A method of finding nearly-optimal solutions to a problem. Generally, this terminology is applied to algorithms and heuristics for solving NP-Hard problems in computer science.

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35 views

Finding optimal point distance to get desired number of random points in an area

I have a random point generator which takes a distance $d$ and fills an area with points such that distance between any two points is no less that $d$: I need to control the number of points in the ...
6
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1answer
77 views

Polynomial order of an approximation of a section of sine and numerical accuracy

I was playing with the idea that a sine function is periodic. But even within one period there are symmetries, namely the second fourth of a period is the mirror image of the first fourth and the ...
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1answer
181 views

imaginary time propagation to find ground state wavefunction

I understand the basic idea of imaginary time propagation method: The wavefunction $\psi(x,t)$ as a superposition of energy eigenstates $\phi_m(x)$: $$ \psi(x,t)=\sum_m \phi_m(x)e^{-iE_mt/\hbar} $$ ...
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0answers
25 views

Alternative to two “for” loops in finding best neighborhoods for TSP?

I am trying to solve Travelling Salesman Problems using tabu search. I have been able to successfully find "near enough" optimal solutions (as well as one optimal, yay!). For the moment I am using ...
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2answers
122 views

Interpolation with the roots of orthogonal polynomials & Spectral expansion

I'm a bit confused about the relationships between these two approximation methods mentioned in the title. Does this kind of interpolation also belongs to the field of spectral methods? Are the ...
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2answers
211 views

Methods for fast approximation of convolution

What are the state of the art methods for fast 2D convolution approximation? I'm familiar with SVD based multiplication and cross approximation approaches, but would be thankful to get additional ...
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98 views

Are there any benefits of computable analysis to numerical algorithms

Computers can work only with computable numbers, while most of the algorithms are based on analysis of real numbers (real analysis). When I heard of the existence of computable analysis I ...
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1answer
48 views

Expected number of steps before a global optimum is found with Simulated Annealing

I'm reading a technical report on Simulated Annealing: On the Convergence Time of Simulated Annealing, by Sanguthevar Rajasekaran. You may find it following this link. Given $G=(V, E)$ is the graph ...
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2answers
249 views

Solving a system of linear equations with only an approximate solution

I have a system of linear equations that is derived partially from experimental data. Theoretically, the system should have a single, exact solution; however, experimental error causes it to not have ...
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136 views

Reference Request: Variational Problem

I want to solve approximately the following variational problem: Given a function $c:[-1,1]^2\rightarrow [0,1]$, constants $p_1...p_n\in \mathbb{R}^+$, $\alpha_1...\alpha_n\in \mathbb{R}$, and $\...
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0answers
42 views

Approximation by sin waves using Fast Fourier Transform

I'm trying to approximate a timeseries by sin waves.I've written a program on python with scipy. But the program yields the wrong result. Please tell me what may be wrong in the code or in my math ...
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1answer
75 views

Efficient Implementation of Taylor Series for Sine

I am trying out a few forms of polynomial expression optimization, and I'd like to improve of what I've got, if anyone has anything they know is better. Implementation 1: $$x-\frac{x^3}{3!}+\frac{x^...
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32 views

diffusion approximation time inhomogeneous Poisson process

I'm trying to approximate a time inhomogeneous Poisson process by means of a diffusion process. the process is defined as: $X(t)=X(t-1)+Y(N_{t-1,t}); X(0)=X_0$ where $N_{t-1,t} = \int_{t-1}^t \...
3
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1answer
85 views

Fast algorithm for computing matrix square root using randomized linear algebra?

Is there a fast algorithm for computing the matrix square root of a real symmetric matrix using random matrices or randomized algorithms?
2
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1answer
61 views

Matching/Assignment Problem

I'm not sure how I can represent and solve the following problem. I have a list of sales (timestamp and quantity) and a list of corresponding inventory draws (timestamp and quantity). What I ...
3
votes
1answer
386 views

Efficiency of an algorithm

I designed an algorithm to find the global minimum of a function and implemented it in MATLAB. And I also implemented the "Tunneling algorithm" for the global minimum of a function in MATLAB. But now,...
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1answer
94 views

Problems Implementing the Remez Algorithm

So first off: *** This code is not being used in production software. It is a personal project of mine, trying to understand approximation theory and advanced curve fitting. In other words, I'm ...
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0answers
42 views

A better way to compute a double integral involving a infinite series?

