Most Popular
1500 questions
7
votes
2 answers
Is it possible to represent non-linear ranking type constraints as equivalent linear constraints?
I have formulated a linear program with binary indicator variables $z_i(a)$ which is equal to $1$ if the $i^{th}$ document is of rank $a$ and $0$ otherwise.
The other variables in the linear program, $z^1_{ij}(a), z^2_{ij}(a)$ are defined as…
stressed_geek
- 319
- 1
- 2
- 5
7
votes
1 answer
What is the scaling or order of molecular dynamics (MD) simulations?
Often in computational science, we talk about the scaling or order of a particular method ($\mathcal{O}(N)$, $\mathcal{O}(N^2)$, $\mathcal{O}(N \log N)$, etc.).
I am having a really difficult time finding a resource (book, journal article, or…
Andrew
- 711
- 1
- 5
- 12
7
votes
1 answer
Examples of problems that cannot be formulated as optimization problems
An optimization problem has 3 main components: decision variables, constraints, and an objective function. Such a problem can be mathematically modelled and solved using an optimization solver. For example, a SUDOKU puzzle can be modelled as such…
progammer
- 171
- 3
7
votes
2 answers
Algebraic multigrid for complex valued matrices
Assume one uses the classical AMG with Ruge-Stuben coarsening and direct interpolation for solving real valued problems. How can this approach be recycled to also solve complex valued problems like every harmonic simulation?
I have read in papers…
vydesaster
- 840
- 4
- 11
7
votes
1 answer
Lack of quadratic convergence in Newton's method
It is well-known that Newton's method can converge quadratically, if initial guess is close enough and if the arising linear systems are solved accurately.
I am applying Newton's method to highly ill-conditioned systems (with condition number around…
computational_scientist
- 178
- 1
- 7
7
votes
0 answers
Quadrature methods for peaky integrands
Consider
$$
I = \int_{-L}^L f(x)dx,
$$
where $f(x)$ is real-valued and analytic on $[-L,L]$, but it has a pole in the complex plane whose real part lies in $[-L,L]$. Call it $z_0$, and assume it is a simple pole. If the imaginary part of $z_0$ is…
Endulum
- 735
- 3
- 9
7
votes
0 answers
Solving a coupled eigen value problem
I have the following problem:
$$\begin{bmatrix}A &B \\C& D\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}\lambda I_m & 0 \\ 0& \mu I_n\end{bmatrix}\begin{bmatrix}x \\y\end{bmatrix}
$$
where $A$, $B$, $C$, $D$ are matrices, $I_m,I_n$…
xadu
- 171
- 2
7
votes
1 answer
Putting N hard spheres randomly in given volume
I need to put $N$ spheres with given radius $R$ randomly in a Volume $[-0.5,0.5]^3$, without any overlap of spheres.
If I choose values so that all the spheres will occupy ~57% of the total volume, I find it difficult to get results.
The basic…
reloh100
- 153
- 6
7
votes
1 answer
Termination Criterion of Golden Section Search
Reading an implementation of the golden section search, I came across the following termination test:
$$| a - d | < \varepsilon ( |b| + |c| )$$
where $a < b < c < d$ are four points at which the function has been probed and $\varepsilon$ is a…
Christopher Johnson
- 471
- 3
- 9
7
votes
1 answer
Automatic differentiation of barycentric rational functions
By a barycentric rational interpolant we understand a function of the form
\begin{align*}
r(t) := \frac{\sum_{i=0}^{n-1} \frac{w_i y_i}{t-t_i} }{ \sum_{i=0}^{n-1} \frac{w_i}{t-t_i}}
\end{align*}
In Schneider and Werner, (proposition 12) a stable…
user14717
- 2,155
- 13
- 14
7
votes
1 answer
Least Squares with Dense-Block Diagonal Structure
I need to solve a least squares problem that takes the following form:
$$p = \arg \min_{x}\Vert J V x - y \Vert_2, $$
where $J \in \mathbb{R}^{N \times N}$ is a general dense matrix, and $V \in \mathbb{R}^{N \times m}$ is a block-diagonal matrix…
Spenser
- 73
- 6
7
votes
2 answers
Can all eigenvalues of a Hermitian Toeplitz matrix be computed in $\tilde{O}(n)$ time?
I know there are "superfast" $O(n \log^p n)$ algorithms for solving Toeplitz linear systems. Is it possible to compute all eigenvalues of such a matrix with the same complexity?
Geoffrey Irving
- 3,969
- 18
- 41
7
votes
2 answers
Recommendations for a usable, fast GPL-compatible derivative-free numerical optimization library that can be interfaced to C++
I am dealing with optimization of functions for which I do not have derivatives available, and the optimization is not constrained. I am searching for a high quality GNU Public License-compatible optimization library compatible with the C++…
tmaric
- 1,916
- 1
- 11
- 22
7
votes
2 answers
Is large condition number good measure of nearness to singularity for a matrix?
I am new to numerical linear algebra, so i came to know that condition number in 2-norm case will be ratio of largest to smallest singular value.
Another concept "Nearness To Singularity" is measured using this number being large.
But consider the…
Siddharth Shakya
- 173
- 4
7
votes
2 answers
What good are hard-sphere event-driven molecular dynamics simulations in the face of chaos?
Simple hard-sphere dynamical systems can exhibit chaotic dynamics. Due to finite-precision arithmetic when implemented on a computer, the presence of chaos implies that for a given set of initial data, simulations of such systems on different…
joshphysics
- 171
- 4