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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25 views

Model to reward adaptative communication

I'm currently working in different simulations focused on the study of artificial evolution of communications in robots, and I have stumbled against the problem of its mathematic formulation. The ...
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2answers
117 views

floating point binary arithmetic

Prove that the decimal number $\displaystyle \frac{1}{5}$ cannot be represented by a finite expansion in the binary system.
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2answers
195 views

Finite representation in the binary $\implies$ finite representation in the decimal system

Any number that has a finite representation in the binary system have a finite representation in the decimal system. Why?
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1answer
134 views

Simplified way of finding a complex number raised to another complex number

This question here has the answer but I'm still in school and I don't understand any of it. I'm writing a computer program that takes a complex number a + ib and raises it to c + id and I need to ...
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1answer
381 views

Exponential Probability Monte Carlo simulation

I need to write a Matlab program to estimate the quantity $\theta = \mathrm{Pr}(X < 1)$, where $X$ is an exponential random variable with mean $1$. I am doing this for multiple monte carlo ...
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0answers
185 views

Power sums, fast algorithm

I know some schemes to compute power sums (I mean $1^k + 2^k + ... + n^k$) (here I assume that every integer multiplication can be done in $O(1)$ time for simplicity): one using just fast algorithm ...
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1answer
63 views

Is regularity is preserved under reversal?

When talking about languages and regular languages. Can I say that reversal preserved regularity since if the language L is regular, we can generate it by right linear grammar. Therefore, the ...
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2answers
111 views

Solve right linear set equations

Trying to solve this set of equations. I'm feeling like I'm making it so complicated. Of course + is union. Am I on the right track? A = 0B + 1D B = 0C + 1A C = 0A + 1B + λ D = OD + 1C + λ A ...
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1answer
47 views

Show Two f.s.a. machines accept different input

I'm trying to solve it for two hours already. I know it somehow related to the pumping lemma Let $M_1 = \langle Q_1,S,f_1,s_1,F_1\rangle$ and $M_2 = \langle Q_2,S,f_2,s_2,F_2\rangle$ be two machines, ...
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1answer
1k views

Show two finite state machines are equivalent

Suppose $M_1 = \langle Q_1,S,R,f_1,g_1\rangle$ and $M_2 = \langle Q_2,S,R,f_2,g_2\rangle$ are two strongly connected machines. I need to show that $M_1 \equiv M_2$ iff there exist a state $p \in Q_1$ ...
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1answer
580 views

Computational Complexity of Modular Exponentiation

The following was posted from a lecture: "($a^n \bmod N$) has a runtime complexity of $\mathcal{O}(n*|a|*|N|)$ using the brute force method. $Z_1 = a \bmod N$ $Z_2 = (aZ_1) \bmod N$ $Z_3 = (aZ_2) ...
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1answer
703 views

radial basis function and neural networks

actually i need a simple explanation consider it for dummies about what is Radial basis function are?and what is the relation between radial basis function and neural networks ?and is there's any ...
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2answers
124 views

Approximating $\pi$ in Binary

I am interested in creating a Java program that generates digits of $\pi$ (in Binary though). To be clear, the number I'm looking for begins: $11.00100100 \dots$ I am unsure of the most efficient way ...
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0answers
382 views

Logistic regression algorithm in Casio and Texas Instruments calculators

When using logistic regression on a Casio or Texas Instruments calculator, the output is of the form $$f(x) = \frac{c}{1+ae^{-bx}} $$ The problem I have (when teaching in a class where both types of ...
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1answer
191 views

Is all mathematics computable?

