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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25
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3answers
7k views

What algorithm is used by computers to calculate logarithms?

I would like to know how are logarithms calculated by computers. The GNU C library, for example, uses a call to the fyl2x() assembler instruction, which means that ...
23
votes
1answer
458 views

Computing (on a computer) the first few (non-trivial) zeros of the zeta function of a number field

If $K$ is a number field, whose Galois closure over the rationals has degree 24 or so, and whose discriminant is around $163^4$, then what is a numerically efficient way of computing the first few ...
18
votes
8answers
2k views

Rapid approximation of $\tanh(x)$

This is kind of a signal processing/programming/mathematics crossover question. At the moment it seems more math-related to me, but if the moderators feel it belongs elsewhere please feel free to ...
17
votes
5answers
869 views

What interesting open mathematical problems could be solved if we could perform a “supertask” and what couldn't?

If we had a computer that could perform a countably infinite number of steps of a Turing machine, what currently open problems could we solve? I guess a lot of number theory problems could be solved ...
13
votes
2answers
206 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 $\alpha$ = ...
13
votes
1answer
1k 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 ...
11
votes
1answer
312 views

Constructing a finite field

I'm looking for constructive ways to obtain finite fields, for any given size $q=p^n$. For example, I know it suffices to find an irreducible polynomial of degree $n$ over $\mathbb{Z}_p$ (and then ...
11
votes
1answer
137 views

Evaluation of a slow continued fraction

Puzzle question... I know how to solve it, and will post my solution if needed; but those who wish may participate in the spirit of coming up with elegant solutions rather than trying to teach me how ...
10
votes
1answer
459 views

Why are there mathematicians that do not use computers?

I was watching a video on Andrew Wiles and his proof of Fermat's Last Theorem and I quite liked the video, especially the complexity of the proof only to prove a simple concept which can be understood ...
10
votes
2answers
882 views

How do I prove the partial denominators formula of the Bauer-Muir transformation of a generalized continued fraction?

Notation: $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ is the Gauss Notation for generalized continued fractions. Description of the Bauer-Muir transformation ...
9
votes
2answers
189 views

Efficient computation of $\sum_{k=1}^n \lfloor \frac{n}{k}\rfloor$

I realize there is probably not a closed form, but is there an efficient way to calculate the following expression? $$\sum_{k=1}^n \left\lfloor \frac{n}{k}\right\rfloor$$ I've noticed $$\sum_{k=1}^n ...
9
votes
1answer
617 views

How else can we be nauty?

The graph canonical labelling package nauty is widely regarded as one of the best (if not the best) around. Unfortunately, it's quite a large package, and making a GPU version seems to be a highly ...
9
votes
2answers
1k views

How to check whether an ideal is a prime (or maximal) ideal?

I have a ring $R$ which is known to be a Dedekind domain, but not necessarily a Euclidian domain, and a nonzero ideal generated by one or two elements in this ring. How can I check if this ideal is a ...
9
votes
1answer
262 views

Krylov-like method for solving systems of polynomials?

To iteratively solve large linear systems, many current state-of-the-art methods work by finding approximate solutions in successively larger (Krylov) subspaces. Are there similar iterative methods ...
8
votes
2answers
386 views

Computing the “lying over”, “going up”, “going down” ideals.

For any commutative unital ring $R$ and an ideal $\mathfrak{a}$ of $R$, we shall denote $$\begin{align*} \mathrm{Spec}(R)&:=\{\text{prime ideals of }R\},\\ ...
6
votes
3answers
193 views

Mathematical Limitations of Computer Experiments

One problem that has always bothered me is the limitations of computers in studying math. With a chaotic dynamical system, for example, we know mathematically that they possess trajectories that never ...
6
votes
1answer
51 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 ...
6
votes
2answers
399 views

Accelerating Convergence of a Sequence

Suppose I had a monotonically increasing sequence $\{d_{n}\}$ which is also bounded above. The $d_{n}$'s satisfy a given recurrence, however computationally they tend very slowly to the limit. What ...
6
votes
1answer
351 views

How to find an expression whose value is 190

Given a set of numbers (in this case): 3, 7, 7, 100, 50 Either: prove it is impossible to form the number k = 190 using ( ) + - * / operators between sub set of the these numbers ex: 1000 = ((3 + ...
6
votes
1answer
128 views

Fractional part of exp(x)

I have a real number $x$ (for concreteness, say $10^4<x<10^6$) and would like to find $e^x-\lfloor e^x\rfloor$ to reasonable precision (10-20 decimal places). What is the most efficient method? ...
6
votes
1answer
194 views

how to calculate this complementary Bessel function?

I am trying to calculate this complementary Bessel function $$\Psi(a,b,\gamma)=\int_0^\infty\Phi({a\over \sqrt{u}}+b\sqrt{u}){u^{\gamma-1}e^{-u}\over \Gamma(\gamma)}du$$ where $\Phi$ is the standard ...
6
votes
2answers
127 views

What is the average weight of a minimal spanning tree of $n$ randomly selected points in the unit cube?

Suppose we pick $n$ random points in the unit cube in $\mathbb{R}_3$, $p_1=\left(x_1,y_1,z_1\right),$ $p_2=\left(x_2,y_2,z_2\right),$ etc. (So, $x_i,y_i,z_i$ are $3n$ uniformly distributed random ...
5
votes
2answers
116 views

How was the $3x+1$ problem checked up to $5 \times 2^{60}$?

