Refined Iterative Methods for Computation of the Solution and the Eigenvalues of Self-Adjoint Boundary Value Problems

I: The Self-Adjoint Boundary Value Problem.- 1. Problems of Dirichlets and Poissons type.- 2. Better approximations.- 3. Energy on the boundary.- 4. Eigenvalue problems.- 5. Biharmonic problems.- 6. Adaption for practical purposes; the test example.- 7. Modes of oscillation of the plate.- II: Theory of Gradient Methods.- 1. Introduction.- 2. The residual polynomial.- 3. Methods with two-term recursive formulae.- 4. Methods with three-term recursive formulae.- 5. Combined methods.- 6. The cgT-method.- 7. Determination of eigenvalues.- III: Experiments on Gradient Methods.- 1. Introduction.- 2. Survey of the plate experiments.- 3. Solution of the system A x + b = 0 (Plate problem with coarse grid).- 3.1 Steepest descent.- 3.2 Tchebycheff method.- 3.3 Conjugate gradient methods.- 3.4 The cgT-method.- 3.5 Combined method.- 3.6 Elimination.- 3.7 Computation of the residuals.- 4. Determination of the eigenvalues of A.- 4.1 Conjugate gradient methods with subsequent QD-algorithm.- 4.2 cgT-method with subsequent QD-algorithm (spectral transformation).- 5. Solution of the system A x + b =0 and determination of the eigenvalues of A; fine grid.- 6. Second test example: the bar problem.- 7. Appendix: The first three eigenvectors of A.- IV: Overrelaxation.- 1. Theory.- 1.1 Principles.- 1.2 General relaxation.- 1.3 Overrelaxation.- 1.4 Property A.- 1.5 Youngs overrelaxation.- 1.6 Different methods.- 2. Numerical results (Plate problem).- 2.1 Overrelaxation.- 2.2 Symmetric relaxation.- 2.3 Block relaxation.- 3. The bar problem.- 3.1 Overrelaxation.- 3.2. Block relaxation.- 3.3 Symmetric overrelaxation.- V: Conclusions.- 1. The plate problem.- 2. The bar problem.- 3. Computation of eigenvalues.- 4. Recollection of the facts.- References.