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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2answers
259 views

Numerical evaluation of partial derivatives

I need to evaluate the following derivative: $$\frac{1}{\prod_i \xi_i!}\frac{1}{\prod_j \eta_j!} \left.\frac{\partial^{\xi_1 + \cdots + \xi_m}}{\partial\alpha_1^{\xi_1}\ldots\partial\alpha_m^{\xi_m}} ...
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3answers
6k views

Algorithm for Principal Eigenvector of a Real Symmetric 3x3 Matrix

I have a 3x3 covariance matrix (so, real, symmetric, dense, 3x3), I would like it's principal eigenvector, and speed is a concern. Is there a fast algorithm for this specific problem? I've seen ...
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4answers
2k views

Efficient solver for a symmetric tridiagonal system where the upper/lower diagonals are offset

I'm looking for an efficient way to solve a symmetric tridiagonal system $Mx = d$, where the upper and lower diagonals of $M$ are offset from the main diagonal by $k$ rows/columns: $$ \begin{bmatrix} ...
5
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1answer
659 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 ...
5
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3answers
180 views

Algorithms for radiation treatment planning

I have a medical physics problem - I want to maximise the dose absorbed by a brain tumour whilst minimising the dose in the rest of the brain, especially certain organs, such as the pituitary gland, ...
5
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2answers
118 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 ...
5
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2answers
205 views

Is there a special algorithm for computing the convex hull ordering when the candidate points are on the hull?

I'm dealing with a set of points which are already placed on the 2D hull boundary: a convex polygon. I know this for sure. However, the point set is not ordered, and I need the polygon points to be ...
5
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2answers
10k views

What is the difference between O(n) and o(n)? [closed]

I was studying Big-Oh notation, and there is apparently a difference between $O(n)$ and $o(n)$. What is it? I think $f(n)$ is $o(n)$ means that $$\lim_{n \to \infty} \frac{f(n)}{g(n)} = 0$$ but what ...
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2answers
479 views

How to parallelize a banded direct solver?

I have a linear system whose matrix that is diagonally dominant, non-symmetric, but banded. Since the band-radius is 2 (producing only 5 variables per equation), a banded direct solver (gaussian ...
5
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2answers
180 views

Fast algorithm for computing the similarity between two arrays

Suppose there are two arrays (They have the same length), I want to give a quantitative description about the similarity between them. I define a formula like this, which means we can shuffle them ...
5
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1answer
391 views

Fast algorithm for computing cofactor matrix

I wonder if there is a fast algorithm, say ($\mathcal O(n^3)$) for computing the cofactor matrix (or conjugate matrix) of an $N\times N$ square matrix. And yes, one could first compute its determinant ...
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2answers
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$k$ nearest neighbors for $n$ points with periodic boundary conditions in R^3

Take $n$ points ${\bf r} =\{ r_1 = (x_1,y_1,z_1), r_2=(x_2,y_2,z_2) \ldots r_n=(x_n,y_n,z_n) \}$ enclosed in a periodic box of length $L$, such that that the point $(0,0,0)=(L,0,0)$, $(0,0,0)=(L,L,L)$,...
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1answer
216 views

Striking examples of success of local search algorithms

In N queens problem https://en.wikipedia.org/wiki/Eight_queens_puzzle, trying to find solution by backtracking encounters difficulties quite fast (even for SWI-Prolog, http://swish.swi-prolog.org/...
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1answer
50 views

Transform from linear index of a packed triangular matrix to dense indices

Given indices $i,j$ s.t. $0\leq i \leq j <n$, the function $f(i,j)=i+j(j+1)/2$ maps 2d indices to linear indices in column major order. What is the fastest way to invert this function? My first ...
5
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2answers
1k views

How to implement adaptive mesh refinement using conformal triangles

I am trying to implement adaptive mesh refinement for a finite element code. The code uses (at least for now) linear triangles and so when I do the mesh refinement I want the triangular mesh to remain ...
5
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2answers
1k views

How to find all complex roots of an equation in a domain

I am facing a problem where I want to find the complex roots of $f(z)=z-sin(z)=0$ numerically. There are infinitely many roots of the function, but I am only interested in the $N$ closest to the ...
5
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1answer
58 views

correct complexity notation

I have written an algorithm where the 2 input arguments are a file and a list of values. I would like to say the algorithm complexity is: ...
5
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1answer
198 views

