Iterative processes for solving a class of incompatible systems of linear algebraic equations

Молчанов, И. Н.; Yakovlev, M. F.

doi:10.1016/0041-5553(75)90068-3

Cited by 5 publications

(3 citation statements)

References 1 publication

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“…holds, then it yields condition (13) and at the same time estimate (14). This completes the proof of the theorem.…”

Section: Termination Conditions For Iterative Processes Of the Solutimentioning

confidence: 51%

“…In [13], convergence theorems are proved for them, the rate of their convergence in the subspace of images of some operator is determined, the ways of finding a normal solution or a normal pseudosolution are specified, and, in particular, it is shown that the following relation holds for the residual: lim…”

Section: Solving Slae With Positive Semidefinite Matrices By Iterativmentioning

confidence: 99%

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Methods for obtaining reliable solutions to systems of linear algebraic equations

2011

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The general case of incompatible systems of linear algebraic equations with matrices of arbitrary rank is considered. The estimates for total errors are obtained for all the considered cases under conditions of approximate data. In solving systems by iterative methods, the conditions of completion of iterative processes that provide solutions with a prescribed accuracy are considered in detail. A special attention is given to the solution of incompatible systems with symmetric positive semidefinite matrices by the method of three-stage regularization in which an algorithm for choosing the regularization parameter is proposed that allows finding solutions with the required accuracy.Keywords: approximate initial data, incompatible systems, total error of the solution, iterative methods.

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“…holds, then it yields condition (13) and at the same time estimate (14). This completes the proof of the theorem.…”

Section: Termination Conditions For Iterative Processes Of the Solutimentioning

confidence: 51%

Section: Solving Slae With Positive Semidefinite Matrices By Iterativmentioning

confidence: 99%

Methods for obtaining reliable solutions to systems of linear algebraic equations

2011

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“…Iterative methods for systems with singular matrices are considered in a number of papers (see e.g. [10,12,14,16,17,22]). The corresponding methods for the Neumann problem could be found in [4,5,7,15,21,23].…”

Section: Introductionmentioning

confidence: 99%

On finite element methods for the Neumann problem

Молчанов¹,

Galba²

1985

Numer. Math.

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Summary.The Neumann problem for a second order elliptic equation with self-adjoint operator is considered, the unique solution of which is determined from projection onto unity. Two variational formulations of this problem are studied, which have a unique solution in the whole space. Discretization is done via the finite element method based on the Ritz process, and it is proved that the discrete solutions converge to one of the solutions of the continuous problem. Comparison of the two methods is done.

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On the solution of singular linear systems of algebraic equations by semiiterative methods

Eiermann

Marek²,

Niethammer

1988

Numer. Math.

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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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Iterative processes for solving a class of incompatible systems of linear algebraic equations

Cited by 5 publications

References 1 publication

Methods for obtaining reliable solutions to systems of linear algebraic equations

Methods for obtaining reliable solutions to systems of linear algebraic equations

On finite element methods for the Neumann problem

On the solution of singular linear systems of algebraic equations by semiiterative methods

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