Numerical Methods With Engineering Applications and Their Visual Analysis via Polynomiography

Abstract

Polynomiography is a fusion of Mathematics and Art, which as a software results in a new form of abstract art. Rendered images are through algorithmic visualization of solving a polynomial equation via iteration schemes. Images are beautiful and diverse, yet unique. In short, polynomiography allows us to draw unique and complex-patterned images of polynomials which be re-colored in different ways through different iteration schemes. In the modern age, polynomiography covers a variety of applications in different fields of art and science. The aim of this paper is to present polynomiography using newly constructed root-finding algorithms for the solution of non-linear equations. The constructed algorithms are two-step predictor corrector methods. For reducing computational cost and making the algorithm more effective, we approximate the second derivative via interpolation technique. These methods have been derived by employing Househölder's method, interpolation technique and Taylor's series expansion. The convergence criterion of the newly developed algorithms has been discussed and proved their sixth-order convergence which is higher than many existing algorithms. To analyze the accuracy, validity and applicability of the proposed methods, several arbitrary and engineering problems have been tested and the obtained numerical results certify the better efficiency of the suggested methods against the other well-known iteration schemes given in the literature. Finally, we present polynomiography through the constructed iteration schemes and give a detailed comparison with the other iteration schemes which reflects the convergence properties and graphical aspects of the constructed algorithms.

Efficient Families of Multi-Point Iterative Methods and Their Self-Acceleration with Memory for Solving Nonlinear Equations

“…which confirms the optimal eight order for the Modified Method 8a (MM a 8 ) (21). Similarly, we can prove the optimal eighth order convegence for the modified method MM b 8 (22).…”

“…Here, we introduce new derivative-free with-memory methods which are extensions of the newly suggested modified eighth order derivative-free families of without-memory methods MM a 8 (21) and MM b 8 (22). It is evident from error Equation ( 23) that the convergence order of the methods MM a 8 (21), and MM b 8 (22) can be increased from 8 to 16 if we take γ = − 1…”

“…Now, applying the approximations of the four accelerating parameters γ n , β n , λ n , and θ n from (43) in the modified methods MM a 8 (21) and MM b 8 (22), we obtain the following new derivative-free with-memory methods.…”

“…Example 3. Let us consider the conversion of the fraction of Nitrogen-Hydrogen feed into Ammonia, called fractional conversion, at a pressure of 250 atm and temperature of 500 • C (see [21] for details). When reduced to the polynomial form, the problem has the following expression:…”

Graphical and Numerical Study of a Newly Developed Root-Finding Algorithm and Its Engineering Applications

“…Multiple second derivative free root-finding methods for nonlinear equations were developed by Naseem et al [8]- [10] and then used for various problems in chemical and civil engineering.…”