NUMERICAL METHODS

AIM

The aim of this course is to equip students with a solid foundation in numerical methods used to solve engineering and computational problems that cannot be addressed analytically. The course develops an understanding of error sources and propagation in numerical computations, and introduces systematic techniques for solving nonlinear equations, performing optimization, fitting and interpolating data, and approximating derivatives and integrals. Emphasis is placed on algorithmic formulation, convergence and stability analysis, and the practical implementation of numerical methods relevant to computer engineering applications. By the end of the course, students will be able to select, analyze, and apply appropriate numerical techniques to real-world engineering problems with an awareness of accuracy, efficiency, and computational limitations.

CONTENT

This course contains; Introduction & Error Analysis: Role of numerical methods in computer engineering. ,Introduction & Error Analysis: Round-off vs. truncation errors. Taylor series approximations.,Roots of Equations: Bracketing methods: Bisection, False Positionç,Roots of Equations: Open methods: Newton-Raphson, Secant. Convergence analysis.,Optimization: One-Dimensional Unconstrained Optimization,Optimization: One-Dimensional Unconstrained Optimization,Optimization: Multidimensional Unconstrained Optimization ,Midterm,Optimization: Constrained Optimization ,Curve Fitting & Interpolation: Least-squares regression,Curve Fitting & Interpolation: Polynomial/Spline interpolation. ,Numerical Integration: Finite differences Trapezoidal/Simpson’s rules,Numerical Integration: Adaptive quadrature,Numerical Differentiation: High – accuracy differentiation formulas ,Numerical Differentiation: Richardson extrapolation .

1. 1

   1- Explain the role and importance of numerical methods in computer engineering and analyze the sources and effects of round-off and truncation errors in numerical computations.

   Taught by: Problem Solving Method, Question - Answer Technique, Simulation Technique, Lecture Method · Assessed by: Traditional Written Exam, Project Task

2. 2

   2- Apply Taylor series approximations to develop and analyze numerical algorithms with respect to accuracy and stability.

   Taught by: Problem Solving Method, Question - Answer Technique, Simulation Technique, Lecture Method · Assessed by: Traditional Written Exam, Project Task

3. 3

   3- Implement and compare bracketing and open methods for solving nonlinear equations and assess their convergence characteristics.

   Taught by: Problem Solving Method, Question - Answer Technique, Simulation Technique, Lecture Method · Assessed by: Traditional Written Exam, Project Task

4. 4

   4- Solve one-dimensional and multidimensional optimization problems using appropriate numerical techniques, including constrained and unconstrained methods.

   Taught by: Problem Solving Method, Question - Answer Technique, Simulation Technique, Lecture Method · Assessed by: Traditional Written Exam, Project Task

5. 5

   5- Perform curve fitting and interpolation using least-squares regression and polynomial or spline-based approaches and evaluate their suitability for given data sets.

   Taught by: Problem Solving Method, Question - Answer Technique, Simulation Technique, Lecture Method · Assessed by: Traditional Written Exam, Project Task

6. 6

   6- Approximate definite integrals using numerical integration techniques such as finite differences, Trapezoidal and Simpson’s rules, and adaptive quadrature methods.

   Taught by: Problem Solving Method, Question - Answer Technique, Simulation Technique, Lecture Method · Assessed by: Traditional Written Exam, Project Task

7. 7

   7- Compute numerical derivatives using high-accuracy differentiation formulas and improve results through Richardson extrapolation.

   Taught by: Problem Solving Method, Question - Answer Technique, Simulation Technique, Lecture Method · Assessed by: Traditional Written Exam, Project Task

8. 8

   8- Select and apply suitable numerical methods to practical engineering problems while considering accuracy, efficiency, and computational cost.

   Taught by: Problem Solving Method, Question - Answer Technique, Simulation Technique, Lecture Method · Assessed by: Traditional Written Exam, Project Task

WEEKLY PLAN

1. WEEK 1

   Introduction & Error Analysis: Role of numerical methods in computer engineering.

2. WEEK 2

   Introduction & Error Analysis: Round-off vs. truncation errors. Taylor series approximations.

3. WEEK 3

   Roots of Equations: Bracketing methods: Bisection, False Positionç

4. WEEK 4

   Roots of Equations: Open methods: Newton-Raphson, Secant. Convergence analysis.

5. WEEK 5

   Optimization: One-Dimensional Unconstrained Optimization

6. WEEK 6

   Optimization: One-Dimensional Unconstrained Optimization

7. WEEK 7

   Optimization: Multidimensional Unconstrained Optimization

8. WEEK 8

   Midterm

9. WEEK 9

   Optimization: Constrained Optimization

10. WEEK 10

    Curve Fitting & Interpolation: Least-squares regression

11. WEEK 11

    Curve Fitting & Interpolation: Polynomial/Spline interpolation.

12. WEEK 12

    Numerical Integration: Finite differences Trapezoidal/Simpson’s rules

13. WEEK 13

    Numerical Integration: Adaptive quadrature

14. WEEK 14

    Numerical Differentiation: High – accuracy differentiation formulas

15. WEEK 15

    Numerical Differentiation: Richardson extrapolation

ASSESSMENT

• Rate of Midterm Exam to Success30%
• Rate of Final Exam to Success70%

WORKLOAD

READING

• 1. Numerical Methods for Engineers (8th Ed.) – Steven C. Chapra & Raymond P. Canale 2. Numerical Methods in Engineering with Python 3, Jaan Kiusalaas

TEACHING STAFF

• Prof.Dr. Mehmet Kemal ÖZDEMİRCOORDINATOR
• Assist.Prof. Tuğba ASLAN KHALİFA
• Prof.Dr. Selim AKYOKUŞ