Let $D_{\nu}(.)$ is the parabolic cylinder function (http://mathworld.wolfram.com/ParabolicCylinderFunction.html) And $\Gamma(.)$ is the Gamma function. Define $s_y(\mu,\nu,t,z)=2^{\nu}\...
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2answers
107 views

Looking for a particular algorithm for numerical integration

Consider the following differential equation \begin{equation} p(t) = \frac{\partial q(t)}{\partial t} \end{equation} where $t \in (0,\infty)$. I have a build a code that spits out values of the ...
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94 views

Finite Difference for Fourth-Order PDE

How to discretize the following 4th order PDE using finite difference method? $$\frac{\partial^{2} y}{\partial t^{2}}+\frac{\partial^{4} y}{\partial x^{4}}=0$$ thanks
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56 views

Bijection between polyhedrals

Does there exist a bijection between a general axis-parallel polytope in $\mathbb{R}^n$ and a polytope embedded in a unit hypercube in $\mathbb{R}^n$? This means that the bijection must preserve the ...
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0answers
56 views

Finding most efficient route (distance/number of nodes) that uses nodes at least X amount away from another node

I'm trying to find the "looped" route with the lowest value of D/n, where D=Distance, and ...
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0answers
29 views

Find out the expression for angular speed in terms of time

The orbit of a small mass orbiting to a much larger mass (e.g. a small planet relative to a fixed star) is described by $$ u=\dfrac{1+e\cos{\theta}}{l} $$ where $u = 1/r$, $r$ and $\theta$ are the ...
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31 views

Piecewise linear fitting of a curve in loglog plot

Iam basically trying to fit a curve similar to that shown in fig (a) with 7 straight line segments (ie. 6 knots) as shown in fig (b) and I need to get the optimum knot positions. This is a tripartite ...
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1answer
72 views

What are good parametrizations of rational functions for response surface models?

For fitting a response surface model to a physical process, I have 3-4 relevant "signals", like a feature density, a signal based on a feature width, or a signal based on a distance to the next ...
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56 views

approximation of nonlinear time-dependent system with history

I have two time-dependent coupled equations. One of which is several orders of magnitude more computationally demanding than the other. I am trying to use machine learning to reproduce the behavior of ...
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117 views

Computation of multipole expansion of potential not converging

According to Beatson and Greengard's short course on FMM: ( Eq. 5.15 & 5.16 setting k=1, q=1 ) We can approximate a potential $\phi = 1/(r-R)$ using: $$ {1\over |\vec{r}-\vec{R}|} = \sum_{n=0}^{...
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1answer
91 views

Computing the (non-convex) boundary of a set of paths between two points

I have a set of paths between two fixed points (marked in red below). Each of these paths consists of an ordered series of $\{x, y\}$ points (marked in blue). I am trying to find the ordered set of ...
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1answer
166 views

What is the algorithm that matlab used in its built-in function 'pca'?

Do anyone know what is the algorithm that MATLAB used in its built-in function "pca"? I have the following data set: 148.9820 55.8438 210.2150 149.3030 56.8891 208.4280 151.4400 ...
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0answers
61 views

Algorithm for optimizing graph interconnectivity

I have a partiuclar kind of graph problem and (not having a background in graph algorithms) I would like to know how this kind of problem is called in the literature and what algorithms exist for ...
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1answer
491 views

Effect of Initial guess B (approximate Hessian) on BFGS algorithm

I am trying to implement BFGS. The purpose is to approximate Hessian matrix only (not using the quasi-newton optimization steps), so i am using steepest ascent for optimization. What I observe is that ...
8
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1answer
162 views

Efficient Gravitational Field Implementation

I asked a similar question on physics.stackexchange, being ignorant about this website. I am basically looking for an efficient way to implement gravitational fields. I have a huge 2D space, with ...
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1answer
114 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 ...
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195 views

closed form approximation of matrix inverse with special properties

I'm trying to find some theory to help me explicitly express the inverse of a matrix (or a close approximation of the inverse). My matrix has the following properties: invertible positive definite ...
2
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1answer
114 views

What is the more than 3rd order Taylor series approximation for a multi-variate function?

