7
votes
1answer
224 views

Compute an arbitrary decimal place of $\pi$

Is there a method to find the value of the $n$-th decimal place of $\pi$ which is more efficient than having to compute all decimal places before as well?
9
votes
1answer
301 views

Fast checking that overdetermined polynomial system does not have a solution

As a result of some inductive procedure for each $n$ I have a system of about $n^2$ polynomial equations with $n$ variables with integer coefficients, which can be precisely computed. As the system is ...
6
votes
2answers
267 views

Rigorous numerics for maxima and minima (one variable)

Let $f:\mathbb{R}_0^+\to \mathbb{R}$ be defined by some combination of the four basic operations and square roots. (The argument of square-roots is assumed is to be non-negative, and the value of ...
2
votes
1answer
166 views

Reducing the error of Algorithms by assigning variables formulas instead of values

Let me first give the intuition for my question: Suppose that you want to use a ruler to mark $n$ points in a line on a page, with 1 cm distance between neighbor points. There are two ways: 1- Mark ...
2
votes
1answer
312 views

Complexity of computing derivatives

Sorry if this is too simple. This is my first question here. Suppose $f : R^n \to R$ is a differentiable function. Say that we can compute in $T$ arithmetic operations the value $f(x)$ at any point ...
3
votes
1answer
235 views

Numerical Beta Function

Anyone know a fast and concise way of calculating the Beta $B(a,b)$ function for smallish (<10) real $a$ and $b$. For integer $a$ and $b$ I have: $B(a,b) = \prod\limits_{j=1}^b \frac{j}{a+j}$ ...
6
votes
6answers
3k views

Fast evaluation of polynomials

Hello everybody ! I was reading a book on geometry which taught me that one could compute the volume of a simplex through the determinant of a matrix, and I thought (I'm becoming a worse computer ...
5
votes
3answers
522 views

Appropiate models of numerical computation

Hello, in contrast to the more discrete part of computational mathematics (cryptography, combinatorial computation), numerical mathematics seems to ignore typical questions of theoretical computer ...
1
vote
2answers
734 views

Verifying a sequence that converges to pi [closed]

A computer program ouputs the digits of $\pi$ by evaluating the recurrence relation $a_{n+1} = a_n + sin \ a_n$ with $a_0 = \frac{6}{5}$ Does the sequence actually converge or is this just ...
5
votes
1answer
3k views

Meaning of \Subset notation

The symbol $\Subset$ (occurring in places where $\subseteq$ could occur syntactically) comes up frequently in a paper I'm reading. The paper lives at the intersection of a few areas of math, and I ...
1
vote
2answers
775 views

What is the right citation for the power iteration method to find eigenvalues?

What is the right citation for the power iteration method to find eigenvalues, if I want to cite the method in a paper? I've seen some Google PageRank references in this context. But Brin and Page ...
31
votes
7answers
2k views

What is the time complexity of computing sin(x) to t bits of precision?

Short version of the question: Presumably, it's poly$(t)$. But what polynomial, and could you provide a reference? Long version of the question: I'm sort of surprised to be asking this, because ...
-2
votes
4answers
10k views

How to write IF…ELSE as mathematic equation? [closed]

I'm writing a computer program and I need to fit some IF..ELSE condition into mathematic model, so I can't use regular programming constructs. For example, how would I turn this into mathematic ...