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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30 views

summation of even and odd numbers

In my video processing algoritm, I do some processing even and odd frames seperately. F = E(x) + O(x) where F is the video, E and O contains its even and odd ...
9
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1answer
146 views

How do I develop numerical routines for the evaluation of my own special functions?

This question has been cross-posted to ComputationalScience.SE here. When performing computational work, I often come across a univariate function, defined in terms of an integral or differential ...
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0answers
40 views

How to solve the Allen-Cahn equation with finite element method?

$$\frac{\partial\phi(\mathbf{x},t)}{\partial t}=\varepsilon^{2}\Delta\phi-F^{'}(\phi),\ \ \ \mathbf{x}\in \Omega,t>0$$ $$\frac{\partial \phi}{\partial\mathbf{n}}=0\ \ \text{on} \ \partial\Omega$$ $$...
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0answers
25 views

Integrating Wishart density

I have several points $\textbf{s} = s_1,...,s_n$ which follow Wishart distribution. In one of my problem, I have to integrate this Wishart pdf over a ball of radius $r$ at origin in $\mathbf{R}^2$ i.e....
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0answers
20 views

Obtaining the weak form of Allen-Cahn equation

\begin{equation} \frac{\partial\phi(\mathbf{x},t)}{\partial t}=g(\mathbf{x})(\varepsilon^{2}\Delta\phi-F^{'}(\phi)),\ \ \ \mathbf{x}\in \Omega,t>0\ \ (*) \end{equation} \begin{equation} \frac{\...
0
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0answers
11 views

A computational problem with implicit function

Given a function $$ H(x) = \sum_{i=0}^{\infty}a_i x^i $$ where $x_i\in[0,1]$ and $a_i \ge0$. Also, $$ F(r) = \sum_{i=0}^{\infty}a_i G(r)^i $$ G(r) is a cdf function on $[r_1,r_2]$ so that $G(r_1)=0,...
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3answers
225 views

Plotting $\left(1+\frac{1}{x^n}\right)^{x^n}$.

When I plot the following function, the graph behaves strangely: $$f(x) = \left(1+\frac{1}{x^{16}}\right)^{x^{16}}$$ While $\lim_{x\to +\infty} f(x) = e$ the graph starts to fade at $x \approx 6$. ...
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0answers
32 views

How to calculate derivative with respect to time for Optical Flow

Suppose we have 2 images in motion for detecting the object in movement according to Lucas and Kanade [u, v] = inv(H)*[dxdt, dydt] where H is the Hessian for partial derivatives for image x and y ...
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0answers
18 views

Efficiently compute the Determinant of a Banded Matrix

So I've got a large (~ 2 million x 2 million) positive semi-definite, banded, square matrix that I need to find the determinant of. What is the correct way to efficiently compute the determinant of a ...
2
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2answers
48 views

evaluating limit using binomial series

I am trying to evaluate the following limit by using binomial series $(1+x)^{1\over 2}=\left ({1/2}\over n\right) x^n$ $$\lim_{x\to 0}{{(1+x)^{1\over 2}-1-{1\over 2}x +{1\over 8}x^2}\over {x^3+x^5}}$$...
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3answers
72 views

solve differential equation $f'(x)=f(x)$

I want to solve the differential equation $f'(x)=f(x)$ using power series of the form $$f(x)=\sum_{n=0}^{\infty}{c_nx^n}$$ From my previous knowledge I know that the solution is $f(x)=c_0e^x$ I can ...
1
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1answer
51 views

explain convergent using power series

I want to use the power series of $(x+1)^{1/2}$ to explain why $$\sum_{n=1}^{\infty}\left({\sqrt{1+{1\over n}}-{1\over n}}\right).$$ converges. I was able to get the expansion of the series using ...
1
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1answer
32 views

Are there any errors in the computation of the earth surface area [closed]

The surface area of earth might be completed using the formula A=4pe *r^2. For the surface area of a sphere of radius r. Are there any errors in the computation of the earth surface area using the ...
0
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1answer
23 views

test for convergence or divergence

So I am looking at the following series: $$\sum_{n=1}^{\infty}{{ln(n)+n^p+r^n+n!}\over{n^n-n!-r^n-n^p-ln(n)}}$$ Before testing, I wanted to look at some series that I can compare this to but haven'...
1
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1answer
47 views

equivalence of theory of reals and Rationals

Present a sentence φ that is in theory of reals but not in thoery of Rationals Following up from this question what is the approach to show that both the theories are equivalent Th(R, 0, 1, +, ≤) ...
1
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1answer
33 views

How to find the closest bounded rational approximation to a rational number?

