Stability of barycentric interpolation formulas

The barycentric interpolation formula defines a stable algorithm for evaluation at points in [−1, 1] of polynomial interpolants through data on Chebyshev grids. Here it is shown that for evaluation at points in the complex plane outside [−1, 1], the algorithm becomes unstable and should be replaced by the alternative modified Lagrange or "first barycentric" formula dating to Jacobi in 1825. This difference in stability confirms the theory published by N. J. Higham in 2004 (IMA J. Numer. Anal., v. 24) and has practical consequences for computation with rational functions

Orthogonal Systems with a Skew-Symmetric Differentiation Matrix

Funder: University of ManchesterAbstract In this paper, we explore orthogonal systems in $$\mathrm {L}_2({\mathbb R})$$L2(R) which give rise to a real skew-symmetric, tridiagonal, irreducible differentiation matrix. Such systems are important since they are stable by design and, if necessary, preserve Euclidean energy for a variety of time-dependent partial differential equations. We prove that there is a one-to-one correspondence between such an orthonormal system $$\{\varphi _n\}_{n\in {\mathbb Z}_+}$${φn}n∈Z+ and a sequence of polynomials $$\{p_n\}_{n\in {\mathbb Z}_+}$${pn}n∈Z+ orthonormal with respect to a symmetric probability measure $$\mathrm{d}\mu (\xi ) = w(\xi ){\mathrm {d}}\xi$$dμ(ξ)=w(ξ)dξ. If $$\mathrm{d}\mu$$dμ is supported by the real line, this system is dense in $$\mathrm {L}_2({\mathbb R})$$L2(R); otherwise, it is dense in a Paley–Wiener space of band-limited functions. The path leading from $$\mathrm{d}\mu$$dμ to $$\{\varphi _n\}_{n\in {\mathbb Z}_+}$${φn}n∈Z+ is constructive, and we provide detailed algorithms to this end. We also prove that the only such orthogonal system consisting of a polynomial sequence multiplied by a weight function is the Hermite functions. The paper is accompanied by a number of examples illustrating our argument.</jats:p

DPP-4 inhibitor dose selection according to manufacturer specifications:A Contemporary Experience From UK General Practice

Recently, 2 dipeptidyl peptidase-4 (DPP-4) inhibitors, sitagliptin and saxagliptin, adjusted dosing specification from creatinine clearance to glomerular filtration rate, more typically reported in routine laboratory tests. This cross-sectional study examines all DPP-4 inhibitor initiations that require dose adjustment and the dose selection using data from UK general practice. Results indicate that 34% of patients taking a nonlinagliptin DPP-4 inhibitor were given a higher dose and 11% a lower dose than specified in the Summary of Product Characteristics. This reinforces the deviation from Summary of Product Characteristics prescription of DPP-4 inhibitors identified in earlier studies despite improvement in compatibility with routine reporting. (C) 2019 The Authors. Published by Elsevier Inc

Pointwise and uniform convergence of Fourier extensions

Fourier series approximations of continuous but nonperiodic functions on an interval suffer the Gibbs phenomenon, which means there is a permanent oscillatory overshoot in the neighbourhoods of the endpoints. Fourier extensions circumvent this issue by approximating the function using a Fourier series which is periodic on a larger interval. Previous results on the convergence of Fourier extensions have focused on the error in the L2 norm, but in this paper we analyze pointwise and uniform convergence of Fourier extensions (formulated as the best approximation in the L2 norm). We show that the pointwise convergence of Fourier extensions is more similar to Legendre series than classical Fourier series. In particular, unlike classical Fourier series, Fourier extensions yield pointwise convergence at the endpoints of the interval. Similar to Legendre series, pointwise convergence at the endpoints is slower by an algebraic order of a half compared to that in the interior. The proof is conducted by an analysis of the associated Lebesgue function, and Jackson- and Bernstein-type theorems for Fourier extensions. Numerical experiments are provided. We conclude the paper with open questions regarding the regularized and oversampled least squares interpolation versions of Fourier extensions

Are sketch-and-precondition least squares solvers numerically stable?

Sketch-and-precondition techniques are efficient and popular for solving large least squares (LS) problems of the form ⁢= with ∈ℝ× and ≫. This is where is “sketched” to a smaller matrix ⁢ with ∈ℝ⌈⁢⌉× for some constant >1 before an iterative LS solver computes the solution to ⁢= with a right preconditioner , where is constructed from ⁢. Prominent sketch-and-precondition LS solvers are Blendenpik and LSRN. We show that the sketch-and-precondition technique in its most commonly used form is not numerically stable for ill-conditioned LS problems. For provable and practical backward stability and optimal residuals, we suggest using an unpreconditioned iterative LS solver on (⁢)⁢= with =⁢. Provided the condition number of is smaller than the reciprocal of the unit roundoff, we show that this modification ensures that the computed solution has a backward error comparable to the iterative LS solver applied to a well-conditioned matrix. Using smoothed analysis, we model floating-point rounding errors to argue that our modification is expected to compute a backward stable solution even for arbitrarily ill-conditioned LS problems. Additionally, we provide experimental evidence that using the sketch-and-solve solution as a starting vector in sketch-and-precondition algorithms (as suggested by Rokhlin and Tygert in 2008) should be highly preferred over the zero vector. The initialization often results in much more accurate soluti

Are sketch-and-precondition least squares solvers numerically stable?

Sketch-and-precondition techniques are popular for solving large least squares (LS) problems of the form $Ax=b$ with $A\in\mathbb{R}^{m\times n}$ and $m\gg n$. This is where $A$ is ``sketched" to a smaller matrix $SA$ with $S\in\mathbb{R}^{\lceil cn\rceil\times m}$ for some constant $c>1$ before an iterative LS solver computes the solution to $Ax=b$ with a right preconditioner $P$, where $P$ is constructed from $SA$. Popular sketch-and-precondition LS solvers are Blendenpik and LSRN. We show that the sketch-and-precondition technique is not numerically stable for ill-conditioned LS problems. Instead, we propose using an unpreconditioned iterative LS solver on $(AP)y=b$ with $x=Py$ when accuracy is a concern. Provided the condition number of $A$ is smaller than the reciprocal of the unit round-off, we show that this modification ensures that the computed solution has a comparable backward error to the iterative LS solver applied to a well-conditioned matrix. Using smoothed analysis, we model floating-point rounding errors to provide a convincing argument that our modification is expected to compute a backward stable solution even for arbitrarily ill-conditioned LS problems.Comment: 22 page