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
1answer
64 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 ...
0
votes
2answers
23 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: $$ ...
1
vote
0answers
31 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 ...
2
votes
0answers
64 views
+50

A conjecture about the prime function $p_n$

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 } ...
1
vote
0answers
31 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 ...
4
votes
2answers
683 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\\ ...
0
votes
1answer
23 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 ...
2
votes
2answers
504 views

Space spanned by matrices

I have a set of $5$ by $5 $matrices, $M_1,M_2,...,M_{19} ,M_{20}$. I want to try to find a basis from this set and also to find relationships between these matrices. This is how I think I should ...
2
votes
5answers
110 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
547 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 ...
0
votes
0answers
10 views

Determine an algorithm for $LU$ factorization and determine the number of operations [duplicate]

Suppose that $A\in\mathbb{R}^{n\times n}$ is a nonsingular matrix and that $A = LU$ is its $LU$ factorization. Give an algorithm that can compute, $e_i^{T}A^{-1}e_j$, i.e., the $(i,j)$ elements of ...
0
votes
0answers
7 views

Determine an efficient algorithm and describe the computational/storage complexity

Recall that a unit lower triangular matrix $L\in\mathbb{R}^{n\times n}$ is a lower triangular matrix with diagonal elements $e_i^{T}L e_i = \lambda_{ii} = 1$. An elementary unit lower triangular ...
2
votes
1answer
41 views

For what value of k onwards is it pointless for a computer to compute the probability mass function of the Poisson distribution

I am asking my computer to compute the probability mass function of $X \sim \text{Pois}(\lambda)$, a Poisson random variable. The function is: $$\Pr(X = k) = \frac{{e^{-\lambda} \lambda^k }}{k!}$$ ...
1
vote
2answers
34 views

Could we find an element on finite field? [closed]

Let $F$ be a finite field. Given an element $a^x$ in $F\setminus\{0\}$, could we find $a$?? I know that finding an integer $x$ is very hard problem (Discrete Logarithm Problem). However, I don't know ...
6
votes
1answer
79 views

Given two algebraic conjugates $\alpha,\beta$ and their minimal polynomial, find a polynomial that vanishes at $\alpha\beta$ in a efficient way

Inspired by this question, I was wondering about the following problem. $\alpha,\beta,\gamma,\ldots$ are the roots of an irreducible polynomial over $\mathbb{Q}$. How to compute the coefficients ...
0
votes
0answers
27 views

What's best way to determine optimal solution given 2 objectives to maximize [closed]

I have 3 machines A, B and C. I would like to rank the machines based on which machine maximizes Score1 and Score2. Score1 and Score2 are performance measures that rang from 0-100%. Below is some ...
1
vote
1answer
41 views

Generating all prime powers $\leq N$

Some very good algorithms exist to generate all primes $p$ up to some bound $N$, like the sieve of Erastothenes and the sieve of Atkin. However, suppose I want to generate a (sorted) list of all prime ...
2
votes
1answer
45 views

Mysterious functions

I originally asked the following question in stackoverflow, but the question is closed because some members meant that the question is about math(see the following thread) So I will give a try here: ...
0
votes
1answer
19 views

Number system question

Is it true that if we are given a number system with base 3, mantissa 2, $-1 \le p \le 1$, determined by $$\pm 0.d_1 d_2 \times 3^p$$ where each number is normalized, unless it is zero, then the ...
3
votes
1answer
47 views

Conjecture about odd primes

For each odd prime $p$ there exist $n\in\mathbb{N}$ such that $p\equiv n^2 \text{ (mod }\varphi(n^2))$, where $\varphi$ is Euler's totient function. I'm developing my Forth based computational ...
7
votes
5answers
384 views

What is a nice way to compute $f(x) = x / (\exp(x) - 1)$?

I want it to be stable near $f(0) = 1$. Is there a nice function that does this already, like maybe a hyperbolic trig function or something like expm1, or should I just check if $x$ is near zero and ...
1
vote
0answers
24 views

Ordering of elements in the base of a group

In section 4.6.7 of HANDBOOK OF COMPUTATIONAL GROUP THEORY, the authors use an ordering $\prec$ for the elements in a coset. That ordering, $\prec$, was defined in section 4.6 as follows. Throughout ...
0
votes
2answers
39 views

A matrix of a single 1 in each row and 0 elsewhere

Is there a particular name given to a matrix of m rows and n columns such that it must have one and only one 1 in each row and 0 elsewhere? For instance: ...
0
votes
0answers
13 views

Nature of state - recursion?

I always wondered how mathematicians define state (or rather: where it comes from?). This is tricky, because I always thought that in math there is only one "thing" - a pure, stateless function. Few ...
0
votes
0answers
14 views

Getting the most weekend for a specific date

Is there a formula that can find all the date(s) that fall on weekend the most times given a range of years. Example: Let say that it is given that the min year as 2000 and max year as 2005. and ...
0
votes
0answers
34 views

Positivity of the last component of non negative least squares based on active set method

I have followed the instructions given in Lawson and Hanson book for non-negative least squares using active set method. I am having a trouble in justifying one of the statements they have made about ...
0
votes
1answer
184 views

Multi layer perceptron activation function

How can you show that the Fourier series approximation of a function (so $f(x)=\sum\limits_{n=0}^{\infty} (a_n cos(nx) + b_n sin(nx))$ can be approximated to arbitrary precision by a feedforward ...
4
votes
0answers
529 views

Two-layer Perceptron for XOR

I'm reading Neural Networks for Pattern Recognition by Christopher M. Bishop. It's for a physics class, but I think the problem is closer to mathematics so I'm asking here instead of PSE. Chapter 4 of ...
0
votes
0answers
12 views

Finding a quadratic function the bounds any function below.

