MTH 209 — Introduction to Numerical Analysis: Lecture Note

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LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Series 1: Numerical Solutions of Algebraic and Transcendental Equations Series Outline Part 1.1 Introduction to Algebraic and Transcendental Equations Definition and classification of equations Importance of numerical solutions Part 1.2 Iterative Methods for Solving Non-linear Equations Bisection method Newton–Raphson method Secant method Part 1.3 Numerical Methods for Transcendental Equations Fixed-point iteration Applications in science and engineering Part 1.4 Systems of Linear Equations Direct methods (Gaussian elimination, LU decomposition) Iterative methods (Jacobi, Gauss–Seidel) Introduction Equations are central to mathematics, science, and engineering, yet many of them cannot be solved exactly using analytical methods. In practice, we often encounter algebraic equations of higher degree and transcendental equations involving functions such as sine, cosine, or exponential, where closed-form solutions are either unavailable or impractical. Numerical analysis provides systematic techniques to approximate these solutions with sufficient accuracy, making it an indispensable tool in applied contexts. The aim of this Series is to introduce you to the numerical methods used for solving algebraic and transcendental equations, and to equip you with the ability to apply these techniques to real-world problems where exact solutions are not feasible. We shall commence this Series by examining the nature and classification of algebraic and transcendental equations, and why numerical solutions are necessary.

LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Next, we will explore iterative methods for solving non-linear equations, including the bisection, Newton–Raphson, and secant methods. Subsequently, we will consider numerical approaches to transcendental equations, focusing on fixed-point iteration and applications in science and engineering. Finally, we will wrap up by studying systems of linear equations, where we will discuss both direct methods such as Gaussian elimination and iterative methods such as Jacobi and Gauss–Seidel. Series Learning Outcomes Upon completing this Series, you should be able to: SLO 1.1 define algebraic and transcendental equations and explain their significance in numerical analysis, SLO 1.2 apply iterative methods such as the bisection method, Newton–Raphson method, and secant method to solve non-linear equations, SLO 1.3 demonstrate numerical techniques for solving transcendental equations, including fixed-point iteration, SLO 1.4 analyse and solve systems of linear equations using direct methods such as Gaussian elimination and iterative methods such as Jacobi and Gauss–Seidel. 1.1 Introduction to Algebraic and Transcendental Equations Algebraic and transcendental equations are central to numerical analysis because they frequently arise in mathematics, science, and engineering. An algebraic equation is one that involves only algebraic operations such as addition, subtraction, multiplication, division, and raising to powers. For example, x3-2x+1=0is algebraic. A transcendental equation involves transcendental functions such as exponential, logarithmic, or trigonometric functions. For instance, sin⁡(x)=xis transcendental because it contains the sine function. Fill in the blank: An equation that contains trigonometric or exponential functions is called a __________________ equation.

LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Examples of algebraic and transcendental equations 1.1.1 Importance of numerical solutions In many cases, equations cannot be solved exactly using analytical methods. Numerical methods provide approximate solutions that are sufficiently accurate for practical purposes. For example, while quadratic equations can be solved exactly, higher-order polynomials or transcendental equations often require iterative numerical approaches. Fill in the blank: Numerical methods provide __________________ solutions when exact analytical solutions are not possible. Why numerical solutions are important Short description: A box highlighting reasons for using numerical methods.

LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS 1.1.2 Classification of equations Equations can be classified based on their form: Linear equations: Equations of the first degree, e.g., 2x+3=0. Non-linear equations: Equations involving higher powers or transcendental functions, e.g., x3-x=0or cos⁡(x)=x. Systems of equations: Sets of equations involving multiple variables, which may be linear or non-linear. Fill in the blank: Equations of the first degree, such as 2x+3=0, are classified as __________________ equations. Classification of equations with examples Short description: A table showing linear, non-linear, and transcendental equations with examples.

LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Pilot Questions 1.1 An equation that involves only algebraic operations such as addition and powers is called: A. Transcendental equation B. Algebraic equation C. Linear system D. Numerical equation (Linked to SLO 1.1) An equation that involves trigonometric or exponential functions is called: A. Algebraic equation B. Transcendental equation C. Linear equation D. Polynomial equation (Linked to SLO 1.1)

LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Numerical methods are used when: A. Exact solutions are always possible B. Approximate solutions are sufficient C. Equations are linear only D. Equations are quadratic only (Linked to SLO 1.1) Equations of the first degree, such as 2x+3=0, are classified as: A. Non-linear equations B. Linear equations C. Transcendental equations D. Systems of equations (Linked to SLO 1.1) Which of the following is a transcendental equation? A. x2-4=0B. sin⁡(x)=xC. 3x+2=0D. x3-x=0(Linked to SLO 1.1) 1.2 Iterative Methods for Solving Non-linear Equations Non-linear equations are those that involve variables raised to powers greater than one, products of variables, or transcendental functions. Unlike linear equations, they often cannot be solved exactly, so we rely on iterative methods. An iterative method is a step-by-step numerical procedure that generates successive approximations to the solution until the desired accuracy is achieved. Fill in the blank: A step-by-step numerical procedure that generates successive approximations to a solution is called an __________________ method. 1.2.1 The bisection method The bisection method is a simple iterative technique used to find the root of a continuous function. It works by repeatedly halving an interval bwhere the function changes sign, ensuring that the root lies within the interval. At each step, the midpoint is calculated, and the interval is updated depending on the sign of the function at the midpoint.

LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Illustration of the bisection method Fill in the blank: The bisection method requires that the function changes __________________ within the chosen interval. 1.2.2 The Newton–Raphson method

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