# Course Details for A.Y. 2015/2016

#### Name:

**Numerical Analysis II / Numerical Analysis II**

### Basic information

##### Credits:

*:* Master Degree in Mathematics 6 CFU (b)

##### Degree(s):

Master Degree in Mathematics 2^{nd} anno curriculum Generale Compulsory

##### Language:

English

### Course Objectives

Goals of the course:
Give the mathematical instruments for numerical optimization and numerical integration
of differential equations.
The course is of 6 credits and consists of 60 hours.
Expected learning:
Being able ti solve numerically and develop codes for general problems in optimization
and in differential equations.

### Course Content

- Iterative methods for the solution of nonlinear systems. Bisection method for nonlinear
equations. General theory of iterative methods. Convergence and order of convergence.
Newton method for a single equation. Extension to nonlinear systems. Newton Kantorovich
theorem. Quasi-Newton methods.
Laboratory session: Newton method.
- Optimization methods. Minimization problems for general functions: conditions for relative
minima; necessary first and second order conditions; sufficient second order conditions;
convexity. Quadratic functions with symmetric positive definite structure; steepest descent
gradient method (SD); convergence estimate; Kantorovich inequality; convergence speed of
steepest gradient method.
Laboratory session: steepest gradient method.
- Conjugate directions methods. Q-conjugacy and implications; expansion lemma; conjugate
gradient method (CG); characterization theorem for the CG method; properties of the method;
CG method optimality; comparison between CG and SD; general convergence estimates and
spectral relationships; partial CG method; application to structured matrices; preconditioned
gradient method. Nonlinear extension of CG method.
Laboratory sessions (2); application of CG to boundary value elliptic problems.
- Penalty methods: properties and convergence; application of partial CG method to penalty
methods; optimal scaling of penalty function.
Laboratory session: penalty method applied to few test problems.
- Numerical methods for the numerical approximation of initial falue problems for ordinary differential
equations. Euler explicit method; convergence; uniform convergence lemma for the numerical solution;
convergence of the generated piecewise linear functions; a priori error estimates under C2-regularity;
a posteriori error estimates; one step methods; examples; Taylor expansion based methods; Runge-Kutta
methods.
Laboratory session: implementation of an embedded Runge-Kutta method with stepsize control.
- Convergence and order conditions for one step schemes; discretization error; exact relative
increment; numerical relative increment; order of a one step method; convergence; consistence
theorem; a priori error estimates; a posteriori error estimates; general form of Runge-Kutta
methods; Butcher notation; order conditions; implicit methods; existence of numerical solution
for an implicit Runge-Kutta scheme.
Laboratory sessions (2): implementation of an implicit integrator.
- Stability; dissipative problems and stability; stiff problems; A-stability; more general stability
definitions; dense output; continuous methods; collocation methods; Gaussian collocation; implementative
issues; stepsize control.
Laboratory session: integration of a stiff problem arising in chemical kinetics.
- Numerical methods to approximate the siolution of boundary value problems for ordinary
differential equations. Shooting method; finite difference method; variational methods;
application to a second order elliptic problem.
Laboratory sessions (2): implementation of shooting method; implementation of the variational
method.

### Prerequisites and Learning Activities

Basic numerical analysis. Differential equations.

### Assessment Methods and Criteria

Oral examination and a term paper with implementation issues.

### Textbooks

- J. Stoer e R. Bulirsch, Introduction to numerical analysis. , Springer Verlag. 2002.
* *
- D.G. Luenberger, Linear and nonlinear programming , Kluwer Academic Publishers. 2003.
* *
- E. Hairer, S.P. Norsett e G. Wanner, Solving ordinary differential equations. I. Nonstiff problems. Second edition. , Springer Verlag.. 1993.
* *

### Course page updates

This course page is available (with possible updates) also for the following academic years:

** To read the *** current * information on this course, if it is still available, go to the university course catalogue .
*Course information last updated on: 15 settembre 2015, 17:53*