| Preface |
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v | |
| Acknowledgments |
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ix | |
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xvii | |
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1 | (4) |
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1.1 Brief introduction to generalized inverses |
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1 | (2) |
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1.2 Nonlinear optimization and generalized inversion |
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3 | (1) |
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1.3 Importance of generalized inverses |
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4 | (1) |
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2 Basic notions from matrix theory, optimization and theory of generalized inverses |
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5 | (50) |
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2.1 Basic notions from matrix theory |
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5 | (13) |
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2.1.1 LU, Cholesky and Jordan decomposition |
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8 | (2) |
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2.1.2 Singular value decomposition |
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10 | (3) |
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2.1.3 (M, N) singular value decomposition |
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13 | (1) |
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2.1.4 Reduced row echelon forms |
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14 | (1) |
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2.1.5 Householder transformations and QR decomposition |
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15 | (3) |
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2.1.6 Idempotent matrices and projectors |
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18 | (1) |
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2.2 Properties and representations of generalized inverses |
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18 | (20) |
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2.2.1 Matrix equations and {t, j, ..., k}-inverses |
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20 | (5) |
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2.2.2 Basic properties of the Moore-Penrose inverse |
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25 | (5) |
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2.2.3 Basic properties of the weighted Moore-Penrose inverse |
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30 | (3) |
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2.2.4 Definition and basic properties of the Drazin inverse |
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33 | (3) |
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2.2.5 Basic properties of outer inverses |
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36 | (2) |
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2.3 Least-squares properties of the Drazin-inverse solution |
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38 | (13) |
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2.3.1 The system of linear equations is Drazin consistent, i.e., b e R(AP) |
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39 | (2) |
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2.3.2 The system of linear equations is arbitrary |
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41 | (2) |
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2.3.3 Least-squares properties of the Drazin inverse |
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43 | (8) |
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2.4 Least-squares properties of the A(2)T,S-inverse solution |
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51 | (4) |
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3 Direct methods for computing generalized inverses |
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55 | (108) |
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3.1 Pull-rank representations of generalized inverses |
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56 | (10) |
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3.1.1 Full-rank representations of outer inverses |
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57 | (2) |
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3.1.2 Full-rank representations of {2, 4} and {2, 3}-inverses |
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59 | (3) |
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3.1.3 Full rank representations of important generalized inverses |
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62 | (1) |
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3.1.4 Factorization based on Gaussian eliminations |
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63 | (3) |
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3.2 Methods based on singular value decompositions and (M, N) singular value decompositions |
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66 | (4) |
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3.2.1 Computing the Moore-Penrose inverse by SVD factorization |
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66 | (2) |
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3.2.2 Computing generalized inverses by SVD like factorizations |
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68 | (2) |
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3.3 Greville's partitioning method for constant matrices |
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70 | (7) |
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3.3.1 Computing Moore-Penrose inverse by partitioning method |
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71 | (1) |
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3.3.2 Computing {1} inverses by Partitioning method |
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72 | (1) |
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3.3.3 Computing the weighted Moore-Penrose inverse by Partitioning method |
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73 | (1) |
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74 | (3) |
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3.4 Generalized inverses as a limit |
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77 | (3) |
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3.5 Le Verrier-Faddeev method |
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80 | (11) |
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80 | (4) |
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3.5.2 Algorithms based on Le Verrier-Faddeev method |
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84 | (7) |
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3.6 Methods based on the Gauss-Jordan elimination |
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91 | (11) |
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3.6.1 Preliminary results |
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93 | (2) |
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3.6.2 The Gauss-Jordan algorithm for computing outer inverses |
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95 | (5) |
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3.6.3 Numerical experiments |
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100 | (2) |
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3.7 Block matrix algorithms |
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102 | (6) |
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3.7.1 Block representations of generalized inverses |
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102 | (2) |
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3.7.2 Generalized inverses of 2 × 2 block matrices |
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104 | (4) |