Given a function $f(x)$ I need to find another function $g(x)$, where $g(x) = Ax^2 + C$, where $A$ and $C$ are constants, such that $g(x) \le f(x)$, for all $x$. I can use a computer to do this for me ...
3
votes
1answer
1k views

Radial Basis Function and Neural Networks

I need a simple explanation about what is the radial basis function? And what is the relationship between the radial basis function and neural networks? And are there any simple examples to explain ...
0
votes
0answers
14 views

Help with Hidden Markov model and SMC methods

So its quite a long background i don't really know where to start but here goes. The background is as follows: Background Observation model As the target is moving, it measures the signal (RSSI) ...
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 ...
2
votes
1answer
24 views

What is the algorithm used by Matlab for computing the Bessel function?

I am quite curious about the algorithm behind. It is definitely not the power series expansion, right? So, what is the trick? I cannot find it in the help file of Matlab.
1
vote
0answers
13 views

Good numerical method for finding the eigenvalues and eigenfunctions of the Dirichlet-Laplacian?

Let us confine ourselves to 2D. What is the best numerical method for solving the eigenvalues and eigenfunctions of the Dirichlet-Laplacian operator? Possibly, it depends on the shape of the domain? ...
-2
votes
0answers
56 views

What are the Correct Conditions for Akra-Bazzi Master Theorem?

The Akra-Bazzi method solves recurrences of the form: $$T(n) = g(n) + \sum\limits_{i=1}^k a_iT(b_in + h_i(n))$$ In the Wikipedia article about the topic, it says that the condition on $g(n)$ is: ...
-1
votes
0answers
18 views

Rejection method in R

I need to generate pseudo random numbers using the rejection method in R X is a random variable normally distributed N(6,1). I did as follows, but I get a error message. Can someone give me a hint? ...
5
votes
1answer
608 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: ...
1
vote
0answers
25 views

How to use CVXOPT to solve an semidefinite programming problem

I'm using Sage to solve a problem and would like to use cvxopt to solve a sdp problem. Specifically, I have a list of expressions of the form $$c + \sum_{i,j} a_{i,j} q_{i,j}$$ where each $c$ and all ...
1
vote
1answer
24 views

Can gauss quadrature integrate this function exactly $f(x) = \frac{2x}{\sqrt{x^3 + 2x + 5}}$?

Suppose I had a function $$f(x) = \frac{2x}{\sqrt{x^3 + 2x + 5}},$$ that I wanted to integrate on the interval $[\pi, 2\pi]$. Can Gauss quadrature of order $2$ (ie. with two points ...
0
votes
1answer
30 views

Finding distance from valid number

I have a game related problem that is pretty complex. Here is the simplified version of the problem. I have a list of "good" numbers. ...
0
votes
1answer
35 views

Is the problem decidable with Turing machine M that inputs x,y,z does M halts on these 3 instances

Is the following problem is decidable? Given a Turing machine M inputs x,y,z does M halts on these 3 instances? Hint: make y and z any two artificial inputs that the program stops with these inputs. ...
5
votes
3answers
571 views

The J programming language: is it useful for mathematics?

I just stumbled upon the J programming language, which has the description: J is particularly strong in the mathematical, statistical, and logical analysis of data. It is a powerful tool in ...
0
votes
0answers
20 views

Computationally check for roots/positiveness of a big polynomial in a given interval

For a proof, I need to check that given a little interval $(0, 0.28)$ some concrete polynomials $\in \mathbb{Q}[w]$ (polynomials in one variable ranging over the real numbers, with degrees around 50) ...
0
votes
0answers
20 views

Help with a proof of the computability of the monus function by recursion

Reading a text on computability by a guy called Cutland, and he basically asserts the following, which is suppose to be a proof by recursion that x ∸ 1 is a computable function: (1) 0 ∸ 1 = 0 (2) ...
1
vote
1answer
24 views

compute probability density function of a bivariate function without sampling

Suppose $X_1 \sim f_{X_1}(x_1)$, $X_2 \sim f_{X_2}(x_2)$ are random variables with known probability density function. Is there any way to compute the probability density function of a bivariate ...
1
vote
1answer
44 views

How to find b ( the most efficient way) in $ax^2+bxy+ cy^2$?

I know the very basic way to find the b in this quadratic expression: $$P(x,y)=ax^2+bxy+ cy^2$$ I can first evaluate $P(0,1)=c$. Similarly, I can do $P(1,0)=a$ and then I can do ...
9
votes
2answers
3k views

Fast Matlab Code for hypergeometric function $_2F_1$

I am looking for a good numerical algorithm to evaluate the hypergeometric function $_2F_1$ in Matlab (hypergeom in Matlab is very slow). I looked across the ...
1
vote
1answer
32 views

n-th number with given prime divisors

I would like to compute th $n$-th positive integer whose prime divisors are among numbers $2$, $3$ and $5$. $n$ is at most $12500$. My first approach was sieving but i found out that there are less ...
9
votes
1answer
218 views

How many numbers $ N \le 10^{10}$ are the product of $3$ distinct primes?

How many numbers $ N \le10^{10}$ are the product of $3$ distinct primes? I can realistically calculate any $\pi(n), n < 10^{15} $ but I don't think it's possible to list all primes $>10^8$ in ...
2
votes
1answer
45 views

Converting Integers from One Base to Another Digit by Digit

So I’ve done some hands-on work with converting integers from one base to another using the well-known method of division and taking the remainder. The most generic algorithm involves dividing the ...