Questions tagged [algorithms]
A description of the specific steps needed to solve a particular problem in an unambiguous way, expressed in an abstract form.
533
questions
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1answer
51 views
Fast algorithm to compute chi-square
I would like to evaluate the chi-square of the form $\chi^2=v^{T}C^{-1}v$ where $v$ is a column vector and $C$ is a covariance matrix. Both $v$ and $C$ are known and $C$ is a $740\times740$ matrix. ...
1
vote
1answer
36 views
Why does this implementation for Eisenstein integer pairs of Euclid's method for finding greatest common denominators get stuck for this one point?
My Math SE question determining if a coincident point in a pair of rotated hexagonal lattices is closest to the origin? explains the problem I have. I won't reproduce the whole thing in detail here, ...
2
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0answers
35 views
Cellular automata - can scaling factors that influence probability of spread be applied to rate of spread?
This is a follow-up question to my earlier question Stochastic cellular automata - algorithm limited by 1 cell per timestep. I am considering blending two approaches to a cellular automata model of ...
2
votes
1answer
51 views
Finding all valid combinations of numeric inputs and operators in a Reverse Polish Notation expression
An arithmetic expression written in Reverse Polish (postfix) Notation is an ordered list of numbers and algebraic operators, which are sequentially evaluated as a stack would process them to return a ...
1
vote
1answer
146 views
Schrodinger's Equation differential
I am working on a modified version of Schrodinger's equation (time-independent) where $\frac{d^2ψ}{dx^2}=-2(E-V)ψ$, where I have to consider $V = 0$ at all times.
I have been asked to use Python in ...
1
vote
2answers
105 views
Find the smallest convex hull that enclose an arbitrary point
Given a set of 2D points, I am trying to find the smallest convex hull that encloses an arbitrary point (which, in the general case, is not part of the set). By 'smallest convex hull' I am ideally ...
4
votes
1answer
80 views
Worst Case complexity of a search engine algorithm
Computer make it possible to find information in large databases. However, the results are often too large to be returned in their entirety to the user who requests them. Computer therefore sort the ...
12
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2answers
1k views
How do I find the minimum-area ellipse that encloses a set of points?
I have a set of points that resembles more of an ellipse than a circle. I implemented the optimization formulation below and the solution gives a circle. I tried with various initial values, still to ...
0
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1answer
150 views
Could we train an AI to find (only) Mersenne primes and beat the current record?
Could we train an AI to find only Mersenne primes and beat the current record $2 ^ {82 589 933} - 1$?
A Mersenne number is a number of the form $2 ^ n - 1$ (where n is a non-zero natural number), a ...
2
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1answer
153 views
How to interpret if $\displaystyle \sum_{j = 0}^{n} \frac {1}{j!}$ is a stable algorithm for computing $e$?
I am trying to solve problem $15.1$ from Numerical Linear Algebra by Trefethen and Bau, which reads
Determine whether the algorithm is backward stable, stable but not backward stable, or unstable.
...
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1answer
695 views
Why is subtraction a stable operation?
In Numerical Linear Algebra by Trefethen & Bau, it is claimed that subtraction is backward stable. Here is the proof:
Let $f(x, y) = x-y$ and let $\tilde f(x,y)$ be the answer you get when doing $...
6
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1answer
149 views
General approach to infinite sums
My question is specific to algorithms and models of computation.
I would like to write code to evaluate the following expression quickly and accurately:
$$\log \left( \sum_{i=1}^{\infty}{I_{\nu+i}(2\...
1
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0answers
110 views
Fast evaluation of trigonometric polynomials
Suppose you have a trigonometric polynomial of the form
\begin{equation*}
x(t) = \sum_{k = 0}^N a_k \cos(2 \pi k f_0 t).
\end{equation*}
Using Clenshaw algorithm, one can evaluate this polynomial in $...
2
votes
0answers
66 views
Multigrid method: linear solver and modified residual
I am trying to better understand the FAS multigrid algorithm for Euler equation in FV discretization. The usage of the modified residual (the residual with forcing) inside the different cases:
...
1
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1answer
50 views
Resampling an array of objects
Context
I have an array of objects (or a list of dictionaries), sorted in order based on a property of each object, say, time. In JSON, it would look something ...
3
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0answers
107 views
Help with restart functionnality in sef-made GMRES solver in python
I am new to this forum and to computational science in general.
I started to learn numerical liner algebra on my own and would like to code a GMRES solver in python (no preconditioner for the time ...
2
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1answer
40 views
Simulate circular mold spread using cellular automata - square emerges instead
I am trying to simulate the spread of mold in a petri dish using a cellular automata based approach. Thanks to the answer in my other question Stochastic cellular automata - algorithm limited by 1 ...
