equation aims to describe describes the erosion by describing it as an advection phenomena: the advection ...
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1
vote
1
answer
188
views
Understanding quadrature rule of a function multiplied by another $C^{\infty}$ function
Define a function $f \in C^m[-1,1],m \in \mathbb{N}$ and $g\in C^{\infty}[-1,1]$. Also define a quadrature rule $Q$ for approximating the integral $\int_{-1}^1 h(x)dx$ for some function integrable ...
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2
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212
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Optimization algorithms for Kronecker approximation of high-dimensional covariance matrices
I'm working with a high-dimensional covariance matrix and exploring Kronecker product approximations to make it computationally manageable.
Here's the setup:
I have a graph $G$ represented by a $D\...
- 1
5
votes
0
answers
107
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What is the maximal advantage of randomized over deterministic algorithms for approximation in the worst-case?
Let $X\subset Y$ be Banach spaces and $B_X:=\{x\in X: \|x\|_X\le1\}$ be the unit ball of $X$.
The goal is to find an approximation of every element from $B_X$ with error measured in $Y$ by using at ...
1
vote
1
answer
457
views
Linear interpolation vs L2-projection
I'm reading the book "The Finite Element Method: Theory, Implementation, and Applications" by Larson and Bengzon. In the first chapters there are presented two methods for approximating ...
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4
votes
1
answer
323
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Reference to formal approach to homotopy analysis method
I'm currently reading a book about the Homotopy Analysis Method (HAM), but it isn't very rigorous (it explains most things with a single example), which is bothering me.
I'm searching for papers where ...
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3
votes
1
answer
227
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Handling absolute value and other discontinuities in numerical optimization methods that use gradients
Suppose we have difficult peak fitting problems where the the users wish to fit asymmetric peaks to their experimental data by the least squares method. One such function is illustrated below:
Here
$...
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4
votes
0
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288
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Pricing zero coupon bonds through PDE
I'm currently studying Paul Wilmott on quantitative finance and saw an interesting idea for an interest rate model that went unexplored in the book.
The idea is to model the market price of risk as a ...
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0
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2
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397
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Inversion formula for discrete sine and cosine transforms
$\newcommand{\wh}[1]{{\widehat{#1}}}
\newcommand{\R}{{\mathbb{R}}}
$I am looking for a proof of the inversion formulas for the discrete sine and cosine transforms, i.e. a proof of the fact that these ...
- 113