Function Secant method

The Wegstein method is a secant method applied to g(x) = x - Fix). In Microsoft Excel, roots are found by using Goal Seek or Solver. Assign one cell to be x, put the equation for/(x) in another cell, and let Goal Seek or Solver find the value of x that makes the equation cell zero. In MATLAB, the process is similar except that a function (m-file) is defined and the command fzeroCf .xO) provides the solution x, starting from the initial guess xO. 

Therefore the method has excellent convergence properties near the root (with order of convergence p = 2), but may result in meaningless estimates otherwise. In addition, the number of equivalent function evaluations is usually larger than in the secant method, which does not require the derivative but has almost the same convergence rate. Neither the Newton-Raphson, nor the secant method are recommended if the function f has an extremum near the root. You can easily construct pathological cases to understand this rule. 

The partial pressure of C02 dissolved in surface waters is proportional to its concentration in the water and inversely proportional to its solubility. This dependence is established by solving the system of Equations (3.12) and (3.13), which describe the functioning of the ocean carbonate system. For the quantitative solution of this system we can use, for instance, the secant method. As a result, we obtain [C02] and P . Based on data on the temperature dependence of the equilibrium constants for the respective chemical reactions, we find ... 

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. 

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. 

Use a nonlinear equation solver (e.g. the bounded-secant method) to And the smallest (real) Y >0 such that the objective function... 

In this method, we attempt to find the root of a function y = f(x) using the tangent lines to functions. This is similar to the secant method, except it "cuts loose" from the old point... 

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). 

The secant method has a convergence speed raised to the power of 1.618. It is slower than Newton s method, but the first derivative does not need to be evaluated. When the computational effort involved in evaluating the derivative is in the order of the computational time required to calculate the function, the secant method... 

The secant method has the same pros and cons as Newton s method, except for the need to provide the analytical expression of the first derivative of the function. 

The regula falsi algorithm is very similar to the previous one. The difference is in the support points adopted to linearize the function the last two values at each iteration are used in the secant method, whereas the boundaries of the interval of uncertainty are adopted in the regula falsi method. 

This device has the advantage of a better convergence speed, similar to the one of secant method. Many programs adopt this strategy as a basic algorithm (often combined with Bolzano s method to guarantee convergence even with complex and/or nonmonotone functions). 

Secant method in zero search of a function - http //twt.mpei.ac.ru/MAS/Work-sheets/secantmcd... 

In Chapter 4, methods for approximating the derivative of a function using finite differences are presented. The secant method uses the idea of finite differences to approximate the derivative in the Newton method formula. Starting with two initial guesses x° and x which need not bracket the root of interest, the approximation to f (x) can be written as follows ... 

Exercise 1.3 Set up a spreadsheet that implements the secant method and then solve each of the problems from Exercise 1.1. Use the graph of each function to select an initial guess. Recall the iteration formula for the secant method ... 

Put the formula for the function under the headings f (xk-l) and f (xk). In the cell under xk+1, put the secant method iteration formula. In the second row, replace the previous xk-l with xk and then xk with xk+1. Now copy the two formulas down one row. At this point, one iteration of the secant method is displayed. To see more iterations, just copy the second row down for as many iterations as desired. If too many iterations are copied and the function difference (the denominator of the iteration formula) becomes exactly zero, a divide by zero error will appear. 

Exercise 1.5 Use the secant method as described in Exercise 1.3 to find the root(s) of the functions given in Exercise 1.4. Carefully choose the two initial guesses so that the function values have opposite signs. The roots found may or may not correspond to those found using Goal Seek in Exercise 1.3— it depends on the initial guesses. 

Shown below is a MATLAB script (no function heading) file that prompts the user for two initial guesses for f(0). These guesses are used by the secant method to converge the right and boundary conditions, which is 7(1) = 0. 

Quasi-Newton methods start out by using two points xP and jfl spanning the interval of jc, points at which the first derivatives of fix) are of opposite sign. The zero of Equation (5.9) is predicted by Equation (5.10), and the derivative of the function is then evaluated at the new point. The two points retained for the next step are jc and either xP or xP. This choice is made so that the pair of derivatives / ( ), and either/ (jc ) or/ ( ), have opposite signs to maintain the bracket on jc. This variation is called regula falsi or the method of false position. In Figure 5.3, for the (k + l)st search, x and xP would be selected as the end points of the secant line. 

A brief review is made of the methods that are currently being used to determine the density (p) and compressibility (6) of electrolyte solutions as a function of pressure. The high pressure equations of state used to represent these properties are also discussed. The linear secant bulk modulus [K = Ppp/(pP - p )] equation of state... 

Secant. The method uses a linear approximation of the Jacobian. It may be implemented with some enhancements, as half interval option. It is recommended for single variable, discontinuous or flat convergence functions. 

The full Maxwell-Bloch equations are solved by using the iterative predictor-corrector finite-diference time-domain method[76,77,78]. In what follows, we assume that the system is initially in the lowest subband. We consider a hyperbolic secant functional form for the initial... 

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... 

The capacity spectrum method of equivalent linearization assumes that the equivalent damping of the system is proportional to the area enclosed by the capacity curve. The equivalent period, Tgq, is assumed to be the secant period at which the seismic ground motion demands, reduced by the equivalent damping, intersect the capacity curve (FEMA-440). Since the equivalent period and damping are both a function of the displacement, the solution to determine the maximum inelastic displacement (i.e., performance point) is iterative. 

Iterative methods utilize the secant stiffiiess, which is defined as Flu. One of the commonly used iterative techniques is the Newton-Raphson method. The first stage in this method is the same as the first increment in the incremental method. After the increment, as the stiffness as a function of displacement is known, the internal forces can be calculated from ... 

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