Abstract
This paper deals with a new numerical iterative method for finding the approximate solutions associated with both scalar and vector nonlinear equations. The iterative method proposed here is an extended version of the numerical procedure originally developed in previous works. The present study proposes to show that this new root-finding algorithm combined with a stationary-type iterative method (e.g., Gauss-Seidel or Jacobi) is able to provide a longer accurate solution than classical Newton-Raphson method. A numerical analysis of the developed iterative method is addressed and discussed on some specific equations and systems.
Highlights
Solving both nonlinear equations and systems is a situation very often encountered in various fields of formal or physical sciences
For Example 1 in the case of guest start point x10 = 10−2, we can see that approximate solutions x1k provided by Adaptative Geometric-based Algorithm” (AGA) with conditions [BC4], [BC5], and [BC6] in the first iterations are accurately better than Newton-Raphson Algorithm (NRA)/Standard Newton’s Algorithm (SNA) and Third-order Modified Newton Method (TMNM)
For Example 3 with guest start point couple (x10, x10) = (20, 20), we can observe that approximate solutions (x1k, x2k) given by AGA using Gauss-Seidel (GS) or Jacobi (J) procedure with: (i) condition [BC1] are more accurate numerically than NRA, TMNM, and SNA; (ii) conditions [BC2] and [BC3] are accurately better than NRA and SNA
Summary
Solving both nonlinear equations and systems is a situation very often encountered in various fields of formal or physical sciences. For Example 3 with guest start point couple (x10, x10) = (20, 20), we can observe that approximate solutions (x1k, x2k) given by AGA using Gauss-Seidel (GS) or Jacobi (J) procedure with: (i) condition [BC1] are more accurate numerically than NRA, TMNM, and SNA; (ii) conditions [BC2] and [BC3] are accurately better than NRA (only in the first iterations) and SNA.
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