Efficient algorithm for a matrix product

Recall that a unit lower triangular matrix $L\in\mathbb{R}^{n\times n}$ is a lower triangular matrix with diagonal elements $e_i^{T}L e_i = \lambda_{ii} = 1$. An elementary unit lower triangular ...
5
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1answer
325 views

Finding closed equipotential surfaces on a 3D grid

In short, I'm looking for either: (1) Publications or other sources dealing with contour/isosurface finding algorithms, so that I can write my own implementation (and parallelize as best I can), or (...
5
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1answer
79 views

Is there an efficient algorithm for calculation of continued fraction expansion from decimal digits?

Suppose to calculate the continued fraction expansion of $\pi$, the common-sense algorithm would be to take the decimal part, perform inversion, which will give the next term as integer part, and the ...
5
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1answer
819 views

Eigenvalue Decomposition of Hermitian Matrix in Scala

I'm working on helping my friend create code to perform the Overlap Dirac Operator and have come across one part that I'm not sure what to do. I need to compute the Eigenvalues and corresponding ...
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0answers
76 views

Efficient way to find eigenvalues of complex symmetric matrix with real off-diagonal elements

My goal is to find all eigenvalues (and eigenvectors) in a given range of magnitudes of a complex symmetric matrix with real off-diagonal elements (only diagonal elements are complex). Currently I'm ...
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0answers
86 views

What Derivative-free optimization method should I use when my initial guess is very good?

I am trying to minimize a function where my initial guess is quite close to the minimum. I'm trying to minimize $$f(q) = \text{angle}(qw_1q*, v_1) + \text{angle}(qw_2q*, v_2) + \text{angle}(qw_3q*, ...
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0answers
67 views

Padua-type pointset for functions singular on line $x=y$

The Padua points $\mathrm{Pad}_{n} \subset [-1,1]^{2}$ are a unisolvent pointset with optimal growth of Lebesgue constant, described in detail here. With some work they can be used to generate a ...
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0answers
238 views

Sound Waves Simulation in 3D Environment

I want to do a simulation of sound waves including wave propagation, absorption, and reflection in 3D space. I did some research and I found this question in stackoverflow but it talks about ...
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0answers
2k views

Two-chordless cycle extraction from a failed comparability graph recognition

I have implemented a comparability graph recognition algorithm from M.C. Golumbic's "Algorithmic graph theory and perfect graphs" book. It is hinted in Fekete, Schepers, and van der Veen's "Exact ...
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4answers
7k views

Efficient assembly of finite element matrix in MATLAB

Question What is the most efficient algorithm for finding a row of a matrix which matches a given row? This is the same as a table lookup based on multiple criteria. Context Finite Element Matrices ...
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2answers
436 views

Why is computational cost measured in Floating Pt. Ops. in times of parallel computing?

In times of parallel computing, it seems to me that algorithms (also basic ones, like matrix-vector multiplication) should be measured by their dependent steps (that use results from steps before) ...
4
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4answers
234 views

Determining the algorithmic complexity

A few of the iterative matrix algorithms (CG,GMRES etc.) I have authored are acting rather funny. They converge to the right answers but take abnormally long time to run. I am in the process of ...
4
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1answer
99 views

How to evaluate a series of derivatives?

Consider the function $$f(\mathbf{x}) = \sum_{n=0}^{N} a_n \left( (\mathbf{b}-\mathbf{x})\cdot \nabla \right)^n \frac{1}{r}$$ where $r = |\mathbf{x}| = \sqrt{(x-x_0)^2 + (y-y_0)^2}$ and $a_n$ and $\...
4
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2answers
206 views

Efficient algorithm to decide if a graph is a cactus?

A cactus is a connected graph in which every edge belongs to at most one simple cycle. How should one modify the Depth First Search algorithm to obtain an efficient algorithm that determines if a ...
4
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1answer
616 views

What is the case of trade-off in different Runge Kutta methods

There are so many Runge Kutta methods, including Dormand-Prince 45 Cash-Karp 54 Fehlberge 78 Is there any comparison between them? What is each approach sacrificing? What is the general trade-off ...
4
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2answers
4k views

Calculate contour line length

I would like to know the algorithm to calculate contour line length. Suppose we have numerical data set of an function $f(x,y)$. How could I calculate the length of the line from $(x_1,y_1)$ to $(x_2,...
4
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3answers
387 views

Propagating Schrodinger equation

My task is to simulate quantum evolution. To do that I need to perform this operation $$w = e^{-itH}v$$ where $H$ is a sparse matrix and $v$ is the initial column vector. I am wondering if there is ...
4
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2answers
190 views

most adequate edit distance for misspellings in names?

