A survey of quasi‐Newton methods with reduced storage

A family of non-linear solution methods is investigated based on limited-memory quasi-Newton updates. Depending upon the number of updates and the updating formula, a number of solution schemes may be constructed such as the conjugate- and secant-Newton methods, the conventional and modified Newton-Raphson methods and a variety of quasi-Newton updates. Under the proposed implementation, the preconditioned truncated Lanczos method is used for the solution of the linearized problem in each non-linear iteration. The complete factorization of the stiffness matrix is avoided and large-scale problems can be solved efficiently both in terms of computing time and storage. The non-linear iterative scheme is properly modified to account for loading variation inside the increment in order to be able to trace post-critical equilibrium paths.

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