Secant function

The functions versed sine, coversed sine, and exterior secant are defined as vers 0=1- cos 0 covers 0=1- sin 0 exsec 0 = sec 0 - 1... 

If a vector it is a function of a single scalar quantity s, the curve traced as a function of s by its terminus, with respect to a fixed origin, can be represented as shown in Fig. 8. Within the interval As the vector AR = R2 - Ri is in the direction of the secant to the curve, which approaches the tangent in the limit as As - 0. Tins argument corresponds to that presented in Section 2.3 and illustrated in Fig. 4 of that section. In terms of unit vectors in a Cartesian coordinate system... 

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. 

For the OS calculation, (6.58), the same optimal solution is reached by choosing a Gaussian-like hyperbolic secant function for w(Au) ... 

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. 

List three reasons why a quasi-Newton (secant) search for the minimum of a function of one variable will fail to find a local minimum. 

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... 

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. 

Three further trigonometric functions,cosecant, secant and cotangent, are provided by the reciprocals of the basic functions ... 

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. 

For shear thinning fluids, it is usual to define an apparent viscosity rja, or r)a(y where the symbol in parentheses means that the viscosity is a function of the strain rate (Figure 13-61). The apparent viscosity at a particular strain rate is defined not as the tangent to die slope of the curve, as you might first guess, but as the slope of the secant drawn from the origin to that point (We don t know why—its probably something to do with the... 

Uniaxial tensile tests were carried out to determine the stress-strain curves and document the damage growth on a computer-controlled Instron model 8516 servo hydraulic testing machine operating at a strain rate of 5% min . The macroscopic tensile yield stress was considered equal to the maximum stress on the loading curve. The Young s modulus was determined as the plateau value of a plot of the secant modulus as a function of the strain. 

These correlations can be transferred to continuous SMB or TMB chromatography where disperse as well as shock fronts are also present. The dimensionless flow rate ratios m,- can then be described as function of either the initial slope or the secant of the isotherm, depending on the situation in every zone. 

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. 

Neglecting relaxation, the envelope of a strongly radiation-damped free induction decay (FID) is given by a hyperbolic secant function, ... 

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

The independent particle function (curve a) is secant to both curves b and c and it has a constant slope of + 2 r,(Af> v) + which is expression (14) with... 

The ordinary trigonometric functions include the sine, the cosine, the tangent, the cotangent, the secant, and the cosecant. These are sometimes called the circular trigonometric Junctions to distinguish them from the hyperbolic trigonometric functions discussed briefly in the next section of this chapter. 

The other hyperbolic trigonometric functions are the hyperbolic tangent, denoted by tanh(x) the hyperbolic cotangent, denoted by coth(x) the hyperbolic secant, denoted by sech(x) and the hyperbolic cosecant, denoted by csch(x). These functions are given by the equations... 

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. 

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... 

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