Operator trigonometry of the model problem (Q2713563)

Though a continuation of the author's work [Numer. Linear Algebra Appl. 4, 333-347 (1997; Zbl 0889.65030)], the paper can be read independently. The concept of operator trigonometry is applied to iterative methods solving a discretized Poisson equation on the unit square, i.e., to the problem \(Ax=b\), where \(A\) is a symmetric positive definite matrix. NEWLINENEWLINENEWLINEFirst, basic operator trigonometry notions as \(\cos A\), \(\sin A\), anti-eigenvalue, and anti-eigenvector are defined. Next, relations between them and eigenvalues and eigenvectors are briefly listed. Then the author pays attention to various iterative methods (Jacobi, Gauss-Seidel, successive overrelaxation, etc.) and reformulates such quantities as spectral radii, acceleration parameters, or convergence rates in terms of the operator trigonometry. Thus a new view of iterative methods is presented.