syllabus
1. Information regarding the programme
1.1 Higher education institution 	Babeş Bolyai University	
1.2 Faculty	Faculty of Mathematics and Computer Science	
1.3 Department	Department of Computer Science	
1.4 Field of study	Computer Science	
1.5 Study cycle	Bachelor	
1.6 Study programme / Qualification 	Computer Science	

2. Information regarding the discipline 
2.1 Name of the discipline	Numerical Calculus	
2.2 Course coordinator 	Lecturer Ph.D. Teodora Catinas	
2.3 Seminar coordinator	Lecturer Ph.D. Teodora Catinas	
2.4. Year of study	3	2.5 Semester	6	2.6. Type of evaluation	C	2.7 Type of discipline	Compulsory	

3. Total estimated time (hours/semester of didactic activities) 
3.1 Hours per week 	4	Of which: 3.2 course	2	3.3 seminar/laboratory	2 lab	
3.4 Total hours in the curriculum 	48	Of which: 3.5 course	24	3.6 seminar/laboratory	24	
Time allotment:	hours	
Learning using manual, course support, bibliography, course notes	20	
Additional documentation (in libraries, on electronic platforms, field documentation) 	10	
Preparation for seminars/labs, homework, papers, portfolios and essays	20	
Tutorship	7	
Evaluations	20	
Other activities: ..................	-	
3.7 Total individual study hours 	77	
3.8 Total hours per semester	125	
3.9 Number of ECTS credits	5	

4. Prerequisites (if necessary)
4.1. curriculum	•		
4.2. competencies	•	Knowledge of main notions and procedures of numerical analysis and the ability to work with them.  Programming skills in MATLAB for implementing numerical algorithms.	

5. Conditions (if necessary)
5.1. for the course	•		
5.2.  for the seminar /lab activities	•	Laboratory with computers.	

6. Specific competencies acquired 
Professional competencies	•	Knowledge, understanding and use of basic concepts of Numerical Analysis 
•	Ability to work with main procedures of Numerical Analysis in order to solve practical problems.
•	Good programming skills in Matlab.	
Transversal competencies	•	Modelling some problems from different fields of science as physics, chemistry, biology, etc.
•	Ability to apply numerical algorithms to solve different real life problems.
•	Improved programming abilities in Matlab. 

7. Objectives of the discipline (outcome of the acquired competencies)
7.1 General objective of the discipline
	•	Be able to understand  and use basic concepts of Numerical Analysis
•	Be able to implement  numerical algorithms  in order to solve practical problems.	
7.2 Specific objective of the discipline
	•	Acquire theoretical and practical knowledge about the basic numerical algorithms regarding approximation of functions, numerical integration of functions,  numerical  solving of  linear/nonlinear systems of equations and differential equations.
•	Ability to apply numerical algorithms to solve practical and real life problems.

