Numerical calculus
Code: 100120 ECTS Credits: 6| Degree | Type | Year | Semester |
|---|---|---|---|
| 2500149 Mathematics | OT | 4 | 1 |
Contact
- Name:
- Judit Chamorro Servent
- Email:
- judit.chamorro@uab.cat
Teaching groups languages
You can check it through this link. To consult the language you will need to enter the CODE of the subject. Please note that this information is provisional until 30 November 2023.
Teachers
- Carles Barril Basil
Prerequisites
It is advisable to have successfully completed all mandatory courses, and also to know a programming language.
Objectives and Contextualisation
Systems of linear and non-linear equations, and ordinary differential equations are present in many mahtematical models of physical processes. In this course we will study numerical techniques for the approximate solution of systems of linear and non-linear equations, and both initial and boundary value problems for ordinary differential equations. We will also study algorithms for the computation of eigenvalues of matrices.
The main goal of the course is that students learn all these methods from their mathematical foundation by studying their convergence properties, ant also that they are able to program them. Computer sessions are a fundamentel part of the course, and they are intended for students to valuebetter understand the features of the different numerical methods.
Competences
- Assimilate the definition of new mathematical objects, relate them with other contents and deduce their properties.
- Calculate and reproduce certain mathematical routines and processes with agility.
- Generate innovative and competitive proposals for research and professional activities.
- Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
- Students must have and understand knowledge of an area of study built on the basis of general secondary education, and while it relies on some advanced textbooks it also includes some aspects coming from the forefront of its field of study.
- When faced with real situations of a medium level of complexity, request and analyse relevant data and information, propose and validate models using the adequate mathematical tools in order to draw final conclusions
Learning Outcomes
- Control for errors produced by machines when computing.
- Generate innovative and competitive proposals for research and professional activities.
- Know how to program algorithms for mathematical calculation.
- Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
- Students must have and understand knowledge of an area of study built on the basis of general secondary education, and while it relies on some advanced textbooks it also includes some aspects coming from the forefront of its field of study.
- Understand the internal workings of computers and be critical of the results received.
Content
- Initial value problems for ordinary differential equations
- One-step methods: Euler and Taylor.
- Local discretization error.
- Runge-Kutta methods.
- Convergence of one-step methods.
- Fehlberg step-size control.
- Comments on multistep methods.
- Stiff problems.
- Numerical solution of systems of non-linear equations
- Matrix norms.
- Fixed point methods: convergence and error estimation.
- Newton's method in several variables.
- Boundary value problems for ordinary differential equations
- Single shooting method.
- Multiple shooting method.
- Finite difference methods.
- Numerical linear algebra
- Perturbation analysis of linear systems.
- QR method for square and over-determined systems.
- Iterative methods for linear systems. Convergence and error estimation.
- Power and inverse power methods for the computation of eigenvalues and eigenvectors.
- QR method for the computation of eigenvalues and eigenvectors.
Methodology
The theoretical and problem sessions will be carried out in a classroom. These sessions will be devoted to the presentation of theoretical aspects of numerical methods, their basic properties and the solution of problems, some of them of theoretical nature and some of them requiring the use of a calculator. Problem lists will be supplied along the course.
The seminar sessions with will be carried out in a computer room. In these sessions, studens will solve an applied problem through the implementation in the C or MATLAB programming languages of methods studied in the course. These practical sessions will be evaluated from the delivery towards the end of the course (a date will be announced) of the C/MATLAB code and a report.
Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.
Activities
| Title | Hours | ECTS | Learning Outcomes |
|---|---|---|---|
| Type: Directed | |||
| Computer sessions | 12 | 0.48 | 1, 6, 2, 5, 4, 3 |
| Problem classes | 8 | 0.32 | 2, 5, 4 |
| Theoretical classes | 30 | 1.2 | 2, 5, 4 |
| Type: Autonomous | |||
| Personal study | 50 | 2 | 2, 5, 4, 3 |
| Problem solving and computer work | 44 | 1.76 | 1, 6, 2, 5, 4, 3 |
Assessment
This subject does not have a single assessment.
The course will be graded from three evaluation activities:
- Parcial exam (PE): an exam that exists in the resolution of problems that deal with the first part of the course (details will be announced at Campus Virtual)
- Final exam (FE): an exam of all the course, with theoretical questions and problems.
- Practical work (PR): delivery of Matlab/C programs and a report
In addition, the following criteria will be considered:
- The students will have the option of taking an additional recovery exam (RE) with the same format as FE.
- Nor the partial exam nor the practical work will be recoverable.
- In order to succeed in this course, it is mandatory that max(EF, ER)>=3.5 and PR>=3.5. If one student does not reach the minimum required in any of the evaluation activities, and the calculation of the final grade is equal to or greater than 5, a grade of 4 will be placed in the file.
- A student is considered "Non-Evaluable" (NE) only if he / she has not done any assessment activity. Remember that the NE grade means that one examination sitting is lost.
- The final grade will be calculate following: max(0.1 EP+0.5EF+0.4PR, 0.5 ER+0.5PR).
- Honor grades will be granted at the first complete evaluation. Once given, they will no be withdrawn even if another student obtains a larger grade after consideration of the RE exam.
Assessment Activities
| Title | Weighting | Hours | ECTS | Learning Outcomes |
|---|---|---|---|---|
| Final exam | 0.5 | 3 | 0.12 | 2, 5 |
| Partial exam | 0.10 | 0 | 0 | 2, 5 |
| Practical work | 0.40 | 0 | 0 | 1, 6, 2, 5, 4, 3 |
| Recovery exam | 0.5 | 3 | 0.12 | 2, 5 |
Bibliography
General references:
- J. Stoer and R. Burlisch, Introduction to numerical analysis, 3a ed, Springer, 2002.
- A. Ralston and P. Rabinowitz, A first curse in numerical analysis, McGraw-Hill, 1988.
- G. Dahlquist and A. Björck, Numerical methods, Englewood Cliffs (N.J.) : Prentice-Hall, 1974.
- A. Aubanell, A. Benseny y A. Delshams, Eines bàsiques del càlcul numèric, Manuals de la U.A. B., 1991.
- A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, TAM, Springer, 2000.
Specialized references:
- R. L. Burden and J. D. Faires, Análisis Numérico, Grupo Editorial Iberoamérica, México D. F., 1985.
- G. W. Gear, Numerical initial value problems in ordinary differential equations, Prentice-Hall, 1971.
- E. Hairer, S.P. Nørsett, G. Wanner, Solving ordinary differential equations. Vol. 1, Springer-Verlag, 1987.
- E. Hairer, S.P. Nørsett, G. Wanner, Solving ordinary differential equations. Vol. 2, Springer-Verlag, 1991.
- L. Elden, L. Wittmeyer-Koch, & H. B. Nielsen, Introduction to Numerical Computation, Studentlitteratur AB, 2004.
- P. C. Hansen. Discrete inverse problems: insight and algorithms, Society for Industrial and Applied Mathematics, 2010.
Software
We will use MATLAB (matrix laboratory) or C during the practical work.
Regarding MATLAB: The UAB has a MATLAB license "for all of its university community members to use the software products made available by this platform, without limitations". See: https://www.uab.cat/web/newsroom/news-detail/university-community-can-now-access-the-matlab-platform--1345668003610.html?noticiaid=1345797909672