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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4
votes
2answers
384 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
1answer
43 views

Numerical evaluation of a complex integral

I have to evaluate numerically $f(z)$ via the Cauchy representation (so via a complex integral), in other words, I have to calculare $f(z)$ performing a complex integral: $\dfrac{1}{2\pi ...
0
votes
1answer
42 views

Change of variables in function $T(n)$.

I've been given this recurrence to solve: $T(n) = T(\sqrt n) + \theta(lglgn)$ And I'm told that the way to solve it is to let $m = lgn$, so that the recurrence can be rewritten as follows: $S(m) = ...
3
votes
1answer
171 views

Changing streams in PhD

I've a masters degree from a reputed Indian university in pure mathematics, with a specialization in Algebraic Number Theory. However, I'd like to apply for a PhD in computational math/theoretical ...
1
vote
1answer
69 views

minimize distance

consider a two dimensional system. two points are given whose co-ordinates are $(h1,h2)$ and $(k1,k2)$. I want to minimize the distance between these two points with the condition that person has to ...
1
vote
1answer
68 views

another counting problem

There are $k$ warriors that participate in the Wars, which have happened for the past $n$ years. Each year there has been a victor. Further, a particular warrior $W$ has won the Wars an even number of ...
1
vote
1answer
67 views

A problem on GCD

I want to calculate $f(n)$ where $f(n)$ is given by $$f(n) = \sum_{i=1}^n \dfrac{n}{gcd(n,i)}$$ and $2\leq n\leq 10^{12}$. Can someone tell me the fastest algorithm to calculate this. thanks
0
votes
1answer
64 views

Deleting subsets in the list of sets

If we have a list like $[[2,1],[5,2,1],[6,5,2,1],[9,10]]$ and I want to find the result in the form of $[[6,5,2,1], [9,10]]$ and want to delete all those list that are contained in another. How may I ...
1
vote
2answers
107 views

How can I solve this problem without having to do it by hand?

I'm dealing with the following problem in computational programming. I'm trying to find a way to build an algorithm that can quickly resolve the following problem statement without forcing me to do it ...
0
votes
1answer
76 views

How can I solve this problem without doing it by hand? [duplicate]

I'm dealing with the following problem in computational programming. I'm trying to find a way to build an algorithm that can quickly resolve the following problem statement without forcing me to do it ...
1
vote
2answers
94 views

Is there any way to solve this problem without having to do it by hand? [duplicate]

I'm dealing with the following problem in computational programming. I'm trying to find a way to build an algorithm that can quickly resolve the following problem statement. Is there any way to group ...
3
votes
0answers
26 views

Has there been work on computational group theory applications to computing colimits of crosses n-cubes of groups?

I'm trying to compute homotopy groups of a few spaces using crossed n-cubes of groups. I'm able to describe a few colimits in terms of quotients of induced crossed modules and nonabelian tensor ...
3
votes
0answers
323 views

Nerve Theorem: Is the finite union of closed convex sets triangulable?

My Question: Let $A_1, \ldots, A_k \subseteq \mathbb{R}^n$ be closed convex sets. Is the union $\bigcup_{i=1}^k A_i$ triangulable$^1$? If so, why? Background: I'm trying to better understand the ...
0
votes
1answer
140 views

Is there any algorithm (if possible, I need the codes) for Jordan normal form decomposition for large matrices in practice?

Although it is an ill-posed problem as B Kågström said in "An algorithm for numerical computation of the Jordan normal form of a complex matrix", I wonder what people do when they need to do Jordan ...
0
votes
1answer
36 views

Eliminate $p$ from these 2 equations.

$$ X \ = \ 2 \left[ \dfrac {h_1pv_1} {(1-p^2v_1^2)^{1/2}} + \dfrac {h_2pv_2} {(1-p^2v_2^2)^{1/2}} \right] \\ T_2 \ = \ 2 \left[ \dfrac {h_1/v_1} {(1-p^2v_1^2)^{1/2}} + \dfrac {h_2/v_2} ...
2
votes
1answer
102 views

Going through all Bit Strings with no 11 in it (no consecutive 1s)

My question is very simple: How can i (efficiently) go through all Bitstrings which don't contain two consecutive 1s? So for instance, all Bitstrings of length 3 with no consecutive 1s are: 000, 001, ...
1
vote
3answers
109 views

For how many consecutive numbers Collatz conjecture was checked?

