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Questions tagged [computational-mathematics]

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

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Calculation of special subsets in high-dimensional binary matrices

I need to solve a rather specific problem related to binary matrices. The task is to count the number of specific "combinations", where "combination" means the following: this is ...
Disciple's user avatar
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Looking for a description of "Ostrowski’s square root technique"

While eagerly searching for iterative methods to approximate $f(x)=0$ (commonly known as root-finder) for the quantile function or probit, I stumbled over "Ostrowski’s square root technique"....
m-stgt's user avatar
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1 answer
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Higher Dimensional Spherical Harmonics in Cartesian Form

Are there any tables in the literature or computer software for computing higher dimensional spherical harmonics in Cartesian form, like this Wikipedia article, which lists them for three dimensions. ...
Andrew's user avatar
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Fast way to calculate (AAᵀ)⊙(BBᵀ)

I need a computationally efficient way to calculate $AA^\top \odot BB^\top$, where $A$ and $B$ are tall, real-valued matrices of the same size ($\text{nrows} \approx \text{ncols}^3$) $\odot$ stands ...
lehoj's user avatar
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Reformulate an algorithm as a sequence of standard matrix operations

Consider the following code snippet ...
lehoj's user avatar
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2 votes
1 answer
93 views

Adequate Root Finder To Compute The Quantile Function

The cumulative distribution function of the standard normal distribution $\Phi(z)=\displaystyle\frac{1}{\sqrt{2\pi}}\int_{-\infty}^z e^{-t^2/2}dt$ cannot be expressed in terms of elementary functions, ...
m-stgt's user avatar
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Probability distribution of exponential chickens

This problem originates from minecraft chickens. Let's say I start with one ( fully grown ) chicken. This chicken lays one egg every $\left[5, 10\right]$ minutes, ...
jettae schroff's user avatar
1 vote
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Solve problem path of resistance stochastically stable equilibria

I'm working on Stochastically stable equilibria bout Evolutionary Game theory. In the famous paper by Peyton Young there is an example of a matrix 3x3 with the solution of the paths less resistance. I ...
Filippo Scarparo's user avatar
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Domain Truncation Error for Semi-Infinite BVP

Suppose I have a BVP defined on a semi-infinite domain, of a form that looks something like $$ N[f] = 0 \\ f(0) = a \\f'(0) =b \\ f'(\infty) = c$$ where $f$ is some (generically nonlinear) third order ...
Cade Reinberger's user avatar
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Finding the coordinate of four points of imaginary intersecting lines which passes through end points of two intersecting lines

image I want to find the coordinates of the points A,B,C,D where two imaginary lines intersect each other, where this imaginary lines passes through the end points of the two lines L1 and L2, the ...
Basavaraj Kittali's user avatar
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Benchmark Neural Networks on High-Dimensional Functions

For a personal project, I am interested in benchmarking certain neural network architectures in the context of high-dimensional function approximation. Specifically, I am interested in continuous, ...
user82261's user avatar
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3 votes
3 answers
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Book Recommendations for Learning Python for Mathematics.

Lately, I've been finding that I often need to compute various things and graph some pretty complicated functions. I've realized that learning to program, especially in Python, could be really helpful ...
Mathematics enjoyer's user avatar
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Generating Equation for randomisation of a long sequence for finding pairwise distance repeats

I am not mathematician. I am making a python code. I am generating a equation to do permutation. We want to analyze repeats in one sequence of pairwise distance and randomize it and calculate the ...
Amen Shamim's user avatar
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Correlation vs Regression for a simple task

Hello everyone and thank you for taking the time with my issue! I want to apologize in advance if my question would've fit better on stack exchange, but I decided that the question is more related to ...
nicaaa's user avatar
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Is there a website that can locate the nearest triangle center from a point? [closed]