2
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0answers
26 views
Scheduling Algorithm Goal is to Fix Max Number of Appointments
First post. Hi everyone.
I’m trying to develop an algorithm to later code that schedules appointments in a way such that the number of hours occupied by the appointments in a given day is maximized.
...
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0answers
83 views
What is the limit involving `Sum`, `Subsets`, and `RankedMax` as `t` approaches infinity?
Motivation
Suppose we have a countably infinite $A$ with order and group structures and suppose $F_1,F_2,\cdot\cdot\cdot$ are an infinite sequence of finite sets (denoted $\left\{F_n\right\}_{n=1}^{\...
9
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1answer
220 views
Is there a way we can compute my sum involving `subsets` more efficiently?
Suppose we have a countably infinite $A$ and $F_1,F_2,\cdot\cdot\cdot$ are an infinite sequence of finite sets (denoted $\left\{F_n\right\}_{n=1}^{\infty}$) such that $\bigcup\limits_{n=1}^{\infty}F_n=...
4
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1answer
72 views
Stochastic cellular automata - algorithm limited by 1 cell per timestep
Context
Let's say I am trying to model the spread of mold in a petri dish, using a stochastic cellular automata approach. The petri dish can be thought of as a grid of 1mm x 1mm squares, each called ...
2
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1answer
72 views
Dividing a continuous domain into small squares; how to perform storage and querying?
I recently had a software engineering interview and was asked a series of questions that was a bit outside of knowledge realm, and I feel like there's some scientific computing principles here (I took ...
0
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1answer
69 views
Looking for Partinioning Algorithms allowing for Constraints
What algorithms exist that partition the domain according to a black box evaluation function (possibly subject to some assumptions)?
Examples
Simple Example
To better exaplain we consider as our ...
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1answer
79 views
In Lanczos algorithm, can we choose the staring vector to be the first eigenvector of the input matrix A?
In Lanczos algorithm, can we choose the staring vector $v$ to be the first eigenvector of the input matrix $A$? How can we select it? and why $v$ need to have norm 1?
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24 views
Do the class of Construction Heuristic that does or does not change previous decisions have a name?
I'm writing a paper where I am discussing different types of construction heuristics.
One type does not change previous decisions when adding new elements to the solution. I'd call them "...
1
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0answers
53 views
Ising model in Python (Magnetization Scaling)
I am trying to implement the Ising Model in Python for Gibbs Distribution:
$$\pi(x) = \frac{1}{Z(\beta)} e^{-\beta H(x)}$$
\begin{align*}
p(x,y)&=r(x,y) \cdot \min \left( \frac{\pi(y)}{\pi(x)},1 \...
4
votes
3answers
185 views
Algorithms to generate spherical codes
A spherical code, specified by the parameters $(n,N,t)$, is a set of $N$ coordinates on the $n$-dimensional unit hypersphere such that the set of dot products between any two unit vectors from the ...
0
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1answer
239 views
Trouble Implementing 1d Wave Equation Finite Difference Solver
Im trying to solve the 1d Wave Equation on $x \in \mathbb{R}, t > 0$: $$u_{tt} = c^2u_{xx}, \hspace{5mm} u(x,0) = \cos(4 \pi x), \hspace{5mm} u_t(x,0) = 0$$ with $c = 1$ and a periodic boundary ...
3
votes
1answer
83 views
Maintain unitary time evolution for a nonlinear ODE
I want to solve a nonlinear ODE of matrix $A(t)$
$$\mathrm{i}\dot A = A(t)M(t),\:\mathrm{with}\: M(t)=A^\dagger(t)H(t)A(t)$$ where $H(t)$ and hence $M(t)$ are Hermitian. Therefore, I presume the time ...
0
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2answers
49 views
Technique or Pattern to calculate conditional statement
I am attempting to create a conditional statement that compares four (4) true/false conditions. Depending on the state of these four conditions (either true or false) the conditional statement will ...
4
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1answer
126 views
Algorithm to merge two polygons (using connectivities)?
I am struggling with implementing an algorithm that does one simple thing:
Consider two polygons (one can just draw any two polygons and number their vertices), whose connectivities in a node list are:...
5
votes
2answers
134 views
Exponent log to compute reciprocal power?
A MATLAB library seems to overcomplicate a computation:
exp( (log(a) - log(b))/b )
which is mathematically equivalent (assuming real & positive ...
3
votes
1answer
305 views
Understand the need for Welford's online algorithm
I am puzzled by the Wikipedia entry discussing many online algorithms for computing the sample variance, including the Welford's online algorithm.