I heard here that Collatz conjecture was checked at least for every first $5 \cdot 10^{18}$ natural numbers, but I cannot find any source or actual information about this. Can anyone help to find out ...
1
vote
2answers
162 views

Questions about the field scientific computing

I have heard about the field of Applied and Computational Mathematics, Scientific Computing and want to get some information. Is this a combination of computer science and mathematics? What subjects ...
1
vote
2answers
58 views

How to compute the sine of a complex number in floating-point arithmetic?

What is the most efficient way to numerically compute the sine of a complex number? Suppose I want to calculate the sine of a complex number a + bi on a computer. Suppose that a and b are both ...
1
vote
1answer
35 views

Approximating zeros on an interval

I'm writing a program for my AP Calculus class, and I'm trying to write an equation solver that approximates the zeros of functions. Right now it can take symbolic derivatives and evaluate functions. ...
1
vote
0answers
80 views

QR decomposition: Same results for Classical Gram-Schmidt and Modified Gram-Schmidt

I am implementing QR decomposition (in Fortran) for a complex-valued matrix, using Classical Gram-Schmidt and Modified Gram-Schmidt (and Householder). I was expecting that the Classical Gram-Schmidt ...
1
vote
1answer
122 views

Rational approximation of $\tanh\,(\sqrt[4]{s}$)

I'd like to find a rational representation of $$f(s) = \frac{\tanh\,\sqrt[4]{s}}{\sqrt[4]{s}}= \frac{a_0 + a_1 s + a_2 s^2 + ... + a_n s^n}{b_0 + b_1 s + b_2 s^2 + ... + b_m s^m} $$ For the case ...
0
votes
1answer
55 views

Practical differences between a PRNG and a Markov chains

In computer programming you can easily find people describing both a PRNG, like a Mersenne Twister, and a Markov / Stochastic process as "pseudo random generators". I honestly never liked this ...
0
votes
0answers
28 views

Numerical calculation of an $x$ for which $\pi(x) > li(x)$

Littlewood 1914 proved that there are an infinite number of $x$ for which $\pi(x) > li(x)$. Skewes 1933 provided the first numerical upper bound on $x$ ...
-1
votes
1answer
85 views

Fermat Last theorem on Poly-Euler numbers

The poly-Euler numbers, denoted as $E_{n}^{(k)}$, are defined by the following generating functions :$${2\operatorname{Li}_k(1-e^{-x}) \over 1+e^{-x}}=\sum_{n=0}^\infty E_n^{(k)}{x^n\over n!}$$ The ...
0
votes
2answers
29 views

Design a DFA with following condition.

A DFA that accepts a language in which every odd position of a string is a 1 with inputs as {0,1}
1
vote
1answer
83 views
0
votes
1answer
43 views

How is the status of a bigger number known while the smaller not?

In this wolfram mathworld article in the table below the status of $2059$ is known before the some smaller numbers, or they choose to aim the computation for that specific number?
7
votes
1answer
495 views

Why is Householder computationally more stable than modified Gram-Schmidt?

I'm having trouble nailing down why using Householder transformations yields a more stable/accurate result than the modified Gram-Schmidt method when computing the QR decomposition of a matrix. Can ...
0
votes
2answers
39 views

How can I find $x$ such that $ax \equiv 1 \pmod{bx+c}$, given $a,b,c$?

Everything I've read about modular arithmetic generally concerns doing things in some "mod m" world where "m" is some constant. But I'm perplexed how to tackle modular arithmetic problems where the ...
1
vote
0answers
812 views

Change MATLAB code from Lax-Wendroff to Leapfrog

I want to see how leapfrog would look using this code, but I'm having issues implementing it. I think my biggest problem is adding in the $ U_j^{n-1}$ term, I just don't get the logic. Here's what ...
1
vote
1answer
35 views

Pumping lemma contradiction

I have to prove that the language $A_{1}= \{\alpha \in \Sigma^{*}|c^{a}(\alpha)>c^{b}(\alpha) \}$ where $\Sigma=\{a,b\}$, where $c^{a}(a)$ means the number of $a$ in $\alpha$, and $c^{b}(\alpha)$ ...
2
votes
2answers
95 views

Very slow convergence of a particular series?