In this question, I conjecture that the sequence of points will converge Given $A_1= (0,0), \ B_1=(11,2), \ C_1(3,5)$ , $C_{25}(3.3556952569262, 2.9821465573228)$ which should be very close to the ...
pie's user avatar
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Clarification Needed on Potential Function Term in Discrete Klein-Gordon Equation Transformation

I am studying the PhD thesis Dynamics of nonlinear lattices: asymptotic behavior and study of the existence and stability of tracked oscillations and I have observed a mistake at a function on page 69....
Athanasios Paraskevopoulos's user avatar
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Closest edge to a rectangle in a plane of rectangles

I have a 2D plane (with discrete points) that contains arbitrary-sized rectangles and all rectangles are axis aligned. I have their coordinates (upper-left) and sizes (length and breadth). Suppose I ...
Harsh's user avatar
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Numerically solving the Advection-diffusion equation with no-flux boundary condition results in violation of mass conservation

I am trying to solve numerically the advection-diffusion equation of the following form $$\frac{\partial C}{\partial t}=\alpha\frac{\partial^2 C}{\partial x^2}+\beta \frac{\partial C}{\partial x}$$ ...
Ornate's user avatar
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0 answers
49 views

Simplest prime test algorithm possible: what's its complexity?

We got the brute force algorithm: Input: natural number $n$. Set $k_{i+1}=k_i+1$, $k_0=2$, $i=0$ If $\frac{n}{k_i}$ is an integer, return $'\mathrm{Composite}'$, else, set $i=i+1$. If $k_i^2\leq n$ ...
Simón Flavio Ibañez's user avatar
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Why does the denominator of the Pad\'e Approximation to Exponential function approximate $e^{-x/2}$?

In the Paper "Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later", there is a statement that "For large $q$, $D_{qq}(A)$ approaches the series for $e^{...
ddk's user avatar
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Transformation of random variables to get a sample for another probability density function

Question: Let $\mathcal{D}$ be a two dimensional unit disk, given by $\mathcal{D}=\{(x,y):x^2+y^2\leq 1\}$. Using rejection sampling algorithm, I managed to generate a two dimensional random vector $(...
Nothing's user avatar
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1 vote
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Correctness of algorithm to find the number of elements of order $x$ in Symmetric Group $n$?

To find the number of elements of order $x$ in $S_n$ Generate all possible partitions of $n$ by divisors of $x$. For each partition, check if the LCM of the part lengths matches $x$. Calculate the ...
A. Random's user avatar
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2 votes
0 answers
32 views

Function increasing slower than any composition of logarithmic functions [duplicate]

I came up with a question of finding $f(n)$ it satisfies following property: $$\forall k \in \mathbb{N} \lim_{n \to \infty} \frac{f(n)}{\log^k(n)} = 0 \;\mathrm{and} \lim_{n \to \infty} f(n) = \infty,$...
Leo Son's user avatar
  • 29
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0 answers
15 views

Analyze the probability distribution of a specific sequence $S(x)$ with compensation mechanism

I'm developing a theoretical model for a sequence $S(x)$ equally spaced in the time dimension where each element is randomly preselected from set $\{1,2,...,L\}$, but the real selection(when it's turn)...
WxxW's user avatar
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0 answers
43 views

Minimizing a function involving a gamma function

In one of the proofs about some estimates of upper bounds of zeros of the Riemann zeta function I am studying, I have a function that looks like $V(\alpha, x, \delta,T)= \alpha e^{2\alpha(\frac{1}{T} +...
Josh's user avatar
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1 vote
0 answers
49 views

Does series acceleration improve computational efficiency?

There are several methods of series acceleration. For example, Euler's transform is: $$\sum_{n=0}^\infty (-1)^na_n=\sum_{n=0}^\infty \frac{(-1)^n}{2^{n+1}}\sum_{k=0}^n (-1)^k {n \choose k} a_{n-k}$$ ...
user46190's user avatar
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Improving the condition of a system of equations

I am trying to understand by hand how balancing a matrix increases the stability of procedures like solving a system of equations. The balance procedure that I am following is here, I also found ...
user3116936's user avatar
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0 answers
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Let L = {1^s| such that there exist s consecutive 1s in the (decimal) expansion of π}. Is the language L Turing-decidable?

