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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0
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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
95 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
98 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
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2answers
115 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
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2answers
52 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
25 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
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0answers
51 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
111 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
51 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
26 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
72 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
24 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
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1answer
62 views
0
votes
1answer
41 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
306 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
36 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
436 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
33 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
74 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
33 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 ...
0
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0answers
24 views

Am I converting this Context Free Grammar correctly?

My homework problem is to convert this context free grammar into Chomsky Normal Form. ...
2
votes
2answers
95 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
345 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
81 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
31 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
votes
0answers
25 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
vote
1answer
39 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
30 views

Computational Maths - Normalised mantissa

What does overflow and underflow of an normalised mantissa mean?
1
vote
1answer
67 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
70 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
109 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
38 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
39 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
31 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
43 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
182 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
58 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
votes
1answer
77 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? ...
1
vote
2answers
52 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
83 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
votes
2answers
214 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 $\alpha$ = ...
4
votes
1answer
357 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
votes
3answers
76 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
votes
0answers
34 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
votes
2answers
112 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
votes
0answers
35 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
vote
1answer
69 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
votes
0answers
28 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
240 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 ...