On Cline’s Direct Method for Solving Overdetermined Linear Systems in the $L_\infty $ Sense

1978 ◽  
Vol 15 (2) ◽  
pp. 255-270 ◽  
Author(s):  
Richard H. Bartels ◽  
Andrew R. Conn ◽  
Christakis Charalambous
Keyword(s):  
Linear Systems ◽  
Direct Method ◽  
Overdetermined Linear Systems

Particularities of overdetermined linear systems arising in particle size analysis

1992 ◽  
Vol 2 (8) ◽  
pp. 1547-1555
Author(s):  
D. M. Calistru ◽  
R. Mondescu ◽  
I. Baltog
Keyword(s):  
Particle Size ◽  
Linear Systems ◽  
Particle Size Analysis ◽  
Size Analysis ◽  
Overdetermined Linear Systems

DMRG Approach to Fast Linear Algebra in the TT-Format

2011 ◽  
Vol 11 (3) ◽  
pp. 382-393 ◽  
Author(s):  
Ivan Oseledets
Keyword(s):  
Linear Systems ◽  
Linear Algebra ◽  
Direct Method ◽  
Fast Algorithms ◽  
Approximate Solutions ◽  
Vector Product ◽  
The Matrix ◽  
Minimization Scheme

AbstractIn this paper, the concept of the DMRG minimization scheme is extended to several important operations in the TT-format, like the matrix-by-vector product and the conversion from the canonical format to the TT-format. Fast algorithms are implemented and a stabilization scheme based on randomization is proposed. The comparison with the direct method is performed on a sequence of matrices and vectors coming as approximate solutions of linear systems in the TT-format. A generated example is provided to show that randomization is really needed in some cases. The matrices and vectors used are available from the author or at http://spring.inm.ras.ru/osel

scholarly journals Orthant–Monotonic Norms and Overdetermined Linear Systems

1997 ◽  
Vol 88 (2) ◽  
pp. 209-227
Author(s):  
K. Zietak
Keyword(s):  
Linear Systems ◽  
Overdetermined Linear Systems

Direct method for a class of symmetric linear systems

1993 ◽  
Vol 9 (9) ◽  
pp. 721-727 ◽  
Author(s):  
Awono Onana ◽  
H. A. Mbala
Keyword(s):  
Linear Systems ◽  
Direct Method

Primal Methods are Better than Dual Methods for Solving Overdetermined Linear Systems in the $l_\infty $ Sense?

1989 ◽  
Vol 26 (3) ◽  
pp. 693-726 ◽  
Author(s):  
Richard H. Bartels ◽  
Andrew R. Conn ◽  
Yuying Li
Keyword(s):  
Linear Systems ◽  
Dual Methods ◽  
Primal Methods ◽  
Overdetermined Linear Systems ◽  
Better Than

scholarly journals Roots of Characteristic Polynomial Sequences in Iterative Block Cyclic Reductions

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3213
Author(s):  
Masato Shinjo ◽  
Tan Wang ◽  
Masashi Iwasaki ◽  
Yoshimasa Nakamura
Keyword(s):  
Linear Systems ◽  
Reduction Method ◽  
Direct Method ◽  
Coefficient Matrix ◽  
Matrix Transformations ◽  
Cyclic Reduction ◽  
Polynomial Sequences ◽  
Finite Step ◽  
Point Arithmetic ◽  
Repeated Block

The block cyclic reduction method is a finite-step direct method used for solving linear systems with block tridiagonal coefficient matrices. It iteratively uses transformations to reduce the number of non-zero blocks in coefficient matrices. With repeated block cyclic reductions, non-zero off-diagonal blocks in coefficient matrices incrementally leave the diagonal blocks and eventually vanish after a finite number of block cyclic reductions. In this paper, we focus on the roots of characteristic polynomials of coefficient matrices that are repeatedly transformed by block cyclic reductions. We regard each block cyclic reduction as a composition of two types of matrix transformations, and then attempt to examine changes in the existence range of roots. This is a block extension of the idea presented in our previous papers on simple cyclic reductions. The property that the roots are not very scattered is a key to accurately solve linear systems in floating-point arithmetic. We clarify that block cyclic reductions do not disperse roots, but rather narrow their distribution, if the original coefficient matrix is symmetric positive or negative definite.

A novel direct resolution method for coupled systems in finite element analysis

2020 ◽  
Vol 39 (5) ◽  
pp. 1271-1280
Author(s):  
Youcef Boutora ◽  
Noureddine Takorabet
Keyword(s):  
Finite Elements ◽  
Linear Systems ◽  
Symmetric Matrix ◽  
Direct Method ◽  
Three Dimensional ◽  
Coupled Systems ◽  
Resolution Method ◽  
Element Analysis ◽  
Content Type ◽  
Magnetostatic Problem

Purpose This paper aims to propose a novel direct method for indefinite algebraic linear systems. It is well adapted for sparse linear systems, such as those of two-dimensional (2-D) finite elements problems, especially for coupled systems. Design/methodology/approach The proposed method is developed on an example of an indefinite symmetric matrix. The algorithm of the method is given next, and a comparison between the numbers of operations required by the method and the Cholesky method is also given. Finally, an application on a magnetostatic problem for classical methods (Gauss and Cholesky) shows the relative efficiency of the proposed method. Findings The proposed method can be used advantageously for 2-D finite elements in stepping methods without using a block decomposition of matrices. Research limitations/implications This method is advantageous for direct linear solving for 2-D problems, but it is not recommended at this time for three-dimensional problems. Originality/value The proposed method is the first direct solver for algebraic linear systems proposed since more than a half century. It is not limited for symmetric positive systems such as many of direct and iterative methods.

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