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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2
votes
0answers
31 views

How to exactly determine whether a sum of n-th roots of unity is zero

Define the set $R = \{e^{2\pi i k/n} | k=0,1,\ldots,n-1\}$ of $n$-th roots of unity. Let $S \subseteq R$ be a subset. How can I (algorithmically?) determine whether $\sum_{s\in S} s = 0$? I'm ...
2
votes
0answers
77 views

Even natural numbers are sums of two primes with twins or of two primes without twins

I seems to be very few even numbers that can't be written as a sum of two primes with twins or as a sum of two primes without twins. That is, suppose that $\mathbb P'$ is the set of the primes not ...
1
vote
0answers
17 views

Relative error in perturbed linear system

The linear system $A\vec{x}=\vec{b}$, where $$A_{(n\times n)}=\begin{bmatrix} 1 &-1 &\dots &\dots &-1 \\ 0& 1 &-1 &... &-1 \\ \vdots & \ddots &\ddots &...
0
votes
1answer
1k views

Fast methods to check linearity of differentials? Generalizing linearity?

The L1 Mat-1.1010 -course here has taught me the linearity conditions $f(a x)=a f(x)$ and $f(a+b)=f(a)+f(b)$. I want to generalize it, some quite irrelevant slow investigation here. It requires time ...
8
votes
2answers
300 views

A conjecture about the prime function $p_n$: $p_m \cdot p_n >p_{m \cdot n}$

While testing my system Zet for computational mathematics I find possible relations now and then. The latest is: Conjecture: For all $(m,n)\in\mathbb Z_+^2$ except $(3,4),(4,3) \text{ and } (4,4)$...
1
vote
0answers
50 views

Finding generators for a polynomial ideal given some polynomials belonging to it

Let $k$ be a finite field, $n$ a positive integer and $R := k[x_1,\ldots,x_n]$ the polynomial ring in $n$ variables. Let $f_1,\ldots,f_n\in R$ be polynomials with the following property: $f_i$ has ...
1
vote
1answer
37 views

Computing pi with maps on the rationals

There are numerous ways to compute approximations to $\pi$. Is it possible to find a mapping $f:A \to A$ where $A \subseteq \mathbb{Q}$ such that the iterates $f^{n}(x)$ tend towards $\pi$ for any ...
0
votes
1answer
48 views

Analysis of bisection search

http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-00sc-introduction-to-computer-science-and-programming-spring-2011/unit-1/lecture-3-problem-solving/ In the following video i'm ...
1
vote
0answers
36 views

Sifting algorithm for group generated by a set. [closed]

On page 38 of "Lecture Notes in Computer Science" by Christoph M. Hoffmann, there is an algorithm (ALGORITHM 2). I have some confusions. The algorithm needs to go to all column element indexed by ...
0
votes
0answers
78 views

Counterexample 3n + 1 problem (Collatz) Exponential and linear Diophantine equation

So, I have found a sufficient condition (not necessary) for finding a counterexample to the 3n + 1 problem, namely the existence of solutions for the following two-parameter family of Diophantine ...
0
votes
0answers
23 views

FFT butterfly diagram

I got confused in the FFT butterfly diagram. Can someone please help me understand it? If I have the vector $x = (-3, -2, -1, 0, 1, 2, 3, 4)$, and I want to apply FFT to it using the Butterfly ...
0
votes
0answers
8 views

Reassembling a matrix from submtrices in MATLAB

Here I have an image X defined as follows: X=imread('sunset.jpg'); So X is a matrix. Now I subdivide the matrix into $8\times 8$ submatricse as follows: $$B = mat2cell(X(:,:,1), 8*ones(1,...
3
votes
0answers
28 views

Finding all real roots of an equation

I am looking for a computational method to find real roots of a function. The function looks like this: $$F(x) = \sum_{i=0}^n \frac{k_i}{\sqrt {(x - x_i)^2 + c_i^2}}.$$ I would like to use something ...
2
votes
0answers
28 views

Is a possible use of a Mill's constant the encapsulation/encryption of messages?

I wonder if the way that Mill's constant is defined could provide a good data encapsulation and encryption method if instead of encapsulating primes, for instance a simple ASCII message is ...
5
votes
1answer
641 views

Cylinder-ray intersections equation

I found an article involving infinite cylinder-ray intersections, and I don't know how they develop this equation: $$(q - p_a - (v_a, q - p_a)v_a)^2 - r^2 = 0$$ In the end of the first page I quote: ...
2
votes
1answer
39 views

Book that helps recognizing underlying mathematical expression of numerical results

I strongly remember having seen a couple of years ago a huge book that only consisted out of an enormous table over hundreds of pages containing approx. 10⁵ numerical results in ascending order with ...
4
votes
2answers
710 views

Software for numerical solution of a non-linear ODE system?

