On the Convergence of Some Generalized Preconditioned Iterative Methods

Convergence domains for the symmetric successive overrelaxation method

“…The iterative scheme (1.7) is known as the Preconditioned Simultaneous Displacement (PSD) method (see [2,4,5]) which for r = w(2 -o) reduces to the SSOR method (see [I, 3,6-9]), which has the following scheme:…”

Convergence theory of extrapolated iterative methods for a certain class of non-symmetric linear systems

“…In this paper we presented the theoretical analysis of the rate of convergence of the first order iterative methods defined in [3] under the assumptions that the coefficient matrix A is consistently ordered, and the Jacobi iteration matrix B possesses purely imaginary eigenvalues. By extending the SOR theory [11] we were able to find sufficient and necessary conditions for the iterative schemes being considered to converge.…”

“…Finally, we consider the Extrapolated Successive Overrelaxation (ESOR) method defined by [3] (see also [2, 6-9])…”

“…Such algebraic systems frequently arise in much the same contexts as in the symmetric case, e.g., the numerical solution of the biharmonic equation [1], the computation of cubic splines [10] and the solution of some integral equations of conformal mapping [43. where D=diag(A), -C L, -C v are the strictly lower and upper triangular parts of A, respectively. For the numerical solution of (1.1) we consider the following first order stationary iterative methods [3]. The Extrapolated GaussSeidel (EGS) method defined by …”

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Summary.A variety of iterative methods considered in [3] are applied to linear algebraic systems of the form A u=b, where the matrix A is consistently ordered [12] and the iteration matrix of the Jacobi method is skew-symmetric. The related theory of convergence is developed and the optimum values of the involved parameters for each considered scheme are determined.

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