Algorithms for edit distance give a measure of the distance between two strings. Question: which of these measures would be most relevant to detect two different persons names which are actually the ...
4
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1answer
240 views

constrained minimization in N dimensions

I am looking to create an algorithm to minimize an N dimensional problem. I am unsure how to write it in its generic form, so I will show it in 1, 2 and 3 dimensions Minimize $ \sum_{i} x_i\left [ f\...
4
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1answer
2k views

Implementing the $\theta$-method for solving an IVP

I also posted this question on the Math SE site, but it was suggested I post here also. I want to implement the $\theta$-method to solve an IVP in MATLAB. The $\theta$-method is: $y_{j+1} = y_j + h[\...
4
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1answer
123 views

Efficient way to find max height repetitive sub-trees in an object tree

I am trying to solve a problem of finding a max repetitive sub-tree in an object tree. By the object tree I mean a tree where each leaf and node has a name. Each leaf has a type and a value of that ...
4
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2answers
1k views

Algorithmically selecting colors that will contrast with their background

I am trying to write an implementation of a color quantization algorithm in order to find an image's dominant colors (say 5) and then find contrasting colors based on the colors found in the image. I'...
4
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2answers
1k views

Discrete-time Algebraic Riccati Equation (DARE) solver in C++

I need to use a Discrete-time Algebraic Riccati Equation (DARE) solver for an embedded controller (with limited processing power) in a research project and sadly, I can't find any implementation of it ...
4
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2answers
179 views

Optimization algorithm selection for 3 variable integer

I have a cost function: $f(x,y,z) \rightarrow \mathbb{R}$ it is very expensive to evaluate $x,y,z \in \mathbb{Z}$ 0 < x < 10 0 < y < 30 0 < z < 100 I thought it was convex, not sure ...
4
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2answers
202 views

Problem Condition and Algorithm Stability

Consider 2 mathematical problems: $$ f_1(x) = a - x \\ f_2(x) = e^x -1 $$ The condition number for a function is defined as follows: $$ k(f) = \left| x \cdot \frac{f'}{f} \right| $$ Lets analyze ...
4
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1answer
69 views

How to use a web-embedded model in a computational workflow?

There is a model embedded in a web browser (Caprio 1998) that I would like to use in an MCMC algorithm. What is the best way to do this? I could implement the model in my favorite language but I ...
4
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1answer
45 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:...
4
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1answer
184 views

Question on speed and accuracy comparisons of different 2D curve fitting methods

This may be a trivial question, and I apologize if so. Consider the following simple problem: We have a 2D, regular grid of points (say $X = [0,5000] \times [0,5000]$) spaced uniformly by units of 1 (...
4
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1answer
115 views

Numerically compute one-to-one mapping

I have a set of points defined by their coordinates $(x_1,y_1)$ after changing some parameters of the problem I obtain a second set of points defined by their coordinates $(x_2, y_2)$. There exists a ...
4
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2answers
589 views

Finding the minimum hamming distance between a bit vector and any pairwise intersection of multiple bit vectors

I'm looking for a way to optimize this procedure. This is the problem: I have a list of bit vectors $\mathbf{A} = [ a_1, a_2, a_3, ..., a_n ]$ I have a list of bit vectors $\mathbf{B} = [ b_1, b_2, ...
4
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1answer
172 views

Updating an approximate solution to a linear system in response to a small change

This question was original posted on SO but it was suggested that I post it here. I'm working on a program in which I have a banded matrix M and a vector b, and I want to maintain an approximate ...
4
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1answer
1k views

Shipping algorithm

I originally posted this in the mathematics StackExchange, but I was told that this is more of an operations research question. I would like to know if anyone has any thoughts on how to go about ...
4
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1answer
2k views

Multi-objectives fitness function with hard and soft constraints

Background I am using genetic algorithm to solve the multiple-traveling-salesman problem. It work fine when I have only one constraint (distance) for my fitness function: ie: the lower the total ...

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