ACCELERATION OF THE CONVERGENCE OF THE KACZMARZ METHOD AND ITERATED HOMOGENEOUS TRANSFORMATIONS.
Technical rept. (Doctoral thesis),
CALIFORNIA UNIV LOS ANGELES NUMERICAL ANALYSIS RESEARCH
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The Kaczmarz method is a well known and computationally easily constructed iterative method for solving a system of linear equations. Convergence is normally slow due to the presence of an eigenvalue with modulus close to unity. Two related methods of accelerating this convergence are considered. The accelerated transformations are nonadditive and homogeneous. In a natural way arises the question of defining suitable finite and asymptotic rates of convergence for certain convergent nonlinear homogeneous transformations. The rate of convergence problem is re-examined for linear transformations and a new average rate for a finite number of iterations is defined. Aspects of Ergodic theory are discussed in connection with an important class of linear transformations. Classes of homogeneous transformations are defined to which the theory of rate of convergence can be extended. Ergodic theorems are proved concerning certain homogeneous transformations in the plane. The relationship of the Kaczmarz operator to the Gauss-Seidel operator is recalled and some aspects of the Young overrelaxation theory are applied to the former. The dissertation concludes with some graphed results showing the effect of accelerating the convergence of the Kaczmarz method. Author