Topic outline
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Semester: 03
Methodological Teaching Unit: UEM
Subject: Numerical Methods
Credits: 4
Coefficient: 2
Learning Objectives: This course will allow students to explore the field of numerical methods necessary for solving problems.
Recommended Prerequisites: Basic mathematics.Numerical analysis is fundamentally intertwined with computer science. The advent of high-performance computing has revolutionized the field, enabling the solution of complex mathematical problems that were previously intractable. Numerical methods are at the heart of computational software, from scientific simulations to data analysis tools. Computer scientists design efficient algorithms and data structures to implement these methods, while numerical analysts develop the mathematical foundations. This interdisciplinary collaboration has led to groundbreaking advancements in fields such as artificial intelligence, machine learning, and data science, where numerical techniques are essential for tasks like training neural networks, processing large datasets, and solving optimization problems.
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Introduction.Numerical analysis is a dynamic field that bridges the gap between abstract mathematics and practical applications. It provides the tools and techniques necessary to solve complex mathematical problems that arise in science, engineering, and many other fields. This chapter introduces the fundamental concepts and techniques of numerical analysis, including number representation, error analysis, and algorithm stability. By understanding these core ideas, we can effectively apply numerical methods to solve real-world problems and advance scientific discovery.
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Chapter 1 Basics of Numerical Analysis and Scientific Computing
1.1. Motivations.
1.2. Floating-Point Arithmetic and Rounding Errors.
1.2.1 Representation of Numbers in Machine.
1.2.2 Rounding Errors.
1.3. Stability and Error Analysis of Numerical Methods and Problem Conditioning.
1.3.1 Algorithm Selection.
1.3.2 Refinement and Adaptation.
1.3.3 Validation and Testing -
Here’s an ensemble of choice questions (MCQs) based on chapiter 1
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Introduction. In this chapter, we will explore direct methods for solving linear systems of equations. These methods aim to provide exact solutions (up to the precision of the computer's arithmetic) by performing a finite sequence of operations on the system's matrix and right-hand side. We will begin by discussing the efficient solution of triangular systems, which serve as building blocks for many direct methods. Next, we will delve into the Gaussian elimination method, a fundamental algorithm for solving general linear systems. Finally, we will interpret Gaussian elimination from a matrix factorization perspective, leading to the concept of LU factorization, which is a powerful tool for solving linear systems and other matrix computations.
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Chapter 2 Direct Methods for Solving Linear Systems
2.1 Remarks on Solving Triangular Systems.
2.1.1 Solving Upper Triangular System.
2.1.2 Solving Lower Triangular System.
2.2 Gaussian Elimination Method.
2.3 Matrix Interpretation of Gaussian Elimination: LU Factorization -
Here’s an ensemble of choice questions (MCQs) based on chapiter 2
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Students are encouraged to complete the survey to provide valuable feedback that will help us enhance the course
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Introduction. Iterative methods are powerful numerical techniques used to approximate solutions to large systems of linear equations. Unlike direct methods, which aim to find the exact solution in a finite number of steps, iterative methods
produce a sequence of approximations that converge to the solution. -
Chapter 3 Iterative Methods for Solving Linear Systems
3.1 General Considerations.
3.2 Jacobi and Relaxation Methods.
3.2.1 Jacobi Method.
3.2.2 Relaxation Method.
3.3 Gauss-Seidel and Successive Relaxation Methods.
3.3.1 Gauss-Seidel Method.
3.3.2 Successive Relaxation Method (SOR).
3.4 Remarks on the Implementation of Iterative Methods.
3.5 Convergence of Jacobi and Gauss-Seidel Methods.
3.5.1 Sufficient Conditions for Convergence of Iterative Methods.
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Here’s an ensemble of choice questions (MCQs) based on chapiter 3
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Introduction. This chapter focuses on the fundamental techniques for calculating eigenvalues and eigenvectors of matrices. Eigenvalues and eigenvectors are crucial in various fields of mathematics, physics, and engineering. The chapter begins by discussing methods for localizing eigenvalues, including analytical calculations and numerical techniques. It then delves into the power method, a widely used iterative algorithm for computing the dominant eigenvalue and corresponding eigenvector of a matrix.
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Chapter 4 Computation of Eigenvalues and Eigenvectors.
4.1 Localization of Eigenvalues.
4.1.1 Finding Eigenvalues: Analytical calculation.
4.1.2 Localization Techniques.
4.2 Power Method.
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Here’s an ensemble of choice questions (MCQs) based on chapiter 4
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Chapter 5 Matrix Analysis.
5.1 Vector Spaces.
5.2 Matrices.
5.2.1 Matrix Operations.
5.2.2 Relationships between Linear Mappings and Matrices.
5.2.3 Inverse of a Matrix.
5.2.4 Trace and Determinant of a Matrix.
5.2.5 Eigenvalues and Eigenvectors.
5.2.6 Similar Matrices.
5.2.7 Some Special Matrices.
5.3 Norms and Inner Products.
5.3.1 Definitions.
5.3.2 Inner Products and Vector Norms.
5.3.3 Matrix Norms
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Attached, please find some sample exam questions with solutions from previous years
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