Computational mathematics involves mathematical research in areas of science where computing plays a central and essential role, emphasizing algorithms, numerical methods, and symbolic methods.

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2answers
39 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
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1answer
15 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 ...
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0answers
7 views

Am I converting this Context Free Grammar correctly?

My homework problem is to convert this context free grammar into Chomsky Normal Form. ...
0
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2answers
52 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
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2answers
169 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
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1answer
53 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
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0answers
24 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, ...
0
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0answers
19 views

finding the least non-zero of a multivariable polynomial

Let $P(x_1,x_2,...,x_m)$ be a homogeneous polynomial of degree n, with integers coefficients. How can you find the least* $a=(a_1,a_2,...,a_m)$, where $a_i$ are positive integers and $P(a)!\neq 0$? ...
1
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1answer
21 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
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1answer
21 views

Computational Maths - Normalised mantissa

What does overflow and underflow of an normalised mantissa mean?
1
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1answer
28 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
32 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
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1answer
55 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
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1answer
20 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
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0answers
31 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
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1answer
25 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
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1answer
51 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
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2answers
24 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
30 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
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1answer
43 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 ...
0
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1answer
30 views

Newton Cotes Rule Derivation

I have this question Derive an open two-point Newton-Cotes quadrature rule for the interval $[a,b]$. I need to find what the resulting weights and nodes are. What is the degree of the resulting rule? ...
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2answers
39 views

Computing an induced matrix norm

Assume I have a $n \times n$ matrix and a norm defined as $\|A\| = \max \limits_{x \not = 0}\frac{\|Ax\|}{\|x\|}$, where $\|x\| = \sqrt{\sum x_i^2}$. How can I compute this norm?
2
votes
1answer
53 views

No advantage to the closed form for Fibonacci numbers?

The closed forms for the Fibonacci sequence, such as: $$F_n=\frac{\varphi^n-\widehat\varphi^n}{\sqrt5}=\frac{\varphi^n}{\sqrt5}-\frac{\widehat\varphi^n}{\sqrt5}\;,$$ the Binet formula, do not seem ...
13
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2answers
203 views

$e$ popping up in topic I'm unfamiliar with

I programmed up a little algorithm that goes like this: Fix two positive, real numbers, call them $\alpha$ and $\beta$. Generate a new, random, real number, $x \in [0,1]$ Set $\alpha$ = ...
3
votes
1answer
88 views

“Opposite” of idempotent operation?

What is the adjective given to a mathematical operation/expression on a variable whose new value can only be described in terms of that variable's existing value? Sequential operation? Example: i = ...
0
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3answers
38 views

How would I convert from frequency back to percent?

I'm working on a web application that uses an oscillator. It has a parameter that goes from 0 to 100 percent. I translate that percent into hertz using this equation: $$ \text{frequency} = ...
0
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0answers
31 views

Is there a way to expand Re(Li(a^z)) in series?

I'm searching a way to expand $ f(z) = Re(Li(a^z)), a \in R, z \in C $ in series. The computer-friendly, quickly convergent series is a huge plus. For being 'computer-friendly' I mean a relatively ...
0
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2answers
65 views

Existing Algorithm / Code to calculate exact values of the Riemann Zeta function at even natural numbers?

I wanted to know if there's any existing algorithm to compute exact values of the Riemann Zeta function at even natural numbers? For example, it should compute $\zeta(4)$ as exactly $\frac{\pi^4}{90}$ ...
0
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0answers
28 views

Properties of linear systems involving band matrices

Let $N$ be a positive integer, $A$ be a square matrix of size $2N+1$, and $x$ and $b$ be vectors of size $2N+1$. All the elements of $b$ are nonzero except for the middle element, $b_{N+1}$. Also, $A$ ...
1
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1answer
52 views

mortage with monthly payment - mathematical modeling

$Question:$ Suppose that $x_n$ is the amount owed on a mortgage after n years, $\$m$ is the monthly repayment and $r$ is the annual percentage interest rate charged on the amount of the mortgage ...
0
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0answers
11 views

Echo State Network learning Mackey-Glass function, but how?

I got this example of a minimal Echo State Network (ESN) which I analyse while trying to understand Echo State Networks. Unfortunately I have some problems understanding why this really works. It all ...
2
votes
1answer
77 views

How To Generate Random Points on the Positive Side of a Plane in 3-D

Edit: The question can also be interpreted as: How to generate random coplanar points in a cube? Here is what I am struggling with: I have a cube, whose origin is $(0,0,0)$ and one edge length ...
2
votes
1answer
36 views

Find regular expression for a binary sequence dividable by 3

I am trying to find a short regular expression that matches to all binary sequences that are dividable by 3. This is homework. It would be great if I could only get some hints before the final ...
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0answers
26 views

Find allowed error of an argument regarding the allowed error of a function.

