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A quadratic equation is an algebraic expression where a variable is raised to the second power and combined with its first power and a constant; all equated to zero. These equations are frequently used to model relationships involving area, motion, and optimization. The general representation of a quadratic equation iswhere a, b, and c are real values, and a is nonzero to ensure the presence of the squared term.One method for solving a quadratic equation involves rewriting it as a product of...
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A quadratic equation in the form ax2+bx+c=0 can have solutions that vary in nature depending on the value of the discriminant, b2−4ac. In this expression, a is the coefficient of the quadratic term x2, b is the coefficient of the linear term x, and c is the constant term. When the discriminant is negative, the equation has no real number solutions. However, by introducing complex numbers through the imaginary unit i, defined by i=-1, these equations can still be solved.The square root of...
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Linear equations form the foundation of many algebraic and real-world applications, characterized by their simplicity and utility. A linear equation is an algebraic statement in which each term is either a constant or a product of a constant and a single variable. These equations represent straight lines when plotted on a Cartesian coordinate plane, reflecting a constant rate of change between two quantities.A typical linear equation in one variable has the form: ax + b = c, where a, b, and c...
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The integrating factor method provides a systematic way to solve first-order linear differential equations, especially those that cannot be handled by separation of variables. This method is particularly useful in modeling time-dependent physical systems influenced by both constant inputs and resistive forces. A common example is the motion of a car subjected to a constant engine force while experiencing air resistance proportional to its velocity.In such scenarios, Newton’s second law...
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The application of the linear momentum equation can be used to analyze the forces needed to hold a 180-degree pipe bend in place with flowing water. In this case, water flows through the bend with a constant cross-sectional area of 0.01 square meters and a flow velocity of 15 meters per second. The pressure at the entrance is 0.2 Megapascals and the pressure at the exit is 0.16 Megapascals.
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Solving a system of linear equations is a fundamental concept in algebra. A system of equations consists of two or more linear equations involving the same set of variables. One of the most efficient algebraic methods for solving such systems is the substitution method. This technique involves expressing one variable in terms of the other from one equation and substituting it into the second equation. This method is particularly useful when one of the equations is easily rearranged.Consider the...
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Extracting Metrics for Three-dimensional Root Systems: Volume and Surface Analysis from In-soil X-ray Computed Tomography Data
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Quadratically convergent algorithm for computing real root of non-linear transcendental equations.

Srinivasarao Thota1, Vivek Kumar Srivastav2

  • 1Department of Applied Mathematics, School of Applied Natural Sciences, Adama Science and Technology University, Post Box No. 1888, Adama, Ethiopia. srinithota@ymail.com.

BMC Research Notes
|December 22, 2018
PubMed
Summary

A new algorithm for solving non-linear transcendental equations combines the guaranteed convergence of Regula-Falsi with the speed of Newton-Raphson. This novel method offers improved convergence and reliability for finding roots.

Keywords:
Newton–Raphson methodQuadratic convergenceRegula-Falsi methodRoot of transcendental equations

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Area of Science:

  • Numerical Analysis
  • Computational Mathematics

Background:

  • Regula-Falsi method offers guaranteed convergence but is slow.
  • Newton-Raphson method converges faster but lacks guaranteed results and fails if the derivative is zero.
  • Solving non-linear transcendental equations is crucial in many scientific and engineering fields.

Purpose of the Study:

  • To develop a new algorithm for finding roots of non-linear transcendental equations.
  • To combine the strengths of Regula-Falsi and Newton-Raphson methods.
  • To achieve faster convergence with guaranteed results, addressing Newton-Raphson's limitations.

Main Methods:

  • A novel algorithm is proposed, integrating the guaranteed convergence of Regula-Falsi with the speed of Newton-Raphson.
  • The algorithm specifically addresses and implements a solution for the failure condition of the Newton-Raphson method when the first derivative is zero or near-zero.
  • Error analysis and comparisons were performed using Bisection, Regula-Falsi, and Newton-Raphson methods.

Main Results:

  • The proposed method demonstrates improved convergence compared to Regula-Falsi.
  • The new algorithm provides guaranteed results, overcoming a key limitation of the Newton-Raphson method.
  • Empirical results show the proposed quadratically convergent method outperforms Bisection, Regula-Falsi, and Newton-Raphson in terms of convergence speed and reliability for real-life examples.

Conclusions:

  • The developed algorithm successfully merges the advantages of Regula-Falsi and Newton-Raphson methods.
  • The new method offers a robust and efficient approach for solving non-linear transcendental equations.
  • This enhanced algorithm provides a reliable and faster alternative for root-finding problems in computational mathematics.