# Tagged Questions

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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### hyperoperation sequence with non-integer values of n

This probably has a very simple answer of some sort, but I'm not a mathematician. For the hyperoperation sequence: $$G(n,a,b)$$ ...there are obvious defined values for positive integer values of $n$ ...
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### deconvolution of exp($x^2$)

I would like to know whether we can get the function of type exp($x^2$) by convoluting any functions. That is which function convolution gives exp($x^2$). Thanks in advance
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### computing a fixed interest rate

I've been struggling for hours now with understanding a Topcoder problem, Autoloan , but i cannot grasp the way of computing it from a mathematical point of view. The excerpt goes as follows: A ...
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### Analytical expressions for the orthogonalization of a specific set of vectors

I would like to know whether analytical or closed-form expressions could be obtained for the orthogonalization of a set of vectors in the following setting. Let $x_t$ be a vector indexed as a time ...
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### Calculate height for rows of rectangles within a given width

I have an array of rectangles, all of the same height, but with different widths. Imagine they are on a single line with a uniform gap between them as shown below... |XXX| |X| |XXXXX| |XXX| ...
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### Can car traffic be managed by mathematical formula? [closed]

How car traffic is managed? Is it managed by mathematical algorithm? Or by human(operator)? If it's by operator, can it be managed mathematically? Or is it by physics? By what theories/formula? ...
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### How to solve the Helmholtz equation in a triangular region?

Suppose we take the Dirichlet boundary condition, namely the function must vanish on the boundary of the triangle. How about a general n-polygon?
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### What is transpose multiplier and forward multiplier?

For linear system X = A*s, we define the forward and transpose multiplies Af and At as follows: Af = @(s) A*s; At = @(s) A'*s; I want to know what is forward ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
### 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 = ...