Search

Results on the rational approximation of functions containing singularities are presented. We build further on the "lightning method," recently proposed by Trefethen and Gopal [SIAM J. Numer. Anal., 57 (2019), pp. 2074-2094], based on exponentially clustering poles close to the singularities. Our results are obtained by augmenting the lightning approximation set with either a low-degree polynomial basis or partial fractions with poles clustering toward infinity in order to obtain a robust approximation of the smooth behavior of the function. This leads to a significant increase in the achievable accuracy as well as the convergence rate of the numerical scheme. For the approximation of xα on [0, 1], the optimal convergence rate as shown by Stahl [Bull. Amer. Math. Soc., 28 (1993), pp. 116-122] is now achieved simply by least-squares fitting. [ABSTRACT FROM AUTHOR]

This article is about both approximation theory and the numerical solution of partial differential equations (PDEs). First we introduce the notion of reciprocal-log or log-lightning approximation of analytic functions with branch point singularities at points {zk} by functions of the form g(z) = ∑ k ck/(log(z-zk)-sk), which have N poles potentially distributed on different sheets of a Riemann surface. We prove that the errors of minimax reciprocal-log approximations decrease exponentially with respect to N and that exponential or near-exponential convergence (i.e., at a rate O(exp(-CN/ logN))) also holds for near-best approximations constructed by linear leastsquares fitting on the boundary with suitably chosen preassigned singularities. We then apply these results to derive a ``log-lightning method"" for the numerical solution of Laplace and related PDEs in two-dimensional domains with corner singularities. The convergence is near-exponential, in contrast to the root-exponential convergence for the original lightning methods based on rational functions. [ABSTRACT FROM AUTHOR]

A new method is introduced for solving Laplace problems on 2D regions with corners by approximation of boundary data by the real part of a rational function with fixed poles exponentially clustered near each corner. Greatly extending a result of D. J. Newman in 1964 in approximation theory, we first prove that such approximations can achieve root-exponential convergence for a wide range of problems, all the way up to the corner singularities. We then develop a numerical method to compute approximations via linear least-squares fitting on the boundary. Typical problems are solved in < 1s on a laptop to 8-digit accuracy, with the accuracy guaranteed in the interior by the maximum principle. The computed solution is represented globally by a single formula, which can be evaluated in tens of microseconds at each point. [ABSTRACT FROM AUTHOR]

Let E(n)kk denote the minimax (i.e., best supremum norm) error in approximation of xn on [0, 1] by rational functions of type (k, k) with k < n. We show that in an appropriate limit E(n)kk ~ 2Hk+1/2 independently of n, where H ≈ 1/9.28903 is Halphen's constant. This is the same formula as for minimax approximation of ex on (-∞, 0]. [ABSTRACT FROM AUTHOR]

Let E(n)kk denote the minimax (i.e., best supremum norm) error in approximation of xn on [0, 1] by rational functions of type (k, k) with k < n. We show that in an appropriate limit E(n)kk ~ 2Hk+1/2 independently of n, where H ≈ 1/9.28903 is Halphen's constant. This is the same formula as for minimax approximation of ex on (-∞, 0]. [ABSTRACT FROM AUTHOR]

We introduce a new algorithm for approximation by rational functions on a real or complex set of points, implementable in 40 lines of MATLAB and requiring no user input parameters. Even on a disk or interval the algorithm may outperform existing methods, and on more complicated domains it is especially competitive. The core ideas are (1) representation of the rational approximant in barycentric form with interpolation at certain support points and (2) greedy selection of the support points to avoid exponential instabilities. The name AAA stands for "adaptive Antoulas-Anderson" in honor of the authors who introduced a scheme based on (1). We present the core algorithm with a MATLAB code and nine applications and describe variants targeted at problems of different kinds. Comparisons are made with vector fitting, RKFIT, and other existing methods for rational approximation. [ABSTRACT FROM AUTHOR]

Algorithms and underlying mathematics are presented for numerical computation with periodic functions via approximations to machine precision by trigonometric polynomials, including the solution of linear and nonlinear periodic ordinary differential equations. Differences from the nonperiodic Chebyshev case are highlighted. [ABSTRACT FROM AUTHOR]

The amplitude of the gradient of a potential inside a wire cage is investigated, with particular attention to the 2D configuration of a ring of n disks of radius r held at equal potential. The Faraday shielding effect depends upon the wires having finite radius and is weaker than one might expect, scaling as |log r|/n in an appropriate regime of small r and large n. Both numerical results and a mathematical theorem are provided. By the method of multiple scales, a continuum approximation is then derived in the form of a homogenized boundary condition for the Laplace equation along a curve. The homogenized equation reveals that in a Faraday cage, charge moves so as to somewhat cancel an external field, but not enough for the cancellation to be fully effective. Physically, the effect is one of electrostatic induction in a surface of limited capacitance. An alternative discrete model of the effect is also derived based on a principle of energy minimization. Extensions to electromagnetic waves and 3D geometries are mentioned. [ABSTRACT FROM AUTHOR]

