Interpolation on arbitrary regions in the complex plane (Q1362899)

A fast and stable numerical algorithm is presented for the interpolation from an arbitrary Jordan curve in the complex plane to the region bounded by that curve. Computation of such stable interpolation points is based on the solution of the modified Symm's integral equation [cf. \textit{V. Rokhlin}, J. Comput. Phys. 60, 187-207 (1985; Zbl 0629.65122)]. The algorithm combines the classical analytical apparatus with the fast multipole method to obtain an order \(O(N^{1.5})\) procedure for the determination of interpolation nodes with \(N\) the number of nodes in the discretization of the given curve and an order \(O(M+K)\) procedure for the actual interpolation from the \(M\) nodes on the boundary of the region to \(K\) nodes inside the region. The performance of the algorithm is illustrated with several numerical examples.