Universitatea Babeş-Bolyai Cluj-Napoca
Facultatea de Matematică şi Informatică
Ciclul de studii: Licenţă
Domeniul: Informatică
Programul de studii: Informatică - în limba engleză
Limba de predare: Engleză
SYLLABUS

I. General data
Code	Subject	
MMN0003	Numerical Calculus	

Semester	Hours: C+S+L+P	Category	Status	
6	2+0+2+0	speciality	compulsory	

II. Full status faculty members
Name and	Scientific	Didactic	Chair	Type of activity	
surname	title	title		C	S	L	
CATINAS Teodora Maria	Ph.D.	Lect.	 Mathematics	*			

Associated faculty members
Name and	Scientific	Institution	Type of	Type of activity	
surname	title		position	C	S	L	

III. Course objectives
Being a basic course in numerical analysis, the objectives are: knowledge of main notions and procedures from numerical analysis and the ability to work with them. The course is oriented to numerical algorithms and implementation of numerical software.	

IV. Course contents
1. Introductive notions. Linear spaces. Orthogonal polynomials. Finite and divided differences.
2. Approximation of functions. Introductive notions. Polynomial interpolation. Lagrange, Hermite and Birkhoff interpolation. Least square approximation.
3. Approximation of linear functionals. Introductive notions. Numerical differentiation and integration. Newton-Cotes formulas. Repeated quadrature formulas. 
4.  Gauss type quadrature formulas. Romberg’s algorithm. Adaptive quadratures formulas.
6. Numerical methods for solving linear systems: direct methods (Gauss, LU methods) and iterative methods (Jacobi, Gauss-Seidel, SOR).
7. Numerical methods for solving nonlinear equations. One-step methods (Newton’s method) and multi-step methods (secant, bisection, false position methods). Lagrange, Hermite and Birkhoff inverse interpolation.
8. Numerical methods for solving nonlinear systems: successive approximation and Newton’s methods.
9. Numerical methods for solving differential equations: Taylor interpolation, Euler and Runge-Kutta methods.	

V. Bibliography
1.	O. Agratini, I. Chiorean, Gh. Coman, R.T. Trîmbitaş, Analiză Numerică şi Teoria Aproximării, vol. III, Ed. Presa Univ. Clujeană, 2002;
2.	R. L. Burden, J. D. Faires, Numerical Analysis, PWS Publishing Company, 1985.
3.	I. Chiorean, T. Cătinaş, R. Trîmbitaş, Analiză numerică, Ed. Presa Univ. Clujeană, 2010.
4.	Gh. Coman, T. Cătinaş, şi alţii, Interpolation operators, Ed. Casa Cărţii de Ştiinţă, Cluj-Napoca, 2004.
5.	Gh. Coman, I. Chiorean, T. Cătinaş, Numerical Analysis. An Advanced Course, Ed. Presa Univ. Clujeană, 2007.
6.	D.D. Stancu, Gh. Coman, O. Agratini, R. Trimbitas, Analiză Numerică şi Teoria Aproximării, vol. I, Ed. Presa Univ. Clujeană, 2001;
7.	D.D. Stancu, Gh. Coman, P. Blaga, Analiză Numerică şi Teoria Aproximării, vol. II, Ed. Presa Univ. Clujeană, 2002; 
8.	A. Kharab, R. B. Guenther, An introduction to numerical methods. A Matlab approach, Taylor&Francis Group, 2006.
9.	R. Trîmbitaş, Numerical Analysis, Ed. Presa Univ. Clujeană, 2007.	

VI. Thematic of didactic activities per weeks
•	Course 1 : Introductive notions. Linear spaces. Orthogonal polynomials. Finite differences (definitions and properties).
•	Lab 1: Introductory examples and problems in Matlab. 
•	Course 2 : Divided differences (definitions and properties). Lagrange interpolation: interpolation polynomial, interpolation formula, study of the error.
•	Lab 2: Problems for orthogonal polynomials and computation of finite and divided differences.
•	Course 3 : Lagrange interpolation: Aitken’s algorithm and Newton’s formula.
•	Lab 3: Computation of Lagrange polynomial using baricentric formula.
•	Course 4 : Hermite interpolation: interpolation polynomial, interpolation formula, study of the error. Hermite interpolation with double nodes.
•	Lab 4: Applied problems to Lagrange interpolation using Aitken’s algorithm and Newton’s method. 
•	Course 5: Birkhoff interpolation: interpolation polynomial, interpolation formula, study of the error. 
•	Lab 5: Applied problems to Hermite interpolation.
•	Course 6:  Least square approximation. Numerical differentiation and integration (introductive notions). Newton-Cotes quadrature formulas. Repeated quadrature formulas
•	 Lab 6: Problems for least square approximation.
•	Course 7:  Gauss type quadrature formulas. Romberg’s algorithm and adaptive methods for quadratures formulas.
•	Lab 7: Problems with simple and repeated integration formulas and with Romberg’s algorithm.
•	Course 8: Numerical methods for solving linear systems - direct methods (Gauss, LU).
•	Lab 8: Problems applied to Gauss type quadrature formulas and adaptive quadratures.
•	Course 9: Numerical methods for solving linear systems - iterative methods (Jacobi, Gauss-Seidel, SOR).
•	Lab 9: Problems for solving linear systems with direct methods. 
•	Course 10:  Methods for solving nonlinear equations in R:  one-step methods (Newton method) and multi-step methods (secant, bisection and false position methods). 
•	Lab 10: Problems for solving linear systems with iterative methods.
•	Course 11: Methods for solving nonlinear equations in R: Lagrange, Hermite and Birkhoff inverse interpolation. Methods for solving nonlinear systems: successive approximation and Newton methods.
•	Lab 11: Problems with Newton, secant and bisection methods for solving nonlinear equations. Problems with Newton method for solving nonlinear systems.
•	 Course 12: Numerical methods for solving differential equations: Taylor interpolation, Euler and Runge-Kutta methods.
•	 Lab 12: Problems for solving some differential equations using Euler and Runge-Kutta methods. Ending of evaluation for laboratories.	

VII. Didactic methods used
Lectures, interactive presentations, exercises, homework assignments.	

VIII. Assessment
The score is computed as (70%score of final test+30%score of laboratories).	

IX. Additional references
1.	T. Cătinaş, Interpolation of scattered data,  "Casa Carţii de Ştiinţă",  2007.
2.	Gh. Coman, Analiză numerică, Ed. Libris, Cluj-Napoca, 1995.	

Dean,							Chair head,
Prof. Leon Timbulea, Ph.D.                                     Prof. Octavian Agratini, Ph.D.

	Date,							Course responsible,
            27.10.2011						Lect. CATINAS Teodora Maria, Ph.D.