BFGS Algorithm

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This command is used to construct a Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm object. The BFGS method is one of the most effective matrix-update or quasi Newton methods for iteration on a nonlinear system of equations. The method computes new search directions at each iteration step based on the initial jacobian, and subsequent trial solutions. The unlike regular Newton-Raphson does not require the tangent matrix be reformulated and refactored at every iteration, however unlike ModifiedNewton it does not rely on the tangent matrix from a previous iteration.

algorithm BFGS

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REFERNCES:

1. Denis, J.E "A Brief Survey of Convergence Methods for Quasi_Newton Methods", SIAMS-AMS Proceedings, Vol (9), 185-200, 1976.
2. K.J. Bathe and A.P.Cimento "Some Practical Procedures for the Solution of Nonlinear Finte Element Equations", Computer Methods in Applied Mechanics and Engineering, Vol(22) 59-85, 1980.

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THEORY:

In the BFGS method the following is employed at each time step:

<math> \Gamma_{n+1} = R(U_{n+1}) - R(U_n),\!</math>

<math> \Delta U_{n+1} = U_{n+1} + \Delta U\,\!</math>

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Code Developed by: fmk