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3.8 Generalized inverses of rank-one modified matrix |
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108 | (30) |
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3.8.1 The Moore-Penrose inverse of rank-one modified matrix |
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110 | (6) |
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3.8.2 Computing {2, 4} and {2, 3}-inverses using rank-one updates |
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116 | (1) |
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3.8.3 Computing {2, 4} and {2, 3}-inverses by means of Sherman-Morrison formula |
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117 | (14) |
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3.8.4 Computing the pseudo-inverse of specific Toeplitz matrices using rank-one updates |
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131 | (7) |
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3.9 Full-rank representations of outer inverses based on the QR decomposition |
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138 | (3) |
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138 | (1) |
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3.9.2 Representations of outer inverse based on QR decomposition |
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138 | (3) |
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3.10 Generalized inversion is not harder than matrix multiplication |
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141 | (10) |
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3.10.1 Strassen matrix multiplication and inversion |
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141 | (4) |
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3.10.2 Recursive Cholesky factorization |
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145 | (4) |
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3.10.3 Rapid computation of generalized inverses |
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149 | (2) |
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3.11 Determinantal representation of generalized inverses and adjoint mappings of matrices |
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151 | (6) |
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3.11.1 Notation and preliminaries about determinantal representations |
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151 | (2) |
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3.11.2 The existence of {2}-inverses |
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153 | (3) |
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3.11.3 Characterizations and representations of {2}-inverses |
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156 | (1) |
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3.12 Full rank representations and properties of the W-Weighted Drazin inverse |
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157 | (6) |
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3.12.1 About the W-Weighted Drazin inverse |
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157 | (2) |
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3.12.2 Basic representations of W-weighted Drazin inverse |
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159 | (1) |
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3.12.3 Additional representations of WDI |
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160 | (3) |
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4 Iterative methods for computing generalized inverses |
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163 | (142) |
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4.1 Modified SMS method for computing outer inverses |
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163 | (7) |
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4.1.1 SMS method for computing {2, 3} and {2, 4}-inverses of matrices |
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167 | (3) |
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4.2 SMS method appropriate for Toeplitz matrices |
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170 | (11) |
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4.2.1 Displacement rank and displacement operator of a Toeplitz matrix |
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170 | (2) |
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4.2.2 Modified SMS method for computing outer inverses of a Toeplitz matrix |
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172 | (9) |
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4.3 Hyperpower iterative method |
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181 | (6) |
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4.3.1 Basic properties of hyperpower iterative methods |
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182 | (5) |
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4.4 On hyperpower family of iterations for computing outer inverses possessing high efficiencies |
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187 | (5) |
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4.4.1 Factorized forms of the hyperpower family |
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188 | (4) |
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4.5 General higher-order iterative methods |
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192 | (26) |
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4.5.1 Generalized Schulz iterative methods for computing outer inverses |
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192 | (4) |
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4.5.2 Convergence to outer inverse |
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196 | (2) |
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4.5.3 The choice of the initial value |
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198 | (2) |
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4.5.4 Improvement of the 9th order method |
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200 | (1) |
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4.5.5 Rapid Hyper-power based iterative methods for computing outer inverses |
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201 | (5) |
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4.5.6 General scheme for construction of the generalized Schulz iterative methods for outer inverses |
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206 | (7) |
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4.5.7 Generalized Schulz methods - revisited |
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213 | (5) |
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4.6 Scaled matrix iterations for computing outer inverses |
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218 | (3) |
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4.7 Iterative method for computing Moore-Penrose inverse based on Penrose equations |
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221 | (20) |
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221 | (2) |
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4.7.2 The iterative method arising from Penrose equations |
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223 | (6) |
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4.7.3 Numerical experience |
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229 | (2) |
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4.7.4 Factorizations of hyperpower family of iterative methods via least squares |
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231 | (4) |
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4.7.5 An accelerated iterative method for computing weighted Moore-Penrose inverse |
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235 | (3) |
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4.7.6 Stability of a pth order iteration for finding generalized inverses |
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238 | (3) |
