The method of alternation projections (MAP) is an iterative procedure for finding the projection of a point on the intersection of closed subspaces of an Hilbert space. The convergence of this method is usually slow, and several methods for its acceleration have already been proposed. In this work, we consider a special MAP, namely Kaczmarz' method for solving systems of linear equations. The convergence of this method is discussed. After giving its matrix formulation and its projection properties, we consider several procedures for accelerating its convergence. They are based on sequence transformations whose kernels contain sequences of the same form as the sequence of vectors generated by Kaczmarz' method. Acceleration can be achieved either directly, that is without modifying the sequence obtained by the method, or by restarting it from the vector obtained by acceleration. Numerical examples show the effectiveness of both procedures.
Summary.Using the theory of Euler methods from summability theory, we investigate general iterative methods for solving linear systems of equations. In particular, for a given Euler method, a region S of the complex plane is determined such that a k-step iterative method converges if the eigenvalues of an iteration operator T are contained in S. For a given S, optimal methods are described, and upper and lower bounds are derived for the associated asymptotic rate of convergence. Special attention is given to two-step methods with complex parameters.
Summary. Given a nonsingular linear system Ax=b, a splitting A = M-N leads to the one-step iteration (1)xm=Tx,,_l+e with T:=M-IN and c, = M-lb. We investigate semiiterative methods (SIM's) with respect to (1), under the assumption that the eigenvalues of T are contained in some compact set Q of r with 1 r There exist SIM's which are optimal with respect to O, but, except for some special sets f2, such optimal methods are not explicitly known in general. Using results about "maximal convergence" of polynomials and "uniformly distributed" nodes from approximation and function theory, we describe here SIM's which are asymptotically optimal with respect to f2. It is shown that Euler methods, extensively studied by Niethammer-Varga [NV], are special SIM's. Various algorithms for SIM's are also derived here. A 1-1 correspondence between Euler methods and SIM's, generated by generalized Faber polynomials, is further established here. This correspondence gives that asymptotically optimal Euler methods are quite near the optimal SIM's.
Summary. For a square matrix T~"'", where (l-T) is possibly singular, we investigate the solution of the linear fixed point problem x = Tx + e by applying semiiterative methods (SIM's) to the basic iteration Xo6~U", Xk 9 "=Txk-1 +c(k>l). Such problems arise if one splits the coefficient matrix of a linear system A x=b of algebraic equations according to A =M-N (M nonsingutar) which leads to x=M-~Nx+M-lb=:Tx+e. Even if x = Tx+e is consistent there are cases where the basic iteration fails to converge, namely if T possesses eigenvalues 2 + 1 with ]21 > 1, or if 2= 1 is an eigenvalue of T with nonlinear elementary divisors. In these cases -and also if x= Tx +e is incompatible -we derive necessary and sufficient conditions implying that a SIM tends to a vector ~ which can be described in terms of the Drazin inverse of (I -T). We further give conditions under which is a solution or a least squares solution of (I -T) x = c.
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