No matter how good a computer is, it will never compute the whole sequence of PI, but we can approximate it to arbitrary degeree. We can also implement programs that can do calculus and linear ...
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1answer
115 views

Finding if two machines can be equivalent

I have this problem: Consider the following machines M1 and M2. M1 has initial state A and the initial state of M2 is unspecified. Can the machines be made equivalent by the correct choice of ...
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0answers
199 views

Confusion related to state equivalence of finite state machines

I have this confusion if there are two states of a machine p and q. Let x be an input string such that length of x = k, g be the output function and let g(p,x) and g(q,x) be the output when the input ...
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1answer
581 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 ...
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308 views

“Green Globs” question

When I was in high school geometry, we had a fun little game on the computer called Green Globs (the website for the software is http://www.greenglobs.net/index.html). A number of targets (globs) are ...
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1answer
98 views

Sieve higher powers with logarithmic optimization

I am factoring number $N = 90283$ using quadratic sieve. Bound is $B = 44$. I find factor base to be $\{2, 3, 7, 17, 23, 29, 37, 41\}$. I have $50$ element sieving interval: $\{318, 921, 1526, ...
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0answers
69 views

Plot randomly oriented gaussian kernel

I would like to plot with scipy randomly oriented gaussian kernels. For a gaussian kernel along x and y axis (with an angle 0 w.r.t. coordinate system), I simply plot function ...
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1answer
2k views

Quadratic sieve algorithm

I am stuck with the sieving stage of Quadratic Sieve algorithm. I've read lots of papers to this point but I can't find any guidlines how to choose sieving interval or how sieving is actually done ...
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4answers
251 views

Prime factorization, Composite integers.

Describe how to find a prime factor of 1742399 using at most 441 integer divisions and one square root. So far I have only square rooted 1742399 to get 1319.9996. I have also tried to find a prime ...
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2answers
234 views

Given two sets of vectors, how do I find a change of basis that will convert one set to another?

Given two sets of dimension $n$ vectors $\lbrace v_1 , v_2 , \ldots , v_m \rbrace$, $\lbrace u_1, u_2, \ldots , u_m \rbrace$, where $m > n$, is there a computational method (in particular, using ...
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2answers
2k views

How to handle big powers on big numbers e.g. $n^{915937897123891}$

I'm struggling with the way to calculate an expression like $n^{915937897123891}$ where $n$ could be really any number between 1 and the power itself. I'm trying to program (C#) this and therefor ...
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1answer
312 views

How to compute efficiently the norm of a cyclotomic integer

Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$. Let $A$ be the ring of algebraic integers in $K$. Let $\alpha \in A$. How ...
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1answer
257 views

Using sympy im Python to substitute values into a 10x10 matrix

'm using the symbolic package sympy to store a 10X10 antisymmetric matrix in terms of 10 variables. and then at every iteration step, i substitute numerical values into the entries of the matrix. ...
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514 views

Obtain a contradiction

Motivation : The motivation is to show that the equation $x^{2b}.x^{2a} +(3-x^{2b}) x^{a} + (1-s^2)=0 $ has no solutions in integers for any values of $x,b,a,s$ ( choosen as per the constraints ...
4
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1answer
145 views

Hilbert curve scheduling not theoretically well defined?

In high-dimensional task scheduling, it is common to use a Hilbert-curve ordering. Given a set of points $\{p_i\}_{i=1}^N \subset \mathbb{R}^d$ the goal is to linearly order them such that points ...
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1answer
111 views

Help understanding a $3n+1$ problem programming project

I was going through this website where I found 3n + 1 problem. I was not able to understand what the output should be. I ...
2
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1answer
212 views

Asymptotic expansion of a special integral

I need an asymptotic expansion of J(n) $J(n)=\frac {2} {\pi} \int_{0}^{\pi/n} \prod_{k=1}^n \frac {\sin kx} {\sin x} dx$, $n=2,3,4,\dots$ Can anybody help to find the asymptotic analytically or at ...
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232 views

Computational efficiency of Machin-like formulae

From what I have read, it appears that the most efficient methods of calculating $ \pi $ are Machin-like formulae. And it is known that certain formulas are more efficient than others. Are there any ...
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1answer
103 views

Minimizing the norm related with iteration method

I am working on a iteration method to compute the generalized inverse of a matrix $A$ of rank $r$ Iteration method is $X_{k+1} = X_{k} + \beta X_{k} (I - A X_{k}) $ where notations are as follows ...
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0answers
37 views

the leading term of a module

I'm reading CLO and I have a question about the following Prop: Let I be an ideal in a polynomial ring $k[x_1,\ldots,x_n]$. Then $k[x_1,\ldots,x_n]/I$ is isomorphic as a $k$-vector space to $S = ...
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2answers
108 views

