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Second‐ and higher‐order convergence in linear and nonlinear multiconfigurational Hartree–Fock theory
Author(s) -
Olsen Jeppe,
J⊘rgensen Poul,
Yeager Danny L.
Publication year - 1983
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560240104
Subject(s) - hessian matrix , convergence (economics) , mathematics , sequence (biology) , newton's method , nonlinear system , local convergence , hessian equation , rate of convergence , iterative method , mathematical analysis , mathematical optimization , physics , computer science , chemistry , differential equation , quantum mechanics , channel (broadcasting) , biochemistry , first order partial differential equation , economics , economic growth , computer network
We discuss how the local convergence of Newton–Raphson and fixed Hessian MCSCF iterative models may be rationalized in terms of a total order of convergence in an error vector and a corresponding error term. We demonstrate that a sequence of N Newton–Raphson iterations has a total order of convergence of 2 N and that a sequence of N fixed Hessian iterations has a total order of convergence of N + 1. We derive the error terms of a Newton–Raphson and a fixed Hessian sequence of iterations. We discuss the implementation of the fixed Hessian and the Newton–Raphson approaches both when linear and nonlinear transformations of the variables are carried out. Sample calculations show that insight into the structure of the local convergence of Newton–Raphson and fixed Hessian models can be based on an order of convergence and an error term analysis.