Browsing by Author "MEZOUAGHI Abdelheq"

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    Numerical Analysis I
    (University Hassiba Benbouali of Chlef, 2026) MEZOUAGHI Abdelheq
    Numerical analysis is a fundamental branch of applied mathematics that focuses on the design, analysis, and implementation of algorithms to obtain approximate solutions to mathematical problems when exact solutions are difficult or impossible to determine. In a context where exact computation is often replaced by numerical values, a rigorous study of errors, stability, and the accuracy of the methods used becomes essential. This Numerical Analysis 1 course material, intended for second-year undergraduate (LMD) mathematics students, aims to provide the theoretical foundations and practical tools necessary to understand and master the basic methods of numerical computation. It is structured into five chapters, each addressing a key area of numerical analysis and supported by examples and exercises designed to encourage progressive learning. Chapter 1: Numerical Errors This chapter introduces the concepts of numerical errors, particularly truncation and rounding errors, decimal notation of approximated numbers, and absolute and relative error analysis. These notions are crucial for evaluating the reliability of numerical results. Chapter 2: Solving Algebraic Equations This chapter deals with the solution of algebraic equations, presenting iterative methods such as the bisection method, the fixed-point method, and the Newton-Raphson method, with particular attention paid to convergence and error estimation. Chapter 3: Interpolation and Approximation This chapter is dedicated to interpolation and approximation. It covers the Lagrange and Newton interpolation methods, the study of interpolation errors, and least-squares approximation techniques for fitting data. Chapter 4: Numerical Differentiation This chapter focuses on numerical differentiation, particularly useful when the function in question is only known through discrete data points. Chapter 5: Numerical Integration This chapter addresses numerical integration, discussing classical methods such as the rectangle rule, trapezoidal rule, and Simpson’s rule, with emphasis on their accuracy and applications. This educational resource seeks to combine mathematical rigor with the practical aspects of numerical computation, while fostering in students a critical approach to interpreting numerical results. It provides an essential introduction to numerical analysis and prepares students for more advanced modules in their mathematics curriculum.
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    Numerical Analysis II
    (University Hassiba Benbouali of Chlef, 2026) MEZOUAGHI Abdelheq
    Numerical analysis forms the cornerstone of modern scientific computing, enabling the solution of complex mathematical problems where analytical methods fail. This discipline is crucial for physics simulations, engineering design, financial modeling, and artificial intelligence applications. Our course material equips second-year LMD mathematics students with fundamental computational techniques, beginning with linear system solutions (Gaussian elimination, iterative methods) in Chapter 1. Chapter 2 covers eigenvalue problems, essential for stability analysis and machine learning algorithms. First-order differential equations (Chapter 3) model dynamic systems in biology and economics, solved numerically via Euler and Runge-Kutta methods. The final chapter tackles nonlinear algebraic systems using Newton-Raphson iterations, vital for optimization and control theory. Through algorithmic rigor and MATLAB examples, we emphasize error analysis and computational efficiency. Progressive exercises develop both theoretical understanding and practical implementation skills, preparing students for research and industry challenges in applied mathematics.