scholarly journals A note on generalizations of strict diagonal dominance for real matrices

1979 ◽  
Vol 26 ◽  
pp. 3-14 ◽  
Author(s):  
M. Neumann
Keyword(s):  
Diagonal Dominance ◽  
Strict Diagonal Dominance ◽  
Real Matrices

scholarly journals Construction of an Effective Preconditioner for the Even-odd Splitting of Cubic Spline Wavelets

2021 ◽  
Vol 20 ◽  
pp. 717-728
Author(s):  
Boris M. Shumilov
Keyword(s):  
Band System ◽  
Cubic Spline ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Diagonal Dominance ◽  
Sweep Method ◽  
Spline Wavelets ◽  
Linear Algebraic Equations ◽  
Initial Scale ◽  
Strict Diagonal Dominance

In this study, the method for decomposing splines of degree m and smoothness C^m-1 into a series of wavelets with zero moments is investigated. The system of linear algebraic equations connecting the coefficients of the spline expansion on the initial scale with the spline coefficients and wavelet coefficients on the embedded scale is obtained. The originality consists in the application of some preconditioner that reduces the system to a simpler band system of equations. Examples of applying the method to the cases of first-degree spline wavelets with two first zero moments and cubic spline wavelets with six first zero moments are presented. For the cubic case after splitting the system into even and odd rows, the resulting matrix acquires a seven-diagonals form with strict diagonal dominance, which makes it possible to apply an effective sweep method to its solution

scholarly journals Simplex algebras and their representation

1973 ◽  
Vol 14 (2) ◽  
pp. 136-144
Author(s):  
M. S. Vijayakumar
Keyword(s):  
Left Ideal ◽  
Maximal Ideal ◽  
Banach Algebras ◽  
Principal Component ◽  
Nonempty Interior ◽  
Regular Elements ◽  
Maximal Left Ideal ◽  
The Relationship ◽  
Order Identity ◽  
Real Matrices

This paper establishes a relationship (Theorem 4.1) between the approaches of A. C. Thompson [8, 9] and E. G. Effros [2] to the representation of simplex algebras, that is, real unital Banach algebras that are simplex spaces with the unit for order identity. It proves that the (nonempty) interior of the associated cone is contained in the principal component of the set of all regular elements of the algebra. It also conjectures that each maximal ideal (in the order sense—see below) of a simplex algebra contains a maximal left ideal of the algebra. This conjecture and other aspects of the relationship are illustrated by considering algebras of n × n real matrices.

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