Say I have a rational number $a/b$ and I want to find its closest rational approximation $x/y$ where $$x_- \leq x \leq x_+$$ $$y_- \leq y \leq y_+$$ for some constants $x_\pm$, $y_\pm$. How can I ...
1
vote
1answer
73 views

Particle swarm optimization

I don't know where to start. Like, I don't know how to plug the info into the algorithm. Show two iterations of particle swarm optimization (neighborhood approach) method. Mathematically show two ...
0
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2answers
56 views

growth rate of n! versus $r^n$

How do you show that $\lim_{n\to \infty} {n!\over {r^n}} $ approaches $\infty$? the growth rate of $r^n$ is slower than $n!$, so the latter one is increasing faster, but how do you show the above ...
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4answers
1k views

Newton form vs. Lagrange form for interpolating polynomials

I'm just wondering, what are the advantages of using either the Newton form of polynomial interpolation or the Lagrange form over the other? It seems to me, that the computational cost of the two are ...
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0answers
17 views

I need help with this Standard Uniform random variable problem

Click here for problem image I need help on how to do b,c and d. Can I please have an idea of how to start it because i have no idea how to do this.
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0answers
55 views

The solution of Allen-Cahn equation?

$$\frac{\partial\phi(\mathbf{x},t)}{\partial t}=\varepsilon^{2}\Delta\phi-F^{'}(\phi),\ \ \ \mathbf{x}\in \Omega,t>0$$ $$\frac{\partial \phi}{\partial\mathbf{n}}=0\ \ \text{on} \ \partial\Omega$$ $$...
1
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0answers
37 views

for what value of p does the integral converge

For what value of p does $\int_0 ^1 x^p dx$ converge? So I used improper integral value for this, looking at the result $$\lim_{p\to 0+} {1\over p+1}(x^{p+1}-p^{p+1})$$ *** I made a mistake, I got ...
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0answers
63 views

Find the different values of this integral when all paths of integration are possible

This is the question.. I only know how to do the question from infinity to infinity.. enter image description here Find the different values of this integral when all paths of integration are ...
0
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1answer
40 views

An unusual limit

I a reading an article in neural coding. It cintains a limit which I cannot understand. (It remindes the benomial sum but it didn't work for me) I am talking about $(22) \Rightarrow (23)$ in the ...
1
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1answer
168 views

Is there a word for an “infinite algorithm”?

According to Knuth's notes (see Slide 3), an algorithm, by definition, satisfies the following five properties: Finiteness: Terminates after a finite number of steps. Definiteness: Each step is ...
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0answers
46 views

Constructing triangulations algorithmically

I am developing a Python package for computations in algebraic topology (namely: cohomology and Massey products on manifolds). Basically all the stuff I'm doing requires an explicit triangulation of ...
2
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0answers
50 views

A recursive problem with GAP concerning lists and an iterator loop

I have the following question concerning a list algorithm in GAP: Let $L_1$ be a non-empty list with certain objects as entries. I wrote a program and called it helping_program_1. The Input for ...
1
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0answers
25 views

Algorithm detect simple curves using Voronoi diagram or Delaunay triangulation?

I wonder if there is algorithm/method to determine if closed (or even non closed) curve is simple or not, using the mathematics from the field of computational geometry? Especially I wonder if exist ...
0
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1answer
23 views

How can I find an approximation for y(1) using MATLAB without ODE solvers? (Euler's Method & Matrices)

Task: Given \begin{align*} y(t+h)&\approx \underbrace{\left(\begin{matrix}1-h & 5h & h\\3h & 1-h & 0\\0 & -th & 1+h\end{matrix}\right)}_{F(t,h)}y(t)+\underbrace{\left(\...
1
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1answer
43 views

Is there a good way to compute this integral?

Sometimes questions like: "How many digits does 2015! have?" become quite trendy. The most reasonable approach would probably use Stirling's formula. However doing this in the same way over and over ...
8
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1answer
552 views

Is there a way to calculate the definite integral of inverse of a 5th degree polynomial?

I want to calculate the definite integral of inverse of a 5th degree polynomial. The problem is that the inverse of the polynomial cannot be calculated (by using Matlab). However without calculating ...
0
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1answer
34 views

Find RHS from LHS.

