On the Local and Superlinear Convergence of Quasi-Newton Methods

This paper presents a local convergence analysis for severalwell-known quasi-Newton methods when used, without line searches,in an iteration of the form to solve for x^* such that Fx^* = 0. The basic idea behind theproofs is that under certain reasonable conditions on x_o, Fand x_o, the errors in the sequence of approximations {H_k} toF'(x^*)^–1 can be shown to be of bounded deterioration inthat these errors, while not ensured to decrease, can increaseonly in a controlled way. Despite the fact that H_k is not shownto approach F'(x^*)^–1, the methods considered, includingthose based on the single-rank Broyden and double-rank Davidon-Fletcher-Powellformulae, generate locally Q-superlinearly convergent sequences{x_k}.     

Error Propagation in Numerical Processes

A method is suggested of analysing the mechanism of error propagationin certain numerical processes, and detailed results are obtainedfor the solution of triangular linear systems, polynomial deflation,LU decomposition and matrix reduction to tridiagonal form usingHouseholder transformations. The analysis is also applicableto the solution of ordinary differential equations.     

A NEW Method of Polynomial Deflation

Previous results on a weighted least-squares approach to polynomialdeflation are exploited to produce a new method. The new methodis shown to preserve zeros exactly. Numerical tests are reportedwhich indicate that this property holds good in practice.     

The Convergence of a Class of Double-rank Minimization Algorithms: 2. The New Algorithm

This paper presents a new minimization algorithm and discussestheoretically some of its properties when applied to quadraticfunctions. Results of comparative testing for a set of non-quadraticfunctions are described and reasons for the observed experimentalbehaviour are suggested.     

An Alternative Derivation of the Simplex Method

This paper seeks to give a rigorous derivation of the SimplexMethod without the assumption that the main variables must necessarilybe non-negative. It is shown that the effects of discardingthis assumption are confined to setting up the initial Tableauand computing the final set of main variables.