Methods of solving the Helmholtz equation have been well-known since the 1970s, but they are either only second-order accurate, or fourth-order accurate but valid only for the two dimensional case.

This paper presents a fast and highly accurate algorithm for the solution of the Helmholtz equation in two- and three-dimensional rectangular domains with boundary conditions of Dirichlet, Neumann, or periodic type. The methods used, recursive cyclic reduction (RCR) and fast Fourier transform (FFT), have been waiting to be used in such algorithms.

In the introduction, the author relates the history of the problem. Part 2 defines the method of discretization and demonstrates a theorem that gives the order of accuracy. Part 3 describes the algorithms. Part 4 contains computational examples, including a comparison of the accuracy and computation times of the two- and three-dimensional algorithms developed using FFT and RCR with those of a previously known analytic solution for the equation.

This is an excellent research paper, with clear explanations and good examples. It will interest specialists in partial differential equations. It should be noted that, while the algorithms described are suitable for parallel computing, the computation times claimed can only be achieved on supercomputers like the Cyber 180/855.