Computable Accurate Upper and Lower Error Bounds for Approximate Solutions of Linear Algebraic Systems

Aird, Thomas; Lynch, Robert E.

doi:10.1145/355644.355648

Cited by 10 publications

(3 citation statements)

References 6 publications

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“…Let us use the fact from [15] that the normal pseudosolution x k of the least-squares problem A x b k = is an orthogonal projection of the normal pseudosolution of the problem onto the main right singular subspace of dimension k for the matrix A. Thus, the rank of matrix (47) is k, i.e., the same as the rank of the matrix of the unperturbed problem.…”

Section: Total Error Estimates Of the Solution Of The Least Squares Pmentioning

confidence: 99%

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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Section: Total Error Estimates Of the Solution Of The Least Squares Pmentioning

confidence: 99%

Methods for obtaining reliable solutions to systems of linear algebraic equations

2011

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“…These include iterative refinement [24] or "check sums" [1]. Both of these methods essentially double the amount of work required at each step, and so they were not used by us.…”

Section: We Estimatementioning

confidence: 99%

“…The RSM (without degeneracy) begins with an n vector X > 0 having exactly m positive components and minimizes the linear function n c'^\= y cx (1) subject to A\ = b, Amxn^ n^ m (2) X > 0. This algorithm is now presented by means of a structured informal flow diagram.…”

Section: Introductionmentioning

confidence: 99%

A mathematical programming updating method using modified Givens transformations and applied to LP problems

Hanson

Wisniewski

1979

Commun. ACM

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An efficient and numerically stable method is presented for the problem of updating an orthogonal decomposition of a matrix of column (or row) vectors. The fundamental idea is to add a column (or row) analogous to adding an additional row of data in a linear least squares problem. A column (or row) is dropped by a formal scaling with the imaginary unit, V^T, followed by least squares addition of the column (or row). The elimination process for the procedure is successive application of the Givens transformation in modified (more efficient) form. These ideas are illustrated with an implementation of the revised simplex method. The algorithm is a general purpose one that does not account for any particular structure or sparsity in the equations. Some suggested computational tests for determining signs of various controlling parameters in the revised simplex algorithm are mentioned. A simple means of constructing test cases and some sample computing times are presented.

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Isotropic distributions of test matrices

Birkhoff

Gulati

1979

Journal of Applied Mathematics and Physics (ZAMP)

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Computable Accurate Upper and Lower Error Bounds for Approximate Solutions of Linear Algebraic Systems

Cited by 10 publications

References 6 publications

Methods for obtaining reliable solutions to systems of linear algebraic equations

Methods for obtaining reliable solutions to systems of linear algebraic equations

A mathematical programming updating method using modified Givens transformations and applied to LP problems

Isotropic distributions of test matrices

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