 | Argyros I.K. Numerical methods for equations and its applications / I.K.Argyros, Y.J.Cho, S.Hilout. - Boca Raton: CRC Press/Taylor & Francis, 2012. - viii, 465 p.: ill. - Bibliogr.: p.415-462. - Ind.: p.463-465. - ISBN 978-1-57808-753-2
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Preface ......................................................... v
1 Introduction ................................................. 1
2 Newton's Method ............................................. 10
2.1 Convergence Under Fr'echet Differentiability ........... 10
2.2 Convergence Under Twice Fr'echet Differentiability ..... 38
2.3 Newton's method on unbounded domains ................... 51
2.4 Continuous Analog of Newton's method ................... 59
2.5 Interior Point Techniques .............................. 63
2.6 Regular smoothness ..................................... 70
2.7 ω-convergence .......................................... 81
2.8 Semilocal Convergence and Convex Majorants ............. 90
2.9 Local Convergence and Convex Majorants ................ 102
2.10 Majorizing Sequences .................................. 110
2.11 Upper Bounds for Newton's Method ...................... 125
3 Secant method .............................................. 135
3.1 Convergence ........................................... 135
3.2 Least Squares Problems ................................ 146
3.3 Nondiscrete Induction and Secant Method ............... 154
3.4 Nondiscrete Induction and a Double Step Secant
Method ................................................ 163
3.5 Directional Secant Methods ............................ 175
3.6 Efficient Three Step Secant Methods ................... 195
4 Steffensen's Method ........................................ 208
4.1 Convergence ........................................... 208
5 Gauss-Newton method ........................................ 218
5.1 Convergence ........................................... 218
5.2 Average-Lipschitz Conditions .......................... 228
6 Newton-type methods ........................................ 250
6.1 Convergence with Outer Inverses ....................... 250
6.2 Convergence of a Moser-type Method .................... 264
6.3 Convergence with Slantly Differentiable Operator ...... 271
6.4 A intermediate Newton Method .......................... 279
7 Inexact Methods ............................................ 293
7.1 Residual control Conditions ........................... 293
7.2 Average Lipschitz Conditions .......................... 306
7.3 Two-step Methods ...................................... 312
7.4 Zabrejko-Zincenko-type Conditions ..................... 326
8 Werner's Method ............................................ 335
8.1 Convergence Analysis .................................. 335
9 Halley's Method ............................................ 344
9.1 Local Convergence ...................................... 344
10 Methods for variational inequalities ....................... 353
10.1 Subquadratic Convergent Method ........................ 353
10.2 Convergence Under Slant Condition ..................... 360
10.3 Newton-Josephy Method ................................. 369
11 Fast two-step methods ...................................... 385
11.1 Semilocal Convergence ................................ 385
12 Fixed Point Methods ........................................ 399
12.1 Successive Substitutions Methods ...................... 399
Bibliography .................................................. 415
Index ......................................................... 463
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