The Secant Method

This means that if we are very close to the solution, Newton s method converges quadrat-ically. For example, assume that we are sufficiently close to a solution for this quadratic convergence to hold and that et = 10 . Then, the sequence of errors in the next few iterations is approximately 

Once Newton s method is close enough to the real solution for the second-order Taylor series approximation to be accurate, the sequence of estimates converges very rapidly (quadrati-caUy) to die solution. 

Each iteration of Newton s method requires not only an evaluation of the function, but also an evaluation of the first derivative. In some cases, the algebraic function may be of such complexity that it is inconvenient to derive the analytical form of the derivative. One alternative would be to use a finite difference approximation, 

In this method, only one new function evaluation per iteration is necessary. In comparison to Newton s method, there is no need to evaluate analytically the value of the first derivative however, convergence is slower than with Newton s method, 

When an analytical expression for the first derivative is available, Newton s method is preferred due to its faster, quadratic rate of convergence otherwise, the secant method is suggested. In practice, the loss of quadratic convergence is not as bad as one might expect, for, in general, it is found only very near the solution. 

4) 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). 

5) When the first derivative is zero, problems arise as the theoretical basis of the model is no longer valid. The convergence speed is no longer quadratic, for 

The advantage of Newton s method is in the fast convergence of the cases in which it effectively converges. Convergence is quadratic in these cases. 

Newton s method is never used in the BzzMaA library classes dedicated to rootfinding. 

In relation (1.31), by approximating the derivative y[ by the line joining the points Yi, ti and yj i, h-i, the following relation takes place  

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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. 

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)... 

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 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 ... 

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. 

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... 

We find the solution to (35) using the secant method with a trust radius of a/4 at each iteration. The algorithm was terminated once the integral on the right-hand side of (35) was less than 10 in absolute magnitude. The results are presented in Table 1, along with the maximum absolute error as defined by... 

The optical transparency of the cemented material allowed visual detection of the moment of crack onset. In this case the load Pi, the shift of the point of its application i p, and the crack face displacement close to the mid-point of its front were recorded (Fig. 3.5). Determination of the critical load Pc from graphs of P- p and P- c using the secant method gave good agreement with the results obtained by visual detection of the crack onset. 

Figure 4.9 Graphical solution of the mixer control problem (a) specifications for the manipulated variable (b) ASPEN PLUS iteration history using the secant method.

If the condition at x = 0 is not satisfied, a different value of a is chosen and the integration process is repeated until the above relation is finally satisfied within some prespecified error. This method is commonly called the shooting method. Use the secant method of Appendix A to show that the next guess of a should satisfy the following equation... 

This formula is often called the Secant method, and to initiate it one needs two initial guesses. Normally, these two guesses are very close to each other. Figure A.4 shows the secant iteration method graphically. 

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 secant method in its pure iterative form is never used in the classes of the BzzMaih library dedicated to root-finding. 

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. 

Its main disadvantage is that its convergence speed is slower than the secant method. 

Heuristic techniques or fimction approximations (i.e., the secant method) can be exploited when the fimction is monotone. 

If necessary, the prediction provided by the secant method can also be exploited on the condition that said prediction is checked it should be in the order of the current value of dt. 

In this context, the value of d calculated using (1.52) assumes the meaning of an error estimation of the solution also for methods other than the secant method. The test ti+i — h < e, is therefore replaced with di < independent of the method adopted. 

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... 

Barnes, J.C.P. (1965) An algorithm For solving nonlinear equations based on the secant method. Computer Journal, 8, 66. 

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. 

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