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CONTRIBUTIONS TO NUMERICAL ALGEBRA

IP.com Disclosure Number: IPCOM000128264D
Original Publication Date: 1976-Dec-31
Included in the Prior Art Database: 2005-Sep-15
Document File: 2 page(s) / 15K

Publishing Venue

Software Patent Institute

Related People

Ned Anderson: AUTHOR [+3]

Abstract

It is fair to say that matrix problems are everywhere dense in the applications of numerical analysis. This is because, speaking very generally, lineax problems often lead directly to matrix problems; also, the standard method of attack for solving non-linear problems is to solve a sequence of linear problems. The most important scalar quantities associated with a matrix are its eigenvalues and singular values. Ill- 151 above are concerned directly with or can be applied to computing these quantities, often in the context of specific applications, which will be mentioned below. A brief description of Ill- 151 above follows.

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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

CONTRIBUTIONS TO NUMERICAL ALGEBRA

Summary of doctoral dissertation Sammanfattning av doktorsavhandling by

Ned Anderson

TRITA-NA

1. Introduction

It is fair to say that matrix problems are everywhere dense in the applications of numerical analysis. This is because, speaking very generally, lineax problems often lead directly to matrix problems; also, the standard method of attack for solving non-linear problems is to solve a sequence of linear problems.

The most important scalar quantities associated with a matrix are its eigenvalues and singular values. Ill- 151 above are concerned directly with or can be applied to computing these quantities, often in the context of specific applications, which will be mentioned below. A brief description of Ill- 151 above follows.

2. Summary of the papers

[1] is concerned with a robust method for the solution of a single nonlinear

(Equation Omitted)

There are many applications for such a method, one of the primary ones is computation of eigenvalues by equation solving (6, sec.5-8-31.

The term robust refers to the desirable quality that the method should be relatively insensitive to poor startirg values. The algorithm in [11 yields, at any stage, a strict error estimate, since it always retains two values of x where the values of the function have opposite signs. Nonetheless its asymptotic order of convergence is shown to be higher than that of the secant method (for, say, f analytic with a simple root a, i.e..f'(a) *0).

[21 deals with the solution of overdetermined linear systems where the system is derived from the approximation of a function by a linear combination of members of a Chebyshev system. The method can be seen as a computationally simpler alternative to the method of least squares. Necessary and sufficient conditions for the existence and uniqueness of a solution to a generalization of' the method of averages are given, complementing the results in [71. A further generalization, the method of local moments, is given which may be useful in applications. [31 will be summarized together with (5].

[41 treats the computation of eigenvalues of matrices with real eigenvalues,e.g. Hermitian matrices. A monotonicity property of the characteristic polynomial of such matrices is proved. It is then shown that the secant method is particularly appropriate for computing the extremal. eigenvalues, since global monotone convergence to these eigenvalues is obtained. A workable

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