Let $L = \{1^s|$ such that there exist $s$ consecutive 1s in the (decimal) expansion of $\pi\}$. Is the language $L$ Turing-decidable? At first glance it seems like it would not be Turing-decidable ...
Noah Hendrickson's user avatar
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0 answers
24 views

Lanczos algorithm for two-dimensional vector space $\mathcal{H}_1\otimes \mathcal{H}_2$

Consider the Hamiltonian $H_1$ (Hermitian matrix) along with initial state $|\psi_1\rangle$ which can be used to generate the vectors $$\mathcal{K}_1 = \{H^n_1|\psi_1\rangle : n=0,1,2,\ldots \}$$ The ...
Young Kindaichi's user avatar
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0 answers
14 views

Properties of a "generalised" little-o notation

According to Wikipedia, the little-o notation is defined this way: $f(x) = o(g(x))$ as $x\to\infty$ if for all $\varepsilon >0$ there exists $N\in \mathbb{R}$ such that $$ |f(x)| \le \varepsilon g(...
Jonathan Huang's user avatar
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0 answers
15 views

Derivation of Q in EM algorith

Trying to learn EM Algorithms. Can someone explain the derivation steps they took to go from step 4.1 - 4.3. I am quite new to computational stats and it feels like I may be missing some fundamental ...
Eyas Ayesh's user avatar
1 vote
0 answers
59 views

Minimal Simplicial Complex from a Sequence of Betti Numbers

I found the following problem in a Computational Topology course that I am following: Write an algorithm that given a sequence $(\beta_0,\ldots,\beta_d)$ of integers builds a simplicial complex whose $...
Pepe's user avatar
  • 126
17 votes
3 answers
1k views

One of the numbers $\zeta(5), \zeta(7), \zeta(9), \zeta(11)$ is irrational

I am reading an interesting paper One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational by Zudilin. We fix odd numbers $q$ and $r$, $q\geq r+4$ and a tuple $\eta_0,\eta_1,...,\eta_q$ of positive ...
Max's user avatar
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0 votes
1 answer
14 views

finite steps to Hessenberg form and/or triangular form

I am learning numerical linear algebra and curious about one thing. It is possible to reduce any matrix to the Hessenberg form in finite steps with a unitary matrix. But why is it impossible to reduce ...
poisson's user avatar
  • 1,015
0 votes
1 answer
23 views

Is the Asymptotic complexity of the find max algorithm O(n) or O(n^2)?

Algorithm pseudocode: 1 def find max(data): 2 biggest = data[0] # The initial value to beat 3 for val in data: # For each value: 4    if val > biggest # if it is greater than the best ...
suryansh_shekhawat's user avatar
1 vote
0 answers
22 views

Prove the optimal property of an orthogonal projection method -- how to show that the approximate solution minimizes the *A-norm* of the error?

Given an orthogonal projection method (when $K = L$) and a symmetric positive definite (SPD) matrix $A$, show that $\tilde{x}$ minimizes $(A(x−x^*),(x−x^*))≡ \|x - x^* \| ^{2} _A = ( A ( x − x^* ) , ...
Sophia 's user avatar
11 votes
1 answer
273 views

Beautiful errors in graph of $\sin(x^2+y^2)$

I was writing a simple program to help visualize inequalities based on 2 variables. The test inequality that I was using was this: $$\sin\left(0.1(x^2+y^2)\right)\geq0$$ Regions that satisfy the ...
Soham Saha's user avatar
  • 1,392
-1 votes
1 answer
45 views

If a matrix is an outer product of two vectors; can I determine the vectors? [closed]

I am working with floating point numbers. There is a 3x3 matrix that has determinant 1e-14. I have reason to believe this matrix is an outer product of two vectors. If the assumption is correct, how ...
Mikke Mus's user avatar
  • 149
0 votes
0 answers
20 views

Hardy-Littlewood maximal function algorithm

There exists implemented algorithm which compute the Hardy-Littlewood maximal function at least for reasonably simple non-trivial cases? E.g. piecewise linear functions?
Giafazio's user avatar
  • 330
1 vote
1 answer
33 views

How do we find the weights for finite differences?