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4.8 Iterative methods based on Groetsch Representation Theorem |
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241 | (9) |
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4.8.1 Representation and approximation of outer generalized inverses |
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241 | (9) |
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4.9 Iterative methods arising from optimization methods |
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250 | (20) |
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4.9.1 Unconstrained optimization |
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251 | (8) |
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4.9.2 Scalar correction method (SC method) |
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259 | (4) |
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4.9.3 Application of the SC method for finding the Moore-Penrose inverse of a matrix |
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263 | (3) |
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4.9.4 Computing the Drazin inverse using optimization methods |
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266 | (3) |
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4.9.5 Computing A(2)T,S-inverse using optimization methods |
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269 | (1) |
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4.10 Another iterative methods |
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270 | (25) |
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4.10.1 Limit representations of generalized inverses and related methods |
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270 | (13) |
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4.10.2 Self-correcting iterative methods for computing {2}-inverses |
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283 | (4) |
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4.10.3 Iterative methods based on matrix splitting |
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287 | (3) |
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4.10.4 A new type of matrix splitting and its applications |
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290 | (5) |
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4.11 Iterative methods for computing matrix functions |
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295 | (10) |
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4.11.1 Approximating the matrix sign function |
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296 | (3) |
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4.11.2 An iterative method for polar decomposition and matrix sign function |
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299 | (6) |
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5 Symbolic computation of generalized inverses |
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305 | (50) |
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5.1 Short overview of algorithms for symbolic computation |
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307 | (1) |
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5.2 Symbolic computation based on Faddeev's algorithms |
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308 | (15) |
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5.2.1 Faddeev's algorithms for constant and rational matrices |
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309 | (2) |
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5.2.2 Faddeev's algorithms for polynomial matrices |
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311 | (2) |
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5.2.3 Computing outer inverses symbolically by the Le Verrier-Faddeev method |
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313 | (2) |
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5.2.4 Computing Moore-Penrose inverse of polynomial matrices by interpolation |
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315 | (2) |
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5.2.5 Le Verrier-Faddeev method for computing Drazin inverse of polynomial matrices |
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317 | (2) |
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5.2.6 Interpolation algorithm for computing various generalized inverses of polynomial matrices |
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319 | (4) |
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5.3 Algorithms based on Greville's Partitioning method |
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323 | (23) |
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5.3.1 Greville's method for constant and rational matrices |
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324 | (4) |
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5.3.2 Greville's method for polynomial matrices |
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328 | (18) |
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5.4 Symbolic computation of A(2)T,S-inverses using QDR factorization |
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346 | (9) |
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5.4.1 Symbolic computation based on QDR factorization |
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348 | (7) |
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6 Applications of generalized inverses |
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355 | (50) |
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6.1 Image restoration based on generalized inverses |
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356 | (34) |
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6.1.1 Preliminaries about image restoration |
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357 | (6) |
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6.1.2 Removal of uniform blur in X-ray images |
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363 | (5) |
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6.1.3 Partitioning method for removing non-uniform blur in images |
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368 | (6) |
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6.1.4 Application of rank-one-updates in image restoration |
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374 | (5) |
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6.1.5 Removal of blur in images based on least squares solutions |
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379 | (6) |
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6.1.6 Image deblurring based on separable restoration methods and least squares |
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385 | (5) |
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6.2 Application of pseudoinverse in balancing chemical equations |
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390 | (5) |
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6.2.1 Matrix iterations for computing generalized inverses and balancing chemical equations |
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391 | (4) |
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6.3 Application of generalized inverses in feedback compensation problem |
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395 | (4) |
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6.4 Application of the PI in solving indefinite least-squares problems |
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399 | (2) |
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6.5 Generalized inverses as solutions of matrix quadratic problems |
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401 | (4) |
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405 | (4) |
| Bibliography |
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409 | (40) |
| Index |
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449 | |
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