# 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 Ability to expand and get in-depth information with scientific researches in the field of mechanical engineering, evaluate information, review and implement.
2 Have comprehensive knowledge about current techniques and methods and their limitations in Mechanical engineering.
3 To complete and apply knowledge by using scientific methods using uncertain, limited or incomplete data; use information from different disciplines.
4 Being aware of the new and developing practices of Mechanical Engineering and being able to examine and learn when needed.
5 Ability to define and formulate problems related to Mechanical Engineering and develop methods for solving and apply innovative methods in solutions.
6 Ability to develop new and/or original ideas and methods; design complex systems or processes and develop innovative/alternative solutions in the designs.
7 Ability to design and apply theoretical, experimental and modeling based researches; analyze and solve complex problems encountered in this process.
8 Work effectively in disciplinary and multi-disciplinary teams, lead leadership in such teams and develop solution approaches in complex situations; work independently and take responsibility.
9 To establish oral and written communication by using a foreign language at least at the level of European Language Portfolio B2 General Level.
10 Ability to convey the process and results of their studies systematically and clearly in written and oral form in national and international environments.
11 To know the social, environmental, health, security, law dimensions, project management and business life applications of engineering applications and to be aware of the constraints of their engineering applications.
12 Ability to observe social, scientific and ethical values in the stages of data collection, interpretation and announcement and in all professional activities.

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