Why Broyden's Nonsymmetric Method Terminates on linear equations

Abstract. The family of algorithms introduced by Broyden in 1965 for solving systems of nonlinear equations has been used quite effectively on a variety of problems. In 1979, Gay proved the then surprising result that the algorithms terminate in at most 2n steps on linear problems with n variables. His very clever proof gives no insight into properties of the intermediate iterates, however. In this work we show that Broyden's methods are projection methods, forcing the residuals to lie in a nested set of subspaces of decreasing dimension. (Also cross-referenced as UMIACS-TR-93-23

Approximating Higher-Order Derivative Tensors Using Secant Updates

Quasi-Newton methods employ an update rule that gradually improves the Hessian approximation using the already available gradient evaluations. We propose higher-order secant updates which generalize this idea to higher-order derivatives, approximating for example third derivatives (which are tensors) from given Hessian evaluations. Our generalization is based on the observation that quasi-Newton updates are least-change updates satisfying the secant equation, with different methods using different norms to measure the size of the change. We present a full characterization for least-change updates in weighted Frobenius norms (satisfying an analogue of the secant equation) for derivatives of arbitrary order. Moreover, we establish convergence of the approximations to the true derivative under standard assumptions and explore the quality of the generated approximations in numerical experiments

Approximating higher-order derivative tensors using secant updates

Quasi-Newton methods employ an update rule that gradually improves the Hessian approximation using the already available gradient evaluations. We propose higher-order secant updates which generalize this idea to higher-order derivatives, approximating, for example, third derivatives (which are tensors) from given Hessian evaluations. Our generalization is based on the observation that quasi-Newton updates are least-change updates satisfying the secant equation, with different methods using different norms to measure the size of the change. We present a full characterization for least-change updates in weighted Frobenius norms (satisfying an analogue of the secant equation) for derivatives of arbitrary order. Moreover, we establish convergence of the approximations to the true derivative under standard assumptions and explore the quality of the generated approximations in numerical experiments