Secant approximation

This is known as the secant approximation.6 In the extension of this idea to higher dimensions and to the second derivatives, we consider the expansion of the gradient... 

The secant approximation of the film thickness profiles allows one to write... 

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. ... 

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

SECANT LAW APPROXIMATION FOR THE CALCULATION OF CURTIS-GODSON QUANTITIES... 

In principle, Curtis-Godson pressures and temperatures have to be computed for each gas, each layer and each limb view of the scan. In practice, only a sub-set of paths (combination of layer and limb view) requires a customised calculation, because, except for the tangent path, the secant law approximation can be applied and consequently the corresponding equivalent quantities are independent on the limb view angle. Therefore equivalent quantities are computed for the paths corresponding to the lowest geometry and only the tangent layers of the other limb views. This is a very effective optimisation because it reduces the number of paths for which cross-sections have to be computed. 

A schematic example of a stress-strain diagram is given in Figure 7.1. From this curve the following properties can be read off The modulus of elasticity is the slope of the first, approximately straight part of the curve. Here the relation E = o/e is valid, or, with an initially curved line, E0 = (do/de) for e= 0. To define the modulus at a higher value of strain, the secant modulus Es = Oi/ can be taken, or the... 

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

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... 

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. 

The advantages of this method are that it offers rapid convergence without requiring the first derivative. Convergence is between linear and quadratic, i.e., a power ranging from 1 to 2. The value is an approximation of the tangent of the secant. The value depends upon the steepness of the curve. Because the denominator always approaches zero near the root, this method is prone to instability. It also requires two initial estimates to start. 

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. 

Propagation error Since the exact value y t ) is unknown in t , we know only its approximation y and thus we have a second source of error independent of the local error. This error can be estimate by means of the secants theorem Given a function d>(a ) and two values 0( va) and 0(xb) obtained in xa and Xb, there is a point within the interval [xa,Xb] where the function s derivative is equal to the secant obtained with the previous points ... 

Alternatively, so-called secant methods can be used to approximate the Jacobian matrix with far less effort (Westerberg et al., 1979). These provide a superlinear rate of convergence that is, they reduce the errors less rapidly than the Newton-Raphson method, but more rapidly than the method of successive substitutions, which has a linear rate of convergence (i.e., the length of the error vector is reduced from 0.1, 0.01, 10 , 10 , 10 , ...). These methods are also referred to as quasi-Newton methods, with Broyden s method being the most popular. 

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

The same procedure is approximately valid using the secant rather than the derivative. In this case, d can be estimated as Yi(ti-tj-i)... 

If the derivative is not readily available, we can approximate this term by a difference between two points, say and x. The next point is x and this results from a secant that is drawn between x and Xj. The secant formula is given by... 

Although, the secant isothermal piezoviscosity coefficient is used in many lubrication problems, we show (Table III) that it doesn t represent exactly the pressure viscosity variation in our experimental investigation. In Table 111, we have also reported the tangent piezoviscosity coefficient at maximum pressure and , the best mean square approximation of our experimental points at all the pressure steps. 

A general comment can be made on mineral fluid values in each case g, the secant coefficient is very close to ojg, the best mean square approximation of experimental results and the tangent coefficient is smaller than otg,. A more precise observation of these values can distinguish between two types of mineral fluids those which show a large difference between and 0(g, and the others. For the first type, we found the naphtenic base oils, especially R 620 15 at all temperatures investigated, 750 PALE and 1300 PALE at low temperatures. At the opposite, the second class is composed of the paraffinic base oils and the compounded oil H 8303. 

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 ... 

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

The secant method is similar to Newton s method but uses two eonsecutive iterative values of the funetion to approximate the derivative. The basic iterative algorithm is ... 

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