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 corequisites: 
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


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

Obtain a wide knowledge about the interpolation and approximation techniques, the existence, uniqueness and the characterization of the approximations.
Contribution to Program Outcomes

Define and manipulate advanced concepts of Mathematics

Acquire scientific knowledge and work independently

Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment

Develop mathematical, communicative, problemsolving, brainstorming skills.

Write progress reports clearly on the basis of published documents, thesis, etc

Demonstrating professional and ethical responsibility.
Method of assessment

Written exam

Use the programing languages (Maple/Matlab) efficiently in order to compute the approximations.
Contribution to Program Outcomes

Define and manipulate advanced concepts of Mathematics

Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems

Acquire scientific knowledge and work independently

Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment

Develop mathematical, communicative, problemsolving, brainstorming skills.

Demonstrating professional and ethical responsibility.
Method of assessment

Laboratory exercise/exam

Use the theoratical results in some application areas.
Contribution to Program Outcomes

Define and manipulate advanced concepts of Mathematics

Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems

Review the literature critically pertaining to his/her research projects, and connect the earlier literature to his/her own results

Acquire scientific knowledge and work independently

Work effectively in multidisciplinary research teams

Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment

Develop mathematical, communicative, problemsolving, brainstorming skills.

Demonstrating professional and ethical responsibility.
Method of assessment

Written exam


Contents


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



Method of assessment 
Week number 
Weight (%) 

Midterms: 
10 
40 
Other interm studies: 

0 
Project: 

0 
Homework: 

0 
Quiz: 

0 
Final exam: 
16 
60 

Total weight: 
(%) 



Workload



Activity 
Duration (Hours per week) 
Total number of weeks 
Total hours in term 

Courses (Facetoface 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 midterm exam: 
20 
2 

Midterm: 
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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