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Pilipchuk, L. A. Sparse Linear Systems and Their Applications

 

Pilipchuk, L. A. Sparse Linear Systems and Their Applications / L. A. Pilipchuk. -Minsk : BSU, 2013. - 235 p.

ISBN 978-985-518-873-6.

This book presents the results of the research of the sparse underdetennined systems of linear algebraic equations and their applications. The methods of decomposition and the theory of graphs partitioning are applied to construct solutions of underdetermined systems with special sparse matrices. Numerous examples of decomposition algorithms for different types of sparsity are considered. Some of these algorithms are implemented in Wolfram Mathematica using new technologies for constructing analytical and numerical solutions.

Table 46. Fig. 64. Bibl. 60.

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CONTENTS

LIST OF REFERENCE SYMBOLS
5
INTRODUCTION
6
1. FRACTAL-LIKE MATRICES
10
1.1.    Introduction
10
1.2.    Definition of Fractal-Like matrices
11
1.3.    Main properties
11
1.4.    Optimal storage
13
2 SPARSE UNDERDETERMINED SYSTEMS
15
2.1.General type of sparsity
15
2.2.Network part of system
16
2.3.Network support criterion
20
2.4.Basis of solution space
22
2.5.Decomposition of system
26
2.6.Properties of support
28
2.7.Sparse systems with extended additional part
29
2.8.Matrix of determinants
33
2.9.Graph theoretical properties
35
2.10.    Implementation of decomposition algorithms
36
3. SENSOR LOCATION PROBLEM
45
3.1.Sensor Location Problem for graphs
45
3.2.Example 1 (Underdetermined system)
47
3.3.Example 2 (Unique solution)
52
3.4.Example 3 (Underdetermined system)
60
3.5.Example 4 (Unique solution)
65
3.5.1.Analytical solution
65
3.5.2.Numerical solution
72
3.6.Example 5 (Underdetermined system)
75
3.7.Example 6 (Unique solution)
80
3.7.1.Analytical solution
80
3.7.2.Numerical solution
91
3.8.    Example of multiple monitored nodes
94
4. NOT FULL RANK SPARSE SYSTEMS
109
4.1.General form of sparse systems
110
4.2.Network part of system
112
4.3.Multigraph support criterion
113
4.4.Characteristic vectors
113
4.5.Decomposition of system
118
4.6.Examples of decomposition of linear systems
122
4.7.Implementation in Wolfram Mathematica
139
5. SLP FOR MULTIGRAPHS
153
5.1.Introduction
153
5.2.Decomposition of the multigraph
157
5.3.Support of the multigraph
159
5.4.Modeling of multigraphs
160
6. FULL RANK SPARSE SYSTEMS
165
6.1.Sparse systems for generalized multinetwork
166
6.2.The basis of the solutions space
169
6.2.1.Support for generalized multinetwork
170
6.2.2.Characteristic vectors
171
6.3.Decomposition algorithms
188
6.4.Example of decomposition of sparse systems
196
6.5.Technology of implementation
210
6.6.Implementation in Wolfram Mathematica
216
BIBLIOGRAPHY
232
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