Estimations from secant method

In cases where the first derivative cannot easily be determined by the Newton-Raphson method, a simple method to approximate the first derivative can be used. The secant method can be determined from two prior estimates. 

It is possible to estimate the error using the two iterations, supposing that they come from the secant method. Through equation (1.52), it results in... 

Table 2.4 shows the SAS NLIN specifications and the computer output. You can choose one of the four iterative methods modified Gauss-Newton, Marquardt, gradient or steepest-descent, and multivariate secant or false position method (SAS, 1985). The Gauss-Newton iterative methods regress the residuals onto the partial derivatives of the model with respect to the parameters until the iterations converge. You also have to specify the model and starting values of the parameters to be estimated. It is optional to provide the partial derivatives of the model with respect to each parameter, b. Figure 2.9 shows the reaction rate versus substrate concentration curves predicted from the Michaelis-Menten equation with parameter values obtained by four different... 

The advantages of this method are that it offers rapid convergence without requiring the first derivative. Convergence is between linear and quadratic, i.e., a power ranging from 1 to 2. The value is an approximation of the tangent of the secant. The value depends upon the steepness of the curve. Because the denominator always approaches zero near the root, this method is prone to instability. It also requires two initial estimates to start. 

In the NZSEE 2006, different inelastic methods are included. The one which is conceptually different from those in other standards is described in this article (see section FEMA-440 ). Contrary to the previously mentioned procedures, the maximum seismic displacement at the reference point is assumed at the beginning of the analysis taking into account the estimated capacity of the structure. It is then used to estimate the properties of an equivalent SDOF model. The dynamic properties of the SDOF model are defined based on the secant stiffness corresponding to the assumed maximum... 

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