Inge Söderkvist (2009) has a nice write-up of solving the Rigid Body Movement Problem by singular value decomposition (SVD). Suppose we are given 3D points $\{x_1,\ldots,x_n\}$ that after ...

See the paper How to crack a Linear Congruential Generator, Haldir ("Reverse Engineering Team", Dec. 2004): In this paper I will present a method which will solve all values of the LCG including ...

MATLAB has a couple of "exact" functions for this, cond and rcond, with the latter returning a reciprocal of the condition number. Matlab approximate function condest is more fully described below. ...

As the Comments to other Answers clarify, the real issue here is not a shortcoming of Householder matrices but rather a question as to why iterative rather than direct ("closed-form") methods are used ...

Doing one-off best rational approximations can often be accomplished by "manual" iterations of the Remez algorithm: interpolate a rational approximation with (relative or absolute) alternating sign ...

Let me add a few remarks explaining why there is no better method for "small" matrices in the range described (order 4 to 64) than the usual approach: tridiagonalize the Hermitian matrix with a ...

A nice portable and open source library for arbitrary precision floating point arithmetic (and much else) is Victor Shoup's NTL, which is available in C++ source form. At a lower level is the GNU ...

There's a GPL'd C library, ANANT - Algorithms in Analytic Number Theory by Linas Vepstas, which includes multiprecision implementation of the polylogarithm, building on GMP. From its README file: ...

Back in the era of Intel 387 math coprocessors I had to maintain an interrupt handler for floating point exceptions. Apart from that, I agree that pretty much everyone ignores denormals (or ...

I have looked at your simple code example, and my suspicion is that what you observe in loss of speed is due to the C-heritage requirement that right-hand side expressions be evaluated using ...

Let's show that we cannot bound the condition number of $X^T A X$ by using only the condition number of $A$ and the norm of $X$. Let $A=I$, so its condition number is exactly $1$. Let $X$ consist of ...

Initially the Question focused on a product computation, f(x)*g(x), where one factor is very small and the other is very large (perhaps large enough to cause overflow in floating point representation)....

With a very large number $N$ of points and a small subset $M$ to be chosen, it may be helpful to consider what is known about continuous versions of the problem in two-dimensions. L. Fejes Tóth ("On ...

Since the outer integral $\int_0^{2\pi} F(\theta) d\theta$ has a periodic integrand, a trapezoidal rule should work nicely, picking up extra accuracy when $F(\theta)$ is smooth. For the inner ...

Yes, that's what it means. This is usually illustrated with $A$ a diagonal matrix having both large and small entries. Clearly such a matrix can be accurately inverted, but a simple measure of ...

In the restated problem both $X$ and $Y$ have rank at most $k \ll n$, and the same is true of Gram matrices $A$ and $B$. Also $C=(X+Y)(X+Y)^T$ will have rank at most $k$. The goal of forming $C$ ...

Linux takes all ten of the top 10 spots in petaflops/supercomputing, but the operating system is perhaps not the only aspect you should focus on. Most developers will never get "time" on the highest ...

The explicit solution in terms of the median is correct, but in response to a comment by mayenew, here's another approach. It is well-known that $\ell^1$ minimization problems generally, and the ...

As the other Answer already touches on the possibility of a symbolic root-solver being applied to this particular equation (by transforming into a polynomial form, albeit of degree $\ge 5$), I'll make ...

As a general rule in developing and testing numerical software, I try to eliminate as many parameters as possible. I'm sure it has occurred to you that the distance $d$ and "floor" $z$ are scalable ...

Without some information about the construction of these $12\times 12$ positive definite real symmetric matrices, the suggestions to be made are of necessity fairly limited. I downloaded the ...

The computation of integer-valued matrix determinants has been a subject of considerable research. Using exact arithmetic the Smith normal form can be computed, and from this diagonal form the ...

The phrase "community detection" is loosely defined as partitioning the vertices of a graph into "communities" such that each has members more densely linked to one another than to members of other "...

Let's use the example (approximate the square root function on $[0.25,1.0]$ with a quartic polynomial) to step through your calculations. I suspect that the code is going to work with only modest ...

Since Doug has pointed out the need for a bigger integer datatype to do the problem you want to tackle, let's talk about the logic of your program and improvements. Your approach is (A) input alpha ...

It is not difficult to determine the complexity of a straightforward elimination/reduction to upper triangular form. Note that the initial matrix:  \begin{bmatrix} a_1&b_1&0&\cdots&...

Normally the author of a work can license it under more than one of the open source licenses you mention (so called dual licensing). However there seems to be an issue with doing so under the ...

I'd guess a QR decomposition is better than solving the normal equations and faster than SVD. There are some class notes that compare the three approaches. Also: The QR decomposition for least-...

For a positive semi-definite matrix such as $A = XX^T$ it may be worth the effort to accelerate convergence with a spectrum shift. That is, a suitable scalar $\mu$ is chosen and the power method is ...