Mathematical Algorithms and Models

Aim

In computational mathematics, numerical methods are developed and analyzed for the solution of commonly occurring mathematical problems, mainly from technology and natural sciences. Important characteristics of the methods are robustness, accuracy and efficiency. Since the methods are meant to be implemented on computer, it is also important to understand how a computer processes numerical information. It is also important to be able to formulate a problem mathematically in an appropriate way.

After completing the course, the student should be able to:

• explain basic concepts in computational science and know how to computer stores numbers and the precision with which arithmetic operations can be performed.
• understand how mathematical problems can be solved approximately by discretization as well as being able to use a selection of numerical algorithms for to solve mathematical problems from applications with a computer.
• discuss possible sources of error in numerical calculations and estimate the accuracy of calculated results.
• formulate a practical problem as a mathematical model in a way that suitable for solution with software.
• develop software to solve practical problems from applications.

From the base group work, the student should be able to:

• independently and in groups, identify learning needs in relation to given problems related to the term's courses
• collaborate with colleagues in both learning and problem solving, as well as lead technical problem solving situations linked to the term's areas
• contribute to discussion and results in a base group

Content

Error analysis and number representation: IEEE standard for number representation in computers. The machine precision. Analysis of calculation errors. Cancellation. Error propagation and Error sources.

Interpolation: Polynomial and Spline interpolation. B-splines. Description of curves and surfaces with Bezier polynomials.

Solving Mathematical Equations Using Software: Fixed Point Iteration. Newton-Raphson method. Methods based on Polynomial interpolation. Implementation of square root and division on computer.

Calculation of Integrals: The trapezoidal method with error estimation. Adaptive methods and Recursion.

Ordinary differential equations: Euler's and Runge-Kutta's methods. Adaptive methods. System of differential equations. Solubility and stability. Simulation of physical phenomena using computer programs

Mathematical modelling: Physical units. Rewriting to dimensionless problem. Discrete and continuous models. Models with an element of randomness. Random number generation on computer. Programming in Python: Implementation of numerical algorithms. Visualization of calculation results.