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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Methods for delaying the "break" in non-linear least squares optimisation when the step size gets too small?
I am using the Levenberg-Marquardt method for calibration purposes. Typically, the RMSE of my calibration looks like:
I want to break the algorithm when the algorithm step-updates start to slow down, ...
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More stable algorithm to calculate `sqrt(a^2 + b^2) - abs(a)` in MatLab
Suppose we want to calculate $\sqrt{a^2+b^2}-|a|$ in MatLab. Using sqrt(a^2 + b^2) - abs(a) will have some problems:
If a or <...
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stupid + stupid = brilliant in scientific computing
I'm interested in examples of very effective methods in scientific computing that are the sum or naive combination of very ineffective or bad ones.
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Algorithm to solve system of nonlinear equations
I would like some tips in figuring out a good algorithm to find the solution of the following system. Let $\theta$ be a constant in $(0,1)$, let $i,l=1,...,N$, let $a_{l}$ and $b_{i,l}$ be some ...
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Code to list all maximal bicliques of a bipartite graph
We are looking for a code to list all maximal bicliques in bipartite graphs efficiently, as we want to run it on (large and sparse) graphs, with up to roughly a million nodes and edges in no more that ...
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Can I find sines or cosines of rational parts of π without using radians? If not, how do I reduce the errors caused solely by the transcendence of π?
this may be irrelevant for people who need fast code. But for me it's just the opposite -- i.e. in the specific situations when I know that the extra time allows me to make my calculations more ...
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Is OCN=DOCN decidable or not?
One-Counter Nets (OCNs) are finite-state machines equipped with an integer counter that
cannot decrease below zero and cannot be explicitly tested for zero.
An OCN $A$ over alphabet $\sum$ accepts a ...
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What do the Max-Cut algorithm graph cuts mean?
The max-cut algorithm divides a graph into 2 subsets, for instance:
While I understand the algorithm, I do not quite understand the meaning of the result. In the above picture, what does the ...
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Algorithm about finding a combination that the sum is closest to a given number
Given a matrix $P\in \mathbb{R}^{n*k}$ (just for ease of notation, no matrix or linear algebra is actually needed; bound to $(0,1)$ if necessary), select one number from each row and compute the sum. ...
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Rayleigh-Ritz under LOBPCG : Nested iteration?
I am trying to understand the basic LOBPCG algorithm, as used in popular python libraries,
primarily from the suggested resource on wiki.
LOBPCG involves a subroutine to compute the coefficients of ...
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Why is the definition of convergence different for root finding algorithms as compared to sequences?
The definition of convergence for root finding algorithms is given in a few sources as: A sequence ${x^k}$ generated by a numerical method is said to converge to the root $\alpha$ with order $p\geq 1$ ...
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General dimensional solution to fast sweeping quadratic equation
I am reading Hongkai Zhao's paper The Fast Sweeping method. I have implemented the method in 2D and now I want to move onto 3D.
However section 2.6 confuses me.
The article says:
But I have no idea ...
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Is it normal for a genetic algorithm to not have mating phase?
I know that is possible for a genetic algorithm to not have a mating phase because some genetic algorithms use a "mutation-only" approach, where new candidate solutions are generated by ...
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On solving a first order nonlinear differential equation
It all starts with this Cauchy problem:
$$
\begin{cases}
\sin(2x(t)) -\cos(3x'(t)) = x(t) + x'(t) \\
x(0) = 1 \\
\end{cases}
\quad \quad \text{with} \; t \in [0,10]\,.
$$
Not knowing which way to turn,...
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Looking for an algorithm to find points where lines intersect most frequently
I have a file that contains the coordinates of two points that make up a line. When I plot these lines, I notice that there are areas where the lines intersect more often. These points are identified ...
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How is the transfer function of a state-space representation numerically computed?
This question is a duplicate of this question that I asked on dsp stack exchange. However, nobody had the answer there but the question seems more appropriate on this forum (If not feel free to tell ...
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What are the benefits of cutting by half the number of multiplications needed to calculate n?
First we need to present the details of what makes that cut possible. The issue is linked to the sum of squares $1^2+2^2+3^2+4^2+...+n^2$.
We consider the two cases:
1-the case of a sum of even ...
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Is it really necessary to solve a system of linear equations in the Finite Element Method?
When we solve some boundary value problem by Finite Element Method, the appropriate system of linear equations is built, $$Ax=b.$$
Usually we use the solution x just for plugging it into some ...
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Simulating Quantum Wave Function/Schrodinger Equation With A Time Varying Potential
I have solved the Time Independent Schrodinger Equation using the Numerov method and diagonalizing the Hamiltonian, in 1 - 3 dimensions. I suppose I could time-evolve it by multiplying every element ...
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Open source implementations of the medial axis transform for vector shapes
Are there any open source implementations of the medial axis transform for vector shapes?
I have searched without finding any useful results. It seems that CGAL library doesn't have it implemented nor ...
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how to generate the Laplacian most efficiently
I have a cubic lattice $L_1\times L_2 \times L_3$ with the periodic boundary condition.
I have to construct the Laplacian matrix in the sparse format. My matlab code is like this
...
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Generating highly gzip-compressible, low entropy text that looks random-ish
I'm searching for an algorithm that can generate a set of 500 to 2000 characters from the alphabet a-z, A-Z, 0-9 (no interpunctuation) where the set has the following properties:
highly compressible ...
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Mass spectrometer atomic species identification
I'm starting to look at ways to analyze the output of a mass spectrometer. My first searches led to some white papers far from what I'm looking for (complex molecules vs simple atoms for me). With ...
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How to find an alignment between two overlapping time series?
Context
I'm working with absorbance spectral data. I have two spectra generated for a single analyte. Each spectrum was collected over two overlapping wavelength regions e.g. 8-10 micron and 9-11 ...
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How do compression algorithms compress data so fast?
I've come across compression algorithms, I tried creating a simple run-length encoding algorithm but I notice that when I searched other algorithms I was shocked to know that some algorithms can ...
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Viterbi algorithm for object tracking
I want to solve a problem of object tracking along time.
The problem is - I have a sequence of images, and I need to find and track the creation of the objects, than their movement, and than their ...
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Is there an efficient way to loop through this problem? [closed]
So I saw this very interesting problem. Let's say you have a length of 2, and a base length of 5
l = 2, b = 5
this would be translated to :
...
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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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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 ...
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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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