Suppose $f$ is a infinite continuously differentiable map: $R^n\to R$, and $x,x_0 \in R^n$, then we have the following second order Taylor expansion of $f(x)$ at $x_0$: $$f(x)\approx f(x_0)+(x-x_0)^T\...
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0answers
108 views

Quadratic programming problem involving permutation matrices

Does anyone know a good algorithm for quickly finding an approximate solution to the following problem? Given two square matrices $A$ and $B$, minimize $\| P A P^\top - B \|$ over all permutation ...
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2answers
891 views

Successive over-relaxation not converging (when not done in-place)

I'm trying to find the potential given some boundary conditions using the successive over-relaxation method. I have 2 solutions: -One iterates over all elements and applies the formula ...
10
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1answer
380 views

Numerical methods for inverting integral transforms?

I'm trying to numerically invert the following integral transform: $$F(y) = \int_{0}^{\infty} y\exp{\left[-\frac{1}{2}(y^2 + x^2)\right]} I_0\left(xy\right)f(x)\;\mathrm{d}x$$ So for a given $F(y)$ ...
5
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3answers
1k views

Fast algorithm for Polar Decomposition

As it known, according to the Polar Decomposition, square matrix can be expressed in the next form $$M=QR$$ ($Q$ - orthogonal matrix $R$ - positive-semidefinite Hermitian matrix) I need to find this ...
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1answer
62 views

Determine low-order polynomial lower bound

I have a function $f$ I'd like to determine numerically and I have a bunch of $(x, y)$ pairs which approximate the function in the following sense: all of the points satisfy $f(x) \leq y$, most of the ...
1
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1answer
269 views

Assigning Groups Based on Preference List [closed]

I am trying to make a system that will sort a list individuals and their preferred list of others. This may not make complete sense, but bear with me. I have a list of people, each with their list of ...
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0answers
103 views

Good approximate solutions for a MILP problem

The company I work for has been developing an application for real-time control of sewer networks. Every 5 minutes, a MILP problem is built or updated, then solved using Gurobi. For mid-sized cities, ...
6
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1answer
245 views

Computing $\sin(\pi/2)$ numerically

I'm trying to understand the types of numerical errors, to do this I want to calculate $\sin(\pi/2)=1$ numerically. To do this I use the Taylor series of $\sin(x)$ in 0: $$\sin(x)=x-\frac{x^3}{6}+\...
5
votes
1answer
319 views

Bin-packing: Maximise number of bins / “Fukubukuro” problem?

I recently encountered a problem that looks like a variation of bin packing or knapsack problem, but with the objective to maximise the number of bins/knapsacks: Consider there is a list of M items ...
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2answers
73 views

Estimate (non-)drift in noisy data

I have a time series representing the result of a complex calculation (physical simulation). Due to round-off errors and approximation errors, there will be some "noise" on the data series. In some ...
7
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1answer
530 views

Polynomial Fitting from Chebyshev Coefficients

I have been reading Numerical recipes about how to create a power series approximation to a function once you have a Chebyshev approximation to the function. However it is still very unclear to me how ...
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3answers
506 views

Choosing subset of vectors to approximate a subspace

Suppose I have a high-dimensional vector space $X$, a subspace $V \subset X$, and a collection of $n$ vectors $\{x_i\}_{i=1}^n \subset X$. My question is: How can I choose a small collection $k < ...
3
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2answers
146 views

How to prove that my problem is np-hard

For an assignment i need to program an application to schedule conversations. Something similar to speeddating or Pta meeting. The problem is that i know that this is hard to solve, but i dont know if ...
2
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1answer
131 views

Numerically designing a periodic 1D curve that maximizes an integral area objective and satisfies value, derivative, and frequency constraints

I need to write MATLAB program (or use an existing one) to obtain Fourier series coefficients. Let's say the series is going to approximate a 1D curve. The boundary conditions are: value of the ...
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1answer
272 views

Quickly computing inversion of a large sparse partial stochastic matrix

Suppose I have a sparse stochastic matrix $M$ (with thousands or millions of stochastic column vectors), possibly encoding some links in a web graph. Now I split it into two matrices: $D$ containing ...