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594 results on '"Golub, Gene H."'

119. Stable Numerical Methods for Obtaining the Chebyshev Solution to an Overdetermined System of Equations.

Author

Bartels, Richard H. and Golub, Gene H.

Subjects
*CHEBYSHEV systems, *ALGORITHMS
Abstract

Presents an implementation of Stiefel's exchange algorithm for determining a Chebyshev solution to an overdetermined system of linear equations. Use of Gaussian LU decomposition with row interchanges; Stability of implementation than those usually given in the literature; Generalization of Stiefel's algorithm which permits the occasional exchange of two equations simultaneously.

Published
1968
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125. 150 years old and still alive: eigenproblems

Author

Vorst, H.A. van der, Golub, Gene H., Duff, I.S., and Watson, G.A.

Subjects
Wiskunde en Informatica
Abstract

Linear eigenproblems continue to be an important and highly relevant area of research in numerical linear algebra. Therefore, it should be no surprise that numerical algorithms for eigenproblems are among the oldest known in the modern literature. In 1846, exactly 150 years ago, Jacobi [33] wrote his famous paper on the computation of solutions for the problem Ax = x, with AT = A. Strictly speaking, this is not correct, since matrix notation was unknown at that time. So Jacobi formulated the problem in terms of systems of equations, written out elementwise. His paper contains computational elements that are still in use, like the plane rotations for making a system more diagonally dominant, and the (Gauss)-Jacobi iteration method for diagonally dominant systems. These elements were combined in his approach for the computation of eigenvalues, together with a clever way of setting up a correction equation for eigenvalue and eigenvector approximations.

Published
1997

127. The Simplex Method of Linear Programming Using LU Decomposition.

Author

Bartels, Richard H., Golub, Gene H., and Traub, J.F.

Subjects
*MATHEMATICAL decomposition, *LINEAR programming
Abstract

Presents the theoretical background for a standard computer implementation based upon the LU decomposition. Criticism of Dantzig's simplex method for linear programming; Round-off error behavior of the LU implementation; Basis of computer implementations using Dantzig's simplex method.

Published
1969
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146. TIKHONOV REGULARIZATION AND TOTAL LEAST SQUARES

Author

Golub, Gene H. and Golub, Gene H.

Abstract

Discretizations of inverse problems lead to systems of linear equations with a highly ill-conditioned coefficient matrix, and in order to compute stable solutions to these systems it is necessary to apply regularization methods. We show how Tikhonov's regularization method, which in its original formulation involves a least squares problem, can be recast in a total least squares formulation, suited for problems in which both the coefficient matrix and the right-hand side are known only approximately. We analyze the regularizing properties of this method and demonstrate by a numerical example that in certain cases with large perturbations, the new method is superior to standard regularization methods. (Also cross-referenced as UMIACS-TR-97-65)

Published
1998

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