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	Category	Status	
6	2+0+2	specialty	mandatory	

II. Full status faculty members
Name and	Scientific	Didactic	Chair	Type of activity	
surname	title	title		C	S	L	
MICULA Sanda	Ph.D.	Lect.	Numerical and Statistical Calculus	*	*	*	

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

III. Course objectives
To acquire basic knowledge of Numerical Analysis, with main focus on applications. To become familiar and be able to work with various approximation methods. To be able to implement some numerical methods.	

IV. Course contents
1. Error analysis. Approximation. Types of errors.
2. Approximation of functions. Least squares approximation. Polynomial interpolation. Spline interpolation.
3. Approximation of linear functionals. Numerical differentiation. Numerical integration.
4. Numerical solutions of algebraic linear systems.
5. Rootfinding for nonlinear equations. The secant method. Newton’s method. 
6. Numerical methods for solving ordinary differential equations.	

V. Bibliography
1. COMAN,GH.: Analiza numerica, Ed. Libris, Cluj-Napoca, 1995.
2. STANCU,D.D.: Analiza numerica, curs si culegere de probleme, Univ. Babes-Bolyai Cluj-Napoca, 1977 (lito).
3. AGRATINI,OCTAVIAN- BLAGA,PETRU- CHIOREAN,IOANA- COMAN,GHEORGHE- STANCU,D.DIMITRIE- TRAMBITAS,RADU: Analiza numerica si teoria aproximarii(vol.I,II,III), Presa Univ.Clujeana, 2002
4. BLAGA,PETRU - COMAN,GHEORGHE, TRAMBITAS,RADU - VASARU,DANIELA- POP,SORIN: Analiza numerica, lucrari de laborator, Univ. Babes-Bolyai Cluj-Napoca, 1995 (lito).
5. GELFAND,A.O.:  Calculul cu diferente finite, Ed. Tehnica Bucuresti 1956.
6. DEMIDOVICI,B.P.- MARON, A.: Elements de calcul numerique, Ed. Mir, Moscou, 1979.	

VI. Thematic of didactic activities per weeks
Lecture 1: Error analysis. Approximation. Types of errors. Sources and propagation of errors. Significant Digits. Floating-point arithmetic. Stability and conditioning
Lab 1: Introduction to Matlab.
Lecture 2: Approximation of functions. The best approximation problem. Least squares approximation. Orthogonal systems. Orthogonal polynomials (Legendre, Chebyshev of 1st and 2nd kind, Laguerre, Hermite, Jacobi).
Lab 2: Error theory, floating-point arithmetic and conditioning.
Lecture 3: Polynomial interpolation. Lagrange interpolation. Hermite interpolation. Using Peano’s theorem to express the remainder.
Lab 3: Lagrange interpolation.
Lecture 4. Aitken’s algorithm. Finite and divided differences. 
Lab 4: Divided differences.
Lecture 5. Birkhoff interpolation. Spline interpolation. Approximation of linear functionals. Numerical differentiation.
Lab 5: Hermite and spline interpolation.
Lecture 6. Newton-Cotes quadratures The rectangle, trapezoidal and Simpson’s (simple and composite) rules. 
Lab 6: Newton-Cotes Quadratures.
Lecture 7: Gaussian Quadratures. Nodes and coefficients of Gaussian quadratures. Jacobi matrices. Numerical integration in Matlab.
Lab 7: Gaussian Quadratures, I.
Lecture 8. Numerical solutions of algebraic linear systems. Matrix analysis, matrix norms. Direct methods: Gaussian elimination with different types of pivoting, factorization-based methods (LU, LUP, QR, Cholesky).
Lab 8: Gaussian Quadratures, II.
Lecture 9. Iterative methods for solving algebraic linear systems. The Gauss-Jacobi, Gauss-Seidel and relaxation methods. Matrices and linear systems in Matlab.
Lab 9: Gaussian elimination, I.
Lecture 10. Rootfinding for nonlinear equations. Iterative methods. Order of convergence. The bisection method. The secant method. Newton’s method.
Lab 10: Gaussian elimination, II.
Lecture 11: One-point iteration methods. Newton’s method for multiple roots.
Lab 11: Factorization and Iterative methods.
Lecture 12: Numerical methods for solving ordinary differential equations. Euler’s method. Single-step and Runge-Kutta methods. 
Lab 12: Completion of lab works. Review.	

VII. Learning methods used
Lecture, description, explanation, conversation, discussion, examples, synthesis, group-based work, individual study, homework. Group discussion over theoretical issues, as well as applications and modeling of the notions and methods presented.	

VIII. Assessment
The final grade will be computed as follows:
  - two written tests:                        70%
  - lab works  during the semester: 30% 
Details about evaluation 
•	There will be two written tests, one during the 6th week, another during the 12th week, consisting of problems similar to the ones discussed in seminar.
•	All laboratories are mandatory, each lab being due the week following the week when it was assigned, unless otherwise specified; each lab will be graded from 1 to 10; there is a “grace period” of one week for each lab, meaning that it can be turned in one week late, for a lower grade; labs that are not turned in within two weeks of being assigned are not accepted anymore, being graded with a 1.
•	Evaluation for lab works will consist of the average of the grades of all the labs.	

IX. Additional references

	Date,							Course responsible,
	April 26, 2007				             Lect. MICULA Sanda, Ph.D.