Spatial frequencies and iterative methods

The detailed knowledge of eigenvalues and eigenvectors of the finite difference representations of differential operators is of great importance in the analysis of computational methods related to differential equations. This includes, inter alia, the connection to Fourier analysis, regularity (smoothness), sampling theory, the behavior of basic iterative methods, and composite error correction methods such as multigrid methods. Here we shall develop the necessary tools for a simple spectral analysis of a multigrid method, based on the previous knowledge of Toeplitz matrices.