Top new questions this week:
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$A$ and $B$ are $n \times n$ matrices and $v$ is a vector with $n$ elements. $Av$ has $\approx 2n^2$ flops and $A+B$ has $n^2$ flops. Following this logic, $(A+B)v$ should be faster than $Av+Bv$.
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According to the survey paper on autodiff (linked) Autodiff works on inputs that cannot be specified in closed form but can be described by a sequence of code, each component of which is ...
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I'm trying to compute efficiently the following
\begin{equation}
A_j = \sum_{l'=1}^{\infty}\sum_{k= 0}^{K-1} L_{l'}T_ke^{2\pi i \frac{k}{K}j}\epsilon_{l',k}
\end{equation}
for $j = 0,1, \ldots, ...
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For the qr factorization using classic Gram-Schmidt algorithm, I found the 2 different implementations below. The first one uses the for loop to compute the upper triangular matrix R, but the second ...
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I write a simple MATLAB code for solving solid FEM problem.
The problem looks like that
(1) (2)
x-------x
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x-------x
(3) (4)
Each node has two degrees of ...
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#include <stdio.h>
#include <stdlib.h>
#include <sys/time.h>
float timedifference_msec(struct timeval t0, struct timeval t1) {
return (t1.tv_sec - t0.tv_sec) * 1000.0f + ...
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My system is a symmetric FE problem with lagrange multipliers:
$Z=\begin{pmatrix}A & C^T \\ C & 0\end{pmatrix}$
The matrix $A$ is positive semi-definite, non-invertible. The whole matrix is ...
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Greatest hits from previous weeks:
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I have a function with a number of arguments, but some of them are optional.
1.) If when the function is called, some are left empty, how do I code it so that it defaults to a specific value?
2.) ...
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I asked this question to help me understand what is going on in one of Maxwell's equations. I am happy with following through the maths on paper now, but would like to use MATLAB to take it one step ...
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So I've got a decent head for what problems I work with are best one in serial, and which can be managed in parallel. But right now, I don't have much of an idea of what's best handled by CPU-based ...
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In the wave equation:
$$c^2 \nabla \cdot \nabla u(x,t) - \frac{\partial^2 u(x,t)}{\partial t^2} = f(x,t)$$
Why do we first multiply by a test function $v(x,t)$ before integrating?
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This question arises from the need I have to prepare a lesson on the limitations of Density Functional Theory as a computational approach. I would like to know not only the limitations, but also ...
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I am familiar with the Courant-Friedrich-Lewy Condition in as far as it applies to the stability of explicit finite difference schemes for standard parabolic and hyperbolic PDEs. However, when ...
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I have several challenging non-convex global optimization problems to solve. Currently I use MATLAB's Optimization Toolbox (specifically, fmincon() with algorithm='sqp'), which is quite effective. ...
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Can you answer these questions?
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Can a subfield such as the representation theory of Lie algebras be studied computationally / numerically -- is there an interplay between the abstract and the concrete? I would be grateful for an ...
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The function $\coth(x) - 1/x$ has a removable singularity at 0.
Its Taylor series is:
$$
\coth(x) - 1/x = \frac{x}{3} - \frac{x^3}{45} + \frac{2x^5}{945} + \ldots
$$
I would like to evaluate the ...
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So, I'm currently trying to implement a MCMC to uniformly sampling hyper-points from the polytope defined as $\mathbb{K}=\{x\in\mathbb{R}^{n}\;\;\text{s.t.}\;\; A\,x=b \}$ in the specific case where, ...
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