Numerical methods secant method

The Wegstein method is a secant method applied to g x) — x — F x). Numerical Solution of Simultaneous Nonlinear Equations... 

Often the melting point and the heat of fusion at the melting point are used as estimates of T and A Hi. It should be noted that the latter equation is nonlinear, since y- on the right-hand side is a function of x . Hence the determination of x calls for an iterative numerical procedure, such as the Newton-Raphson or the secant methods. 

Various Newton, secant J. E. Dennis, and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, 1983, Appendix A, Prentice-Hall, Englewood Cliffs, NJ 07632... 

The secant method is the Newton-Raphson method with a numerical version of the derivative based on the last... 

The Newton-Raphson method requires that you differentiate the function with respect to all the variables. The secant method avoids that mathematical step and uses a numerical difference to calculate the derivative ... 

The value of the functions is kept after each iteration in order to make this calculation. The Wegstein method is essentially a secant method, with some constraints on the parameters as described in Chapter 7. It is also possible to use a numerical derivative. 

The first derivatives of the function should not be calculated numerically as, in this instance, the method is less efficient than the secant method (see Section 1.4.2). 

Solution The system of two equations (Eqs. (4.214) and (4.215)) and two unknowns (that is and P ) for the Baker and Luks formulation can be solved via the secant method. This is just a Newton-Raphson method on numerical derivatives. Computation of the residual in Eq. (4.214) is fairly straight forward, because it only requires expressions for the second derivatives of the Helmholtz free energy/I in terms of V and These derivatives arc provided in Example 4.8. Equation (4.215) is somewhat more complicated its determinant requires derivatives of Eq. (4.214) with respect to V and N. The procedure presented in Problem 4.14 can be used to evaluate the determinant derivatives. After both residuals are computed, the system of two equations and two unknowns are solved by the Newton-Raphson method to convergence. 

Equation (13-14) is solved iteratively for V/F, followed by the calculation of values o(x,anAy, from Eqs. (13-12) and (13-13) and L from the total mole balance. Any one of a number of numerical root-finding procedures such as the Newton-Raphson, secant, false-position, or bisection method can be used to solve Eq. (13-14). Values of K, are constants if they are independent of liquid and vapor compositions. Then the resulting calculations are straightforward. Otherwise, the K, values must be periodically updated for composition effects, perhaps... 

One can solve equation (5.27) numerically using the secant root finding method and select the smallest positive root as an optimal step length, because we have to be conservative and not go too far from the previous iteration. 

Finding the roots of the above equation can be numerically achieved by using the optimized method suggested by T. Dekker, employing a combination of bisection, secant, and inverse quadratic interpolation methods (Forsythe et al., 1976). 

The determination of the switching point can be done, by using Newton s method, the secant rule or by inverse interpolation. We prefer derivative-free methods, because the numerical computation of the time derivative of q involves an extra evaluation of the right hand side function due to... 

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