On the fast solution of a linear system arising in numerical conformal mapping

One step of the Newton method for the discretized Theodorsen equation in conformal mapping requires the solution of a certain 2N×2N system. Application of the Gaussian algorithm costs O(N³) arithmetic operations (a.o.). We present an algorithm which reduces the problem to the solution of three N×N linear Toeplitz systems. These systems can be solved in O(N log²N) a.o. This is also the amount of work required by the whole algorithm.