Secant method

In each following step Ot is the slope of the line joining [xk,fixi)] to the most recently determined point where/(x,) has the opposite sign from that of/(xi). This method is of first order. If one uses the most recently determined point (regardless of sign), the method is a secant method. 

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

Quasi-Newton or Variable Metric or Secant Methods... 

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. 

In the quasi-Newton method (secant method) the approximate model analogous to Equation (5.7) to be solved is... 

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 ... 

In the inner loop, the rate and composition of fractionator bottoms recycle are determined by successive substitution sometimes accelerated by a secant method. The rate and composition of recycle plus makeup gas routed to the reactor inlets and... 

For the outer loop, the preprocessor adjusts isothermal temperature by secant method to satisfy the target octane. 

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. 

Iteration and convergence method explicit equations Monotone sequences and secant method Newton- Raphson Free ion molali-ties by difference Newton- Raphson conti nued fraction Newton- Raphson Newton-Raphson conti nued fraction conti nued fraction for anions only conti nued fraction conti nued fraction conti nued fraction brute force... 

At higher loading rates the load signal is not of sufficient quality to directly determine dK/dt values. Hence, to better compare the results of various specimen configurations the local loading rate defined as Kc/tf (secant method) was used as the relevant rate parameter. Moreover, the dynamic method also yields directly Kp/tf values. 

In the secant method the approximate model analogous to the right hand side of Eq. (4.6) (equated to zero) is... 

Thus the secant method, imitates Newton s method (in this sense the secant method is also a quasi-Newton method (see Figure L.7)... 

Secant methods start out by using two points and Xq spanning the interval of x points at which the values of f x) are of opposite sign. The zero off x) is predicted by... 

The two points retained for the next step are x and either Xq or x, the choice being made so that the pair of values/(Jc), and either/(jc ) or/(jc ), have opposite signs to maintain the bracket on x. (This variation is called regula falsi or the method of felse position.) In Figure L.7, for the k + l)st stage, x and Xq would be selected as the end points of the secant line. Secant methods may seem crude, but they work well in practice. The details of the computational aspects of a sound algorithm to solve multiple equations by the secant method are too lengthy to outline here (particularly the calculation of a new Jacobian matrix from the former one instead refer to Dennis and Schnabel"). 

Figure L.7. Secant Method for the solution oif x) = 0. x is the solution, x the approximate to x, and x, and x the starting points for iteration k of the secant method.

Wegstein s method, which is used in many flowsheeting codes, accelerates the convergence of the method of successive substitutions on each iteration. In the secant method, the approximate slope is... 

The crack growth rate at a particular point was calculated to be the average crack growth rate from the previous to the succeeding point (i.e., modified secant method) that is... 

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. 

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