This tag concerns computational problems central to mathematical and scientific computating. The scope includes algorithms, numerical analysis, optimization, and linear algebra, computational topology, computational geometry, symbolic methods, and inverse problems.

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Simplifying the Generalized Eigenvalue Problem

Let $\Sigma_1$, $\Sigma_2$ be symmetric positive-definite real $n\times n$ matrices. We want to solve the generalized eigenvalue problem $$ \Sigma_1V=\Lambda\Sigma_2V, $$ where $\Lambda$ is the ...
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24 views

Using Orthogonal Collocation to solve Coupled Ordinary Differential Equations

I am trying to solve six first order coupled ODE's, two of these are associated with a heat balance of a catalyst pellet, and four are mass balances. I have been trying to solve these equations using ...
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1answer
56 views

Prime numbers distribution theorem

I'm trying to understand Gauss' theorem: $$ \frac{\pi(x) }{x/\ln x} \to 1 $$ for large $x$. I've taken the list of first 1000 prime numbers from Utah university site, saved them to file ...
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29 views

periodic boundary conditions and the FEM

I am trying to set up the mass matrix for a 1D system which I want to solve using finite elements. So the mass matrix is defined as $$ M = \int{NN^T}dL, $$ where $N$ is the finite element linear ...
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5 views

What is the behavior of the spatial median in high-dimensional spaces?

I am a photographer who is investigating a technique known as image stacking, in which multiple images of the same subject are combined to reduce noise (by CLT). Commonly used techniques are mean and ...
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1answer
31 views

How to make Poisson voronoi diagram

I am facing a problem as follows : I want to make poisson voronoi diagram & for this I have to appropriately choose some generating points. Sources on internet(WIKI) refers that these points have ...
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1answer
24 views

Calculate pairwise cosine distance only returning the lower triangular matrix

I have a matrix, where each row is a feature vector. I would like to find out the pairwise cosine distance between all of these feature vectors. The cosine value between all rows in a matrix could be ...
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2answers
21 views

Special properties in the direct solving of sparse symmetric linear systems

In the area of computational solving of large sparse linear systems, some solvers specialize only on symmetric sparse matrices, be it positive definite or indefinite as compared to general ...
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1answer
28 views

Computational complexity of the algorithm

Make an analysis of the computational complexity of the algorithm below, where it is given by the number of elementary operations that the algorithm performs (assignment is not considered). Where ...
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1answer
48 views

inverse of $AQ^{-1}A'$

Suppose that $A$ is a $m\times n$ full row rank sparse matrix, and $Q$ is an $n\times n$ symmetric positive definite sparse matrix with $m<n$. Besides, $m$ is about $10^5$, and $n$ is about $10^6$. ...
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2answers
94 views

How to determine whether a point is inside a closed region or not?

Take the following parametric equation of an implicit curve as an example: $$ \left\{\quad \begin{array}{rl} x=& 9 \sin 2 t+5 \sin 3 t \\ y=& 9 \cos 2 t-5 \cos 3 t \\ \end{array} \right. $$ ...
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39 views

Cholesky decomposition and rotation matrix inverse

I implemented three methods for inversion of a matrix, all are classic. I wanted to test for the most generalized method, while taking efficiency into account. For Cholesky decomposition, which is ...
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1answer
39 views

Compute finite series

The problem is to count the sum of the finite series $$\sum_{k=0}^{k_0} \frac{a_k}{b_k}$$ I need to count this series in binary with some precision, that would output $n$ correct binary digits after ...
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13 views

Literature Reference for transformations through vector spaces

I am trying to understand the transformations through vector spaces: Problem 1. Let's say we have orthonormal basis $B=\{v_1, v_2, \ldots, v_n\}$ spanning the vector space $V$ and basis $B_1=\{u_1, ...
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39 views

Algorithm for finding zero of an odd function from n-sphere -> R^n

There is a well-known Borsuk-Ulam theorem stating that each continuous mapping $f : S^n \rightarrow \mathbb{R}^n$ that is odd in sence of $f(v) = -f(-v)$ for each $v \in S^n$ (where $-v$ denotes the ...
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1answer
25 views

Analysis of iterative optimization methods using lyapunov analysis

In analysis of iterative methods, is it possible that we have to use two time-lagged version of the time-varying system to analyze its convergence? (that is, we construct the evolution of x^k, ...
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0answers
22 views

Why steepest descent gives a wrong direction search?

I have to minimize the function $ƒ(x_1,x_2)=(x_1-1)^2+x_2^3-x_1x_2$. The initial point is $[1,1]^T$. The gradient of this function is $∇ƒ(x_1,x_2)=[2(x_1-1)-x_2,3x_2^2-x1]$. This gradient evaluated ...
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1answer
97 views

How to find this number, which is probably a very big prime or a product of big primes?

