PRECONDITIONED RICHARDSON AND MINIMAL RESIDUAL ITERATIVE METHODS FOR PIECEWISE HERMITE BICUBIC ORTHOGONAL SPLINE COLLOCATION EQUATIONS

The preconditioned Richardson and preconditioned minimal residual iter ative methods are presented for the solution of linear equations arisi ng when orthogonal spline collocation with piecewise Hermite bicubics is applied to a selfadjoint elliptic Dirichlet boundary value problem on a rectangle. For both methods, the orthogonal spline collocation di scretization of Laplace's operator is used as a preconditioner. In the preconditioned Richardson method, an approximation of the optimal ite ration parameter is computed from knowledge of spectral equivalence co nstants. Such a priori information is not required for the preconditio ned minimal residual method. In each iteration of both methods, orthog onal spline collocation Poisson's problems are solved by a fast direct algorithm which employs fast Fourier transforms. An application of th e preconditioned minimal residual method is also discussed for the sol ution of linear equations arising from the orthogonal spline collocati on discretization of nonselfadjoint elliptic Dirichlet boundary value problems.