Global convergence of quasi-Newton methods based on adjoint Broyden updates
For solving a system of nonlinear equations, the authors propose an iterative quasi-Newton type method. It is known that the Newton method converges quadratically if the Jacobian of the functions from the left hand side of the system is non-singular. However it requires the reapeted evaluation of the whole Jacobian and its factorization or the inverse for each iterate. This computational effort can be reduced by approximating the Jacobian blow-rank updates, for example by projected updates from Broyden's method [see \textit{C. G. Broyden}, Math. Comput. 19, 577--593 (1965; Zbl 0131.13905)]. In this paper the authors propose the nested use of adjoint Broyden updates in combination with monitoring the angle between the quasi-Newton direction and the stepest descent direction. This yields finite convergence after \(n\) steps for affine problems and local \(q\)-superlinear convergence for general nonlinear problems. The local convergence of the quasi-Newton iterations using adjoint Broyden updates is proved. A nested update algorithm is proposed. The basic structure of this algorithm is a quasi-Newton method with adjoint Broyden residual updates. The computational effort of the method is evaluated in terms of function evaluations and linear algebra operations. The nested update algorithm is compared with Newton's method and with Broyden's method. Numerical results are performed in case of three test functions.
scientific article; zbMATH DE number 5543924
scientific article