A New Root‐Finding Algorithm for Solving Real‐World Problems and Its Complex Dynamics via Computer Technology

Abstract

Nowadays, the use of computers is becoming very important in various fields of mathematics and engineering sciences. Many complex statistics can be sorted out easily with the help of different computer programs in seconds, especially in computational and applied Mathematics. With the help of different computer tools and languages, a variety of iterative algorithms can be operated in computers for solving different nonlinear problems. The most important factor of an iterative algorithm is its efficiency that relies upon the convergence rate and computational cost per iteration. Taking these facts into account, this article aims to design a new iterative algorithm that is derivative-free and performs better. We construct this algorithm by applying the forward- and finite-difference schemes on Golbabai–Javidi’s method which yields us an efficient and derivative-free algorithm whose computational cost is low as per iteration. We also study the convergence criterion of the designed algorithm and prove its quartic-order convergence. To analyze it numerically, we consider nine different types of numerical test examples and solve them for demonstrating its accuracy, validity, and applicability. The considered problems also involve some real-life applications of civil and chemical engineering. The obtained numerical results of the test examples show that the newly designed algorithm is working better against the other similar algorithms in the literature. For the graphical analysis, we consider some different degrees’ complex polynomials and draw the polynomiographs of the designed quartic-order algorithm and compare it with the other similar existing methods with the help of a computer program. The graphical results reveal the better convergence speed and the other graphical characteristics of the designed algorithm over the other comparable ones.

A Blended Root-Finding Algorithm for Solving Transcendental Equations

“…with interval [1,4] and ǫ = 10 −14 . We show the proposed algorithm is faster converges than other methods.…”

“…Example 3. [1] Let us consider a real-world problem related to volume using van-der Waal's equation. This equations is used to test the behavior of gases.…”

“…More details about the van-der Waal's equation (7) can be found in [1,17]. We have the following non-linear equation after assuming feasible values as in [6].…”

“…We take the interval [0, 2] with ǫ = 10 −23 . In the Table 2, NRF, TM [5], HM [1] and PM indicate the Newton-Raphson method, Traub's method proposed in [5], hybrid method implemented in [1] and proposed method respectively. From Table 1 and Table 2, we can observe that the proposed algorithm converges faster than the other existing methods, and also it is easy to understand as explained in Example 1.…”

Urban Planning and Design Layout Generation Based on Artificial Intelligence

“…e traditional experience-based urban planning and design methods have certain limitations, and the scheme ideas put forward by designers based on practical experience are sometimes not comprehensive and in-depth [13]. Today, in a digital era, computer technology is widely used in various elds [14][15][16][17][18][19], including architecture and urban areas. Concepts are parametric architectural design [20,21], computational urban design [22,23], urban big data analysis [24,25], and smart city emerge [26] one after another.…”