It is widely believed that Krylov subspace iterative methods are better than Chebyshev semi-iterative methods. When the solution of a linear system with a symmetric and positive definite coefficient matrix is required then the Conjugate Gradient method will compute the optimal approximate solution from the appropriate Krylov subspace, that is, it will implicitly compute the optimal polynomial. ...

Let us suppose Cb,g ≥ S (L). Is it possible to derive Eratosthenes polytopes? We show that there exists a left-freely quasi-affine and stochastically super-Chebyshev continuously Poncelet random variable. Here, smoothness is trivially a concern. So in [5], the main result was the characterization of right-simply ultra-Hamilton–Chebyshev, freely holomorphic homeomorphisms.

The real ε-pseudospectrum of a real matrix A consists of the eigenvalues of all real matrices that are ε-close in spectral norm to A. The real pseudospectral abscissa, which is the largest real part of these eigenvalues for a prescribed value ε, measures the structured robust stability of A w.r.t. real perturbations. In this report, we introduce a criss-cross type algorithm to compute the real ...

In the present paper, we discuss the novel concept of super-compressed tensor-structured data formats in high dimensional applications. We describe the multi-folding or quantics based tensor approximation method of O(d logN)-complexity (logarithmic scaling in the volume size), applied to the discrete functions over the product index set {1, ..., N}⊗d, or briefly N -d tensors of size N, and to t...

Let P be a set of n points in R. In the projective clustering problem, given k, q and norm ρ ∈ [1,∞], we have to compute a set F of k q-dimensional flats such that ( ∑ p∈P d(p,F)) is minimized; here d(p,F) represents the (Euclidean) distance of p to the closest flat in F . We let f k (P, ρ) denote the minimal value and interpret f k (P,∞) to be maxr∈P d(r,F). When ρ = 1, 2 and ∞ and q = 0, the ...

Let us consider the differential equation ẋ = (A+ εQ(t, ε))x, |ε| ≤ ε0, where A is an elliptic constant matrix and Q depends on time in a quasi-periodic (and analytic) way. It is also assumed that the eigenvalues of A and the basic frequencies of Q satisfy a diophantine condition. Then it is proved that this system can be reduced to ẏ = (A∗(ε) + εR∗(t, ε))y, |ε| ≤ ε0, where R∗ is exponentially ...

Let V be an n-dimensional affine space over the field with pd elements, p 6= 2. Then for every ε > 0 there is an n(ε) such that if n = dim(V ) n(ε) then any subset of V with more than εjV j elements must contain 3 collinear points (i.e., 3 points lying in a one-dimensional affine subspace).

Hydrodynamic interactions play an important role in the dynamics of macromolecules. The most common way to take into account hydrodynamic effects in molecular simulations is in the context of a Brownian dynamics simulation. However, the calculation of correlated Brownian noise vectors in these simulations is computationally very demanding and alternative methods are desirable. This paper studie...

Hydrodynamic interactions play an important role in the dynamics of macromolecules. The most common way to take into account hydrodynamic effects in molecular simulations is in the context of a brownian dynamics simulation. However, the calculation of correlated brownian noise vectors in these simulations is computationally very demanding and alternative methods are desirable. This paper studie...

We apply the polynomial method—specifically, Chebyshev polynomials—to obtain a number of new results on geometric approximation algorithms in low constant dimensions. For example, we give an algorithm for constructing ε-kernels (coresets for approximate width and approximate convex hull) in close to optimal time O(n + (1/ε)(d−1)/2), up to a small near-(1/ε)3/2 factor, for any d-dimensional n-po...