$A=\left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array} \right]<==>\left[\begin{array}{cccc} 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ ...
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0answers
43 views

Affine Matrix for Computer Graphics

I was given this assignment task which I have no idea where to start with so I'm hoping that someone can help me with this? Student A: Matrix 4x4 (0.989, 0, -0.148, 0, 0.018, 0.992, 0.121, 0, 0....
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0answers
137 views

Finding The Radix of A Quadratic Equation

I have found previous solutions to finding the radix of a quadratic equation, where both of the provided roots return the same radix or base. However, unless I am some type of arithmetic error of ...
0
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1answer
22 views

Question related to the security of RSA method

I learned about the RSA method, where if B wants to send a message $M$, say $0 \leq M <n = pq$ to A with public key $(n,e)$, then B sends $M'= M^e (mod \ n)$. Then A can decode this message using ...
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0answers
15 views

adding a vertex and edges which do not change lengths of shortest paths (ACM ICPC problem)

I am trying to figure out how to solve the problem "Farm and Factory" from the ICPC archives. For the sake of making this question self-contained, I've reproduced the problem below: All hail ...
0
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0answers
73 views

What is meant by “deterministic error”?

I am reading a scientific paper in computer science and I have found the term of deterministic error. I googled to find any meaning to this notion but I did not find anything. So is there anyone to ...
0
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0answers
26 views

Minimization Optimization

MAPE (Mean Average Percentage Error):Let $\left\{p_{1},p_{2},p_{3},p_{4}\right\} $ be the numbers $1,-4,4,-5$ respectively. Find the number $x$ that minimizes the Mean Average Percentage Error, \...
1
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1answer
76 views

System of ODEs Matrix Representation?!

Task: Consider the system of first-order ODES: \begin{align*} y_1' =&5y_2-y_1+y_3\\ y_2'=&3y_1-y_2+t^2\\ y_3'=&y_3-ty_2 \end{align*} Write out the matrix-vector representation of this ...
1
vote
1answer
29 views

polynomials in terms of elementary symmetric polynomials

Let a polynomial of $2n$-variables be $$ f(x_1,\cdots,x_n,y_1,\cdots,y_n)=\prod_{i,j=1}^n(1+x_i+y_j). $$ Let the elementary symmetric polynomials be $\alpha_1=\sum_{i=1}^n x_i$, $\alpha_2=\sum_{i<j}...
2
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0answers
34 views

terms of taylor expansions of multiple variables at the origin

By the fundamental theorem of symmetric polynomials, $X_1,X_2,\cdots,X_n$ are polynomials of $ e_1,\cdots,e_n$ and $$ \mathbb{Z}[ e_1,\cdots,e_n]=\mathbb{Z}[X_1,X_2,\cdots,X_n]. $$ We define a ...
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1answer
539 views

Can I get multivariable taylor series expansion on wolfram alpha or matlab?

I need something like this : say $a, x, y$ and $t$ are three non-negative real numbers. Now define the complex number, $z = -y + i(a+x-t)$ and consider the function $f(x,y) = z\text{ }tanh (\pi z) log ...
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0answers
43 views

Algorithm for min max of a function

I want to solve a problem of the form $$ \min_{x \in X} \max_{y \in Y} f(x,y) $$ where $X \subset \mathbb{R}^d$ and $Y \subset \mathbb{R}^k$. Typically $d$ is large ($>100$) and $k$ is small (1-3)....
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4answers
133 views

Is the function $y=xe^x$ invertible?

I'm wondering if the equation $re^r=se^s$ has any answer. If there is any answer,and $r=-1+v,s=-1-v$ in which $v$ is a positive real number,what can we say about $v$? Thank you in advance.
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0answers
53 views

Determining N d-points yielding equal sums of Euclidean distances from M s-points

Given M source points (s-points), determine N, the number of destination points (d-points), and their locations (coordinates), such that the sum of the N Euclidean distances from each source point to ...
-1
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3answers
74 views

To solve this numerically : [closed]

$$0.5 = 1 −{0.955}^n − {0.005}^n{0.995}^{n −1}n − {0.005}^2{0.995}^{n −2}\left(\frac{n(n−1)}{2}\right)$$ I'm using MatLab but should I use a for-loop? Can anyone work me through the steps? Thank ...
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0answers
55 views

Represent Dirac Delta function in Finite Difference method

I recently solving $-\Delta u=\delta$ where $\delta$ is dirac delta function using FDM on 2 dimensional space. Since dirac delta function is undefined at origin, and 0 elsewhere, I will use $\delta(...
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1answer
28 views

Integration using Fourier Transform

How to integrate the function $(\sin x)^2/x^2$ using Fourier transform of function $g(x)=1$ if $|x|<1$ else $g(x)=0$ which is $(sin w/w)*2/pi$?!thank you in advance.
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0answers
28 views

Extended Transition Function Equivalent Proof

I encounter a difficult question (for me), and until now I haven't found a solution for it. In this question, I have to proof that these two are equivalent using induction. ...
0
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1answer
119 views

Open Composite Newton–Cotes formula

I'm after an Open Composite Newton-Cotes formula. The reason for this is I have a function that I know at N evenly spaced interior grid points but I do not know it at the two endpoints. I'm after ...