To what precision can $x$ be obtained with logarithmic table (with $5$ digit table) if $x$ lies between $300$ and $400$? Any ideas?
3
votes
4answers
117 views

Computing a large exp(x) in a numerically robust way.

I'm trying to compute $\lfloor e^x \rfloor$, where x is a 64-bit integer. The problem is that the result of the computation may be close to 2^64. In this range, 64-bit floating point numbers will be ...
0
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0answers
23 views

Binary Algorithms

I was wondering. How is it that integrals are done via computers. I can understand basic functions and exponentials but how do you put an integral into binary form so that a computer can do its magic? ...
2
votes
1answer
39 views

The “computability” of fundamental physical constants

I would like to ask if any of the fundamental physical quantities like the speed of light or plancks constant (all measured according to a common standard of of units) can be classified as computable ...
0
votes
1answer
36 views

Discretize an ellipsoid given its semi-major axes and orientation

An ellipsoid centered at the origin can be defined by the solutions to $$ \mathbf{x}^\text{T} A \mathbf{x} = 1 $$ where $A$ is symmetric and positive-definite. The eigenvectors of $A$ define the ...
2
votes
1answer
33 views

Generating Eulerian digraphs/isographs

I would like to be able to quickly generate (all) non-isomorphic isographs (that is, digraphs where each node has the same indegree and outdegree - also called "balanced networks" in the distributed ...
0
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2answers
31 views

logarithm and exponent computation performance

Using glibc on a x86 processor, which takes more CPU time? $a\ log\ b$ or $b^a$? For which values of $a$ is one faster than the other? Optional: Does the base used matter? See also: What algorithm ...
0
votes
1answer
74 views

Calculating eigenvectors and eigenvalues of a 2x2 complex matrix

I've previously asked elsewhere, http://stackoverflow.com/questions/21118820/non-trivial-eigenvectors-of-a-22-matrix-in-code, how to calculate the eigenvectors and eigenvalues of a 2x2 matrix in a ...
0
votes
1answer
45 views

Is it possible to reduce a lambda expression to it's smallest equivalent form?

In the Untyped Lambda Calculus, is it possible to reduce any arbitrary expression to it's smallest equivalent form? (defined as an expression with the smallest number of lambda terms) If so, is there ...
1
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1answer
56 views

Galois group command for Magma online calculator?

I need to test if a family of 7th deg and 13 deg equations are solvable. I'm new to Magma, so my apologies, but what would I type in, http://magma.maths.usyd.edu.au/calc/ to determine the Galois ...
2
votes
0answers
45 views

Computing the decomposition of a representation of $S_n$

I have an explicitly defined representation of the symmetric group that I would like to decompose into irreducibles. How to do this most easily? The best approach I have so far is as follows: Find a ...
3
votes
0answers
41 views

Algorithm for primary decomposition of ideals in a power series ring over a field

Let $K$ be a field such that there exists an algorithm for factoring a polynomial over $K$ into the product of irreducible polynomials. For example, the field of rational numbers $\mathbb{Q}$ is such ...
5
votes
3answers
183 views

FFT for power of 3

Classic FFT works fine, when n is power of 2. How to generalize FFT procedure when n is power of 3? Is it possible to easily ...
1
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2answers
33 views

Subtlety about the definition of B-splines

I came across the following definition for the zero'th order B-spline $$b_0(x) = \left\{ \begin{array}{lr} 0 & |x|>1/2\\ 1 & |x|<1/2\\ 1/2& |x|=1/2. \end{array} ...
1
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0answers
59 views

How calculators compute. [duplicate]

I would like to teach a class on the "magic" behind the calculator, so I would like to generate a list of "algorithms" for how a calculator computes the things we want it to. I will get the ball ...
2
votes
2answers
99 views

Calling GAP's StructureDescription from SAGE

Given a sage group object $G$, I want to obtain its structure description using GAP, as follows: sage: gap.StructureDescription(G) The command works fine in ...
2
votes
3answers
60 views

Satisfying equality between logarithmic expressions

Apologies in advance for any misused terminology, or if this is the wrong place for the question (I think it's okay though). I am given a group of logarithmic expressions such as: $- (a \log(a) + ...