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Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS MAT 5064 ANALYSIS OF NUMERICAL METHODS ELECTIVE 3 0 0 7

Offered By

Graduate School of Natural and Applied Sciences

Level of Course Unit

Second Cycle Programmes (Master's Degree)

Course Coordinator

ASSOCIATE PROFESSOR MELTEM ADIYAMAN

Offered to

Mathematics (English) Mathematics (English)

Course Objective

The course will develop numerical methods to solve algebraic, transcendental equations; to approximate functions, and to calculate derivatives and integrals. The course will also develop an understanding of the elements of error analysis for numerical methods and certain proofs. The course will further develop problem solving skills.

Learning Outcomes of the Course Unit

1   solve an algebraic or transcendental equation using an appropriate numerical method 2   approximate a function using an appropriate numerical method 3   evaluate a derivative at a value using an appropriate numerical method, calculate a definite integral using an appropriate numerical method 4   perform an error analysis for a given numerical method, prove results for numerical root finding methods 5   code a numerical method in a modern computer language

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description 1 Conditioning, Error, Accuracy, Stability, Convergence 2 Iterative solutions of Non linear Equations;Functional iteration for a single equation,Second and higher order iteration methods o The Simple Iteration or Chord Method o Newton's Method o Method of False Position o Aitken's algorithm 3 Functional Iteration for a System of Equations o Some Explicit Iteration Schemes for Systems o Convergence of Newton's Method o A Special Acceleration Procedure for Non-Linear Systems 4 Algebraic Interpolation o Existence and Uniqueness of Interpolating Polynomial o The Lagrange Form of Interpolating Polynomial 5 o The Newton Form of Interpolating Polynomial. Divided Differences o Comparison of the Lagrange and Newton Forms o Conditioning of the Interpolating Polynomial 6 Classical Piecewise Polynomial Interpolation o Definition of Piecewise Polynomial Interpolation o Formula for the Interpolation Error 7 o Approximation of Derivatives for a Grid Function o Estimate of the Unavoidable Error and the Choice of Degree for Piecewise Polynomial Interpolation 8 Examples for midterm, Midterm 9 Trigonometric Interpolation o Interpolation of Periodic Functions o An Important Particular Choice of Interpolation Nodes o Sensitivity of the Interpolating Polynomial to Perturbations of the Function Values o Estimate of Interpolation Error o An Alternative Choice of Interpolation Nodes 10 Interpolation of Functions on an Interval. Relation between Algebraic and Trigonometric Interpolation o Periodization o Trigonometric Interpolation o Chebyshev Polynomials. Relation between Algebraic and Trigonometric Interpolation o Properties of Algebraic Interpolation with Roots of the Chebyshev Polynomial 11 o Numerical Differentiation o Differentiation Using Equidistant Points o Richardson Extrapolation 12 Computation of Definite Integrals. Quadratures o General Construction of Quadrature Formulae o Trapezoidal Rule o Simpson's Formula o Error Analysis of methods 13 o Gaussian Quadratures o Improper Integrals. o Combination of Numerical and Analytical Methods 14 o Multiple Integrals o Repeated Integrals and Quadrature Formulae o The Use of Coordinate Transformations

Recomended or Required Reading

1. Analysis of Numerical Methods, E. Isaacson, H.B. Keller 2. A Theoretical introduction to Numerical Anlaysis, V.S Ryaben kii, S. V. Tysnkov

Planned Learning Activities and Teaching Methods

Lecture notes Presentation Problem solving

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA 1 MTE MIDTERM EXAM 2 ASG ASSIGNMENT 3 FIN FINAL EXAM 4 FCG FINAL COURSE GRADE MTE * 0.20 + ASG * 0.40 + FIN * 0.40 5 RST RESIT 6 FCGR FINAL COURSE GRADE (RESIT) MTE * 0.20 + ASG * 0.40 + RST * 0.40

Further Notes About Assessment Methods

None

Assessment Criteria

Homework, Midterm, Final

Language of Instruction

English

Course Policies and Rules

Attending at least 70 percent of lectures is mandatory.

Contact Details for the Lecturer(s)

meltem.evrenosoglu@deu.edu.tr

Office Hours

To be announced.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours) Lectures 14 3 39 Preparations before/after weekly lectures 14 3 39 Preparation for final exam 1 27 25 Preparation for midterm exam 1 17 15 Preparing assignments 10 4 50 Final 1 3 3 Midterm 1 3 3 TOTAL WORKLOAD (hours) 174

Contribution of Learning Outcomes to Programme Outcomes

PO/LO PO.1 PO.2 PO.3 PO.4 PO.5 PO.6 PO.7 PO.8 PO.9 PO.10 PO.11 LO.1 5 5 5 5 5 5 5 4 3 5 4 LO.2 4 5 5 5 5 4 5 3 5 4 LO.3 4 5 5 5 5 5 4 4 3 5 4 LO.4 3 4 5 5 4 5 5 4 2 4 4 LO.5 3 4 5 5 4 5 5 4 2 4 4