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 desktop 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 a few microseconds at each point.

Numerical algorithms based on rational functions are introduced that solve the Laplace and Helmholtz equations on 2D domains with corners quickly and accurately, despite the corner singularities.

The traditional view in numerical conformal mapping is that once the boundary correspondence function has been found, the map and its inverse can be evaluated by contour integrals. We propose that it is much simpler, and 10-1000 times faster, to represent the maps by rational functions computed by the AAA algorithm. To justify this claim, first we prove a theorem establishing root-exponential convergence of rational approximations near corners in a conformal map, generalizing a result of D. J. Newman in 1964. This leads to the new algorithm for approximating conformal maps of polygons. Then we turn to smooth domains and prove a sequence of four theorems establishing that in any conformal map of the unit circle onto a region with a long and slender part, there must be a singularity or loss of univalence exponentially close to the boundary, and polynomial approximations cannot be accurate unless of exponentially high degree. This motivates the application of the new algorithm to smooth domains, where it is again found to be highly effective.

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Many applications in scientific computing and data science require the computation of a rank-revealing factorization of a large matrix. In many of these instances the classical algorithms for computing the singular value decomposition are prohibitively computationally expensive. The randomized singular value decomposition can often be helpful, but is not effective unless the numerical rank of the matrix is substantially smaller than the dimensions of the matrix. We introduce a new randomized algorithm for producing rankrevealing factorizations based on existing work by Demmel, Dumitriu and Holtz [Numerische Mathematik, 108(1), 2007] that excels in this regime. The method is exceptionally easy to implement, and results in close-to optimal low-rank approximations to a given matrix. The vast majority of floating point operations are executed in level-3 BLAS, which leads to high computational speeds. The performance of the method is illustrated via several numerical experiments that directly compare it to alternative techniques such as the column pivoted QR factorization, or the QLP method by Stewart.