Displaying similar documents to “Remarks on polynomial methods for solving systems of linear algebraic equations”

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...

A new optimized iterative method for solving M -matrix linear systems

Alireza Fakharzadeh Jahromi, Nafiseh Nasseri Shams (2022)

Applications of Mathematics

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In this paper, we present a new iterative method for solving a linear system, whose coefficient matrix is an M -matrix. This method includes four parameters that are obtained by the accelerated overrelaxation (AOR) splitting and using the Taylor approximation. First, under some standard assumptions, we establish the convergence properties of the new method. Then, by minimizing the Frobenius norm of the iteration matrix, we find the optimal parameters. Meanwhile, numerical results on test...

Approximation of a semilinear elliptic problem in an unbounded domain

Messaoud Kolli, Michelle Schatzman (2003)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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Let f be an odd function of a class C 2 such that f ( 1 ) = 0 , f ' ( 0 ) < 0 , f ' ( 1 ) > 0 and x ↦ f ( x ) / x increases on [ 0 , 1 ] . We approximate the positive solution of - Δ u + f ( u ) = 0 , on ℝ + 2 with homogeneous Dirichlet boundary conditions by the solution of - Δ u L + f ( u L ) = 0 , on ] 0 , L [ 2 with adequate non-homogeneous Dirichlet conditions. We show that the error u L - u tends to zero exponentially fast, in the uniform norm.

On realizability of sign patterns by real polynomials

Vladimir Kostov (2018)

Czechoslovak Mathematical Journal

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The classical Descartes’ rule of signs limits the number of positive roots of a real polynomial in one variable by the number of sign changes in the sequence of its coefficients. One can ask the question which pairs of nonnegative integers ( p , n ) , chosen in accordance with this rule and with some other natural conditions, can be the pairs of numbers of positive and negative roots of a real polynomial with prescribed signs of the coefficients. The paper solves this problem for degree 8 polynomials. ...

Using successive approximations for improving the convergence of GMRES method

Jan Zítko (1998)

Applications of Mathematics

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In this paper, our attention is concentrated on the GMRES method for the solution of the system ( I - T ) x = b of linear algebraic equations with a nonsymmetric matrix. We perform m pre-iterations y l + 1 = T y l + b before starting GMRES and put y m for the initial approximation in GMRES. We derive an upper estimate for the norm of the error vector in dependence on the m th powers of eigenvalues of the matrix T . Further we study under what eigenvalues lay-out this upper estimate is the best one. The estimate shows and...