My professor gave us this question to solve, but I don't know have much familiarity with the topic. Question The forward difference is first order accurate and is defined to be $$ D_{+} = \frac{f(x+h) ...
Calum's user avatar
  • 399
2 votes
1 answer
150 views

Confusion about atan vs atan2

I have the following function: $$f(\omega) = \arctan\left(\frac{-\omega\cdot R / L}{-w^2 + 1/(C\cdot L)}\right)$$ When I try to plot it (atan(f(w))), I get the ...
Martel's user avatar
  • 131
0 votes
0 answers
24 views

Investigating the numerical accuracy of a truncated Legendre polynomial expansion of an unknown function

I have an integral equation involving an unknown function $f(x)$, of the most basic form $$ \int_{-1}^{1} e^{iω(t) x} f(x) \ dx = g(t) $$ I am solving for an approximation of $f(x)$ by substituting in ...
Silver Pages's user avatar
0 votes
0 answers
66 views

algorithm guaranteed to converge for convex function

I have a multi-variate convex function, and I want to find its global minimum. We know there is one and only one minimum. Is there any algorithm which is guaranteed to converge to the minimum starting ...
poisson's user avatar
  • 1,015
0 votes
0 answers
19 views

Need help with proving energy stability of forward Euler for heat equation

I have the 1D heat equation $u_t= \alpha u_{xx}$ on $x \in [0,1]$. It has homogeneous Dirichlet boundary conditions. I intend to use the forward Euler numerical scheme. $$\frac{u^{n+1}-u^{n}}{k} = \...
laplacian18's user avatar
0 votes
0 answers
46 views

Is there a way to find if there relationship of numbers

I have a challenge. This may be little tricky or even not possible but wanted to check if anyone has any thoughts on this? PS : This question is in general and not related to only to R. May be I can ...
manu p's user avatar
  • 111
0 votes
1 answer
67 views

verifying Ramanujan constant

The famous Ramanujan constant $ e^{\pi \sqrt{163}} $ is a near-integer. see the link here. I tried to calculate this number with matlab and failed. Matlab cannot even deliver the first 9 apparently ...
S. Kohn's user avatar
  • 1,104
2 votes
1 answer
53 views

Is it possible to find something better than binary search for this problem?

Let's say we have $n$ urns (numbered $1$ through $n$) and the first $k$ urns have a ball in them (for some $k$ unknown to us) and the remaining urns are empty. Our goal is to determine $k$ by looking ...
user23571113's user avatar
  • 1,458
0 votes
0 answers
33 views

Fast solvers for saddle-point problem (linear system)

I would like to speed up my multi body simulator. There, in every time step a linear system, often referred to as saddle-point problem, has to be solved. The system looks like this $$ \begin{bmatrix} ...
freddy90's user avatar
  • 285
0 votes
0 answers
21 views

Finding Maximum Value of Variable Using Only Its Fourier Transform?

Assume I have a variable u, which is an $m\times n$ array filled with exclusively real values. I want to find the maximum value within this ...
Jacob Ivanov's user avatar
0 votes
1 answer
34 views

How to stop a negative exponential from rounding to zero?

I'm doing materials homework and calculating vacancy density, which has some large constants (Na and k), and have to do the function. the right side of the density equation is exp(-Q/(kT)), which ...
Erwin Davinky's user avatar

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