Questions tagged [algorithms]
A description of the specific steps needed to solve a particular problem in an unambiguous way, expressed in an abstract form.
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Does an alias method exist for sampling a discrete distribution that is slighlty modified at each iteration?
I have the following problem. I must sample from a discrete distribution that is changing at each sort.
Let me explain, with a "vivid" description, I draw a color ball from a bag. The ...
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Is there an algorithm or graph theory that allows me to not need to store an intermediate matrix when calculating AT*Y1*A + BT*Y2*B?
I have a system of conductors for which there are two dense matrices of the (complex) mutual admittances, $Y_A$ and $Y_B$, which are symmetric. Then, an equivalent nodal admittance matrix $Y_N$ is ...
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Solving the time dependent Schrödinger equation with leapfrog integration in 1D
To my frustration I am struggling to implement leapfrog integration for the time dependent Schrödinger equation.
To the best of my knowledge this was first explicitly done in "A fast explicit ...
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Recursion relations for integrating Gaussian functions
I'm trying to implement a numerical method used in quantum chemistry from scratch. I'm using this paper as a reference. It's also available on Sci-Hub. So, the method requires calculating integrals of ...
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Solving Symmetric/Hollow Matrix issue
I have a particular issue and need something creative or solution from calculus.
I have Symmetric/Hollow Matrix, a numbers are % of mismatch between them. Ideally, all of them should be 0, but I have ...
3
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1
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Which optimization algorithm to max a single parameter by searching a landscape of five parameters?
Background: We're operating a small betatron which makes use of a vacuum tube where electrons are accelerated circularly. First, they get injected (like inserted) and contracted (like squeezed). After ...
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Quasi-Newton Method with a Transformed Hessian
I've recently came across an implementation of the BFGS algorithm but it has an additional step where the Hessian is transformed after the each update. This transformation is done so that certain ...
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Which algorithms exist to create a tetrahedral volume mesh from an STL file?
Basically just that. I am trying to write a C++ program that reads in an STL and should compute an unstructured tetrahedral volume mesh based on the surface triangulation given by the STL file.
I ...
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How to implement a generic monte carlo algorithm for n-dimensional integration?
A very visual picture for Monte Carlo integration is the approximation of $\pi$, by sampling in a square which contains a quarter of the unit circle.
We can extend this picture to 3 dimensions, by ...
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calculate order difference between 2 arrays (global and local)
Actual Problem: I want to assess the difference in two text rewrites: it can be fine today => today, it can be fine
Simplified Problem: ...
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Solve a large-scale linear system of equations with millions of unknowns
I have a large-scale system of linear equations: $Ax = b$, where $A$ is an $n\times n$ square symmetric positive definite matrix (not sparse), $b$ is an $n \times 1$ vector and $x$ is $n\times 1$ ...
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Generate polynomial basis through a sequence of SVD
I need help to understand how to use the result given by an algorithm for constructing an orthonormal polynomial basis over $L^{2}(X)$, where $X\subset\mathbb{R}^2$, with respect to the inner product $...
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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. ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
user41897
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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 ...
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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 ...
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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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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 $...
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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\...
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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 $...
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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:
...
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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 ...
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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 ...
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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 ...
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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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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}^{\...
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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=...
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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 ...
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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 ...
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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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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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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 "...
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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 ...
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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$$ with $c = 1$ and a periodic boundary ...
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Maintain unitary time evolution for a nonlinear ODE
I want to solve a nonlinear ODE of matrix $A(t)$
$$\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 ...
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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 ...
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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:...
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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 ...
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1
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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 ...
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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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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 ...
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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 ...
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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 ...
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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 \...
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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(...
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