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Theory Manual Version 3.6
 Section 3.8: Newton-Raphson Method Up Section 3.8: Newton-Raphson Method Subsection 3.8.2: BFGS Method 

3.8.1 Full Newton Method

The Newton-Raphson equation (3.1.1-1) can be written in terms of the discretized equilibrium equations that were derived in the previous section as follows: Since the virtual velocities are arbitrary, a discretized Newton-Raphson scheme can be formulated as follows: This is the basis of the Newton-Raphson method. For each iteration , both the stiffness matrix and the residual vector are re-evaluated and a displacement increment u is calculated by pre-multiplying both sides of the above equation by . This procedure is repeated until some convergence criteria are satisfied.
The formation of the stiffness matrix and, especially, calculation of its inverse, are computationally expensive. Quasi-Newton methods do not require the reevaluation of the stiffness matrix for every iteration. Instead, a quick update is calculated. One particular method that has been quite successful in the field of computational solid mechanics is the BFGS method, which is described in the next section.
 Section 3.8: Newton-Raphson Method Up Section 3.8: Newton-Raphson Method Subsection 3.8.2: BFGS Method