The Wikipedia article for the Collatz conjecture states that: The conjecture has been checked by computer for all starting values up to $5 \times 2^{60} \approx 5.764 \times 10^{18}$. It gives ...
5
votes
1answer
1k views

A search for integers which can be written as a sum of two squares in multiple ways

As part of a number theory hobby project, I'm looking for a computational way to enumerate all integers $n$ which can be written as a sum of two integer squares in three or more ways. The range of ...
5
votes
1answer
140 views

What's the best way to detect an algebraic number?

Suppose you calculate the first few (dozen, hundred) digits of a number which you believe to be a rational number. You can calculate the continued fraction for the number and truncate after a large ...
5
votes
3answers
219 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 ...
5
votes
2answers
460 views

What free software can I use to solve a system of linear equations containing an unknown?

Question: What free software can I use to solve a system of linear equations $M\mathbf{x}=\mathbf{y}$ where the entries of $\mathbf{y}$ vary with an unknown quantity $n$? Presumably I could do ...
5
votes
1answer
1k views

How does Knuth's algorithm for calculating logarithm work?

I had a look at Knuth's The Art of Computer Programming, book 1. In chapter 1, section 1.2.2, exercise 25, he presents the following algorithm for calculating logarithm: given $x\in[1,2)$, do the ...
5
votes
2answers
791 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 ...
5
votes
2answers
280 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 ...
5
votes
3answers
419 views

Calculate $\pi$ in an arbitrary base, to arbitrary precision

I need to calculate $\pi$ -- in base: 4, 12, 32, and 128 -- to an arbitrary number of digits. (It's for an artist friend). I remember Taylor series and I've found miscellaneous "BBP" formulas, but so ...
5
votes
1answer
306 views

cylinder-ray intersections equation

Can You Pleas Help with this one I found an article http://www.mrl.nyu.edu/~dzorin/rendering/lectures/lecture3/lecture3.pdf for Infinite cylinder-ray intersections And I don't know how they develop ...
4
votes
4answers
305 views

Computing partition numbers

Today a friend and myself came up with the question of computing partitions of numbers, i.e.: given a number $n$, what is the number $p(n)$ of was of different ways writing $n$ as a sum of non-zero ...
4
votes
1answer
37 views

GAP code to get Multiplication Table.

I have a finite set $S=\{0,1,2,\ldots,n-1\}$ and binary operation $\star$ on $S$ defined by $$x\star y= \left\{ \begin{array}{l l l} \frac{3(x+y)}{2} ~~\text{modulo} ~~n& \qquad \mbox{if $x$ ...
4
votes
1answer
53 views

About parallel time computation

I am studying a paper where it is mentioned that Newton iteration may be used to compute the inverse of $n \times n$, well- conditioned matrix in parallel time $o(\log^2n)$ and that this computation ...
4
votes
1answer
116 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 ...
4
votes
1answer
312 views

Computing with ideals: over $K$ or over $\mathbb{Q}\subseteq K$? does it matter?

I'm beginning to learn to use SINGULAR, the computer algebra system (CAS) for commutative algebra. NOTATION: If $K$ is a field of characteristic $0$, then $\mathbb{Q}\subseteq K$; otherwise ...
4
votes
1answer
128 views

What is the upper bound on the error of a matrix multiplication

When both A and B are n x n upper-triangular matrices, the entries of C = AB are defined as follows: $$ c_{ij} = \begin{cases} \sum _{k=i}^ja_{ik}b_{kj} & 1\leq i\leq j\leq n \\0 & 1\leq j\lt ...
4
votes
1answer
68 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 ...
4
votes
1answer
285 views

Wolfram Alpha error?

I was seeing some equations in WA, and i got with http://www.wolframalpha.com/input/?i=%28k%2B1%29%5E2%3E%3D4%28k-1%29%5E2 Let's manually solve the equation $$(k+1)^2\ge4(k-1)^2$$ ...
4
votes
1answer
179 views

What is the fastest computational graph theory package?

What is the fastest computational graph theory package with respect to executing algorithms and computing graph theoretic data? I am aware of this related question, which requests graph theory ...
4
votes
2answers
99 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 ...
4
votes
0answers
61 views

Finding every $n$ such that $n\times$ ('reverse' number of $n)=m^2$ such as $1584\times 4851={2772}^2$

Let $r(n)$ be the 'reverse' number of $n$ in the decimal system. For example, $r(1234)=4321$. Then, here is my question. Question : Can we find every $n(\in\mathbb N)$, which is not a square ...
4
votes
3answers
1k views

Derivative of Associated Legendre polynomials at $x = \pm 1$

I'm creating meshes for spherical harmonics, and I need a normal at a given point. Whenever I'm at the poles, $\cos{\theta} = \pm 1$, and I do not know how to find the derivative there. All the ...
4
votes
0answers
269 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 ...
4
votes
0answers
497 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 ...
3
votes
8answers
4k views

Fastest Square Root Algorithm

What is the fastest algorithm for finding the square root of a number? I created one that can find the square root of "987654321" to 16 decimal places in just 20 iterations (I'm not ready to release ...
3
votes
4answers
236 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 ...
3
votes
2answers
1k 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 ...
3
votes
4answers
134 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 ...