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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15 views

Is there a way to reverse engineer an already large number to make it smaller? [on hold]

Using the Ackermann function for example, it's quite easy to make massive numbers. My question is whether there's an existing algorithm that can take a large number and reverse engineer it to make an ...
1
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1answer
27 views

How to write new algorithm of root finding by combining 2 or 3 standard algorithms(bisection, fixed, etc)

I just learned about Bisection Method, Fixed-Point Iteration Method, Newton- Raphson Method, and Secant Method. My prof wants us to be able to write new Algorithm of root finding by coming 2 or 3 ...
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0answers
13 views

Statistical calculation for neural firing rates with negative rate on numerical simulation

I am now working on a biological neural network simulation (NEST-Simulator) project with a problem of calculating firing rates. Background: The data set as result of simulation is a set of events in ...
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1answer
30 views

Constructing a specific Rank-One Matrix

Given u $\in \mathbb{R}^{n}$ and v $\in \mathbb{R}^{m}$ with unit $L^{2}$ norm, i.e. $\|u\|_{2}$ = $\|v\|_{2}$ = 1. Construct a rank-one matrix B $\in \mathbb{R}^{mxn}$ such that $Bu = v$ and ...
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2answers
86 views

Use $\log(x)$ to calculate $\log(x+1)$

Given that I know the value of $\log(x)$, I would like to calculate the value of $\log(x+1)$ on a computer. I know that I could use the Taylor expansion of $\log(1+x)$, but that uses $x$ rather than ...
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1answer
49 views

One wants to determine $1/(2+\sqrt 3)^4$ having access to an approximate value of $\sqrt 3$.

Can someone help me with this question please: One wants to determine $1/(2+\sqrt 3)^4$ having access to an approximate value of $\sqrt 3$. Compare the relative errors on direct computation and on ...
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0answers
33 views

A computation from an article in computational neurosciences (from physical review) which doesn't fit

I am reading this article (with this erratum) in computational neuroscience, and there is a computation there that simply doesn't fit.. Maybe one of you can see something that I am missing? For the ...
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0answers
17 views

Hilbert Matrix, Gaussian Elimination with varying pivot strategies, and computation error.

I'm doing a project for my Numerical Analysis class about computational error related to Gaussian elimination, gaussian elimination with partial pivoting, and gaussian elimination with scaled partial ...
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2answers
17 views

Error analysis on numerical solutiol of an equation

Say I am solving an equation numerically -- the derivatives in the equation I find by a finite difference scheme with an accuracy of the grid spacing $h$. Does this imply that the final solution I ...
2
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1answer
165 views

Non-calculator proof that $\pi^\pi -\pi \lt \frac{100}{3}$

I am looking for a few non-computational, non-calculator proof of the following inequality: $$\pi^\pi -\pi \lt \frac{100}{3}$$ I can't really seem to come up with a proof because of that killer ...
2
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5answers
375 views

Proving $\pi^3 \gt 31$

$$\large \pi^3 \gt 31$$ Using a calculator, $\pi^3/31 \approx 1.0002$, so I thought this may be challenging to do by hand. It is extremely easy with the use of any calculator, so I was wondering ...
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0answers
7 views

How do I validate my ARMAX model?

Say I have some ouput $y_1, y_2, \ldots, y_N$ and inputs $x_1, x_2, \ldots, x_N$ which, by various time series methods, I've found to match an ARMAX(2,2,1) model. So I've found the estimations for ...
0
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0answers
11 views

Formula for MLM like system

I'm trying to figure out a formula for a system similar to a MLM system such that all members will receive 50/50 of the shares. So for example, X recieves 50% and A recieves 50%. When A recruits B and ...
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0answers
29 views

Inverse of sum of matrices

Let $A,B$ be invertible positive definite matrices of the same size. My goal is to efficiently compute $(xA + yB + zI)^{-1}$ for many triplets of positive real numbers $(x,y,z) \in \mathbb{R}^3$. ...
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1answer
65 views

Minimizing computations for evaluating two polynomial simultaneously

I want to evaluate two polynomials $f$ and $g$ simultaneously, on the same input (in a computer program). These polynomial have only coefficients $0, 1, a , b$ and their degree is less than 700. I ...
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1answer
44 views

Finite element method books

I know this question has been asked before; I just want to enquire if anybody has any suggestions to learn how to compute finite element problems, including plenty of examples. The topics I would ...
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2answers
94 views

For what values of $k$ does $(1+x)^{500+k}(1-x)^{500-k}$ exceed $10^9$?

Pretty simple question, for what values of $0\leq k \leq 500$ do we have $\max\{(1+x)^{500+k}(1-x)^{500-k}|x\in[0,1]\} \geq 10^9$ ? Some trivial observations: The problem is equivalent to finding ...
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1answer
37 views

More efficient method of computing the square root of $-1 \mod p$

I am currently doing collecting some preliminary data about elliptic curves over finite fields of order $p$ where $p$ is a prime congruent to 1 mod 4. Part of the data collection process requires me ...
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1answer
26 views

c(x) = k for all positive k is primitive recursive [closed]

How can I show this function is whether primitive recursive or not? Do I need to use Godel number?
3
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0answers
53 views

Is there a systematic way of “discovering” an algebra from observations of its universe?

