Outline of Contents 1 Introduction 1.1 Newton-Raphson Method for Scalar Equations 1.2 Newton's Method for General Nonlinear Problems 1.3 A Roadmap of Newton-type Methods 1.4 Adaptive Inner Solvers for Inexact Newton Methods Exercises Part I ALGEBRAIC EQUATIONS 2 Systems of Equations: Local Newton Methods 2.1 Error Oriented Algorithms 2.2 Residual Based Algorithms 2.3 Convex Optimization Exercises 3 Systmes of Equations: Global Newton Methods 3.1 Globalization Concepts 3.2 Residual Based Descent 3.3 Error Oriented Descent 3.4 Convex Functional Descent Exercises 4 Least Squares Problems: Gauss-Newton Methods 4.1 Linear Least Squares Problems 4.2 Residual Based Algorithms 4.3 Error Oriented Algorithms 4.4 Underdetermined Systmes of Equations Exercises 5 Parameter Dependent Systems: Continuation Methods 5.1 Newton Continuation Methods 5.2 Gauss-Newton Continuation Method 5.3 Computation of Simple bifurcations Exercises Part II DIFFERENTIAL EQUATIONS 6 Stiff ODE Initial Value Problems 6.1 Affine Similar Linear Contractivity 6.2 Nonstiff versus Stiff Initial Value Problems 6.3 Uniqueness Theorems for Implicit One-step Methods 6.4 Pseudo-transient Continuation for Steady State Problems Exercises 7. ODE Boundary Value Problems 7.1 Multiple Shooting for Timelike BVPs 7.2 Parameter Identification in ODEs 7.3 Periodic Orbit Computation 7.4 Polynomial Collocation for Spacelike BVPs Exercises 8 PDE Boundary Value Problems 8.1 Asymptotic Mesh Independence 8.2 global Discrete Newton Methods 8.3 Inexact Newton Multilevel FEM for Elliptic PDEs Exercises References Software Index