Cyclic reduction is a numerical method for solving large linear systems by repeatedly splitting the problem.
For example fast solvers for Poisson's equation express the problem as solving a tridiagonal matrix, discretising the solution on a regular grid.
Moreover, to a point where a good approximate solution can be given,[1] but because the special matrix form must be preserved pivoting cannot be performed to improve numerical accuracy.
The method is not iterative, it seeks an exact solution to the linear problem consistent with the given boundary values, contrast that with the similar but computationally cheaper multigrid method which propagates error-correction estimates down and allows for different relaxation parameters at different scales, the iterative aspect allowing better incorporation of non-linear features.
Transforming from the spatial domain and restating the PDE is called a spectral method, Fourier analysis and cyclic reduction are combined in the FACR algorithm[2] which is explained in Numerical Recipes – see 19.4 Fourier and Cyclic Reduction Methods for Boundary Value Problems.