In particular, the sample variance $s_n^2$ can be ...
1
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0answers
37 views
Largest triangle that contains a point
Given the location of $n$ points on a 2D plane ($P_1, P_2, \ldots, P_n$); and the location of a special point $X$.
Find three points $P_i,P_j,P_k$ ($i \neq j \neq k$) such that point $X$ is inside the ...
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0answers
48 views
Storing and retrieving two-dimensional and three-dimensional data
I work on computational geometry.
A huge number of two-dimensional and three-dimensional data are found in my project. Coordinates of polygon and polyhedrons vertices consisted of two-dimensional and ...
3
votes
3answers
326 views
How to determine if 2 rays intersect?
We are given the 2D coordinates of 2 points: the first point is where the ray starts and it goes through the second point. We are given another ray in the same way. How do we determine if they have a ...
0
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0answers
51 views
Finding block structure of a tensor
Are there any well-known algorithms for partitioning a dense tensor into block-sparse form?
In other words, I need to find a set of non-overlapping blocks that contain all non-zero entries of the ...
4
votes
2answers
705 views
Time Reversibility of Velocity Verlet Algorithm
I'm very new to computational Physics and am finding conflicting statements on whether the velocity Verlet algorithm, defined as:
$\begin{align}
x_{n+1} &= x_n + v_n \Delta t + \frac{1}{2} a_n \...
1
vote
1answer
90 views
Asymptotic complexity of fixed-rank SVD
According to the Wikipedia article on Singular Value Decomposition, the asymptotic complexity of computing the SVD of an arbitrary m×n matrix M with m>n by the popular Householder QR methods is O(...
4
votes
2answers
124 views
Which algorithms(paper) should be reproduced by a student to enter the field of computational fluid-structure interaction?
We'd better not to reinvent the wheel. But without some programming, one can hardly understand computational fluid-structure interaction. And I would like to know which papers or algorithms should a ...
1
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0answers
31 views
Bipartite Euclidean Matching simple to implement approximate algorithm
I am looking for a simple to implement algorithm for the bipartite euclidean matching problem (or an implementation of any practical algorithm). I am aware of Agarwal's paper, but I would like to ...
1
vote
1answer
70 views
An efficient algorithm to find Nearest Neighbours
So imagine I have a $m$ vectors each of dimension $d$. Lets call them, $\vec x_{i}$, with $i = 1, 2, 3, 4, 5, \dots, m$. Now the idea is to find the neighbours of $\vec x_{i}$ (calling them $\vec x_{j}...
2
votes
3answers
522 views
On the reordering of sparse matrices
I have been reading on different techniques used to reorder sparse matrices to achieve better performance, the most popular being the Cuthill-McKee or Reverse Cuthill-McKee algorithm. Most of those ...
2
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0answers
146 views
Computing Singular Value Decomposition of small ($4\times 4$) matrices
I need to compute the Singular Value Decomposition (SVD) of many $4 \times 4$ matrices. I'm looking for SVD algorithms specialized for small matrices. I've read that the ...
6
votes
1answer
803 views
Cheap recalculation of eigenvalues and eigenvectors for a low-rank update of the matrix
Suppose I have a correlation matrix, $A$, and I already have the eigenvalues and eigenvectors of this matrix.
For a given vector, $\mathbf{\mathit{v}}$, I want to calculate the eigenvalues and ...
0
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0answers
55 views
Producing Voronoi diagram in three dimensional
A Voronoi diagram is a kind of tesselation that divided the medium into polygons in 2D and polyhedrons in 3D.
Although there are many algorithms to construct a Voronoi diagram, some of them are faster ...
3
votes
3answers
287 views
What are some algorithms to calculate the width of an arbitrary polygon when a bounding box approximation is inaccurate
What are some alternative algorithms to creating a bounding box for finding the max width of a concave, simple winding polygon, like the one in the below image? I prefer solutions that are more ...
0
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0answers
78 views
Derivative-free ill-conditioned non-linear least squares
I am looking for a package which can solve (non-linear) least squares problems without the use of derivatives (because of an expensive model), but which also deals with ill-conditioning well (such as ...
0
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0answers
118 views
Explanation of Givens rotation in Jacobi Rotation SVD
I'm trying to implement Singular Value Decomposition (homework of sorts) via the Jacobi Rotation method (more info here, pages 11 and 12).
I am stuck at the bullet saying (sorry for the picture, but I'...
2
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1answer
48 views
How do you construct a self-similar binary structured-tree?
Please excuse me if this question somehow looks trivial or not really interesting, but I recently have a hard time to convince someone else that my algorithm for constructing a self-similar binary ...
