Secant iterative method

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. 

Although the evaluation of partial derivatives is not usually an insurmountable obstacle in networks involving one-phase flow in pipes, several investigators (C3, L2) have explored alternative iterative methods which do not require direct evaluation of partial derivatives. These methods are generally based on linearized approximations using secants rather than tangents. ... 

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... 

Use an iterative method to find sig (based on ID bounded secant method)... 

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 ... 

Once the bracket becomes sufficiently small that we feel that Newton s method or the secant method should be able to find the solution, we switch to one of those more efficient procedures. If this fails, we continue with bisection until the initial guess is sufficiently close for the iterative method to succeed. In MATLAB, the routine fzero takes such an approach. For further discussion of iterative methods to solve a single equation /(x) = 0, consult Press et al. (1992) and Quateroni et al. (2000). 

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. 

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... 

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... 

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. 

The Newton method uses an estimate of the gradient at each step to calculate the next iteration, as described in Section 1.9.6. Quasi-Newton methods such as Broyden s method use linearized secants rather than gradients. This approach reduces the number of calculations per iteration, although the number of iterations may be increased. 

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 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. 

Let X be the guess for the tear stream variables at the A th iteration. Let F(X ) be the calculated result for that tear stream s values. Treat each variable with a secant method. Do two or more successive substitution iterations to generate F(X ) = X k Then accelerate ... 

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... 

Outer loop. When the sum S stops changing within the inner loop, we test whether that sum equals unity (conservation of mass). If it does not, we adjust the temperature and compute new values for the liquid-phase fugacity coefficients. This step closes the outer search over temperature. In many cases each K-factor responds to a change in temperature in a sufficiently well-behaved way that T can be adjusted by the simple secant method at the end of the iteration of the outer loop, the next guess k + 1) for T is taken to be... 

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.

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. 

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... 

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 solution method involved two iteration loops. The outer loop was a secant method to solve Eq. (94) that was discretized by Simpson s method to give the value of u. The inner loop was Newton s method to compute u, v, T, and p. 

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. 

Substituting this approximation into the Newton formula (Equatiou 1.9), the following iteration formula results for the secant method ... 

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