Computational Mathematics II, 7.5 credits

Aims and objectives

General aims for first cycle education

First-cycle courses and study programmes shall develop:

• the ability of students to make independent and critical assessments
• the ability of students to identify, formulate and solve problems autonomously, and
• the preparedness of students to deal with changes in working life.

In addition to knowledge and skills in their field of study, students shall develop the ability to:

• gather and interpret information at a scholarly level
• stay abreast of the development of knowledge, and
• communicate their knowledge to others, including those who lack specialist knowledge in the field.

(Higher Education Act, Chapter 1, Section 8)

Course objectives

Knowledge and understanding
After the course the student should

• know and be able to use the most important methods for ill posed linear problems, interpolation and approximation in R^n,
• know and be able to use the most imporant stochastic methods for simulation and calculation, and
• know some common application areas for computational mathematics.

Skills
After the course the student should

• be able to identify, analyse and numerically solve linear ill posed linear systems of equations and linear least squares problems,
• be able to use and analyse methods for interpolation and approximation with piecwise polynomials and Bezier curves,
• be able to use and analyse stochastic methods for simulation, calculation of integrals and solving differential equations,
• be able to use and analyse the FFT, and
• be able to use and analyse multistep methods for ODE.

Main content of the course

Regularization of ill posed linear equation systems and linear leasts squares probems. A priori information. Truncated SVD. L-curve. Cross validation. Applications on integral equations. Interpolation and approximation with piecewise polynomials in several variables. Error analysis. Algorithms for constructing Bezier-curves. Applications to CAD. Monte-Carlo methods with applications to integrals. Numerical methods for differential equations using randomwalk. FFT with error analysis and complexity. FFT with error and complexity analysis and applications to signal analysis. Multistep methods for initial value ODE. Numerical analysis of methods for initial value ODE.

Teaching methods

Teaching in the form of lectures and supervised projects.

The teaching methods may be altered, should only a few students take the course.

Students who have been admitted to and registered on a course have the right to receive tuition and/or supervision for the duration of the time period specified for the particular course to which they were accepted (see, the university's admission regulations (in Swedish)). After that, the right to receive tuition and/or supervision expires.

Examination methods

Examination, 7.5 credits (Code: A001)
Written and oral presentation of projects.

For students with a documented disability, the university may approve applications for adapted or other forms of examinations.

For further information, see the university's local examination regulations (in Swedish).

Grades

According to the Higher Education Ordinance, Chapter 6, Section 18, a grade is to be awarded on the completion of a course, unless otherwise prescribed by the university. The university may prescribe which grading system shall apply. The grade is to be determined by a teacher specifically appointed by the university (an examiner).

In accordance with university regulations regarding grading systems for first and second-cycle courses (Vice-Chancellor’s decision ORU 2018/00929), one of the following grades shall be used: Fail (U), Pass (G) or Pass with Distinction (VG). For courses that are included in an international Master’s programme (60 or 120 credits) or offered to the university’s incoming exchange students, the grading scale of A-F shall be used. The vice-chancellor, or a person appointed by the vice-chancellor, may decide on exceptions from this provision for a specific course, if there are special grounds.

Grades used on course are Fail (F), Sufficient (E), Satisfactory (D), Good (C), Very Good (B) or Excellent (A).

Examination
Grades used are Fail (F), Sufficient (E), Satisfactory (D), Good (C), Very Good (B) or Excellent (A).

For further information, see the university's local examination regulations (in Swedish).

Specific entry requirements

Optimization, 7.5 Credits, Differential Equations, 7.5 Credits and Numerical Methods for Differential Equations, 7.5 Credits.

For further information, see the university's admission regulations (in Swedish).

Transfer of credits for previous studies

Students who have previously completed higher education or other activities are, in accordance with the Higher Education Ordinance, entitled to have these credited towards the current programme, providing that the previous studies or activities meet certain criteria.

For further information, see the university's local credit transfer regulations (in Swedish).

Other provisions

All or part of the course can be given in English.

Reading list and other teaching materials

Required reading

Sauer ,Timothy (2013)
Numerical Analysis
Pearson

Reference literature

Heath, Michael T (2002)
Scientific Computing: An Introductory Survey
McGraw-Hill

Material handed out by the Department of Mathematics.