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A direct series solution for the Falkner–Skan equation is obtained by first transforming the problem using the Crocco–Wang transformation. The transformation converts the third-order problem to a second-order two-point boundary value problem. The method first constructs a series involving the unknown skin-friction coefficient $\alpha$. Then, $\alpha$ is determined by using the secant method or Newton’s method. The derivative needed for Newton’s method is also computed using a series derived from the transformed differential equation. The method is validated by solving the Falkner–Skan equation for several cases reported previously in the literature.

In this paper, a class of generalized secant methods was investigated. The convergence of these methods was discussed, and their striking fractal patterns are generated to represent the chaotic behavior of these methods in the complex plane.

We evaluate the initial public offering price of a new DRAM chipmaker in Taiwan in accordance with the compound real call options model of Lin (2002). The worldwide average sales price is the underlying variable, and the average production cost of the new DRAM foundry is the exercise price. The twin security is defined as a portfolio of DRAM manufacturing firms publicly listed in Taiwan stock markets. We estimate the dividend-like yield with two methods, and find that the yield is negative. The negative dividend-like yield results from the negative correlation between the newly constructed DRAM foundry and its twin security, implying the diversification advantage of a new generation of DRAM foundry with a relative low cost of investment opportunity. We solve the critical value for the multivariate normal integral with the secant method, approximating the integral with the lattice method. It has been found that there is only a 4.6% difference between the market IPO price and the estimated one.

Clustering data into natural groupings has important applications in fields such as Bioinformatics. Support Vector Clustering (SVC) does not require prior knowledge of a dataset and it can identify irregularly shaped cluster boundaries. A major SVC challenge is the choice of an important parameter value, the width of a kernel function that determines a nonlinear transformation of the input data. Since evaluating the result of a clustering algorithm is a highly subjective process, a collection of different parameter values must typically be examined. However, no algorithm has been proposed to specify the parameter values. This paper presents a secant-like numerical algorithm that generates an increasing sequence of SVC kernel width values. An estimate of sequence length depends on spatial characteristics of the data but not the number of data points or the data's dimensionality. The algorithm relies on a function that relates the kernel width value to the radius of the minimal sphere enclosing the images of data points in a high-dimensional feature space. Experimental results with 2D and higher-dimensional datasets suggest that the algorithm yields useful data clusterings.