Scientific computations

obligatory courses

introductory programming,

basic linear algebra,

basic calculus

Basics of scientific computations with examples in the python language.

1. Number representation, coputer arithmetic, numerical stability of algorithms

2. Vectors and matrices - representation and basic operations

3. Vector functions, combining functions, plotting one and multidimensional data

4. Systems of Linear Equations - Gauss elimination

5. Eigen values and eigenvectors

6. Linear Least squares

7. Polynomials as a vector space, interpolation.

8. Approximating functions with polynomials and splines

9. basic signal processing, filters, smoothing of data

10. FIlters for 2d and 3d signal processing - pixel and voxel images.

11. Basics of data compression - lossless and lossy compression

12. Numerical differentiation (polynomials, numerical differentiation)

13. Numerical integration - numerical quadratures

14. Symbolic computations

A primer on scientific programming with python, Lagtangen

Scientific Programming, Barone, Marinari, Organtini, Ricci-Tersenghi

Numerical Recipes, Press Teukolsky, Veterling, Flannery

Effects of teaching:

Knowedge and abilities: the student:

- understands the basics of computer arithmetic representation and problems associated with it

- Knows methods to solve non-linear equation problem;

- Understands the direct method of solving a system of linear equations probem by the LU decomposition

- Knows the definition of the linear least squares problem, its solution by the QR decomposition and its application to curve fitting

- knows the power iteration and inverse iteration methods for solving the eigenvalue problem

- Knows the definition of the Lagrange and Hermite interpolation problems

- knows the bases of polynomial vector space proposed by Lagange, and Newton .

- Knows the definition of linear and cubic splines for the purpose of interpolation

- Can perform all of the discussed operations on matrices in python programming language

- knows the basic python operations needed to present data graphically using line graphs, bar charts, boxplots, heatmaps and histigrams

- understands the basic notions of computer image representation and analysis

Social competences:

1. understands the role of numerical sicentific computing in modeling of pehnomena in physical and biological world.

2. understands the ethical implications of proper data visualisation

Written test,

programming project,

homework assigments,

written exam