Introduction
In mathematics, an algebraic expression is a combination of integers, variables, and algebraic operations such as summation, subtraction, product, division, and exponentiation. Equating two such algebraic expressions gives rise to an algebraic equation. For example, Eqns. 8.1–8.2 are algebraic expressions, whereas Eqns. 8.3–8.4 are algebraic equations.
There are different kinds of algebraic equations, such as, linear equations, non-linear equations (quadratic, cubic, and higher order polynomial equations) and exponential equations. The aim of this chapter is to find solutions for these equations by making use of various numerical techniques and to determine a set of values of the unknown variables contained in these equations. In this chapter, the Gauss–Seidel and the Gaussian elimination methods have been used for solving systems of linear equations, which involve ‘n’ equations and an equal number of unknown variables.
Transcendental equations are encountered ubiquitously in theoretical physics. They consist of transcendental functions that cannot be expressed in the form of finite sequences of algebraic operations. Elementary transcendental functions are logarithmic, trigonometric, exponential, and hyperbolic functions. These equations commonly occur while solving heat transfer problems and studying diffraction patterns. This chapter focuses on Scilab programs for determining the approximate numerical solution of non-linear and transcendental equations using the following iterative numerical techniques,
• Bracketing methods
• Open methods
• Secant method
• Newton–Raphson method
This chapter is arranged such that Section 8.2 recapitulates the built-in functions of Scilab that are often used for solving linear, quadratic, and polynomial equations. The reader is advised to revise the chapter on matrices before attempting to go through this section. In Section 8.3, the Gauss–Seidel method has been discussed along with its pitfalls. In Section 8.4, a brief overview of the Gaussian elimination method for solving system of linear equations has been provided. It is followed by an explicit explanation of the Scilab program based on this numerical technique. Section 8.5 discusses a variant of the Gaussian elimination method. This version re-orients the pivot elements of the Gaussian elimination method. The technique of bisection method has been explained in Section 8.6. The other techniques for solving non-linear and transcendental equations, such as the Regula Falsi method, the Secant method, and the Newton–Raphson method have been explained in Sections 8.7–8.9 respectively.