Articles of computational mathematics

Using GAP to compute the abelianization of a subgroup

Let $K$ be the group generated by four elements $x_1,\cdots,x_4$ with relations that each generator commutes with all its conjugates. (An equivalent relation is, any simple commutator with repeated generator is trivial; for example, $[[x_2,[x_1,x_3]],x_3]=1$.) It can be proved that $K$ is finitely presented. Let $A$ be the subgroup of $K$ generated by the following […]

Is there a computer programm or CAS (maybe GAP?) that can calculate with projective (indecomposable) A-modules (A is a finite dimensional k-algebra)?

I have the following question(s): I have an “Algebra-With-One” $R$ as a subalgebra of a full matrix algebra in GAP. Furthermore, I have 5 primitive orthogonal idempotents $e_1,…,e_5$, which sum up to $1_R$ (the identity matrix). I would like to compute the projective indecomposable modules $P_1=e_1R,…,P_5=e_5R$ with GAP (or another computer programm (e.g. SAGE) which […]

An elliptic curve for the multigrade $\sum^8 a_n^k = \sum^8 b_n^k$ for $k=1,2,3,4,5,9$?

I. The first solution to, $$\sum^6_{n=1} a_n^9 =\sum^6_{n=1} b_n^9$$ $$13^9+18^9+23^9-5^9-10^9-15^9 = 9^9+21^9+22^9-1^9-13^9-14^9$$ was found in 1967 by computer search by Lander et al. It stood for 40 years as a “numerical curiosity” until Bremner and Delorme discovered it had the highly structured form, $$\small(u + 9)^k + (u + 14)^k + (u + 19)^k + […]

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 following: L1: [Initialize] Set $y\leftarrow0$, $z\leftarrow x/2$, $k\leftarrow1$. L2: [Test for end] If $x=1$, stop. L3: [Compare] If $x-z<1$, go to L5. L4: […]

Converting Integers from One Base to Another Digit by Digit

So I’ve done some hands-on work with converting integers from one base to another using the well-known method of division and taking the remainder. The most generic algorithm involves dividing the number recursively, and looking up the digits of the target base using the remainder. For example, let’s say I wanted to convert 72310 from […]

How to find a perturbation of coefficients of a linear system to guarantee the error of solutions is small?

Assume $A$ is a $n \times n$ matrix, and $rank(A)<n$. For $b \in \mathbb{R}^n$, assume $AX=b$ has a solution $X=(x_1, \cdots, x_n)$, then clearly there exist infinitely many solutions. By the structure of the solutions, we may assume $\sum_{i=1}^n x_i=1$. Now my question is, for any $\epsilon>0$, does there exist invertible $n \times n$ matrix […]

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 nontrivial task. In my research into algorithms for network motif detection, we often require an effective solution to the problem […]

Solving for streamlines from numerical velocity field

Say I have a given numerical velocity field in two dimensions, (u,v). I am trying to find the streamlines from this data set at a particular contour level and I thus have to solve the differential equation $$ dy/dx = v/u=g(x,y) $$ I can rewrite the equation to $$ dy = g(x_i,y_i)dx $$ The subscript […]

Finite element method books

I know this question has been asked before; I just want to enquire if anybody has any suggestions to learn how to compute finite element problems, including plenty of examples. The topics I would like to focus in are as follows: Introduction to finite elements for 1D and 2D problems covering: weak formulation Galerkin approximation […]

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 formulas I have found to describe the derivative have an $1 – x^2$ in the denominator, and I […]