This paper by J.E. Dennis and J.J. More from Cornell University provides an overview of quasi-Newton methods, motivated by their utility in solving nonlinear systems of equations and unconstrained minimization problems. The authors discuss the historical context, including the contributions of Davidon, Broyden, Fletcher, Powell, and others, which led to the development of these methods. They emphasize the theoretical underpinnings and practical advantages of quasi-Newton methods, such as their ability to reduce computational costs compared to Newton's method while maintaining or improving convergence rates. The paper covers key concepts like the quasi-Newton equation, which is central to Broyden's method, and provides a detailed derivation of this method. It also explores local convergence results for quasi-Newton methods, including theorems that guarantee linear and superlinear convergence under certain conditions. The authors discuss various variations of Newton's method for unconstrained minimization, highlighting the trade-offs between local and global convergence properties. Overall, the paper aims to provide a comprehensive understanding of quasi-Newton methods, their theoretical foundations, and their practical applications in solving nonlinear problems.This paper by J.E. Dennis and J.J. More from Cornell University provides an overview of quasi-Newton methods, motivated by their utility in solving nonlinear systems of equations and unconstrained minimization problems. The authors discuss the historical context, including the contributions of Davidon, Broyden, Fletcher, Powell, and others, which led to the development of these methods. They emphasize the theoretical underpinnings and practical advantages of quasi-Newton methods, such as their ability to reduce computational costs compared to Newton's method while maintaining or improving convergence rates. The paper covers key concepts like the quasi-Newton equation, which is central to Broyden's method, and provides a detailed derivation of this method. It also explores local convergence results for quasi-Newton methods, including theorems that guarantee linear and superlinear convergence under certain conditions. The authors discuss various variations of Newton's method for unconstrained minimization, highlighting the trade-offs between local and global convergence properties. Overall, the paper aims to provide a comprehensive understanding of quasi-Newton methods, their theoretical foundations, and their practical applications in solving nonlinear problems.