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Syllabus ( MATH 547 )


   Basic information
Course title: Numerical Approximation Theory I
Course code: MATH 547
Lecturer: Assoc. Prof. Dr. Hülya ÖZTÜRK
ECTS credits: 7.5
GTU credits: 3 (3+0+0)
Year, Semester: 1/2, Fall and Spring
Level of course: Second Cycle (Master's)
Type of course: Area Elective
Language of instruction: English
Mode of delivery: Face to face
Pre- and co-requisites: None
Professional practice: No
Purpose of the course: Function approximation in mathematics is concerned with either how to fit a function with a rather complicated form by some simpler functions (polynomials, rational functions) or how to find an analytical model for a given data. The aim of this course is to give the students the ability to decide which approximation techniques to use in which situations, to compute these approximations, to conclude the results and to analyze the error of the approximations.
   Learning outcomes Up

Upon successful completion of this course, students will be able to:

  1. Obtain a wide knowledge about the interpolation and approximation techniques, the existence, uniqueness and the characterization of the approximations.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Acquire scientific knowledge and work independently
    3. Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment
    4. Develop mathematical, communicative, problem-solving, brainstorming skills.
    5. Write progress reports clearly on the basis of published documents, thesis, etc
    6. Demonstrating professional and ethical responsibility.

    Method of assessment

    1. Written exam
  2. Use the programing languages (Maple/Matlab) efficiently in order to compute the approximations.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Acquire scientific knowledge and work independently
    4. Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment
    5. Develop mathematical, communicative, problem-solving, brainstorming skills.
    6. Demonstrating professional and ethical responsibility.

    Method of assessment

    1. Laboratory exercise/exam
  3. Use the theoratical results in some application areas.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Review the literature critically pertaining to his/her research projects, and connect the earlier literature to his/her own results
    4. Acquire scientific knowledge and work independently
    5. Work effectively in multi-disciplinary research teams
    6. Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment
    7. Develop mathematical, communicative, problem-solving, brainstorming skills.
    8. Demonstrating professional and ethical responsibility.

    Method of assessment

    1. Written exam
   Contents Up
Week 1: Polynomial Interpolation and Lagrange polynomial
Week 2: Cubic Spline Interpolation
Week 3: Least Squares Approximation (Discrete and Continuous)
Week 4: Orthogonal Polynomials and Least Squares Approximation
Week 5: Chebyshev Polynomials and Economization in Power Series
Week 6: Chebyshev series
Week 7: Best Linear Approximation – Existence, Uniqueness and Characterization
Week 8: Best Linear Approximation – Different Norms
Week 9: Best Linear Approximation – Algorithms
Week 10: Midterm Exam
Week 11: Rational Approximation
Week 12: Rational Interpolation
Week 13: Best Rational Approximation – Existence, Uniqueness and Characterization
Week 14: Best Rational Approximation – Different Norms and Algorithms
Week 15*: Best Rational Approximation – Different Norms and Algorithms
Week 16*: Final exam
Textbooks and materials:
Recommended readings: M. J. D Powell, Approximation Theory and Methods
- E. W. Cheney, Introduction to Approximation Theory
- J. R. Rice, The Aproximation of Functions
- R. L. Burden and J. D. Faires, Numerical Analysis
  * Between 15th and 16th weeks is there a free week for students to prepare for final exam.
Assessment Up
Method of assessment Week number Weight (%)
Mid-terms: 10 40
Other in-term studies: 0
Project: 0
Homework: 0
Quiz: 0
Final exam: 16 60
  Total weight:
(%)
   Workload Up
Activity Duration (Hours per week) Total number of weeks Total hours in term
Courses (Face-to-face teaching): 3 14
Own studies outside class: 4 14
Practice, Recitation: 0 0
Homework: 0 0
Term project: 0 0
Term project presentation: 0 0
Quiz: 0 0
Own study for mid-term exam: 20 2
Mid-term: 2 1
Personal studies for final exam: 20 2
Final exam: 2 1
    Total workload:
    Total ECTS credits:
*
  * ECTS credit is calculated by dividing total workload by 25.
(1 ECTS = 25 work hours)
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