Let f(z) be an analytic or meromorphic function in the closed unit disk sampled at the nth roots of unity. Based on these data, how can we approximately evaluate f(z) or f(m)(z) at a point z in the disk? How can we calculate the zeros or poles of f in the disk? These questions exhibit in the purest form certain algorithmic issues that arise across computational science in areas including integral equations, partial differential equations, and large-scale linear algebra. We analyze some of the possibilities and emphasize the distinction between algorithms based on polynomial or rational interpolation and those based on trapezoidal rule approximations of Cauchy integrals. We then show how these developments apply to the problem of computing the eigenvalues in the disk of a matrix of large dimension. Finally we highlight the power of rational in comparison with polynomial approximations for some of these problems. [ABSTRACT FROM AUTHOR]

An object-oriented MATLAB system is described that extends the capabilities of Chebfun to smooth functions of two variables defined on rectangles. Functions are approximated to essentially machine precision by using iterative Gaussian elimination with complete pivoting to form "chebfun2" objects representing low rank approximations. Operations such as integration, differentiation, function evaluation, and transforms are particularly efficient. Global optimization, the singular value decomposition, and rootfinding are also extended to chebfun2 objects. Numerical applications are presented. [ABSTRACT FROM AUTHOR]

Padé approximation is considered from the point of view of robust methods of numerical linear algebra, in particular, the singular value decomposition. This leads to an algorithm for practical computation that bypasses most problems of solution of nearly-singular systems and spurious pole-zero pairs caused by rounding errors, for which a MATLAB code is provided. The success of this algorithm suggests that there might be variants of Pad#233; approximation that are pointwise convergent as the degrees of the numerator and denominator increase to ∞, unlike traditional Padé approximants, which converge only in measure or capacity. [ABSTRACT FROM AUTHOR]

The barycentric interpolation formula de?nes 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 [11], 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., 24 (2004), pp. 547-556] and has practical consequences for computation with rational functions. [ABSTRACT FROM AUTHOR]

Chebfun is an established software system for computing with functions of a real variable, but its capabilities for handling functions with singularities are limited. Here an analogous system is described based on sinc function expansions instead of Chebyshev series. This experiment sheds light on the strengths and weaknesses of sine function techniques. It also serves as a review of some of the main features of sinc methods, including construction, evaluation, zerofinding, optimization, integration, and differentiation. [ABSTRACT FROM AUTHOR]

We compare the convergence behavior of Gauss quadrature with that of its younger brother, Clenshaw-Curtis. Seven-line MATLAB codes are presented that implement both methods, and experiments show that the supposed factor-of-2 advantage of Gauss quadrature is rarely realized. Theorems are given to explain this effect. First, following O'Hara and Smith in the 1960s, the phenomenon is explained as a consequence of aliasing of coefficients in Chebyshev expansions. Then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log((z + 1)/(z - 1)) in the complex plane. Gauss quadrature corresponds to Padé approximation at z = ∞. Clenshaw-Curtis quadrature corresponds to an approximation whose order of accuracy at z = ∞ is only half as high, but which is nevertheless equally accurate near [-1, 1]. [ABSTRACT FROM AUTHOR]

An object-oriented MATLAB system is described for performing numerical linear algebra on continuous functions and operators rather than the usual discrete vectors and matrices. About eighty MATLAB functions from plot and sum to svd and cond have been overloaded so that one can work with our "chebfun" objects using almost exactly the usual MATLAB syntax. All functions live on [-1, 11 and are represented by values at sufficiently many Chebyshev points for the polynomial interpolant to be accurate to close to machine precision. Each of our overloaded operations raises questions about the proper generalization of familiar notions to the continuous context and about appropriate methods of interpolation, differentiation, integration, zerofinding, or transforms. Applications in approximation theory and numerical analysis are explored, and possible extensions for more substantial problems of scientific computing are mentioned. [ABSTRACT FROM AUTHOR]

Pseudospectra of rectangular matrices vary continuously with the matrix entries, a feature that eigenvalues of these matrices do not have. Some properties of eigenvalues and pseudospectra of rectangular matrices are explored, and an efficient algorithm for the computation of pseudospectra is proposed. Applications are given in (square) eigenvalue computation (Lanczos iteration), square pseudospectra approximation (Arnoldi iteration), control theory (nearest uncontrollable system) and game theory. [ABSTRACT FROM AUTHOR]