Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT

Code	Subject	
MMC1001	Numerical Methods for Operatorial Equations	

Section	Semester	Hours: C+S+L	Category	Type	
Applied Mathematics	1	2+2+0	speciality	compulsory	

Teaching Staff in Charge	Lect. MICULA Sanda, Ph.D., smicula@math.ubbcluj.ro	
Aims	To train students in finding numerical solutions of operatorial equations and to extend their knowledge from the Numerical Analysis course; at the end of this course, the students will have the necessary skills to solve numerically practical problems, modeled by operatorial equations. To be able to implement approximating processes.	
Content	PART I. Numerical Linear Algebra.
1. Introduction to Numerical Linear Algebra. Matrix Analysis.
2. Numerical methods for linear systems. LU, LUP, Cholesky decomposition.
3. QR decomposition. Householder transformation matrices. Givens rotation matrices.
4. The conjugate gradient method.. A-orthogonality. Properties. 
5. Eigenvalues and eigenvectors. Canonical forms. Eigenvalue location, error and stability results.
6. The power method and the QR method. Convergence, stability.
7. The calculation of eigenvectors. Inverse iteration.
8. Least squares solutions. SVD decomposition.

PART II. Numerical Methods for Ordinary Differential Equations.
9. Euler&#8217;s method. Derivation, convergence, stability. Modified Euler&#8217;s method.
10. Single-step methods. Runge-Kutta, Runge-Kutta-Fehlberg methods. Butcher tables.
11. Multistep methods. Definition, truncation error, consistency, order of convergence. The method of undetermined coefficients. 
12. Methods based on numerical integration. Adams-Bashforth, Adams-Moulton methods, predictor-corrector methods. 
13. General description of multistep methods. Linear difference equations. The root condition. Stability and convergence.
14. Stiff differential equations. Single- and multistep methods. A-stability, Padé approximation, A(&#945;)-stability.

Examples, applications and MATLAB implementation of the above methods.
	
Bibliography	1. K. E. Atkinson: An Introduction to Numerical Analysis, John Wiley and Sons Inc., 1988
2. K. E. Atkinson: Elementary Numerical Analysis, Second Edition, John Wiley and Sons Inc., 1993
3. John Dormand: Numerical Methods for Differential Equations,. A Computational Approach, CRC Press, Boca Raton New York, 1996
4. Hairer E., Norset S.P., Wanner G.: Solving Ordinary Differential Equations I (Nonstiff Problems), Second Revised Edition, Springer Verlag, 1993
5. Hairer E., Wanner G.: Solving Ordinary Differential Equations II (Stiff and Differential-Algebraic Problems), Springer Verlag, 1991
6. Sanda Micula, R. Sobolu, M. Micula : Analiză Numerică cu Maple, Editura Academic Press, Cluj-Napoca, 2008
7. O. Agratini, I. Chiorean, G. Coman, R. Trîmbiţaş: Analiză numerică şi Teoria Aproximării, III, Presa Universitară Clujeană, Cluj-Napoca,  2002
8. W. Gautschi: Numerical Analysis. An Introduction, Birkhaeuser, Boston 1997
9. Al. Quarteroni, R. Sacco, F. Saleri: Numerical Mathematics, Springer Verlag, 2000
10. R. Trîmbiţaş: Analiză numerică &#8211; o introducere bazată pe MATLAB, Presa Universitară Clujeană, Cluj-Napoca 2005
	
Assessment	Activity during the semester (attendance, class participation in seminar): 25%
Written midterm test (at the end of the first part):                       25%
Final written exam :                                                       50%