8. Content
8.1 Course	Teaching methods	Remarks	
1.	Introductive notions. Linear spaces. Orthogonal polynomials. Finite and divided differences (definitions and properties).	Exposure: description, explanation, examples.		
2.	Lagrange interpolation: interpolation polynomial, interpolation formula, study of the error.	Exposure: description, explanation, examples, proofs.		
3.	Lagrange interpolation: Aitken’s algorithm and Newton’s formula.	Exposure: description, explanation, examples.		
4.	Hermite interpolation: interpolation polynomial, interpolation formula, study of the error. Hermite interpolation with double nodes. 	Exposure: description, explanation, examples, proofs.		
5.	Birkhoff interpolation: interpolation polynomial, interpolation formula, study of the error. Least square approximation.	Exposure: description, explanation, examples, proofs.		
6.	Numerical differentiation and integration (introductive notions). Newton-Cotes quadrature formulas. Repeated quadrature formulas.	Exposure: description, explanation, examples, proofs, dialogue.		
7.	General quadrature formulas. Romberg’s algorithm. Adaptive quadratures formulas. Gauss type quadrature formulas (definition, rectangle formula, Romberg’s algorithm).	Exposure: description, explanation, examples.		
8.	Numerical methods for solving linear systems - direct methods (Gauss, Gauss-Jordan, LU-methods). Perturbations of a linear system.	Exposure: description, explanation, examples.		
9.	Numerical methods for solving linear systems - iterative methods (Jacobi, Gauss-Seidel, SOR).	Exposure: description, explanation, examples.		
10.	Methods for solving nonlinear equations in R:  one-step methods (Newton (tangent) method) and multi-step methods (secant, bisection and false position methods). 	Exposure: description, explanation, examples.		
11.	Methods for solving nonlinear equations in R: Lagrange, Hermite and Birkhoff inverse interpolation. Methods for solving nonlinear systems: successive approximation and Newton methods.	Exposure: description, explanation, examples, proofs.		
12.	Numerical methods for solving differential equations: Taylor interpolation, Euler and Runge-Kutta methods.	Exposure: description, explanation, examples.		
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.	A. Kharab, R. B. Guenther, An introduction to numerical methods. A Matlab approach, Taylor&Francis Group, 2006.
7.	R. Plato, Concise Numerical Mathematics,  Amer. Math. Soc., 2003. 
8.	D.D. Stancu, Gh. Coman, O. Agratini, R. Trimbitas, Analiză Numerică şi Teoria Aproximării, vol. I, Ed. Presa Univ. Clujeană, 2001;
9.	D.D. Stancu, Gh. Coman, P. Blaga, Analiză Numerică şi Teoria Aproximării, vol. II, Ed. Presa Univ. Clujeană, 2002;
10.	R. Trîmbitaş, Numerical Analysis, Ed. Presa Univ. Clujeană, 2007.	
8.2 Laboratory	Teaching methods	Remarks	
1.	Introductory examples and problems in Matlab. 
	Explanation, dialogue.		
2.	Problems for orthogonal polynomials and computation of finite and divided differences.	Explanation, dialogue, examples.		
3.	Lagrange interpolation.
Computation of Lagrange polynomial using baricentric formula.	Explanation, dialogue, practical examples. 		
4.	Applied problems to Lagrange interpolation using Aitken’s algorithm and Newton’s method.	Explanation, dialogue, practical examples. Evaluation.		
5.	Applied problems to Hermite interpolation.
	Explanation, dialogue, practical examples. Evaluation.		
6.	Problems for least square approximation.
	Explanation, dialogue, practical examples. Evaluation.		
7.	 Problems with simple and repeated integration formulas and with Romberg’s algorithm.	Explanation, dialogue, practical examples.		
8.	 Problems applied to Gauss type quadrature formulas and adaptive quadratures.	Explanation, dialogue, examples. Evaluation.		
9.	Problems for solving linear systems with direct methods and study of perturbations of a linear system.	Explanation, dialogue, examples.	
	
10.	Problems for solving linear systems with iterative methods.	Explanation, dialogue, examples. Evaluation.		
11.	Problems with Newton, secant and bisection methods for solving nonlinear equations. Problems with Newton method for solving nonlinear systems.	Explanation, dialogue, practical examples. Evaluation.		
12.	Problems for solving some differential equations using Euler and Runge-Kutta methods. Ending of evaluation for laboratories	Explanation, dialogue, practical examples. Evaluation.		
Bibliography
1	R. L. Burden, J. D. Faires, Numerical Analysis, PWS Publishing Company, 1985.
2	A. Kharab, R. B. Guenther, An introduction to numerical methods. A Matlab approach, Taylor&Francis Group, 2006.
3	R. Trîmbitaş, Numerical Analysis, Ed. Presa Univ. Clujeană, 2007.	

9. Corroborating the content of the discipline with the expectations of the epistemic community, professional associations and representative employers within the field of the program

•	The course exists in the studying program of all major universities in Romania and abroad;
•	The content of the course is important for seeing the application of mathematical knowledge in solving practical and real life problems.  	

10. Evaluation
Type of activity	10.1 Evaluation criteria	10.2 Evaluation methods	10.3 Share in the grade (%)	
10.4 Course	- know the basic principle of the domain;
- apply the course concepts
- problem solving	Written exam
	70%	
10.5 Seminar/lab activities	- be able to implement course concepts and algorithms
- apply techniques for different practical problems	Evaluation and continuous observations during the semester.	30%	
10.6 Minimum performance standards	
•	At least grade 5 (from a scale of 1 to 10) at both written exam and laboratory work.	

Date		Signature of course coordinator 		            Signature of seminar coordinator
...............	Lecturer Ph.D. Teodora Cătinaş                                Lecturer Ph.D. Teodora Cătinaş 

Date of approval  				                       	  Signature of the head of department	...........................................					    ...........…............................