Numerical Methods

General

• Course Code: 1641
• Semester: 6th
• Course Type: Scientific Area (SA)
• Course Category: Compulsory Optional (CO-OP)
• Scientific Field: Programming and Algorithms (PA)
• Lectures: 4 hours/week
• ECTS units: 6
• Course webpage: https://aetos.it.teithe.gr/~gouliana/aa_theory.html
• Teching and exams language: Greek, English
• Recommended prerequisite courses: (1101) Mathematics Ι, (1102) Structured Programming
• Coordinator: Goulianas Konstantinos
• Instructors: Tzekis Panagiotis

Educational goals

The purpose of the course is for students to acquire basic knowledge of Numerical Analysis. The main objectives of the course are: a) Introduction to error theory by presenting the definitions of rounding and clipping errors, the errors of converting real decimal numbers into floating point numbers on the PC and transmitting these errors into operations between floating point numbers. b) The approximate calculation of mathematical series and the simulation of the mathematical functions in the mathematical libraries of the programming languages. c) The presentation and study of approximate methods of finding the roots of nonlinear equations and polynomials and the creation of corresponding algorithms for their implementation on PC. d) The presentation and study of direct and approximate methods for solving linear equation systems. e) The study and presentation of methods for finding polynomial interpolations from a table of values ​​of an unknown function. g) The presentation of approximate methods for finding some integrals and the development of corresponding algorithms for implementing illustrative examples of the above methods and their programming on a computer. Upon successful completion of the course the student will be able to:

• Understand how errors affect storage, computation, and operations between real numbers on the PC.
• Apply the Mac Laurin expansions to simulate the mathematical functions that exist in the mathematical libraries of the programming languages and understand the resulting clipping errors.
• Apply the methods of finding root equations and polynomials and distinguish the advantages of each method in terms of speed and approximation of solutions.
• Apply linear system solving methods and distinguish the advantages of each method in terms of speed and computational cost of the operations required to approximate the solutions.
• Apply the interpolation methods and estimate the error transmission in the difference tables.
• Apply Numerical Integration methods and distinguish the advantages of each method in terms of speed and approximation of solutions.

Course Contents

• Error Theory: Errors, Floating Arithmetic, Error Transfer.
• Calculation of Series of Mathematical Functions: Series Calculation, Clipping Error, Correction.
• Numerical Solution of Equations: Isolation of Roots of Nonlinear Equations, Value Calculation, Polynomial Derivatives (Horner Scheme), Methods of Solving Nonlinear Equations (Convergence, Convergence Speed), Partitioning Method, Misfit, New, Sequential
• Solving Linear Equation Systems: Direct Methods (Diagonal Solution, Upper-Lower Triangular System, Gauss Deletion), Repetitive Methods (Gauss-Seidel, Jacobi Method).
• Ascending Differences: Forward, Backward, Central Differences, Error Transfer, Difference Rulers.
• Linear Interpolation: Newton-Gregory Interpolation types, Lagrange Interpolation types, correction to Interpolation types.
• Numerical Integration: Tables Method, Newton-Cotes Method, Simpson Method, Gauss Method

Teaching Methods - Evaluation

Teaching Method

• Person-to-person theoretical teaching (delivery, discussion, problem solving)

Use of ICT means

• Using the moodle platform

Teaching Organization

Students evaluation

Written project
Written examination
Laboratory Exercises

Recommended Bibliography

Recommended Bibliography through "Eudoxus"

1. (Ελληνικά) "Αριθμητική ανάλυση", Ζήτη Πελαγία & Σια Ι.Κ.Ε., 1η έκδ., 2008, ISBN: 978-960-456-084-4, Κωδικός Βιβλίου στον Εύδοξο: 10987
2. "Αριθμητικές Μέθοδοι για Μηχανικούς", ΕΚΔΟΣΕΙΣ Α. ΤΖΙΟΛΑ & ΥΙΟΙ Α.Ε., 7η Έκδοση Βελτιωμένη, 2018, ISBN: 978-960-418-763-8, Κωδικός Βιβλίου στον Εύδοξο: 77106818
3. "Εισαγωγή στην Αριθμητική Ανάλυση", ΕΚΔΟΣΕΙΣ Α. ΤΖΙΟΛΑ & ΥΙΟΙ Α.Ε., 2η Έκδοση, 2015, ISBN: 978-960-418-572-6, Κωδικός Βιβλίου στον Εύδοξο: 50657724

Complementary greek bibliography

1. (Ελληνικά) Σημειώσεις για το Θεωρητικό μέρος του μαθήματος «Αριθμητική Ανάλυση & Προγραμματισμός Επιστημονικών Εφαρμογών – Θεωρία, Παραδείγματα και Άλυτες Ασκήσεις». Γουλιάνας Κωνσταντίνος, Τμήμα Πληροφορικής, ΑΤΕΙ-Θ, 2011.
2. Σημειώσεις για το Εργαστηριακό μέρος του μαθήματος «Εργαστηριακές Ασκήσεις Αριθμητικής Ανάλυσης στη Γλώσσα Προγραμματισμού C». Γουλιάνας Κωνσταντίνος, Τμήμα Πληροφορικής, ΑΤΕΙ-Θ, 2007.
3. Κυτάγιας Δημήτρης, Βρυζίδης Λάζαρος, “Αριθμητική Ανάλυση/Αλγοριθμική Προσέγγιση”: Εκδόσεις Ίων, 1991.
4. Χατζηδήμος Απόστολος, “Εισαγωγή στην Αριθμητική Ανάλυση”: Πανεπιστημιακές Εκδόσεις Ιωαννίνων, 1977.
5. Χατζηδήμος Απόστολος, “Αριθμητική Ανάλυση Ι και ΙΙ”: Πανεπιστημιακές Εκδόσεις Ιωαννίνων, 1979.

Complementary international bibliography

1. Stoer, Josef, Bulirsch, R., Introduction to Numerical Analysis, Springer-Verlag New York, 3, 2002, 978-0-387-21738-3