I am faced with the following situation: I have a finite set of some $m$ positive integers $Q^m \in \mathbb{N}$ These integers go through a series of $N$ possible black boxes that transform them. ...
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1answer
32 views

function computed by programs

I have two questions: Which is the function computed by the program $o^1_1(Succ, Succ)$? Which is the function computed by the program $\mu^1(\pi^2_1)$? where $o^n_m$ for the composition rule ...
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0answers
16 views

Solve Lotka-Volterra by hand? [duplicate]

I learn Lotka-Volterra model in computing mathematics textbook and solve it by different numerical methods. $$ \frac{1}{x}\frac{dx}{dt} = a - by$$ $$ \frac{1}{y}\frac{dy}{dt} = cx - d$$ where, ...
3
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2answers
91 views

Squeezing primes

Any positive odd number $n$ can be coded one binary digit smaller by the rule $\frac{n-1}{2}$ and that's obviously the best squeeze: a bijection from $\mathbb N$ such that $f(n)\geq n$. I'm looking ...
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3answers
81 views

Digits of $\pi$ using Integer Arithmetic

How can I compute the first few decimal digits of $\pi$ using only integer arithmetic? By 'integer arithmetic' I mean the operations of addition, subtraction, and multiplication with both operands as ...
5
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5answers
549 views

A valid floor function trick?

Given $x\in\mathbb R_+$ and $m,n\in\mathbb Z_+$, is it true that $$\bigg\lfloor\frac{\lfloor \frac{x}{m}\rfloor}{n}\bigg\rfloor=\bigg\lfloor \frac{x}{mn}\bigg\rfloor?$$ Thanks for at least three ...
8
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3answers
309 views

Three pythagorean triples

Are there any solutions for $a, b, c$ such that: $$a, b, c \in \Bbb N_1$$ $$\sqrt{a^2+(b+c)^2} \in \Bbb N_1$$ $$\sqrt{b^2+(a+c)^2} \in \Bbb N_1$$ $$\sqrt{c^2+(a+b)^2} \in \Bbb N_1$$
4
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2answers
50 views

What are all the concordant forms $n$ such that $a^2+b^2 = c^2,\,a^2+nb^2=d^2$ for $n<1000$?

Part I. The list of congruent numbers $n<10^4$ such that the system, $$a^2-nb^2 = c^2$$ $$a^2+nb^2 = d^2$$ has a solution in the positive integers is known (A003273) $$n = 5, 6, 7, 13, 14, 15, ...
1
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1answer
24 views

When is ${n \choose k} > (n-k)(k+1) + (n-k-1)k$?

I have two algorithms that output the same result for an input value of a non-negative integer k and a list of n elements, where $1 \leq k \leq n$. However, the two algorithms are very different in ...
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0answers
22 views

Trying to learn about kernel PCA but cannot understand some math.

I'm trying to learn about kernel PCA by reading through the paper of it's creators (I assume) "Nonlinear Component Analysis as a Kernel Eigenvalue Problem", Bernhard Schölkopf, Alexander Smola, ...
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1answer
77 views

How can I optimise a scalar function over a matrix?

I need to optimise the following scalar function with respect to a matrix $S$. $$ f(S) = \boldsymbol{y}^{T}\boldsymbol{X}w - \boldsymbol{1}_{n}^{T} \exp \left\{ \boldsymbol{X}w + \frac{1}{2} ...
2
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1answer
59 views

Which continued fraction for $e$ is the most computationally efficient?

I know that famous numbers like $\pi$ and $e$ have multiple representations as continued fractions and I'm fascinated with the variety of representations. My question: What continued fraction for ...
2
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3answers
67 views

Looking for fractals which are computationally demanding and preferrably parallelizable.

Oh hello guys. I am in the middle of challenging myself to putting my computer and math skills together, trying to build a small hobby computational cluster. Being interested in fractals for a long ...
0
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1answer
25 views

How to obtain fair competition between two teams

Consider a class with $4$ students having min goals as $\big\{ 1, 3, 4, 5 \big\} $ and max goals as $\big\{ 2, 5, 8, 6 \big\}$. Find the best way to divide the class in such a way that the match is ...
0
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0answers
17 views

Finite element boundary conditions

I have a boundary condition given by $\mathbf{n}\cdot \nabla m=\phi$, where $n$ is a vector normal to a surface, $m$ is a physical quantity (say mass) and $\phi$ is a constant. The boundary condition ...
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0answers
15 views

How can I implement Newton-Raphson's method with a function of one vector and one matrix?

I have a function $f(\mathbf{u}, \Sigma)$ where $\mathbf{u}$ is a $p \times 1$ vector and $\Sigma$ is a $p \times p$ real symmetric matrix (positive semi-definite). I somehow successfully computed ...
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2answers
72 views

Converting Java Code to Mathematic formula

I have algorithm , but I don't know how to convert it to mathematic formula. ...
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0answers
13 views

bio heat equation modification

I have the bio heat equation as described .here And the solution to it is, But to this I am trying to include the effect from exercise intensity as well. So the modified bio heat equation is ...
2
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0answers
18 views

Given a set of integers $S$, what is the maximum integer that is a product of one or more integers from $S$ not exceeding $X$?

Given a set of integers $S$, which will contain no more than $100$ integers. Now, what would be the fastest approach to find $M$ which is a product of one or more integers from $S$ (and multiple usage ...
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0answers
28 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 ...
8
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1answer
110 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
19 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
17 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 $\mathbf{R}^2$ ...
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0answers
10 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} ...
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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 ...
15
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3answers
212 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
19 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
10 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
42 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 ...
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3answers
69 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
46 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 ...