## Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive AlgorithmsThis book deals with the efficient numerical solution of challenging nonlinear problems in science and engineering, both in finite dimension (algebraic systems) and in infinite dimension (ordinary and partial differential equations). Its focus is on local and global Newton methods for direct problems or Gauss-Newton methods for inverse problems. The term 'affine invariance' means that the presented algorithms and their convergence analysis are invariant under one out of four subclasses of affine transformations of the problem to be solved. Compared to traditional textbooks, the distinguishing affine invariance approach leads to shorter theorems and proofs and permits the construction of fully adaptive algorithms. Lots of numerical illustrations, comparison tables, and exercises make the text useful in computational mathematics classes. At the same time, the book opens many directions for possible future research. |

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### Contents

7 | |

Part I ALGEBRAIC EQUATIONS | 42 |

Local Newton Methods | 43 |

Global Newton Methods | 109 |

GaussNewton Methods | 173 |

Continuation Methods | 232 |

Part II DIFFERENTIAL EQUATIONS | 283 |

6 Stiff ODE Initial Value Problems | 285 |

7 ODE Boundary Value Problems | 315 |

8 PDE Boundary Value Problems | 369 |

405 | |

Software | 416 |

419 | |

### Other editions - View all

Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms Peter Deuflhard No preview available - 2011 |

Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms Peter Deuflhard No preview available - 2010 |

### Common terms and phrases

adaptive trust region affine contravariant affine covariant affine similar algorithm Assume assumption asymptotic Bibliographical Note CGNE collocation computational estimates continuation method convergence analysis convergence mode convergence theorem convex criterion damping factors damping strategy defined denote derive Deuflhard discrete discretization error error oriented F(xk finite dimensional function space Gauss-Newton method GBIT global convergence homotopy inexact Newton methods initial guess inner iteration invariance Jacobian Jacobian matrix Kantorovich least squares problems Lemma level function linear convergence linear system Lipschitz condition Lipschitz constant matrix mesh monotonicity test multigrid methods multiple shooting natural monotonicity Newton corrections Newton path nonlinear nonlinear least squares nonsingular notation numerical obtain ordinary Newton method parameter PDEs perturbation preconditioner proof quadratic convergence quasi-Newton quasi-Newton method realized replace require residual based residual norm Section solution point solved solver stepsize tion trust region strategy upper bound value problems xk+1