Let $\mathcal{N}(n)$ be the next prime greater than $n$. Which is the smallest natural number $n>0\;$ such that: $\mathcal N(2\cdot 3\cdot 5\cdot 7\cdot 11\cdot n)−2\cdot 3\cdot 5\cdot 7\cdot ...
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1answer
48 views

Consider the recursively defined language, L2

Consider the recursively defined language, $L_2$ i) $x \cap L_2$ and $y \in L_2$ ii) if $w \in L_2$, then so is $wxw \in L_2$ Find all strings in L_2 with length less than $7$ ...
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1answer
13 views

Logistic Regression - malty classification

I want to understand why the probability of P(D|p) is presented as a product of mentioned probabilities. I read a lot of texts, but everywhere the explanations are full of terminologies to confuse ...
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1answer
162 views

Accelerating approximations for arccos

I have recently built a method to accelerate drastically the accuracy of the following approximation of $\arccos(x)$ : $f_n(x)=2^n\sqrt{2-2g^{n-1}(x)}$ where $g(x)=\frac{1}2\sqrt{2+2x}$ and ...
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37 views

Mathematical equivalent to curve fit between polynomials

I am adapting a calculation done in an Excel workbook to code. Right now, we are predicting a variable based on three other variables, say $x,y,z$. We are creating six functions of $x$ and $y$ at ...
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7 views

Modulo 3 operation on days/seconds

Simple question.. I want to do a modulo 3 operation on the number of days in a month (28/30/31). and based on that i want to put my user into 3 different groups.. i am also willing to use seconds ...
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1answer
88 views

Carmichael numbers of form $m^3+1$ and Ramanujan's $1729$

While researching for a post on tetranacci pseudoprimes I came across a list of Carmichael numbers, $$C_n = 561,\, 1105,\, 1729,\, 2465,\, 2821,\dots$$ Of course, Ramanujan's taxicab number $1729 = ...
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64 views

What are the next few “tetranacci-like” pseudoprimes?

Starting with $n=0$: $k=2$ Given the roots $x_i$ of $x^2-x-1=0$. Then, we have the Lucas numbers, $$A_n = x_1^n+x_2^n = 2, 1, 3, 4, 7, 11, 18,\dots$$ The $n$ that divides $A_n-1$ are all the ...
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3answers
30 views

Let $x$ be an integer and $n$ be a positive integer. Find the smallest $n$ such that $x^4+n^2$ is not a prime for any $x$.

I need help proving the following: Let $x$ be an integer and $n$ be a positive integer. Find the smallest $n$ such that $x^4+n^2$ is not a prime for any $x$. I know that the smallest $n$ is 8 by ...
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1answer
25 views

Entropy Calculation and derivation of logarithm

I have probabilities as $$p_1 = 0.4,\ p_2 = 0.3,\ p_3=0.2,\ p_4=0.1$$ hence entropy is given by: $$H(x) = -\big(0.4\cdot \log_2(0.4) + 0.3\cdot \log_2(0.3) + 0.2\cdot \log_2(0.2) + 0.1\cdot ...
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38 views

If a computer can check 1 million colorings per second, about how long would it take to check all possible three-colorings on 100 vertices?

If we imagine a graph G with 100 vertices, how would we find all possible colorings for G if G(v) = 100? I think that to solve this problem we would start with vertex 1 with 99 edges for the first ...
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1answer
55 views

Prove that sets don't intersect

I am trying to solve a computer algorithm problem that boils down to solving the following. I would appreciate some mathematician assistance on the proof. So here goes: Having: Set $S$ - rational ...
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1answer
51 views

How to find the order of accuracy of this implicit RK method (using Taylor series)?

I want to get the order of accuracy (local truncation error - LTE) of this implicit 2-step method. The first step is Backward Euler to determine an approximation to the value at the midpoint in time, ...
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51 views

Convergence of the Midpoint (Leapfrog) method when applied to $u'(t)=\lambda u(t)$?

So, I am trying to solve this question: where example 7.7 can be found here: http://i.stack.imgur.com/PVCIC.png My approach: Forward Euler (FE) method is given by: ...
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1answer
60 views

Locally evaluate nonlinear dynamic system's stability using eigenvalues

I don't have a large mathematical background, but I'm working with Computational Neuroscience. I have a large Synaptic Matrix (x axis: presynaptic NeuronID, y axis: postsynaptic NeuronID) in a Modular ...
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1answer
40 views

Pollard Rho intuition

I have been reading about pollard rho factorization, however their is something I don't understand if we don't use a polynomial that is pick two random numbers and see the gcd(a-b,n) > 1 if it is ...
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0answers
129 views

Counting problem of combinations of symmetric matrix.