I have been given a nonlinear system of ODEs which has arisen out of a colleague's engineering research: $$\begin{array}{rcl} \dot{x}_0&=&x_1\\ \dot{x}_1&=&-\frac{\lambda}{(x_2)^n-k^2\...
1
vote
0answers
16 views

Discrete Fourier transform - seemingly different definitions

I've noticed that in most sources the definition of the DFT is given by $$F_k = DFT\{f_n\}=\frac{1}{N}\sum\limits_{n=0}^{N-1}f_nW_N^{-kn}$$ where $W_N$ is the $N$th root of unity and $0\le k \le N-1$. ...
0
votes
0answers
21 views

Lyapunov exponent of multiple time series

I have multiple time series data that I have generated by varying the initial conditions infinitesimally. I now want to calculate the Lyapunov exponent to identify the sensitivity to initial ...
0
votes
0answers
40 views

Fastest point-plane distance in $R^3$

Many questions regard computing the point-plane distance, my question in borderline with computer science, though. What is the fastest way of computing in $R^3$ the point-plane distance, with ...
1
vote
0answers
35 views

references for concrete computations in Lie groups for abstract toplogical concepts

A Lie group is a smooth manifold whose tangent space at its origin is its Lie algebra. Taking an example for lie group such as SL(2), and due to above facts we should then be able to translate the ...
-1
votes
0answers
76 views

Number of $N$ formed from the set of points

Given $k$ points on 2d plane, I need to find the number of $N$ shaped figures from these $k$ points. lets consider four different points from the set and name them $A$, $B$, $C$, and $D$ (in that ...
0
votes
1answer
48 views

Compute Christoffel symbols & Riemann tesors in Maple 17

I invented a metric tensor g and now I'm trying to compute my first Christoffel symbol but an error message is popping up "Error, bad index into matrix" Is there a way for maple to compute Christoffel ...
0
votes
0answers
21 views

Find all vectors with magnitude between $n-\delta$ and $n+\delta$

I am working on an program to compute the structure factor of a given configuration of 3D points, and I need an efficient algorithm to generate all the possible 3D vectors with integer coordinates and ...
0
votes
1answer
38 views

How to find a modulus equation? [closed]

Let $x$ and $q$ be an integer.Also, $a$ and $b$ are integers. We know the two modulus equations. i) $x \equiv y$ mod $p^a -1$ ii) $x \equiv z$ mod $p^b -1$ Then how to find $x$? Can we find $x$ ...
1
vote
2answers
56 views

The eventual advantage of a primality test without known exceptions

The primality test of Fermat with base $2$ seems to be as secure as the computer hardware for testing numbers big enough. However, I think there are an infinite numbers of false primes using this ...
1
vote
2answers
165 views

Sequences of a computable function

Is there any computable function $f(n)$, which given any integer $n$ has been proven to return either $0$ or $1$ in finite time, and for which the statement "$f(1), f(2), f(3),\ldots$ contains ...
0
votes
0answers
54 views

Is the cohomology ring of a CW complex computable?

There is a well-developed technology for computing the cohomology groups of a CW complex, cellular cohomology. It reduces the problem of computing cohomology to the two simpler problems of (1) ...
3
votes
4answers
107 views

Is there an alternative way to represent the $\operatorname{diag}$ function?

In optimization, it is common to see the so called $\operatorname{diag}$ function Given a vector $x \in \mathbb{R}^n$, $\operatorname{diag}(x)$ = $n \times n$ diagonal matrix with components of $x$ ...
0
votes
0answers
21 views

How to define a variable which is an integral involving cauchy principal value inside?

How to define a variable which is an integral involving cauchy principal value inside in any computer programming language? I want to know how to break down the procedure step by step from a ...
0
votes
0answers
48 views

To solve multivariate polynomial equations

For a system of multivariate polynomial equations like this: $$ \left( {\begin{array}{*{20}c} {\frac{{124}} {3}} & { - 24} & {\frac{{ - 68}} {3}} & {\frac{{68}} {3}} \\ {32} & {...
0
votes
1answer
29 views

Analyzing Accuracy of a method to compute summation

How does one analyze the accuracy of the following method to evaluate the summation? Also, what is the difference between analyzing the accuracy and proving that it is backward stable? I think I ...
1
vote
1answer
32 views

How to find all positive integer solutions of a Diophantine equation?

Here is the equation $$ 6a+9b+20c=16 $$ To solve this, i follow the below steps : $\gcd(6,9)(2a+3b)+20c = 16$ let, $w = 2a+3b$ So, $3w+20c =16$ then, specific solution of $w = 112+20n$, $c = -16-...
0
votes
0answers
18 views

Composite integration rule derivation

How does one derive a composite integration rule $\tilde{I}$ for $I=\int\limits_a^b f(x)dx$ using the integration rule $\hat{I} = w_0 y_0+w_1y_1+w_2 y_1'$ and $N+1$ equidistant points $x_i = a+i(b-a)$,...
4
votes
0answers
55 views

Why using primes as base in the Rabin-Miller test?

