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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Maximum order of a sum of functions

I'm being introduced to the Big-O notation via Susanna Epp's Discrete Mathematics with Appplications 3rd edition. The following defintion is stated on page 519: Let f and g be real-valued functions ...
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0answers
60 views

is there a computationally efficient formula for computing the mutual information between two continuous variables?

I need to compute the mutual information between two continuous variables. Below is an equation shown to compute the mutual information between a variable $X$ and $Y$. $I(X;Y) = \int_Y \int_X ...
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1answer
90 views

How to compute this set operation?

Suppose there are two sets (spaces) X and Y. Given N subsets of $X \times Y$: $S_1, \dots, S_N \subseteq X \times Y$. I need to compute the following set $S_X \subseteq X$: $$ S_X = \{x \in X : ...
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2answers
389 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\},\\ ...
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3answers
870 views

Modular exponentiation?

I came upon an interesting way to relatively quickly compute modular exponentiation with large numbers. However, I do not fully understand it and was hoping for a better explanation. The method ...
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1answer
89 views

Regarding the definition of a problem

I recently noted from http://rjlipton.wordpress.com/2010/11/07/what-is-a-complexity-class/ that a problem is defined as a mere set of strings. So, here is the point: If I say the following: "Find ...
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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 ...
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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 ...
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1answer
3k views

Plot Y-Range on Mathematica

I have a plot that I would like to slightly manipulate in Mathematica. Here is the code I am entering: Plot[{x, 2^x, log_2(x)}, {x, -1, 3}] As you can see $x$, $2^x$, and $log_2(x)$ are all ...
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5answers
873 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 ...
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1answer
533 views

Area of Union of n circles

I am trying to calculate the area of union of n circles in a plane when it is known that all circles are of equal radii and their centers are also known(of all n circles). I was trying to follow the ...
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1answer
599 views

double exponential point smoothing - what do alpha and gamma do? What is “trend”?

I have some code that does double exponential point smoothing. In addition to the points to be smoothed, it accepts two inputs as "alpha and gamma" values and outputs something called "trend" in ...
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6answers
854 views

Gödel's Proof as a proof strategy for P = NP or P != NP

I have had this thought for quite a while. Gödel proved the incompleteness of arithmetic by creating a one-to-one correspondence with a number and certain numerical relationships to create a statement ...
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2answers
346 views

Graphical-entry knot theory program for Mac?

Is there a good program that runs on Mac OSX, which has a graphical interface for inputting knot or link diagrams, and calculates standard invariants like the Conway and Jones polynomials? I have ...
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1answer
313 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 ...
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2answers
145 views

Sequences of a computable function

Is there any computable function $f(n)$, which given any integer $n$ has been proven to return either $0$ or $1$ in finite time, and for which the statement "$f(1), f(2), f(3),\ldots$ contains ...
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0answers
70 views

lower bounds for maximum computing times for integer factorisation

Supposing that n were known to have two prime factors, and that the computer had a database of all the primes $<\sqrt{n}$. Then, unless n is square, one factor would be $<\sqrt{n}$. If an ...
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0answers
374 views

Clarification on different types of solutions: analytical, closed form, iterative, algorithmic,

When doing computation, there are different types of solutions: analytical, closed form, iterative, algorithmic, exact, approximate, ... (there are probably more than I just listed and please don't ...
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0answers
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calculating multivariable integrals

I having a look at how to calculate using PC a multivariable integrals. I am reading about the Quasi Montecarlo methods using the following (t, m, s)-Nets and (t, s)-Sequences Faure sequences My ...
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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 ...
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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 ...
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1answer
219 views

Structure and Formula encoding for Turing Machine

During my study of Finite Model Theory I found that usually purely relational structure say $\mathcal{M} = \langle A, R_1,\ldots,R_k \rangle$ are encoded as ...
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2answers
279 views

System of linear equations, resulting from a weighted graph. How to solve this numerically?

We have a problem that leads to a system of linear equations which has to be solved numerically. There are thousands of algorithms to solve linear equations, but I haven't found any that fits our ...
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1answer
626 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 ...
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1answer
95 views

Factoring short intervals

There are algorithms (e.g., SIQS) that factor individual numbers. For large ranges of numbers, sieving is more efficient: for example, $(x^2,x^2+x)$ can be factored in time roughly linear in $x$. ...
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286 views

Uniform Load on an Elastic Bar

One applied math question talks about an elastic bar with a displacement equation $du^2/dx^2 = 1$ for a uniform load and fixed on both ends. I solved this using the Toeplitz matrix K and u comes out ...
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1answer
319 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 ...
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1answer
199 views

Finding all results of a permutation group on a set

Given a finite group $G < Sym(\Omega)$; $\Omega$ finite, and $X \subset \Omega$, I can define a by the function $H(g) = \{x^g \| x \in X\}$ for each $g \in G$. Of course, each $H$ has the same ...
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2answers
407 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 ...
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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 ...
23
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1answer
462 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 ...
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0answers
60 views

Computing relations on the columns of a matrix

Given an $m\times n$ (with $n>m)$ matrix $M$ over a polynomial ring $R=k[x_1,...,x_n]$, suppose that every column of $M$ is an $R$-linear combination of $m$ specified columns. I would like to ...
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1answer
102 views

Find $k^{th}$ root of $M \in GL(n,F_2)$

Given $M \in GL(n,F_2)$ which is known to have a $k^{th}$ root. How can I find a root algorithmically? Can I find all roots? Other than being invertible and having a $k^{th}$ root I know nothing of ...
5
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3answers
427 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 ...
9
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1answer
263 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 ...
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1answer
93 views

Decision procedure for the problem of regular expression equivalence with not

As stated in the title, I need a decision procedure for the problem of regular expression equivalence with not. Wikipedia states the problem is in the NONELEMENTARY complexity class. All I really ...
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2answers
883 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 ...