Given, a symmetric $n*n$ matrix $G$ with 0,1 entries. Each row of has same number of 1. $G$ is arranged in a certain order using a rule. The rule is described below- $G$ is partitioned in to two sub ...
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1answer
117 views

Sums of three cubes in arithmetic progression equal to a cube $x^3+(x+y)^3+(x+2y)^3 = z^3$

Using exhaustive search, small positive and primitive integer solutions to, $$x^3+(x+y)^3+(x+2y)^3 = 3 x^3 + 9 x^2 y + 15 x y^2 + 9 y^3= z^3\tag1$$ are, $$x,y = 3,1,\quad x+y =2^2$$ $$x,y = ...
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1answer
36 views

Essential Prime Implicants and Minterm Expressions

I have an exam for a university course shortly, and upon reviewing one of my assignments I have come to realize that I don't understand why I have lost marks/how to do a couple of questions. Hopefully ...
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4answers
3k views

Is “A New Kind of Science” a new kind of science?

A couple of years ago I was reading "New Kind of Science" (NKS) by S. Wolfram, and it presented lot of interesting ideas for a young Physics undergraduate. Now that I am studying Mathematics however, ...
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1answer
77 views

How do I find the sum of first N numbers common to 2 APs?

Here is the question - Certain numbers appear in both arithmetic progressions 17, 21, 25, ... and 16, 21, 26, ... . Find the sum of first 100 numbers appearing in both progressions. The ...
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23 views

numerically solving linear integral equations

I want to solve a 3*3 linear equation system but the equations are integral equations and he coefficients of solutions are to be extracted NUMERICALLY from some other integrals.I do not know how. I ...
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0answers
25 views

Norm (modulus) of the derivative of complex function and Newton Method

I am implementing a function that approximates a root of a complex function, say $f(z)$. As we know, at iteration $i$ we ave $$z_i = z_{i-1} - \frac{f(z_{i-1})}{f'(z_{i-1})}$$ The division of ...
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2answers
64 views

If negative of negative (--) = positive then why not positive of positive(++)= negative

As per my understanding positive and negative are just indicative of direction of number axes with zero at the center. If that is the case we should apply same laws to both positive and negative ...
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Simpson's rule is not good enough for the best approximation in L2 problem

The problem came from my computation methods (practice) class. It was to write a program which does the following: Original problem statement: We have a [0; 1] segment. Let us divide it into $2^n$ ...
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1answer
84 views

(x,y) coordinates from gluing together a sequence of right triangles with arbitrary angles [duplicate]

I have been scratching my head all day over this question for one of my assignments. I haven't made any progress and I'm at the point of giving up. Here's what I need help with. Start by gluing ...
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1answer
48 views

is it possible to find $x$ where $y$ is equal to a whole number in a non iterative fashion

Given the equation $$\frac{635x+326}{637+x} = y$$ where $$x>0$$ Is it possible to find all positive values of $x$ (there is only one) where $x$ is positive and $y$ is a whole number. While I ...
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10 views

Fourier Analysis of a p2 continous Galerkin Scheme for the Laplace & Poisson Equation

Background: I am obtaining residual calculations for the 3D Laplace and Poisson Equation using finite element continuous galerkin scheme with lagrange polynomial basis functions for p1, p2, p3 and ...
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44 views

does any polyhedral partition admit a convex piecewise quadratic surface defined over?

Given a polyhedral partition, i learnt that there exist some conditions for the existence of a convex piecewise affine surface over this partition for example the following study. ...
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1answer
23 views

Golden Section Search termination condition

From textbooks I found that the tolerance for Golden Section Search method should be set to $\sqrt{\epsilon}$, where $\epsilon$ - is the machine epsilon. This can be derived from Taylor series. So, in ...
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22 views

Computational verification request

Let $I(x) = \sigma(x)/x$ be the abundancy index of $x$. Note that $\sigma$ is the classical sum-of-divisors function. Previously, I computed for $u$ in the inequality $$\sqrt{3} < ...
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3answers
312 views

A bug with the WolframAlpha computational search engine?

I think I may have discovered a bug with WolframAlpha. So I was trying to determine all $x$ such that $$\sum_{i=0}^{5}{x^{-i}} < \frac{13}{12}.$$ WolframAlpha spit out $x > 13$ (see this ...
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2answers
82 views

How to calculate the errors of single and double precision

We consider the initial value problem $$\left\{\begin{matrix} y'=y &, 0 \leq t \leq 1 \\ y(0)=1 & \end{matrix}\right.$$ We apply the Euler method with $h=\frac{1}{N}$ and huge number of ...