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Applying approximate LU-factorizations as preconditioners in eight iterative methods for solving systems of linear algebraic equations

Zahari Zlatev, Krassimir Georgiev (2013)

Open Mathematics

Many problems arising in different fields of science and engineering can be reduced, by applying some appropriate discretization, either to a system of linear algebraic equations or to a sequence of such systems. The solution of a system of linear algebraic equations is very often the most time-consuming part of the computational process during the treatment of the original problem, because these systems can be very large (containing up to many millions of equations). It is, therefore, important...

Approximate multiplication in adaptive wavelet methods

Dana Černá, Václav Finěk (2013)

Open Mathematics

Cohen, Dahmen and DeVore designed in [Adaptive wavelet methods for elliptic operator equations: convergence rates, Math. Comp., 2001, 70(233), 27–75] and [Adaptive wavelet methods II¶beyond the elliptic case, Found. Comput. Math., 2002, 2(3), 203–245] a general concept for solving operator equations. Its essential steps are: transformation of the variational formulation into the well-conditioned infinite-dimensional l 2-problem, finding the convergent iteration process for the l 2-problem and finally...

Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in l 2 by the use of finite submatrices

Maria Malejki (2010)

Open Mathematics

We consider the problem of approximation of eigenvalues of a self-adjoint operator J defined by a Jacobi matrix in the Hilbert space l 2(ℕ) by eigenvalues of principal finite submatrices of an infinite Jacobi matrix that defines this operator. We assume the operator J is bounded from below with compact resolvent. In our research we estimate the asymptotics (with n → ∞) of the joint error of approximation for the eigenvalues, numbered from 1 to N; of J by the eigenvalues of the finite submatrix J...

Asymptotics for weakly dependent errors-in-variables

Michal Pešta (2013)

Kybernetika

Linear relations, containing measurement errors in input and output data, are taken into account in this paper. Parameters of these so-called errors-in-variables (EIV) models can be estimated by minimizing the total least squares (TLS) of the input-output disturbances. Such an estimate is highly non-linear. Moreover in some realistic situations, the errors cannot be considered as independent by nature. Weakly dependent ( α - and ϕ -mixing) disturbances, which are not necessarily stationary nor identically...

Avoiding look-ahead in the Lanczos method and Padé approximation

E. Ayachour (1999)

Applicationes Mathematicae

In the non-normal case, it is possible to use various look-ahead strategies for computing the elements of a family of regular orthogonal polynomials. These strategies consist in jumping over non-existing and singular orthogonal polynomials by solving triangular linear systems. We show how to avoid them by using a new method called ALA (Avoiding Look-Ahead), for which we give three principal implementations. The application of ALA to Padé approximation, extrapolation methods and Lanczos method for...

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