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

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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. Посмотреть в электронной библиотеке
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