# Numerical Linear Algebra (MDES621) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Numerical Linear Algebra MDES621 3 0 0 3 5
Pre-requisite Course(s)
MATH 275 Linear Algebra or equivalent
Course Language English N/A Natural & Applied Sciences Master's Degree Face To Face Lecture. This course is designed to give engineering students in graduate level the expertise necessary to understand and use computational methods for the approximate/numerical solution of linear algebra problems that arise in many different fields of science like electrical networks, solid mechanics, signal analysis and optimisation. The emphasis is on methods for linear algebra problems such as solutions of linear systems, least squares problems and eigenvalue-eigenvector problems, the effect of roundoff on algorithms and the citeria for choosing the best algorithm for the mathematical structure of the problem under consideration. The students who succeeded in this course; After successful completion of the course the student will be able to: 1-choose an efficient method to solve (large) linear systems, eigenvalue problems and least squares problems coming from a certain application field, 2-implement the methods and/or algorithms as computer code and use them to solve applied problems, 3-discuss the numerical methods and/or algorithms with respect to stability, applicability, reliability, conditioning, accuracy, computational complexity and efficiency. Floating point computations, vector and matrix norms, direct methods for the solution of linear systems, least squares problems, eigenvalue problems, singular value decomposition, iterative methods for linear systems.

### Weekly Subjects and Releated Preparation Studies

Week Subjects Preparation
1 Introduction to numerical computations. Vector and matrix norms Read related sections in references
2 Condition numbers and conditioning, Stability, Propogation of roundoff errors Read related sections in references
3 Direct methods for linear systems, Gaussian elimination, Pivoting, Stability. LU and Cholesky decompositions Read related sections in references
4 LU and Cholesky decompositions (cont.) Operation counts, Error analysis, Perturbation theory, Special linear systems Read related sections in references
5 Least Squares. Orthogonal matrices, Normal equations, QR factorization Read related sections in references
6 Gram-Schmidt orthogonalization, Householder triangularization, Least Squares problems Read related sections in references
7 Eigenproblem. Eigenvalues and eigenvectors, Gersgorin’s circle theorem, Iterative methods for eigenvalue problems Read related sections in references
8 Power, Inverse Power and Shifted Power methods, Rayleigh quotients, Similarity transformations, Reduction to Hessenberg and tridiagonal forms Read related sections in references
9 QR algorithm for eigenvalues and eigenvectors, Other eigenvalue algorithms. Singular Value Decomposition Read related sections in references
10 SVD(cont.) and connection with Lesat Squares problem, Computing the SVD using the QR algorithm Read related sections in references
11 Iterative Methods for Linear Systems. Basic iterative methods, Jacobi, and Gauss-Seidel methods Read related sections in references
12 Richardson and SOR methods, Convergence analysis of the iterative methods Read related sections in references
13 Krylov subspace Methods, Preconditioning and preconditioners Read related sections in references
14 General Review -
15 General Review -
16 Final exam -

### Sources

Course Book 1. L.N. Trefethen and D. Bau, III, Numerical Linear Algebra, SIAM, 1997. 2. J.W.Demmel, Applied Numerical Linear Algebra, SIAM, 1997 3. G.H. Golub and C.F. van Loan. Matrix Computations, John Hopkin’s University Press, 3rd edition, 1996. 4. A. Greenbaum, Iterative Methods for Solving Linear Systems, SIAM, 1997. 5. C.D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, 2000. 6. O. Axelsson, Iterative Solution Methods, Cambridge University Press, 1996. 7. D.S. Watkins, Fundamentals on Matrix Computations, John Wiley and Sons, 1991. 8. K.E.Atkinson, An Introduction to Numericall Analysis, John Wiley and Sons, 1999.

### Evaluation System

Attendance/Participation - -
Laboratory - -
Application - -
Field Work - -
Special Course Internship - -
Quizzes/Studio Critics 5 10
Homework Assignments 7 9
Presentation - -
Project - -
Report - -
Seminar - -
Midterms Exams/Midterms Jury 2 46
Final Exam/Final Jury 1 35
Toplam 15 100
 Percentage of Semester Work 65 35 100

### Course Category

Core Courses X

### The Relation Between Course Learning Competencies and Program Qualifications

# Program Qualifications / Competencies Level of Contribution
1 2 3 4 5
1 Accumulated knowledge on mathematics, science and mechatronics engineering; an ability to apply the theoretical and applied knowledge of mathematics, science and mechatronics engineering to model and analyze mechatronics engineering problems.
2 An ability to differentiate, identify, formulate, and solve complex engineering problems; an ability to select and implement proper analysis, modeling and implementation techniques for the identified engineering problems.
3 An ability to design a complex system, product, component or process to meet the requirements under realistic constraints and conditions; an ability to apply contemporary design methodologies; an ability to implement effective engineering creativity techniques in mechatronics engineering. (Realistic constraints and conditions may include economics, environment, sustainability, producibility, ethics, human health, social and political problems.)
4 An ability to develop, select and use modern techniques, skills and tools for application of mechatronics engineering and robot technologies; an ability to use information and communications technologies effectively.
5 An ability to design experiments, perform experiments, collect and analyze data and assess the results for investigated problems on mechatronics engineering and robot technologies.
6 An ability to work effectively on single disciplinary and multi-disciplinary teams; an ability for individual work; ability to communicate and collaborate/cooperate effectively with other disciplines and scientific/engineering domains or working areas, ability to work with other disciplines.
7 An ability to express creative and original concepts and ideas effectively in Turkish and English language, oral and written.
8 An ability to reach information on different subjects required by the wide spectrum of applications of mechatronics engineering, criticize, assess and improve the knowledge-base; consciousness on the necessity of improvement and sustainability as a result of life-long learning; monitoring the developments on science and technology; awareness on entrepreneurship, innovative and sustainable development and ability for continuous renovation.
9 Be conscious on professional and ethical responsibility, competency on improving professional consciousness and contributing to the improvement of profession itself.
10 A knowledge on the applications at business life such as project management, risk management and change management and competency on planning, managing and leadership activities on the development of capabilities of workers who are under his/her responsibility working around a project.
11 Knowledge about the global, societal and individual effects of mechatronics engineering applications on the human health, environment and security and cultural values and problems of the era; consciousness on these issues; awareness of legal results of engineering solutions.
12 Competency on defining, analyzing and surveying databases and other sources, proposing solutions based on research work and scientific results and communicate and publish numerical and conceptual solutions.
13 Consciousness on the environment and social responsibility, competencies on observation, improvement and modify and implementation of projects for the society and social relations and be an individual within the society in such a way that planing, improving or changing the norms with a criticism.
14 A competency on developing strategy, policy and application plans on the mechatronics engineering and evaluating the results in the context of qualitative processes.

Activities Number Duration (Hours) Total Workload
Course Hours (Including Exam Week: 16 x Total Hours) 16 3 48
Laboratory
Application
Special Course Internship
Field Work
Study Hours Out of Class 16 2 32
Presentation/Seminar Prepration
Project
Report
Homework Assignments 7 3 21
Quizzes/Studio Critics 5 1 5
Prepration of Midterm Exams/Midterm Jury 2 8 16
Prepration of Final Exams/Final Jury 1 10 10