Displaying similar documents to “Improving the convergence of iterative methods”

Convergence of extrapolation coefficients

Jan Zítko (1984)

Aplikace matematiky

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Let x k + 1 = T x k + b be an iterative process for solving the operator equation x = T x + b in Hilbert space X . Let the sequence { x k } k = o ∞ formed by the above described iterative process be convergent for some initial approximation x o with a limit x * = T x * + b . For given l > 1 , m 0 , m 1 , ⋯ , m l let us define a new sequence { y k } k = m 1 ∞ by the formula y k = α 0 ( k ) x k + α 1 ( k ) x k - m 1 + ... + α l ( k ) x k - m l , where α i ( k ) are obtained by solving a minimization problem for a given functional. In this paper convergence properties of α i ( k ) are investigated and on the basis of the results thus obtainded it is proved that lim k → ∞ x * - y k / x * - x k p = 0 for some...

Combining the preconditioned conjugate gradient method and a matrix iterative method

Jan Zítko (1996)

Applications of Mathematics

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The preconditioned conjugate gradient method for solving the system of linear algebraic equations with a positive definite matrix is investigated. The initial approximation for conjugate gradient is constructed as a result of a matrix iteration method after m steps. The behaviour of the error vector for such a combined method is studied and special numerical tests and conclusions are made.