Search Results

No, research in numerical mathematics is still very relevant today. One of the main challenges is big data: scaling the usual algorithms up to larger dimensions. Today's linear systems may involve sp …

Try NumPy (assuming you're fine with an interpreted language -- otherwise, there is a Python compiler, but I know very little about it). Why languages like Java are not widespread? Well, my view on th …

Ill-conditioning isn't a concept that depends on the precision that you use to compute the solution. "A small change in the data turns into a large change of the solution" isn't a concept that involve …

At least in my field (numerical linear algebra), the current standard is that including the full source code is not mandatory for a publication. That said, there are many reasons why sharing your code …

Disclaimer 1: Treating these topics properly would require a quick course in numerical analysis. Disclaimer 2: If you are using any sane computer system, it's already going to have a library function …

Implementing the QR factorization with Householder rotations is cheaper ($2n^2m$ vs $3n^2m$ for a $m\times n$ matrix), and equally accurate in practice. See Section 19.6 of Higham's Accuracy and Stabi …

Since you speak about mathematical proofs, probably you don't want an arbitrary-precision library, but a verified computation library based on interval arithmetic. Maybe Arb? Or boost-interval? And ma …

Stothers' thesis, from 2010, p. 18, lists 19 and 23 as the current lower and upper bounds. These are the same numbers that I have heard from some researchers in the field, and I don't think there has …

Convergence depends on the ratio between the eigenvalues, not on the difference. Oversimplifying: if $\lambda_1$, $\lambda_2$ are two eigenvalues and you shift by $\mu$, then the magic ratio is $\frac …

The book by Higham and the "nineteen dubious ways" paper deal with the dense case only. For the sparse case, the best way to go is using an algorithm that computes the so-called action, i.e., the map …

Interval/ball arithmetic may help, actually. It can be used to prove existence of solutions to multivariate systems like this one. The main idea is: reformulate your system as a fixed-point system $x …

Not 100% in-field but I'll try an answer. 1) Why are there so few automatic zero-dimensional solvers? As the Wikipedia page says, the problem is intrinsically difficult and slow to solve. Ill-c …

Another method to update the solution is using the Sherman-Morrison formula: http://en.wikipedia.org/wiki/Sherman%E2%80%93Morrison_formula in your case, $u$ and $v$ are canonical basis vectors. So ba …

Just a random idea: The standard method for getting a small part of the spectrum in large and sparse symmetric problems is the restarted Lanczos method. Essentially, you run some iterations of the La …

Another contender that has appeared very recently is Julia. It is a Matlab-like language designed from scratch for scientific computing by compiler and programming language experts, with a special eye …