Distinction between error estimator and error indicator

When solving differential equations numerically one can incur discretization error and one can construct a posteriori error estimates to approximate the true error. There is a distinction often made ...
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0answers
104 views

Collaborative modular exponentiation

EDIT: Rephrased. I have, stored somewhere, the values $a$ , $Q$, $N_1$ (plus its factor) and $a^{2Q} \mod N_1$. I also know $b$, $R$ and $N_2$ (but not its factors). I want to know whether there is ...
4
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1answer
73 views

Need little hint to prove a theorem from a paper

I have an iterative method \begin{eqnarray} X_{k+1}=(1+\beta)X_k-\beta X_k A X_k~~~~~~~~~~~~~~~~~ k = 0,1,\ldots \end{eqnarray} with initial approximation $X_0 = \beta A^*$ ($\beta$ is scalar ...
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0answers
234 views

How a direct method can be compared with an iterative method?

How a direct method can be compared with an iterative method? I have an iterative method to compute Moore- penrose generalized inverse. There are some direct methods available to compute Moore-Penrose ...
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0answers
159 views

Calculation of stopping condition for Conjugate Gradient

I am a person with programming background and need some math help. I am looking at the source code for an implementation of the Conjugate Gradient iterative solver ...
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2answers
90 views

Comparison of two very large necklaces

I came across one problem and I would like to find an answer. It is well-known how to calculate the number of fixed necklaces of length $n$ composed of $a$ types of beads $N(n,a)$. ...
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2answers
252 views

Value of double product

What is $$ \prod_{i=1}^n\prod_{j=1}^{n-i}i^2+j^2 $$ ? It feels like there should be some way to simplify this or calculate it more efficiently than iterating over each of the $\sim n^2/2$ points. ...
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1answer
126 views

more matrix inversion

Related to a previous question: Suppose I want to invert a (sparse) matrix written in block form as \begin{array}{cccc} A_{11} & A_{12} & \ldots & A_{1n}\\ A_{21} & A_{22} & & ...
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2answers
132 views

efficient matrix inversion problem

So I'm trying to invert a matrix of the $\begin{pmatrix} A & B \\ C & D \end{pmatrix}$ where $A$ and $D$ are square, $D$ is much larger than $A$, and $D$ is diagonal. $A$ $B$ and $C$ have no ...
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1answer
156 views

Can someone explain to me the relationship between primorials and factorials and how that relation can be used to compute large factorials?

What I am trying to figure out is a way to compute large factorials, !1000000. For what it's worth luschny's computer algorithms do a very good job of it.
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1answer
160 views

Solve equation on the PC

A friend of mine asked me to help him and make a small application to solve a problem. This problem can be reduced to this equation system: aX = Yb; Y > c; Y < d; X is a whole number (X has ...
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1answer
74 views

Finding the computational complexity of an algorithm

Algorithm: for (int i = 0; i < 2*n; i += 2) for (int j = n; j >i; j--) foo(); I want to find the number of times foo() is called. ...
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1answer
71 views

$T(n) = n^{O(1)}$ iff exists $k > 0$ such that $T(n) = O(n^k)$

I must use O notation to show that: $T(n) = n^{O(1)}$ iff exists $k > 0$ such that $T(n) = O(n^k)$ But, I don't understand what mean: $n^{O(1)}$
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2answers
2k views

Notation : What is the meaning of the (mod n) in factoring algorithms?

Pretty much every thing is in the title, really! I'm trying to come up with an efficient algorithm to factorize large integer as an homework for a parallel programming course. I've seen a few pages ...
3
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1answer
192 views

Method For Constructing Self Referential Formulas Like Tupper's

Can anyone please explain exactly how formulas like Tupper's self referential formula can be constructed? I'll like to see the reasoning behind the derivation of such formulas and the steps required ...
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2answers
288 views

Abuse of big-O notation? (version 2 - simplified and revised)

Given exam question: Algorithms A & B have complexity functions $f(n)=2 log(n^3)+3n$ and $g(n)=1+0.1n^2$ respectively. By classifying each $f$ and $g$ as $\mathcal{O}(F)$ for a suitable ...