Contents Preface PART ONE POINT H-MATRICES Chapter 1 Introduction 3 1.1 Speaking from diagonally dominant matrices 4 1.2 H-matrices 6 1.3 The relationship between diagonally dominant matrices and H-matrices 8 Chapter 2 Nonsingularity/Singularity on H-matrices 10 2.1 Introduction 10 2.2 On critical conditions for nonsingularity of nonstrictly diagonally dominant matrices 11 2.3 Nonsingularity/singularity of nonstrictly diagonally dominant matrices 18 2.4 Further results on nonsingularity/singularity of nonstrictly diagonally dominant matrices 21 2.5 Nonsingularity/singularity of general H-matrices 24 2.6 Conclusion 27 Chapter 3 The Schur Complements of General H-matrices 28 3.1 Introduction 28 3.2 The Schur complement 28 3.3 Some classical results on the Schur complement of H-matrices 31 3.4 The Schur complements of strong H-matrices 33 3.5 The Schur complements of weak H-matrices 36 3.5.1 The Schur complements of degenerate H-matrices 37 3.5.2 The Schur complements of mixed H-matrices 39 3.5.3 Further results on the Schur complements of H-matrices 52 3.6 The generalized Schur complements of weak H-matrices 68 Chapter 4 The Eigenvalue Distribution on H-matrices and Their Schur Complements 72 4.1 Introduction 72 4.2 The eigenvalue distribution on nonstrictly diagonally dominant matrices and general H-matrices 73 4.3 The eigenvalue distribution on the Schur complements of H-matrices 76 4.4 The eigenvalue distribution on the generalized Schur complements of H-matrices 85 4.5 The generalized eigenvalue distribution on H-matrix pair 87 4.5.1 Some notions and preliminary results 87 4.5.2 The generalized eigenvalue distribution of diagonally dominant matrices pairs 88 4.5.3 The Generalized Eigenvalue Distribution of H-matrix pairs 92 4.5.4 The generalized eigenvalue location of some special matrix pairs 95 Chapter 5 Convergence on the Basic Iterative Methods for H-Matrices 97 5.1 Introduction 97 5.2 The Jacobi iterative method 97 5.3 The Gauss-Seidel iterative methods 101 5.3.1 Introduction 101 5.3.2 Some classic results 103 5.3.3 Convergence on Gauss-Seidel iterative methods 104 5.3.4 Convergence on symmetric Gauss-Seidel iterative method 109 5.3.5 Conclusions and remarks 112 5.3.6 Convergence on preconditioned Gauss-Seidel iterative methods 114 5.3.7 Numerical examples 117 5.3.8 Conclusions 120 5.4 The SOR iterative methods 120 5.4.1 Introduction 120 5.4.2 Some classic results 122 5.4.3 Convergence on FSOR and BSOR iterative methods 122 5.4.4 Convergence on SSOR iterative method 126 5.4.5 Numerical examples 130 5.4.6 Further work 133 5.5 The AOR iterative methods 133 5.5.1 Introduction 133 5.5.2 Convergence on FAOR and BAOR iterative methods 135 5.5.3 Convergence on SAOR iterative method 139 5.5.4 Numerical examples 143 5.5.5 Conclusion 149 Chapter 6 Radial Matrices and Asymptotical Stability of Linear Dynamic Systems 150 6.1 Introduction 150 6.2 Some notations and preliminary results 151 6.3 Some necessary and su.cient conditions on ∞-radial matrices (1-radial matrices) 154 6.4 Some properties on ∞-radial matrices (1-radial matrices) 156 6.5 Applications in the linear discrete dynamic systems 158 6.6 Conclusions 160 PART TWO GENERALIZATIONS OF H-MATRICES Chapter 7 Two Generalizations of H-matrices 163 7.1 Introduction 163 7.2 Block Diagonally Dominant Matrices and Block H-matrices 165 7.3 Generalized H-matrices and extended H-matrices 169 Chapter 8 Block Diagonally Dominant Matrices and Block H-matrices 177 8.1 Nonsingularity/singularity on block diagonally dominant matrices and block H-matrices 177 8.2 The Schur complement of block diagonally dominant matrices and block H-matrices 179 8.2.1 The Schur complement of block diagonally dominant matrices 179 8.2.2 The Schur complement of block H-matrices 189 8.3 The eigenvalue distribution of block H-matrices 191 8.3.1 Some generalizations of Taussky’s theorem 192 8.3.2 The eigenvalue distribution of block diagonally dominant matrices and block H-matrices 198 Chapter 9 Generalized H-matrices 204 9.1 Nonsingularity/singularity on generalized H-matrices 204 9.2 Convergence of block iterative methods for linear systems with generalized H-matrices 204 9.2.1 Convergence of block iterative methods for generalized H-matrices 207 9.2.2 Some applications to special cases from the computations of partial di.erential equations 214 9.2.3 Numerical examples 217 9.2.4 Conclusion 220 9.3 On parallel multisplitting block iterative methods for linear systems with generalized H-matrices 220 9.3.1 On parallel multisplitting block iterative methods 221 9.3.2 Main results 223 9.3.3 Applications to special cases from the solution of partial di.erential equations 227 9.3.4 Numerical
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