I have done some computer tests with the Rabin-Miller primality test: To test an odd number $n$, write $n=2^r\cdot s + 1$, where $s$ is odd. Given a number $a$ such that $1<a<n-1$, if $...
2
votes
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 ...
0
votes
1answer
28 views

Finite difference formula to approximate second derivative

I have one question which asks to derive a finite difference formula to approximate $f''(x)$ in the form of $$f''(x)\approx Af(x+2h)+Bf(x+h)+Cf(x)$$ with the method of undetermined coefficients. ...
4
votes
0answers
126 views

How to keep up when converting between bases?

Here is a schematized binary channel that neatly conveys a decimal number. $ \require{begingroup}\begingroup \def\T {{ \cal T }} \def \Ti {{ \T \raise5mu{ \text- \scriptsize 1 } }} \def\Bx #1{{ ~ ...
0
votes
0answers
21 views

First Intersection Of Periodically Repeating Intervals

I have a set of coupled tasks, let's say $M$ of them. The $ith$ coupled task is represented as the following 3-tuple $\{A_i,D_i,B_i\}$ where $A_i$ represents the time it takes to perform the first ...
0
votes
1answer
24 views

Combination with Repetitions Including Duplication of N Value

Is a Combination with Repetition the correct term for the following problem N - Letters a, b, c R - 2 Example Result should equal aa ab ac bb ba bc cc ca cb Total ...
3
votes
1answer
218 views

Conjecture about Rabin-Miller pseudo prime test

I tested the Rabin-Miller pseudo prime algorithm using a single test value and found that the number of false calls depends on the size of the number to test, reducing to a (conjectured) negligible ...
0
votes
1answer
22 views

Product of all Square Roots, taken only Decimal Digits

How and where could I compute the decimal reminder of a product of square roots times ten: $$Dr\left( \prod_{x=1}^{k}x^\frac{1}{2} \right) \times 10$$ Where $k$ is a power of $10$. I would like to ...
0
votes
0answers
69 views

What is the fastest known algorithm for finding eigenvalues?

What is the fastest known algorithm for finding eigenvalues? Second and third fast are also of interest if they are simpler, basically anything better than the standard solve characteristic polynomial ...
2
votes
1answer
67 views

Is this polynomial time for greatest prime factor of odd numbers?

For natural numbers $n$ and $x,$ the number of $n^{th}$ roots that have $x$ in the whole numbers place can be represented as $(x+1)^{n}-x^{n}.$ For $p$ prime, $(x+1)^{n}-x^{n}-1\equiv0\bmod p$ iff $n=...
0
votes
2answers
24 views

Newton Raphson Example help

For h:= $\mathbb{R} \rightarrow \mathbb{R}, x \rightarrow e^{x}-x^2+1$ I know the formula as $$X_{n+1}=X_{n}-\frac{f(X_{n})}{f'(X_{n})}$$ so this would give me: $$ X_{n+1}=X_{n}-\frac{e^{x}-x^{2}+1}{e^...
1
vote
0answers
35 views

R $\subseteq \omega$ recursive iff $\exists m \in \omega$ such that $R=\{n \ | \ \bar{\omega} \models \phi[m,n] \}$.

The queston I'm trying to solve is use Kleene's enumeration theorem to show R $\subseteq \omega$ recursive iff $\exists m$ such that $R=\{n \ | \ \bar{\omega} \models \phi[m,n] \}$ for some $m \in \...
1
vote
0answers
40 views

Normalizing an elliptic curve to find integer solutions

I have an elliptic curve $$ c_1y^2 + a_1xy + a_3 = c_2x^3 + a_2x^2 + a_4x + a_6 $$ with integers $a_1,a_2,a_3,a_4,a_6,c_1,c_2$ and I would like to find all integer solutions of this elliptic curve. I ...
0
votes
1answer
26 views

Write column form elementary matrix in terms of element form elementary matrices

Recall that any unit lower triangular matrix $L\in\mathbb{R}^{n\times n}$ can be written in factored form as \begin{equation} L = M_1 M_2\ldots M_{n-1} \end{equation} where $M_i = I + l_i e_i^{T}$ ...
2
votes
5answers
113 views

Minimize $a^5+b^5+c^5+d^5+e^5 = p^4+q^4+r^4+s^4 = x^3+y^3+z^3 = m^2 + n^2$ with distinct positive integers

Find the minimum value of the following: $$a^5+b^5+c^5+d^5+e^5 = p^4+q^4+r^4+s^4 = x^3+y^3+z^3 = m^2 + n^2$$ where all numbers are different/distinct positive integers. I know the answer (see ...
10
votes
6answers
583 views

Calculating logs and fractional exponents by hand

In view of what we can compute by hand, on a piece of paper, without having to use a computer or a calculator, how far can we go with the evaluation of $\log$-functions and fractional powers? More ...