I've read that $$ \sum_{k=2}^{\infty} \frac{1}{k (\log k)^2} = 2.1097\ldots $$ However when I compute the partial sums it looks like a lot of terms are needed to even get the first decimals right. My ...
1
vote
1answer
43 views

Computational complexity of expanding a MacLaurin/Taylor Series

What methods exist to computationally determine the first $k$ coefficients of a function (possibly polynomial or rational polynomial function)? How do Mathematica/MatLab/Maple/etc. solve this ...
2
votes
2answers
117 views

summation of ceil and floor function

I need a closed solution or a faster algorithm for calculating $$ \sum_{k=1}^{n-1} \left\lceil \frac{n}{k}-1 \right\rceil $$ and $$ \sum_{k=1}^{n-1} \left\lfloor \frac{n}{k} \right\rfloor $$ where $ ...
0
votes
2answers
419 views

Google Code Jam's Cookie Clicker Program…

Today, the Google Code Jam's cookie clicker problem was something like this. Problem In this problem, you start with 0 cookies. You gain cookies at a rate of 2 cookies per second, by ...
3
votes
1answer
87 views

Factors of integers of the form $k^2-k+1$

Factorisation of arbitrary integers is of course a computationally hard problem. But what if the integers I'm interested in factorising are all of the form $k^2-k+1$ ? Is there some way to compute ...
1
vote
0answers
35 views

Generating N-dimensional points acording to given distribution

I searched here on Math.SE and haven't found anything that was exactly what I was looking for, so I'm posting it here. If there is anything, pardon my ignorance. I thought it was my first post here, ...
1
vote
1answer
42 views

Please help with this Discrete fourier transform question

Consider the ODE $\frac {d^2u}{dx^2} + 2\pi\frac {du}{dx} + \frac 54\pi^2u = g(x)$ where g is a periodic fuction with period 1 given by $g(x) = e^{\pi x}$ , $ 0 \le x \lt 1$. It is desired to find ...
1
vote
1answer
32 views

Computational Maths - Normalised mantissa

What does overflow and underflow of an normalised mantissa mean?
1
vote
1answer
110 views

Max value - Mantissa calculation clarification?

I've been reading this article about floating point representation floating point representation is : Where mantissa is : All understood. But they also say : How did they get to ...
1
vote
1answer
108 views

Jacobi Iteration Question

I have a question that says use a relative tolerance of $10^{-3}$ and asks if the estimate errors are in line with the actual errors. What does relative tolerance mean and how do you work out ...
2
votes
1answer
181 views

Bisection Method Question, Multiple Roots

I understand how to do the bisection method and how to do it with a point of intersection. My question is should this not actually have multiple points of intersection? and if you're not given any ...
0
votes
1answer
51 views

Prove a language is Context Free

I am working on the following problem and I do make a little progress. Let me state the question first: Prove that the set of strings consisting of $a$'s and $b$'s with an equal number of $a$'s and ...
2
votes
0answers
40 views

Quantitatively comparing event trains of different lengths for Poissonness

I have a parameterized, effectively black box process that generates a series of events (simulated action potentials). Different parameter values often lead to different numbers of events. How can I ...
0
votes
1answer
34 views

Simple Error Question

I have a question which asks to find the absolute error and then asks if the actual error agrees with the theoretical error bound. Am I missing something or are absolute and actual error the same ...
3
votes
1answer
69 views

Computational Maths

I'm trying to revise for a test and these 2 questions I just don't really understand what I'm meant to do, any pointers would be good. Any help I'd be very grateful for.
1
vote
2answers
56 views

Knapsack variation NP-complete

I have C processors and $C$ items that have to be run on it. I can either run each item on a seperate processor and have a running time of $\sum_{i=1}^{c} c_i$, or divide the $C$ items into $k$ ...
0
votes
1answer
292 views

Computation Method to solving Homogeneous Fredholm Integral Equation of Second Kind with Symmetric Kernel

I am attempting to write a program that will be able to numerically solve a homogeneous Fredholm Integral Equation of Second Kind, with a Symmetric Kernel. I have been looking through textbooks and ...
3
votes
1answer
63 views

How do computers compute the expected value of an infinite distribution?

I was trying to compute: $$E[X] = \int^{\infty}_{-\infty} xP(x)dx$$ but it might be a distribution over a sample space that is infinite. How do computers actually deal with this in an efficient and ...